High-Temperature Superfluidity in an
Ultracold Fermi Gas
byMartin W. Zwierlein
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
October 2006
c Massachusetts Institute of Technology 2006. All rights reserved.
°

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
October 26, 2006

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacAurthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thomas J. Greytak
Professor of Physics, Associate Department Head for Education

High-Temperature Superfluidity in an
Ultracold Fermi Gas
byMartin W. Zwierlein
Submitted to the Department of Physics
on October 26, 2006, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
This thesis presents experiments in which a strongly interacting gas of fermions wasbrought into the superfluid regime. The strong interactions are induced by a Feshbach
scattering resonance that allows to tune the interfermion scattering length via an
external magnetic field. When a Fermi mixture was cooled on the molecular sideof such a Feshbach resonance, Bose-Einstein condensation of up to 107 molecules
was observed. Subsequently, the crossover region interpolating between such a BoseEinstein condensate (BEC) of molecules and a Bardeen-Cooper-Schrieffer superfluidof long-range Cooper pairs was studied. Condensates of fermion pairs were detected
in a regime where pairing is purely a many-body effect, the pairs being stabilized by
the presence of the surrounding particles. Superfluidity and phase coherence in thesesystems was directly demonstrated throughout the crossover via the observation of
long-lived, ordered vortex lattices in a rotating Fermi mixture. Finally, superfluidity
in imbalanced Fermi mixtures was established, and its Clogston limit was observedfor high imbalance. The gas was found to separate into a region of equal densities,
surrounded by a shell at unequal densities.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacAurthur Professor of Physics

To my family,
Erika and Otto, Anne-Julia and Cornel

Acknowledgments
Over the years at MIT I had the good fortune to learn from and the pleasure to workwith an exceptional group of people. These few lines can only be a knowing nod toall of them, while there is no way to fully express my gratitude in writing.
First of all, I would like to thank Wolfgang Ketterle for teaching me the ways ofa physicist. His sparkling enthusiasm for science is well known to be infectious. The
trust he had in every one of us to achieve our goals is what made us believe that we
actually might. I treasure every discussion we had, be it about physics or generalmatters of life.
It is my pleasure to thank my fellow graduate students on the lithium project, its
first members and initiators, Zoran Hadzibabic, Claudiu Stan and Subhadeep Gupta,
the new crew around Christian Schunck and André Schirotzek, and special guest
Jamil Abo-Shaeer. None of the work in this thesis would have been possible withoutthem. I consider myself a lucky guy, as all of them have not only been the greatest
colleagues, but have also become wonderful friends.
When I first came to MIT as a visiting student from Paris, Zoran took me under
his wings. I immediately grew attached to his way of doing science, of figuring things
out instead of taking them for granted, and having tremendous fun at it. He taught
me invaluable lessons about physics and life, and he was a big reason for me to comeback for my PhD. We shared a lot of habits, the bizarre schedule, the preference
for Miracle burgers with extra chutney, later I copied his addiction to diet coke.
Unfortunately I could not copy his wit, but maybe that simply means we need moretime to work together.
From Claudiu I learned that what has to work well has to look good: His precision and care in building apparatus was the pillar for the reliability and successof our experiments. If you have any question about some unknown, weird-looking
and potentially dangerous substance, Claudiu can tell you all about it. I was always
amazed by his ability to keep his cool when everything around us seemed to fall apart.
The only way you could tell that the problem he just fixed could have turned into amajor catastrophe was by the number of cigarettes consumed afterwards.
When lithium atoms and light were ready for the upgrade of the sodium BEC
machine, we teamed up with Deep Gupta, an exceptional scientist and a great teacher.
He introduced me to the sodium BEC machine, the dye(t) laser, the subtleties of thesodium MOT, saddle points (sic) of magnetic trapping and evaporative cooling andto frequent punning. It was wonderful to watch Zoran and Deep bounce ideas offeach other during the optimization of the sympathetic cooling scheme, and see theatom number grow by the minute.
We, the youngens, were thrown into cold water when they both graduated a dayfrom each other, but they had left a miraculous heritage, the brightest source ofultracold fermions in the world.
Christian first joined the crew as a diploma student, while I was back in Paris, andhe must have experienced the same welcoming warmth I had felt, as he was able tocite the relevant puns even before having watched The Big Lebowski. He now makes
the best White Russian in the Northern Hemisphere, as I can certify. He has been adependable friend, office-mate and lab companion through countless days and nights.
Not widely known are his superb musical skills: His harp play is simply outstanding.
I wish him and his newly-wed wife Alessandra all the best for the future.
In a memorable night in February 2001, I heard a tremendous outcry coming fromthe BEC II lab. I jumped inside, fearing something terrible had happened, and saw
a stunning image on the screen: Jamil Abo-Shaeer and Chandra Raman had justcreated a vortex lattice in a Bose-Einstein condensate containing about 100 vortices!
The image stuck in my mind. Over the next years, Jamil had found in me, his poor
office-mate, the ideal subject for his teases. He tricked me, embarrassingly more thanonce, into believing that I was really heavy (which I am not!), next, that his mouse
has a life of its own and that he could solve the Rubik’s cube in less than five minutes.
Jamil was our star in soccer. He needed twelve sure chances to score one goal. At
least he always scored. My biggest success in grad school was to convince Jamil todefect from BEC II, delay his thesis and ”bring the vortices over” to the sodium
condensates in BEC I. Which he did. In a day. Within the two months or so he had
been a member of our lab I learned experimental skills for a lifetime. I pay this backby dedicating Fig. 6-9 to him.
André, from the same village in Germany and almost as well-built as me, hadbeen a close friend from day one. His enthusiasm and fearlessness in the lab are by
now legendary. When we were almost about to give up on vortices in Fermi gases,
but had a beautiful gas cloud on our screen, he asked ”should we try it once m...?”, I
interrupted saying ”yes!”, and the result was Fig. 6-8a. I will always cherish the times
watching Friends in Clinton Street, eating entire cows, ”making gym” and playing
all sorts of existing or invented sports inside building 26. We were all happy to have
André on our soccer defense, where he flattened the other team’s players with grace.
Next time we play squash, I might let him win once or twice.
I am grateful for all things I learned from our postdocs Axel Görlitz, Kai Dieck-mann and Jamie Kerman, and the newest addition to the lithium lab, Yong-Il Shin.
After seminal works on bosons in BEC III, Yong immediately jumped onto the hottest
topic in fermion physics and made his lasting mark. His clear way of thinking is admirable, and I profited enormously from discussions with him.
Thanks also go to Sebastian Raupach, philosopher/physicist, who joined us forhis diploma thesis in the heyday of fermion pair BEC. He competed with Christian
for the title of the nicest guy on earth. Peter Zarth, another diploma student, and
new PhD student Ye-yroung Lee impressed me with their enthusiasm and drive to
get the first potassium MOT at MIT, working in the same little room where Zoran,
Claudiu and myself produced our first laser cooled lithium atoms.
I owe a lot to the crew of BEC II. The past members Roberto Onofrio, Chandra
Raman, Jamil, and Johnny Vogels for developing the robust technique of stirring
ultracold atoms, and the new team Kaiwen Xu, Takashi Mukaiyama, Jit Kee Chin,
Dan Miller, Yingmei Liu, Christian Sanner and Widagdo Setiawan for letting me steal
Jamil for a while. Dan Miller must receive extra thanks for the many parties at his
house that helped foster the group spirit, for buying excellent Dunkin’ Donuts coffee,
for having me park my belongings at his place and generally for being hilarious.
With BEC III we lithium guys are connected in a love-hate relationship, becauseof the illegal ”trade” of optics and other equipment that is constantly going on. They
should have put the wall back up a long time ago. Aaron Leanhardt provided therole model for everyone in our group with his work ethics and deep understanding of
the most diverse areas of physics. I thank him for his excellent advice. Tom Pasquini
I thank for improving my general behavior, especially related to coffee mugs. Michele
Saba was not only an invaluable resource of knowledge and original thoughts but also
the best soccer player we ever had and the best cook, and because of the latter I let
him stay in my apartment during his final month. Thanks also go to Gyu-Boong Jo
and Caleb Christensen, for letting me steal optics behind their backs. Sebastian Will,
diploma student filling the role of a grad student, I thank for the fun we had out inthe city, and I am looking forward to exploring Mainz with him.
I want to apologize to BEC IV for the many times we forgot to ask them tojoin us for lunch, but I thank Micah Boyd, Eric Steed, Gretchen Campbell, Dominik
Schneble, Luis Marcassa, Patrick Medley, and Jongchul Mun for being good sports
about it. Micah paid that back with a good tease towards me every now and then,being second only to Jamil in this profession. It was great having him around when
we were attacked by a huge chunk of sodium, and generally when we needed a goodtime. Gretchen deserves two PhDs, one for physics and one for helping everyone out

with little and big things, holding up the group spirit and having us feel at home
in a big family. I do not see how the gap she will leave behind in the group can be
filled. I thank her for her company during countless coffee breaks I needed at themost abnormal hours, for convincing me that it’s all ok when I laid on her couch andfor handing in this thesis for me.
My thanks go to James Anglin for very useful discussions on vortex physics, toDan Kleppner for advice and help at numerous occasions, and to Dave Pritchard forbeing a great academic advisor, in the fullest sense of the word.
The late Carol Costa has made our hallway in building 26 a warmer place. She
warned me to never write acknowledgements at the end of writing a thesis. I shouldhave listened. She is heavily missed. Ellenor Emery took her role and mothered us
as Carol did. I am very happy for her that she has become a real mum. Joanna
Keseberg is now the good heart of the hallway, and I thank her for all the help overthe last year. Big thanks go to the RLE Support staff, especially Maxine Samuelsand Al McGurl.
I owe many memorable matches to the entire intramural soccer team ”The Balldrivers”, including the above-mentioned as well as Adam Black, Marko Cetina, Andrew Grier (the actual best player), Cort Johnson, Ian Leroux, Yu-ju Lin, Bonna
Newman, Dave Palazzo, Monika Schleier-Smith, Jon Simon (the best player ever),Igor Teper, James Thompson (thanks also for great advice in diverse matters) andMukund Vengalattore (the truly best player), and my room mate and friend from oldtimes at the University of Bonn, Andreas Malmendier. When I told Andreas that I
would come to MIT, he immediately offered that we could share an apartment. The
rest is history. I loved cracking ancient math problems with him, watching the worst
movies ever and holding parties that we will certainly never forget, guaranteed. With
André we found our brother in crime, and these crimes should be left untold.
Finally, I would like to thank my parents, Otto and Erika, as well as my sisterAnne-Julia (Anju) and my brother Cornel for providing me with inspiration. To them
I dedicate this thesis.

Contents
1 Introduction
1.1 Discovery of superfluidity and superconductivity — from BEC to BCS
1.1.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . .
1.1.2 Fermionic superfluidity and Bardeen-Cooper-Schrieffer theory

15
17
17
19

1.1.3 The BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . .
1.1.4 Universal regime at unitarity . . . . . . . . . . . . . . . . . . .
Ultracold atomic gases - model systems for bosonic and fermionic superfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Bosonic gases . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
22
23
24

1.2.2 Fermionic gases . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 High critical temperature . . . . . . . . . . . . . . . . . . . . .
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
26
26

2 Superfluidity in bosonic and fermionic gases
2.1 Quantum statistics at work - bosons vs fermions . . . . . . . . . . . .
2.1.1 Quantum statistics and second quantization . . . . . . . . . .
2.1.2 Bose-Einstein versus Fermi-Dirac distribution . . . . . . . . .

28
28
28
33

2.1.3 Ideal Bose and Fermi gases in a harmonic trap . . . . . . . . .
Interacting gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The interacting Bose gas . . . . . . . . . . . . . . . . . . . . .
2.2.3 Fermi gases with attractive interaction: Cooper instability . .

34
37
38
43
48

The BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Gap and number equation . . . . . . . . . . . . . . . . . . . .
2.3.2 Evolution from BCS to BEC . . . . . . . . . . . . . . . . . . .
2.3.3 Single-particle and collective excitations — Landau criterion .

55
57
61
63

2.3.4
2.3.5

66
69

1.2

1.3

2.2

2.3

Finite temperatures . . . . . . . . . . . . . . . . . . . . . . . .
Definition of the condensate wavefunction . . . . . . . . . . .
8

3 Feshbach resonances
3.1 Scattering resonances . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2

74
75

Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 A simple model . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Feshbach resonances in 6 Li . . . . . . . . . . . . . . . . . . . .
3.2.3 Broad versus narrow Feshbach resonances . . . . . . . . . . .

77
78
82
85

4 Cooling fermions - The apparatus
4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Sympathetic cooling of 6 Li by 23 Na . . . . . . . . . . . . . . . . . . .
4.3 Preparation of an interacting Fermi mixture in an optical trap . . . .

88
89
93
95

4.3.1
4.3.2

Magnetic plus optical trapping . . . . . . . . . . . . . . . . . .
Spin transfer . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Bose-Einstein condensation of fermion pairs
5.1

5.2

5.3

95
96
99

Bose-Einstein condensation of molecules . . . . . . . . . . . . . . . .
5.1.1 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Methods for making molecules . . . . . . . . . . . . . . . . . .
5.1.3 Formation of molecules . . . . . . . . . . . . . . . . . . . . . .

100
100
101
104

5.1.4 Observation of molecular Bose-Einstein condensates . . . . . .
Condensation of fermion pairs close to a
Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The rapid ramp technique . . . . . . . . . . . . . . . . . . . .
5.2.2 Condensate fraction in the BEC-BCS crossover . . . . . . . .

108
112
112
114

5.2.3 Discussion of the condensate fraction . . . . . . . . . . . . . . 116
5.2.4 Discussion of the time scales involved . . . . . . . . . . . . . . 118
Direct observation of the phase transition in a resonantly interacting,equal mixture of fermions . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.1
5.3.2

The normal and the superfluid gas on resonance . . . . . . . . 121
Anomalous features in the density profiles of expanded clouds
at unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6 High-temperature superfluidity:
Observation of vortices in a strongly interacting Fermi gas
6.1 Origin of the vortex phenomenon . . . . . . . . . . . . . . . . . . . .
6.2 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Roadblocks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9

126
128
130
131

6.3

6.2.2 ”Sanding off the bumps” . . . . . . . . . . . . . . . . . . . . . 133
Observation of vortices in a Fermi gas . . . . . . . . . . . . . . . . . . 142
6.3.1
6.3.2
6.3.3

Vortex number and lifetime . . . . . . . . . . . . . . . . . . . 144
A rotating bucket . . . . . . . . . . . . . . . . . . . . . . . . . 146
Pair breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7 Fermionic superfluidity with imbalanced spin populations
7.1 Historic overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The Clogston limit of superfluidity . . . . . . . . . . . . . . . . . . .
7.2.1 The Clogston limit of superconductors . . . . . . . . . . . . .
7.2.2 Translation to dilute Fermi gases in atom traps . . . . . . . .

152
153
154
154
157

7.2.3 Phase separation in the BCS-regime . . . . . . . . . . . . . . .
7.2.4 Critical number imbalance . . . . . . . . . . . . . . . . . . . .
Fermionic superfluidity with spin imbalance . . . . . . . . . . . . . .
7.3.1 Creation of imbalanced Fermi mixtures . . . . . . . . . . . . .

159
160
163
163

7.3.2 Rotating imbalanced mixtures: Vortices . . . . . . . . . . . .
7.3.3 Condensate fraction . . . . . . . . . . . . . . . . . . . . . . . .
Direct observation of the phase transition, imbalanced case . . . . . .
7.4.1 On resonance: Universality . . . . . . . . . . . . . . . . . . . .

165
167
169
169

7.4.2 On the BEC side: strongly interacting Bose-Fermi mixture . .
Observation of phase separation . . . . . . . . . . . . . . . . . . . . .
7.5.1 Experimental method: Phase-contrast imaging . . . . . . . . .
7.5.2 In-situ profiles of the column density difference . . . . . . . .
7.5.3 Two regimes of phase separation . . . . . . . . . . . . . . . .

176
182
183
184
185

7.3

7.4

7.5

7.6

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8 Conclusion

189

A Spectroscopic Insensitivity to Cold Collisions in a Two-State Mixture of Fermions193
B Observation of Bose-Einstein Condensation of Molecules

198

C Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance
203
D Feshbach Resonances in Fermionic 6 Li

10

208

E Formation Dynamics of a Fermion Pair Condensate

213

F Vortices and Superfluidity in a Strongly Interacting Fermi Gas

218

G Superfluid Expansion of a Rotating Fermi Gas

224

H Fermionic Superfluidity with Imbalanced Spin Populations

230

I

Direct Observation of the Superfluid Phase Transition in Ultracold
Fermi Gases
242

J Observation of Phase Separation in a Strongly Interacting Imbalanced Fermi Gas
253
Bibliography

258

11

List of Figures
1-1 The BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . .

21

2-1
2-2
2-3
2-4
2-5

46
49
53
59
61

Excitation spectrum of the weakly interacting Bose gas . . . . . . . .
Bound state wavefunctions in 1D, 2D and 3D . . . . . . . . . . . . .
Cooper problem: Two particles scattering on top of a Fermi sea . . .
Chemical potential and gap in the BEC-BCS crossover . . . . . . . .
Occupation nk of momentum states k in the BEC-BCS crossover . . .

2-6 Evolution of the spatial pair wavefunction φ(r) in the BEC-BCS crossover 62
2-7 From tightly bound molecules to long-range Cooper pairs . . . . . . . 63
2-8 Evolution of the single-particle excitation spectrum in the BEC-BCS
crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2-9 Critical velocity vc in the BEC-BCS crossover . . . . . . . . . . . . . 65
2-10 Superfluid transition temperature TC and pair creation temperature T ∗ 68
3-1 Scattering resonances for the square well potential . . . . . . . . . . .
3-2 Origin of Feshbach resonances . . . . . . . . . . . . . . . . . . . . . .
3-3
3-4
3-5
3-6

76
78

Simple model for a Feshbach resonance . . . . . . . . . . . . . . . . . 79
Bound state energy and scattering length close to a Feshbach resonance 83
Feshbach resonances in 6 Li . . . . . . . . . . . . . . . . . . . . . . . . 83
Bound state energies for 6 Li2 in a magnetic field . . . . . . . . . . . . 84

4-1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23

6

4-2 Hyperfine states of Na and Li . . . . . . . . . . . . . . . . . . . . .
4-3 Magnetic trap for sympathetic cooling . . . . . . . . . . . . . . . . .
4-4 Hyperfine transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91
92
93
97

5-1 Creating molecules via three-body collisions . . . . . . . . . . . . . . 101
5-2 Creating molecules via magnetic field ramps . . . . . . . . . . . . . . 102
5-3 Imaging atoms and molecules . . . . . . . . . . . . . . . . . . . . . . 104

12

5-4 Molecule formation by magnetic field sweep across the Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5-5 Locating the Feshbach resonance by molecule dissociation . . . . . . . 106
5-6 Revival of the atomic signal during expansion and strength of Feshbach
coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5-7 Lifetime of molecules in partially condensed clouds . . . . . . . . . . 109
5-8 Observation of Bose-Einstein condensation of molecules . . . . . . . .
5-9 Bimodal density distribution emerging in a cloud of molecules . . . .
5-10 Comparison between a molecular BEC and a degenerate Fermi sea . .
5-11 The dilemma with strongly interacting Fermi gases . . . . . . . . . .
5-12 Rapid ramp to the molecular side to observe pair condensation . . . .

110
110
111
113
114

5-13 Fermion pair condensates . . . . . . . . . . . . . . . . . . . . . . . . . 115
5-14 Condensate fraction as a function of magnetic field . . . . . . . . . . 116
5-15 ”Phase diagram” for the condensate fraction as a function of magnetic
field and temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5-16 Condensate fraction as a function of the interaction strength . . . . .
5-17 Density profiles of an equal Fermi mixture on resonance . . . . . . . .
5-18 χ2 of the Thomas-Fermi fit as a function of trap depth . . . . . . . .
5-19 Density profiles, their curvature and their fit-residuals on resonance .

119
123
124
125

6-1 Fate of a quadrupole oscillation in a rotating atomic Bose-Einsteincondensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-2 Vortex nucleation for violent stirring in an atomic BEC . . . . . . . .
6-3 Experimental setup for the observation of vortices in a Fermi gas . . .
6-4 Trapping potential for the study of rotating Fermi gases . . . . . . .

130
134
135
136

6-5 Alignment of the optical trap to achieve a radially symmetric potential 138
6-6 Decrease of vortex visibility for a sodium condensate expanding into amagnetic saddle potential . . . . . . . . . . . . . . . . . . . . . . . . 139
6-7 Vortex lattice in a Bose-Einstein condensate of sodium atoms in themagnetic and the optical trap . . . . . . . . . . . . . . . . . . . . . .
6-8 Vortex lattice in a 6 Li2 molecular condensate . . . . . . . . . . . . . .
6-9 Observation of vortices in a strongly interacting Fermi gas . . . . . .
6-10 Lifetime and number of vortices in the BEC-BCS crossover . . . . . .
6-11 Rotating bucket for superfluid Fermi gases . . . . . . . . . . . . . . .

141
142
143
145
146

6-12 Observation of pair breaking in a superfluid Fermi gas . . . . . . . . . 148
6-13 Peak interaction strength during superfluid expansion . . . . . . . . . 14913

6-14 Loss of vortex contrast on resonance . . . . . . . . . . . . . . . . . . 150
7-1 Imbalanced chemical potentials for electrons in the normal state under
the influence of a magnetic field . . . . . . . . . . . . . . . . . . . . . 156
7-2 A weakly interacting, imbalanced Fermi mixture with superfluid component in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . 158
7-3 Critical population imbalance for BCS-type superfluidity in a harmonic
7-4
7-5
7-6
7-7

trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creation of imbalanced Fermi mixtures . . . . . . . . . . . . . . . . .
Vortices in imbalanced Fermi mixtures . . . . . . . . . . . . . . . . .
Number of vortices obtained in an imbalanced Fermi mixture . . . . .
Fermion pair condensates and the Clogston limit in an imbalanced

161
164
165
166

Fermi mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7-8 Superfluid window of imbalanced Fermi mixtures . . . . . . . . . . . 168
7-9 Direct observation of the superfluid phase transition in a resonantlyinteracting, imbalanced Fermi mixture . . . . . . . . . . . . . . . . . 170
7-10 Direct observation of the superfluid phase transition via in-situ phase
contrast imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7-11 Critical temperature for an imbalanced Fermi mixture on resonance . 172
7-12 Constrained radii of the superfluid, the mixed and the fully polarizedphase on resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7-13 Phase transition in a strongly interacting atom-molecule mixture . . . 177
7-14 Condensate fraction of the molecular condensate as a function of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7-15 Effective potential and density of excess fermions in a Bose-Fermi mixture180
7-16 Phase diagram for a strongly interacting Bose-Fermi mixture in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7-17 Phase contrast imaging for the observation of phase separation in an
imbalanced Fermi mixture . . . . . . . . . . . . . . . . . . . . . . . . 183
7-18 In situ imaging of an imbalanced Fermi mixture for various population
imbalances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7-19 Reconstruction of 3D density profiles . . . . . . . . . . . . . . . . . . 185
7-20 Correlating the condensate fraction with the observation of phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7-21 Phase diagram for trapped, strongly interacting, imbalanced Fermi
mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

14

Chapter 1
Introduction

This is how the worlds are formed. In a given section many atoms of allmanner of shapes are carried from the unlimited into the vast empty space.
These collect together and form a single vortex, in which they jostle against
each other and, circling round in every possible way, separate off, by like atoms
joining like. And, the atoms being so numerous that they can no longer revolvein equilibrium, the light ones pass into the empty space outside, as if they werebeing winnowed; the remainder keep together and, becoming entangled, go on
their circuit together, and form a primary spherical system.
Diogenes Laertius, Lives of eminent Philosophers, IX 31(Leucippus, fifth century B.C.)

Superfluidity, frictionless flow, is a most remarkable phenomenon that we come acrossin many different areas of physics. The most prominent examples are superconductors, charged superfluids, in which electrical current flows without resistance. Liquid
helium forms a superfluid at low temperatures. Superfluid phenomena occur in nuclei
and inside a neutron star, whose core is even believed to contain a superfluid of quark
pairs, and particles are thought to owe their very mass to a process similar to a superfluid phase transition: The formation of a Higgs condensate. It is indeed tempting
pto identify the excitation spectrum of a superconductor, ²2p + ∆2 (²p being the free15

particle dispersion, and ∆ the superfluid gap), with the relativistic expression for apparticle’s energy, p2 + m2 . This Higgs mechanism might be verified in the near
future in experiments at the new Large Hadron Collider (LHC).
Evidently, superconductors are of great relevance for technology and the transport
of energy. However, the record critical temperature below which materials would
become superconducting had for many decades been fixed at 23 K, well below any
easily accessible temperature. The discovery of high-temperature superconductors incuprates by Bednorz and Müller [29] in 1986 started an entire new era of research.
The record critical temperature quickly rose above liquid nitrogen temperatures, andis currently 138 K.
Clearly, the ultimate goal would be to find a room-temperature superconductor.
Rather than following a trial and error approach in the discovery of new materials, one
would like to fully understand the mechanism behind the phenomenon and predictmaterials with higher critical temperature. Unfortunately, the mechanism leading
to High-TC superconductivity is not fully understood. The superconducting state
comes about in an intricate compromise between several competing states. Strong
interactions between the electrons obscure our understanding of the normal state
above TC , that cannot be described as a simple collection of (quasi-)particles [147].
The success of ultracold atomic gases over the past 11 years gives new hope thatsome fundamental questions on superfluidity and strongly correlated systems mightfind an answer in the near future. Bose-Einstein condensates (BECs) of atoms provide
the paradigm of the weakly-interacting Bose gas [67, 136]. Atomic gases confined in
optical lattices allow the simulation of condensed matter systems, such as the studyof the superfluid-to-Mott insulator transition [101]. The advent of superfluids offermionic atom pairs, the topic of this dissertation, brings us in direct contact withthe phenomenon of superconductivity. Fermi gases give us access to the crossover froma Bose-Einstein condensate of tightly bound molecules to a superfluid consisting of
long-range Cooper pairs. The fermions are strongly interacting within this crossover.
This allows them to form a superfluid state already at a high critical temperature.
In fact, scaling the density of the atomic gas up to the density of electrons insolids, the critical temperature for superfluidity lies far above room temperature.
In the future, ultracold fermionic gases, confined in optical lattices with tunableinteractions and variable spin imbalance, might provide us with a basic model system
for High-TC superconductivity.
In this dissertation I will describe our work on the creation and direct observationof fermionic superfluidity in strongly interacting atomic gases. In the following I will16

briefly review the history of superfluids and superconductors, showing how ideas anddiscoveries in one field have always brought fruitful insight into the understanding of
the other. Two seemingly distinct ideas, the condensation of point-like bosons andthe condensation of long-range Cooper pairs, leading to vastly different predictions forthe critical temperature, are found to be simply two extremes of the same underlying
phenomenon.

1.1

Discovery of superfluidity and superconductivity — from BEC to BCS

In a remarkable ”accident of history”, the first superfluid was produced (without
noticing its remarkable properties) by the same scientist who later used it as a coolant
to discover superconductivity. Kamerlingh Onnes liquefied helium-4 in 1908, andlowered its temperature below the superfluid transition point (the λ-point) at Tλ =2.2 K. He remarked in his Nobel lecture 1913 ”that the density of the helium, which at
first quickly drops with the temperature, reaches a maximum at 2.2 K approximately,
and if one goes down further even drops again. Such an extreme could possibly beconnected with the quantum theory” [177]. But he first focussed his studies on the
behavior of metals at low temperatures. In 1911, Onnes used helium-4 to cool down
mercury, finding that the resistivity of the metal suddenly dropped to non-measurable
values at TC = 4.2 K, it became ”superconducting”. Tin (at TC = 3.8 K) and lead(at TC = 6 K) showed the same remarkable phenomenon.
The special properties of helium-4 below Tλ went practically unnoticed until the
1930s, when Keesom and Clusius discovered the λ-shaped peak in the specific heat
curve versus temperature at Tλ . In 1938, the vanishing viscosity of the substance wasindependently discovered by Kapitza [132] and Allen & Misener [8], suggesting theterm ”superfluid” in direct analogy with superconductivity. Obviously, the similar
phenomena of resistance-less currents in metals and frictionless flow in helium-4 hint
at a similar underlying mechanism. However, as we will see, the similarities in thetwo critical temperatures Tλ and TC turned out to be a complete accident of Nature.

1.1.1

Bose-Einstein condensation

F. London suggested in 1935 that superconductivity was a quantum mechanical phenomenon occurring on a macroscopic scale, stemming from a certain rigidity of theparticles’ wave functions against perturbations [155]. The lowest quantum state of the17system should be ”separated by a finite distance from the excited ones” and therefore
be protected against the influence of a weak perturbation. This is the essence of what
we now call the superconducting gap. In 1938, he proposed that quantum statistics
might play an important part in the explanation of superfluid helium-4. He was thefirst to make a connection to Bose-Einstein condensation [34, 86], at that time only atheoretical concept, in which a collection of non-interacting, indistinguishable bosons
condenses into one and the same macroscopic wave function. This occurs when thenumber of available quantum states becomes comparable to the number of particles N(a condition called ”degeneracy”), and bosons start to condense into a single quantum
state rather than being distributed over all possible states. As the extent of an average particle’s
wave function at temperature T is given by the de Broglie wavelengthq22π~
λ = mk(with m the particle mass), the number of available quantum states inBTa volume Ω is roughly equal to Ω/λ3 . Setting this equal to N we can deduce thecharacteristic temperature for reaching degeneracy and Bose-Einstein condensation,
2
TBEC ∼ 2π~(N/Ω)2/3 ≈ 3 K. For this estimate, we have inserted the mass m4 He of amhelium-4 atom and a typical density of n = 1022 cm−3 . The similarity of this value
to the observed critical temperature of 2.2K was intriguing. Furthermore, L. Tisza
pointed out that for the Bose gas below TBEC , one can make a natural distinction
between the condensed part of the gas and the thermal, or excited fraction [244].
Only particles in the thermal fraction take part in dissipative processes. This ideaof a two-fluid model is so fruitful that it applies quite generally to superfluids andconventional superconductors [22].
Of course, Einstein had considered a non-interacting Bose gas, a condition thatshould hardly be justified for the liquid helium-4. For this reason, L.D. Landaurejected the concept of Bose-Einstein condensation, and rather approached the problem of superfluidity with a highly successful phenomenological two-fluid model [144].
The important issue of how to combine the idea of BEC with interactions was dealt
with at a later time starting with Bogoliubov’s theory of the weakly-interacting Bosegas [150]. The fact that quantum statistics was crucial to the understanding of superfluid helium-4 was clarified beyond any doubt when liquid helium-3, a fermion,became available and did not show superfluid properties at temperatures on the scaleof Tλ .

18

1.1.2Fermionic superfluidity and Bardeen-Cooper-Schrieffer
theoryHow can the idea of Bose-Einstein condensation, the two-fluid model etc. be carriedover to superconductors? What determines their critical temperature TC ? Clearly,
the superfluid current must somehow be carried by electrons, but the degeneracy
temperature of this Fermi gas lies a factor m4 He /me > 7000 above the condensation
temperature for Helium-4 atoms, and cannot set the scale for TC . Condensation of
the electrons themselves is in any case not allowed. Electrons being fermions, theycannot occupy one and the same quantum state but instead must arrange themselvesin different states. At zero temperature, a gas of N fermions will fill all available
states one by one, starting with the ground state, up to the N th state at the Fermi
energy EF . For an electron density n ∼ several 1022 cm−3 , a similar counting as above~2
gives EF = 2m
(6π 2 n)2/3 ∼ 50 000K. An obvious idea might be to form tightly boundpairs of electrons, which can act as bosons and would form a condensate. Apartfrom the problem that the condensation temperature would still be on the order ofEF /kB , there is no known interaction which could be sufficient to overcome the strong
Coulomb repulsion and form tightly bound electron pairs (Schafroth pairs [213]).
Considering these problems, one can understand why it took more than four decades
from the discovery to the explanation of the phenomenon of superconductivity. It
turned out that the general idea of electrons forming pairs was after all correct.
In 1950, it became clear that there was indeed an effective attractive interaction between electrons, mediated by the crystal lattice vibrations (phonons), thatwas responsible for superconductivity. The lattice vibrations left their mark in the√characteristic variation TC ∝ 1/ M of the critical temperature TC with the isotope
mass M of the crystal ions, the isotope effect [206, 165] that had been predictedpby H. Fröhlich [93]. Vibrational energies in the lattice are a factor me /M smaller
than the typical electronic energy1 EF , on the order of kB × several 100 K (the Debye temperature TD of the metal). While the isotope effect strongly argues for TC
being proportional to TD , the Debye temperature is still one or two orders of magnitude higher than the observed critical temperature. In addition, attempts to explain
superconductivity by including electron-phonon interactions in a perturbative way
failed.
1

The average distance between electrons r0 is on the order of atomic distances (several Bohr
radii a0 ), the Fermi energy EF ∼ ~2 /me r02 is thus on the scale of typical Coulomb
q energies in an
atom. Vibrational energies of the lattice ions are then on the order ~ωD ≈ ~pp~ EF /M r02 ∼ me /M EF .

19

∂ 2 UCoulomb
/M
∂r 2

∼

A breakthrough came in 1956, when L. Cooper realized that fermions interacting
via an arbitrarily weak attractive interaction on top of a filled Fermi sea can form abound pair [66]. In other words, the Fermi sea is unstable towards pair formation.
However, unlike the tightly bound pairs considered before, the ”Cooper” pair is of avery large extent, much larger than the interparticle spacing. That is, a collectionof these pairs necessarily needs to overlap very strongly in space. In this situation,
it was initially far from obvious whether interactions between different pairs could
be simply neglected. But this was what led to the final goal: Bardeen, Cooper andSchrieffer (BCS) developed a full theory of superconductivity starting from a new,stable ground state in which pair formation was included in a self-consistent way [23].
Using the effective electron-electron interaction V , attractive for energies smaller thankB TD (and assumed constant in this regime), the pair binding energy was found tobe ∆ = 2kB TD e−1/N (0)|V | , with N (0) = me kF /2π 2 ~2 the density of states at the
Fermi energy and N (0)|V | assumed small compared to 1. The bound state energy (orpairing gap) depended in the non-analytic fashion e−1/N (0)|V | on the electron-electroninteraction V , explaining why the earlier attempts using perturbation theory had to
fail. Also, this exponential factor can now account for the small critical temperatures
TC ' 5 K: Indeed, it is a result of BCS theory that kB TC is simply proportional to∆(0), the pair binding energy at zero temperature: kB TC ≈ 0.57 ∆(0). Hence, thecritical temperature TC ∼ TD e−1/N (0)|V | is proportional to the Debye temperature
TD , in accord with the isotope effect, but the exponential factor suppresses TC by afactor that can easily be 100.
A remarkable example for such a BCS-type fermionic superfluid is helium-3, forwhich the superfluid transition had been observed in 1972 [183, 182] at TC ∼ 3
mK. Here the necessary attractive interaction between the helium-3 atoms occurs via(p-wave) collisions2 . The low TC can be compared to Tλ = 2.2 K for the bosoniccounterpart 4 He. Clearly, Cooper pairing is a fragile affair, strongly reducing TC fromthe degeneracy temperature scale.

1.1.3

The BEC-BCS crossover

As F. London had already suspected in 1950 [156], the BCS state can be understood asa pair condensate in momentum space, in contrast to a BEC of tightly bound pairs inreal space. The former will occur for the slightest attraction between fermions, while
the latter appears to require a true two-body bound state to be available to a fermion
2

Strong hard-core repulsion between the atoms renders s-wave pairing unfavorable.

20

BEC of Molecules

Resonance Superfluid

BCS state

Figure 1-1: The BEC-BCS crossover. By tuning the interaction strength between thetwo fermionic spin states, one can smoothly cross over from a regime of tightly boundmolecules to a regime of long-range Cooper pairs, whose characteristic size is muchlarger than the interparticle spacing. In between these two extremes, one encounters
an intermediate regime where the pair size is comparable to the interparticle spacing.
This strongly interacting ”soup” of particles is a fascinating state of matter that mightprovide a model for the interior of neutron stars or the Quark-Gluon plasma of theearly Universe.
pair. Eagles [85] noted in 1969 that, in the limit of very high density, the BCS statesmoothly evolves into a condensate of pairs that are smaller than the interparticle
distance (In the language of Fermi gases, he fixed the negative scattering length andvaried the interparticle spacing). Using a generic two-body potential, Leggett showedin 1980 that the limits of tightly bound molecules and long-range Cooper pairs areconnected in a smooth crossover [149] (here it was the interparticle distance that was
fixed, while the scattering length was varied, which allowed the inclusion of a true
two-body bound state). The size of the fermion pairs changes smoothly from being
much larger than the interparticle spacing in the BCS-limit to the small size of amolecular bound state (see Fig. 1-1). Accordingly, the pair binding energy varies
smoothly from its small BCS value (weak, fragile pairing) to the large binding energyof a molecule in the BEC limit (stable molecular pairing). The presence of a paired
state is in sharp contrast to the case of two particles interacting in free (3D) space:
For too weak an attraction, there is no bound state available to the particles. Only ata critical interaction strength does the molecular state become available and a boundpair can form. Leggett’s result shows that in the many-body system, the physics
changes smoothly with interaction strength, also at the point where the two-bodybound state disappears. Nozières and Schmitt-Rink extended Leggett’s model tofinite temperatures and verified that the critical temperature to attain superfluidity
21

varies smoothly from the BCS limit, where it is exponentially small, to the BEClimit where one recovers the value for Bose-Einstein condensation of tightly boundmolecules [172]. These results are presented in chapter 2.
The importance of considering strongly interacting fermions became clear with thediscovery of novel materials. Up to 1986, BCS theory (or more involved variationsof it) was largely successful in explaining the properties of superconductors. The
record critical temperature increased only slightly from 6 K in 1911 to 23 K in 1985.
In 1986, however, Bednorz and Müller [29] discovered superconductivity at 35 K in
the compound La2−x Bax CuO4 , triggering a focussed search for even higher criticaltemperatures. Soon after, materials with transition temperatures above 100 K were
found. Due to the strong interactions and quasi-2D structure, the exact mechanisms
leading to High-TC superconductivity are still not fully understood.
The physics of the BEC-BCS crossover in a gas of interacting fermions does not directly relate to the complicated phenomena observed in High-TC materials. However,some features of the two problems are shared: In the crossover regime, the pair size iscomparable to the interparticle distance. One therefore deals with a strongly correlated ”soup” of particles, where interactions between different pairs can no longer beneglected. Also the normal state above the phase transition temperature is a highly
non-trivial state. Even here, correlations are still large enough to form uncondensed
pairs at finite momentum. This region is sometimes called the ”Pseudo-Gap”.

1.1.4

Universal regime at unitarity

One point in the BEC-BCS crossover is of special interest: When the interparticle
potential is just about strong enough to bind two particles in free space, the bondlength of this molecule tends to infinity (unitarity regime). In the medium, thisbond length can thus not play any role anymore in the description of the many-bodystate. The only length scale of importance is then the interparticle distance n−1/3 ,the corresponding energy scale is the Fermi energy EF . In this case, physics is said
to be universal [119]. The average energy content of the gas, the binding energy ofa pair and (kB times) the critical temperature must be related to the Fermi energy
by universal numerical constants. The size of a fermion pair must be given by auniversal constant times the interparticle distance. This means that results obtainedwith one system at unitarity will directly carry over to another system (sharing thesame symmetry), even if their densities are vastly different. Results obtained in an
ultracold gas at unitarity, at densities of 1012 cm−3 , might thus have relevance for the22physics inside a neutron star3 at densities of 1038 cm−3 .

1.2

Ultracold atomic gases - model systems forbosonic and fermionic superfluids

When our understanding of a physical system is obscured by too many competing
effects, it is a natural approach to consider simpler, well-defined problems and increase
complexity in a controlled way. Instead of dealing with dense, strongly interacting
liquids like for example helium-4, it would be of great advantage to have at one’s
disposition a dilute gas, in which one can enhance interactions starting from the wellunderstood limit of weak interactions - for bosons, Bogoliubov’s weakly interacting
Bose gas. But what are the requirements to reach the regime of degeneracy in a gas?
They seem impossible to fulfill: Essentially all gases (helium being the exception)solidify at low temperatures.
The only way out can be to work at ultralow densities: It turns out that forming
a solid takes a certain amount of time: Three atoms have to meet before a moleculecan form, as the released energy and momentum has to be carried away by a third
particle. At densities of 1015 cm−3 , one has a window of about 1 s before molecules andlarger clusters can form in the gas. If this is long compared to the timescale needed
to reach thermal equilibrium, one can hope to observe a gas in the degenerate regime
before it collapses into its true solid ground state. Thermal equilibrium requires only
two-body collisions, their rate is thus proportional to the density, whereas molecule
formation requires three-body collisions, at a rate proportional to the square of thedensity. For low enough densities, the ”good” collisions occur at a faster rate thanthe ”bad” ones.
There is one major road block: Ultralow densities mean, given the condition fordegeneracy we saw above, ultracold temperatures. For n ∼ 1014 cm−3 , the degeneracy
temperature is on the order of µK! This seems already unrealistically low. But it
appears completely hopeless to even think about observing a BCS-like superfluid ina gas of fermions: The exponential suppression we have seen above would render thisphase transition unobservable.
Laser cooling and evaporative cooling were the key to achieve submicro-Kelvintemperatures in dilute atomic gases. In addition, in the seemingly hopeless case of3
The physics inside neutron stars is very intricate. This simple comparison rests on the observation that the neutron-neutron scattering length of about -18.8 fm is large in absolute terms comparedto the distance between neutrons at these high densities, which is only a few fm.

23

fermions, a ”magic knob” came to the rescue: Feshbach resonances.

1.2.1

Bosonic gases

The realization of Bose-Einstein condensation in ultracold atomic gases in 1995 has led
to a revolution in atomic physics which is still on-going [67, 136]. These condensates
represent the paradigm of Bogoliubov’s weakly interacting Bose gas. As such, they arevery well described by one macroscopic wave function that is the solution to a nonlinear Schrödinger equation, the Gross-Pitaevskii-equation. Beautiful experiments
for example on interference, on solitons and on vortices have directly demonstrated
coherence, the wave-like nature of the gas and superfluid flow [133, 208, 15]. Froma condensed matter perspective, these weakly interacting condensates represent themost fundamental, basic many-body wave functions.
The importance of interactions compared to the kinetic energy of the particles
can be increased in (at least) two ways: The first is to impose a lattice potential
via interfering laser beams and thereby localize particles onto lattice sites, quenching
their kinetic energy. A prominent example is given by the superfluid-to-Mott insulatortransition for bosons confined in a 3D lattice [101, 31]: As the repulsive interactions
start to dominate over the tunneling between adjacent lattice sites, atoms are pinned
to their respective site and long-range coherence and superfluidity is lost. This can beobserved as a loss of coherence peaks in momentum space, revealed after expansion
from the lattice, and more directly by measuring the insulator gap an atom needs toovercome in order to hop to an already occupied lattice site.
A second way to increase the role of interactions is to directly modify the interatomic scattering using a so-called Feshbach resonance [241, 243]. These resonances
occur when the energy of two colliding atoms coincides with the energy of a boundmolecular state in the interatomic potential, tunable via an applied magnetic field.
On resonance, the scattering length describing the interatomic interactions diverges.
Feshbach resonances were first observed in 1998 in a 23 Na Bose-Einstein condensate [126] and in a laser-cooled cloud of 85 Rb [68]. However, while it was shownin [126] that elastic scattering was indeed highly increased, unfortunately inelastic
scattering due to molecule formation was also resonantly enhanced, leading to strong
losses in the BEC. To the very fortunate surprise for the ultracold atom community,
gases of fermionic atoms were found not to decay close to a Feshbach resonance, butinstead to be very stable. This was the key fact that enabled all of the subsequentstudies on fermionic superfluids.

24

1.2.2Fermionic gases

The first degenerate Fermi gas of atoms was created in 1999 by B. DeMarco and
D. Jin at JILA using fermionic 40 K [76]. Until the end of 2003, six more groups had
succeeded in producing ultracold degenerate Fermi gases [247, 214, 100, 112, 207, 129],
one more using 40 K (M. Inguscio’s group in Florence, ’02) and five using fermionic
6
Li (R. Hulet’s group at Rice, C. Salomon’s group at the ENS in Paris, J. Thomas’
group at Duke, our group at MIT in 2001 and R. Grimm’s group in Innsbruck in2003). Already before the discovery of Feshbach resonances, the hopes to observe
fermionic superfluidity in lithium were high due to the unusually large and negative
triplet scattering length [3, 124]. Starting in 2002, experiments began to employ
Feshbach resonances to induce strong interactions between the fermions [78, 154,
175, 175, 130, 204, 106, 36]. As the energy of the Feshbach molecular state was tuned
below that of two free atoms (”molecular” or ”BEC” side of the Feshbach resonance),molecules could be produced [205] that were surprisingly long-lived [69, 128, 239, 202].
Petrov, Shlyapnikov and Salomon [192] pointed out that this remarkable stability of
the gas close to a Feshbach resonance is directly owed to the fermionic nature of theatoms: Inelastic three-body collisions, resonantly enhanced in a gas of bosons, arestrongly Pauli suppressed for a two-state mixture of fermions, as the process would
involve at least two fermions of the same spin state to be in close proximity. The
surprising longevity of the (bosonic!) molecules spurred hopes of their Bose-Einsteincondensation. Indeed, in November 2003 three groups reported the realization of
Bose-Einstein condensation of molecules [102, 271, 129] (also see [26]). In retrospect,
one can say that the binding of these ”molecules” already had a strong many-bodycharacter, as their size was not very small compared to the interparticle spacing. Soon
after, fermion pair condensates were observed via a rapid ramp from the ”BCS” sideof the Feshbach resonance, were no two-body bound state exists for isolated atom
pairs [203, 272]. Here, pairing was necessarily purely a many-body effect. Tuning themagnetic field across the Feshbach resonance thus gave access to the physics of the
BEC-BCS crossover described above.
Over the past two years, the properties of this ”tunable” fermionic superfluid
were studied in thermodynamic measurements [37, 139], experiments on collective
excitations [138, 25], RF spectroscopy revealing the formation of pairs [59], analysis of
the two-body part of the pair wave function [188], and finally, in April 2005, fermionicsuperfluidity and phase coherence was directly demonstrated at MIT through theobservation of vortices [265] (chapter 6).

25

1.2.3High critical temperatureThis new type of fermionic superfluid differs from Helium-3, conventional and even
High-TC superconductors in its high critical temperature TC when compared to the
Fermi temperature TF . Indeed, while TC /TF is about 10−5 . . . 10−4 for conventional
superconductors, about 10−3 for Helium-3 and 10−2 for High-TC superconductors, thestrong interactions induced by the Feshbach resonance allow the fermionic gases toenter the superfluid state already at4 TC /TF ≈ 0.2. It is this large value which allows
us to call this phenomenon ”high-temperature superfluidity”. Scaled to the density ofelectrons in a metal, this form of superfluidity would already occur far above room
temperature (actually, even above the melting temperature).
With gases of strongly interacting fermionic atoms we have a new ”model material” at our disposition, which can serve as a test-bed for many-body theories. All
the defining parameters of this material can be freely controlled, such as the density,
temperature, dimensionality and interaction strength. Already at the present time,
we are learning new facts about fermionic superfluidity in the presence of spin imbalance, which challenge current theories. For the future, one can expect that fermionsconfined in optical lattices will shed light on the workings of real materials, the mostimportant class of which being High-TC superconductors.

1.3

OutlineIn the following chapters, I summarize the work by my colleagues and me on theobservation of superfluidity in strongly interacting Fermi gases. This work was carriedout in the research group of Professor Wolfgang Ketterle at MIT.
I will focus the discussion on fermionic superfluidity. The production of degenerate Fermi gases [112, 111] and strongly interacting Fermi mixtures [106] has been
described in previous dissertations of our group [110, 104, 233] and will be briefly reviewed in chapter 4. The work on the absence of clock-shifts in an interacting Fermi
mixture [266], important for the interpretation of RF spectroscopy experiments on
fermionic superfluids [59], is included in appendix A. The observation of Feshbachresonances between two different atomic species [234], that might allow us to produce
heteronuclear molecules in future experiments, is not included but may be found in
the PhD Thesis of Claudiu Stan [233] who led the effort.
In chapter 2, we will introduce some of the theoretical tools necessary to treat
4

Predictions for the homogeneous case vary between 0.15 and 0.25 [172, 74, 190, 231, 39, 42, 6].

26

strongly interacting Fermi gases with tunable interactions, namely the variational
BCS approach due to Leggett, Nozières and Schmitt-Rink [149, 172]. Chapter 3
presents the phenomenon of Feshbach resonances, which has allowed to freely tune
the scattering length in ultracold atom gas experiments.
Chapter 4 describes the apparatus which was used for our work, a double-species
experiment that cools fermionic lithium atoms by sympathetic cooling with bosonicsodium.
In chapter 5, I present our observation of Bose-Einstein condensates of fermion
pairs, both on the molecular side of the Feshbach resonance ([271], reproduced inAppendix B) and on the ”BCS-side” ([272], Appendix C), where pairing is purely
a many-body effect. It also discusses the relevant time scales for the growth of afermion pair condensate, subject of the publication [270] reproduced in Appendix E.
At the end of this chapter, I include unpublished data on the density profiles of an
equal mixture on resonance that show a clear, direct signature of the condensate.
Chapter 6 discusses in its first part the experimental technique developed for setting optically trapped atomic gases into rotation. In its second part, the observationof vortices and ordered vortex lattices in strongly interacting Fermi gases is described,
which provided the direct demonstration of superfluidity and phase coherence in thesesystems [265]. This work is reproduced in Appendix F. Vortex lattices enabled us to
study the expansion of a rotating Fermi gas [216], Appendix G.
Finally, chapter 7 reviews our work on fermionic superfluidity with imbalanced
spin populations. This has been subject of three publications, on imbalanced superfluidity and the Clogston limit [268] (Appendix H), the direct observation of thephase transition [269] (Appendix I) and the observation of phase separation between asuperfluid core of equal densities and a shell of excess atoms at unequal densities [223].

27

Chapter 2
Superfluidity in bosonic andfermionic gases
This chapter will present some of the theory of superfluidity in bosonic and fermionicgases, relevant to our experiments on strongly interacting gases. We will first study
the physics of the two ”extremes” of the BEC-BCS crossover, the weakly interacting
Bose gas on the one hand and the formation of Cooper pairs on the other. Finally,
we will see how the variational BCS Ansatz is able to smoothly connect these twolimits [85, 149, 172].

2.1

Quantum statistics at work - bosons vs fermions

We will start with a quick reminder of quantum statistics, second quantization notation and the grand canonical ensemble. For details we refer the reader to the many
excellent books on many-body theory, for example by Fetter and Walecka [90] andby Abrikosov, Gorkov and Dzyaloshinski [5]. In section 2.1.1, we closely follow thepresentation in [90].

2.1.1

Quantum statistics and second quantization

We consider a collection of N indistinguishable particles in a system described bysingle-particle states ψEk (x), with x denoting the collection of spatial and spin coordinates and Ek the appropriate set of quantum numbers. For example, for atoms
with spin confined in a box, Ek denotes the definite momentum k and z-component28

of spin σ. Let A(E1 , . . . , EN ) be the probability amplitude to find particle 1 in state
E1 , particle 2 in E2 , etc. Since the particles are indistinguishable, the probability tofind the ath particle in state Ea and the bth particle in state Eb must equal the one
for which the particles a and b are exchanged:|A(. . . Ea . . . Eb . . . )|2 = |A(. . . Eb . . . Ea . . . )|2(2.1)

This leaves two possibilities for the probability amplitudes under exchange:
A(. . . Ea . . . Eb . . . ) = ±A(. . . Eb . . . Ea . . . )(2.2)

Thus, the concept of indistinguishability immediately leads to the grouping of allparticles into two fundamental classes: bosons (upper sign here and in the following)
and fermions (lower sign). Quantum field theory teaches us that particles with integer
spin are bosons, particles with half-integer spin are fermions (Pauli’s spin-statistics
theorem [189, 238]).
Indistinguishability implies that we cannot decide which particle is in what quantum state. The only information which we can hope to measure is how many particles
occupy a particular state. From Eq. 2.2 directly follows that two fermions can neveroccupy the same quantum state (Pauli’s exclusion principle),A(. . . Ea . . . Ea . . . ) = −A(. . . Ea . . . Ea . . . )⇒ A(. . . Ea . . . Ea . . . ) = 0 for Fermions(2.3)

For bosons, no such restriction applies. We can then fully specify a collection of
N bosons by giving the particle numbers nk occupying each quantum state Ek (withPi ni = N ). For example, if we have two bosons, and we know that state E1 andstate E2 is each occupied, the total wavefunction of the system is1
Φ1,1 (x1 , x2 ) = √ (ψE1 (x1 )ψE2 (x2 ) + ψE1 (x2 )ψE2 (x1 ))
2

(2.4)

However, if we know that only state E1 is occupied (n1 = 2), the wavefunction reads
Φ2,0 (x1 , x2 ) = ψE1 (x1 )ψE1 (x2 )(2.5)

Note that for correct normalization, there is no factor √12 in front. For a general stateof N bosons, the basis states specifying state E1 occupied n1 times, state E2 occupied

29

n2 times, etc., isµΦn1 n2 ...n∞ (x1 , . . . , xN ) =n1 !n2 ! . . . n∞
N!

¶1/2

X

ψE1 (x1 )ψE2 (x2 ) . . . ψEN (xN )E1 ...EN(n1 n2 ...n∞ )(2.6)
where the sum is over all different distributions of N particles over the states, with n1
particles in state 1, n2 in state 2 etc. There are N !/n1 !n2 ! . . . n∞ ! such arrangements,
which explains the normalization factor.
For fermions, the Pauli principle states we will never have more than 1 particle
occupying a given state, nk = 0 or 1. Two fermions in two distinct states E1 and E2
are described by the fully antisymmetric wavefunction1Φ1,1 = √ (ψE1 (x1 )ψE2 (x2 ) − ψE1 (x2 )ψE2 (x1 ))
2

(2.7)

For N fermions, the basis states can be conveniently written in terms of a Slater
determinant,¯¯ ψ (x ) ψ (x )¯ E1 1E1
2¯¯
1 ψE (x1 ) ψE2 (x2 )Φn1 n2 ...n∞ (x1 , . . . , xN ) = √ ¯¯ 2...
N! ¯ . . .
¯¯ψEN (x1 ) ψEN (x2 )¯ψE1 (xN ) ¯¯¯
. . . ψE2 (xN ) ¯¯...
. . . ¯¯¯
. . . ψEN (xN )¯
...

(2.8)

This ensures the required antisymmetry under exchange of any two particle coordinates.
Many-particle operators
Consider a typical operator ta (xa ) in standard ”first” quantization (for example thekinetic energy or an external potential), that we constrain to act only on the coordinate xa of the ath particle. The fully symmetric operator with respect to all N
particles is
N
X
T =
ta
(2.9)
a=1

It is not hard to see that this operator only allows transitions of the many-body state
for which the occupation numbers nk are either unchanged or those in which theoccupation of one state is reduced by one and that of another state is augmented
by one (one particle has changed its state). It is a simple exercise to show that for30example for bosons, in the first case,hΦn1 n2 ...n∞ | T |Φn1 n2 ...n∞ i =X

hk| t |ki nk(2.10)k

For the second case, one finds
¯ ¯
­
®
√
Φn1 ...ni ...nj −1... ¯ T ¯Φn1 ...ni −1...nj = hi| t |ji ni nj(2.11)√
One can imagine that keeping track of all the factors ni etc. in calculations
can get extremely tedious, especially when two-particle operators are introduced thatact on two particle coordinates. Second quantization provides a very elegant wayof ”book-keeping” for these combinatoric factors arising in calculations as well as
ensuring the correct full (anti-)symmetry of the many-body state.
Second quantization
For this, a natural set of basis states is defined:|n1 n2 . . . n∞ i ≡ |n1 i |n2 i . . . |n∞ i(2.12)

which describes n1 particles in state 1, n2 in state 2 etc. For bosons, all occupation
numbers are allowed. One associates particle ”creation” and ”annihilation” operatorsa†k and ak to each mode |nk i that satisfy the commutation ruleshi††ak , a j−= 0[ak , aj ]− = 0hi†ak , a j= δij−

for bosons(2.13)

where [A, B]− = AB − BA. These are the same commutation rules as for the creation
and annihilation operator of a harmonic oscillator. Thus, they act just in the sameway, namely by raising and lowering the number of particles in state Ek : a†k |nk i =√√nk + 1 |nk + 1i, ak |nk i = nk |nk − 1i, and especially ak |0i = 0, where |0i is thevacuum state. a†k ak is the number operator.
For fermions, the occupation numbers are necessarily nk = 0 or 1. This is ensured
if the following anticommutation rules are imposed on the fermionic operators c†k , ck :

31

i
h
† †
c k , cj+

= 0

[ck , cj ]+ = 0hick , c†j= δij+

for fermions(2.14)

where [A, B]+ = AB + BA. We see that for fermions, c†k c†k = 0, which is again
Pauli’s exclusion principle. It also follows c†k |0i = |1i, c†k |1i = 0, ck |1i = |0i andck |0i = 0.
As an example, for N non-interacting (and spinless) bosons confined in a box, the(unnormalized) ground state will simply beΨbosons= (a†0 )N |0, 0, 0, . . . i ∝ |N, 0, 0, . . . i
N(2.15)

with a†0 the creation operator for the single-particle ground state with k = 0. Thatis, all bosons occupy the same state: this is the Bose-Einstein condensate. Fermions,on the other hand, cannot occupy one and the same single-particle state and needto arrange themselves in different states. At zero temperature, N fermions will thus
occupy the N lowest single-particle states:Ψfermions=NYc†k |0, 0, 0, . . . i

k<kFwith kF defined such thatP∝

| 1, 1, . . . , 1, 0, 0, . . . i| {z }
N times(2.16)

k<kF = N .

Many-particle operators in second quantization
With these definitions, it turns out that we can now represent the operator T fromabove by the second quantized operatorT̂ =X

hi| t |ji a†i aj(2.17)

i,jIndeed, the matrix elements of T̂ in the states |n1 n2 . . . n∞ i are identical to the onesof the first quantized operator T in Eqs. 2.10 and 2.11, as is easy to check. A manybody hamiltonian involving the kinetic energy operator t(x) and a potential V (xa , xb )32can thus be written asĤ =X

hi| t |ji a†i aj +i,jXa†i a†j hij| V |kli al ak(2.18)

i,j,k,lwhere the ordering in the last term is important if we deal with fermions (replacing
the bosonic operators ak with the fermionic ones ck ).
P
PIntroducing the field operators Ψ̂(x) = k ψk (x)ak and Ψ̂† (x) = k ψk (x)† a†k one
can rewrite Ĥ as1
Z
Z
1
3
†
Ĥ = d xΨ̂ (x)t(x)Ψ̂(x) +d3 xd3 x0 Ψ̂† (x)Ψ̂† (x0 )V (x, x0 )Ψ̂(x0 )Ψ̂(x)(2.19)
2
In this form, the origin of the name ”second quantization” is apparent: The
hamiltonian is ”disguised” as the expectation value of operators t(x) and V (x, x0 ) in astate described by a wavefunction Ψ̂(x). However, here, Ψ̂(x) and Ψ̂† (x) are operators,
not wave functions, and t(x) and V (x, x0 ) are complex numbers, not operators.

2.1.2

Bose-Einstein versus Fermi-Dirac distribution

The postulate of statistical mechanics states that an isolated system in equilibrium
is equally likely to be found in any of its accessible states. Accessible states are thosethat are consistent with the specified number N and total energy Etot of the system(micro-canonical ensemble). However, it is often convenient to consider a system
in contact with a reservoir, with which it can exchange particles and energy (grandcanonical ensemble). The probability that the system is in any state with particle
number N and energy Etot is then given by the Boltzmann factor e−(Etot −µN )/kB T ,where the temperature T and the chemical potential µ are imposed by the large
reservoir.
For non-interacting particles with single-particle spectrum Ei , Etot for the manyP
particle state |n1 n2 . . . n∞ i is thus simply given by i ni Ei . The grand canonicalpartition function Z is then
Z =

P

=
1

{ni } e

P

{ni }

P
−( i Ei ni −µni )/kB T

Q−ni (Ei −µ)/kB Tie

To simplify the notation, we assume here that x denotes a spatial coordinate.

33(2.20)(2.21)

For bosons, all ni run from 0 to ∞, so that
Y

ZB =

i

1
1 − e−(Ei −µ)/kB T(2.22)For fermions, the only possible values for ni are 0 and 1, so that
ZF =Y

(1 + e−(Ei −µ)/kB T )(2.23)i

The average occupation of state i is thenhni i = kB T

∂ ln Z1= (Ei −µ)/k T
B
∂µ
e
∓1

(2.24)

again with the upper sign corresponding to bosons, the lower sign to fermions. These
are the Bose-Einstein and Fermi-Dirac distributions, respectively. Typically, the numPber of particles is fixed, which can be achieved by choosing µ such that N = i hni i.

2.1.3

Ideal Bose and Fermi gases in a harmonic trap

Let us now consider particles confined in a harmonic trap, with trapping potential
1
V (r) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )2(2.25)

Typical values in our experiment are ωz = 2π · 20 Hz and ωx = ωy = 2π · 100 Hz. The
single particle spectrum is
¶
µ
¶
µ
¶ ¸
·µ
1
1
1
Enx ,ny ,nz = ~ nx +
ωx + ny +ωy + nz +ωz2
22(2.26)

We will assume that the thermal energy kB T is much larger than the level spacings~ωx,y,z . In this case, sums over discrete states can be replaced by integrals over phasespace, which is divided in cells of size h. The occupation of phase space cell {r, p} isthen given by the semi-classical distribution1

f (r, p) =ep2
+V (r)−µ
2m

34

/kB T

(2.27)∓1

The density distribution of the gas (for bosons: not including the condensate) isZ

d3 pf (r, p)(2π~)3¡¢
1= ± 3 PolyLog 3/2, ±e(µ−V (r))/kB T
λdBn(r) =

q
where

2π~2
mkB T

(2.28)

is the de Broglie wavelength and PolyLog(n, λ) is the nth -order Poly-

logarithm, defined as
1
PolyLog(n, λ) ≡ n
π

Z

1
1
n6=0
=
d r r2Γ(n)e /λ − 1
2n

Z ∞
0

q n−1
dq q
e /λ − 1(2.29)

where the first integral is over 2n dimensions, r is the radius vector in 2n dimensions,n is any positive half-integer or zero and Γ(n) is the Gamma-function2 . Note that
expression 2.28 is correct for any potential V (r). However, we still need to relate thechemical potential to the given particle numberZN=

d3 r n(r)(2.31)For a harmonic potential 2.25, we obtain using Eq. 2.30 (see footnote below):
µ
N =±

kB T
~ω̄

¶3
PolyLog(3, ± eµ/kB T )(2.32)

with ω̄ = (ωx ωy ωz )1/3 the geometric mean of the trapping frequencies.
In the classical limit at high temperature, both density distributions go over tothe Maxwell-Boltzmann result, that is
ncl (r) =PN− i x2i /2σx2ie(2π)3/2 σx σy σz2with σx,y,z
=

kB T
2
mωx,y,z(2.33)

so density is given by the number of particles times the classical probability to find a2

1, PolyLog(1, λ) = −Ln(1 − x), PolyLog(2, λ) = Li2 (λ),
Special cases: PolyLog(0, λ) = 1/λ−1
p2

the Dilogarithm. We see that f (r, p) can be written as ±PolyLog(0, ±e(µ− 2m −V (r))/kB T ). A usefulformula for integrals over Polylogarithms is
Z ∞
√
2
1
dr PolyLog(n, λ e−x ) = π PolyLog(n + , λ)(2.30)
2−∞
which can be quite directly seen from the definition Eq. 2.29. Limiting values: PolyLog(n, λ) →λ¿1 λ1lnn (λ).
and −PolyLog(n, −λ) →λ→∞ Γ(n+1)

35

particle at point r, which is simply a gaussian for a harmonic potential.
However, as the temperature decreases, the central density grows to the point
where the de Broglie waves of different particles start to overlap and we enter the
regime of quantum degeneracy. For bosons, it is at this point that the groundstate becomes macroscopically occupied, the condensate forms. As the number ofcondensed atoms N0 grows to be significantly larger than 1, from Eq. 2.24 we find therelation between N0 and the chemical potential µ = − kNB0T . For N0 large, µ will bemuch closer to the ground state energy than even the first excited harmonic oscillator
state. We can thus set µ = 0 in the expression for the non-condensed density nth andnumber Nth and obtain
1
PolyLog(3/2, e−V (r)/kB T )λ3dB= N (T /TC )3nth (r) =(2.34)

Nth(2.35)

with the critical temperature
TC ≡ ~ω̄ (N/ζ(3))1/3 = 0.94 ~ω̄N 1/3(2.36)

where ζ(3) ≈ 1.202. The maximum density of the thermal cloud is capped at thecritical value n = 2.612/λ3dB : The gas is a quantum saturated vapor. The density
profile of the thermal cloud is a bit more peaked than a gaussian, a direct effect ofbosonic stimulation at low momenta [232]. The condensate fraction is given by
N0 /N = 1 − (T /TC )3(2.37)

We note that for T /TC = 0.5 the condensate fraction is already almost 90%. The
density profile of the condensate is simply given by the square of the ground state
wave function of the harmonic oscillator:−
N0enc (r) = 3/2
π ax ay azwhere axi =

q

P

x2
i
i a2
xi

(2.38)~mωxi

are the harmonic oscillator lengths. The width of the condenqi¿ 1,sate profile is much smaller than that of the thermal cloud, axi /σxi = k~ωBT
N λ3

and the central density of the condensate is much larger, nc (0)/nth (0) ∝ ax0aydB∝
az√ T 3/2
3
C
N ( T ) (1−(T /TC ) ). Condensation in a harmonic trap thus results in a dramaticchange in the density profile of the bosonic gas, the condensate is clearly visible as a36

central dense feature in the cloud. We will see below that this qualitative signatureof condensation survives in the presence of weak interactions, but is no longer true inthe case of strongly interacting gases (see chapter 5).
For fermions, nothing dramatic happens around the degeneracy temperature.
The occupation of available phase space cells approaches 1 in a smooth fashion.
Accordingly, the density profile smoothly changes from its gaussian form at hightemperatures to its zero temperature shape given by
1
nF (r) = 2 kF3
6π

Ã
1−

X x2

!3/2

i

i

(2.39)

Ri2

2EF2, the Fermi wave vector kF =with the Thomas-Fermi radius Rx,y,z
= mω
2
x,y,z
and the Fermi energy defined by the number of fermions N ,ZN=1
d r nF (r) =
6
3

µ

EF
~ω̄

p

2mEF /~2¶3(2.40)or EF = ~ω̄(6N )1/3 . All these equations can be directly obtained from n(r) =
R
2
2
d3 k
F (r)
= 6π1 2 kF (r)3 using the local Fermi energy at point r, ²F (r) = ~ k2m=
k<kF (r) (2π~)3EF − V (r).

2.2Interacting gases

At low temperatures, the stable phase for practically all substances is the solid. The
only exception to that rule is helium, which remains a liquid even at zero temperature,
due to the large zero-point motion of the very light atoms. But even though the gas
phase is only metastable, at the ultralow densities we deal with in our experiments
it can take minutes until the gas solidifies. The reason is that three-body collisions
are needed to form molecules and larger clusters, and these collisions are extremelyrare at our densities3 of ∼ 1012−15 cm−3 . However, two-body collisions are importanteven at these low densities. Elastic collisions between atoms redistribute momenta,
allowing thermalization of the gas, and let particles experience attraction or repulsion
from the surrounding gas. Inelastic collisions between two particles lead to a change
in the internal state of one or both collision partners, leading to trap loss. These
collisions are unwanted but can be eliminated by working with the lowest internal
3This is only true away from scattering resonances. If a weakly bound molecular state is available
to the atoms, three-body collisions can populate this state efficiently, even at our low densities.

37

energy states of the atoms.

2.2.1

Elastic collisions

If we neglect the weak magnetic dipole interaction between the spins, the interatomic
interaction is described by a central potential V (r). At large distances from eachother, atoms interact with the van der Waals-potential −C6 /r6 as they experienceeach other’s fluctuating electric dipole4 . At short distances on the order of a few
Bohr radii a0 , the two electron clouds strongly repel each other, leading to ”hardcore” repulsion. If the spins of the two valence electrons (we are considering alkali
atoms) are in a triplet configuration, there is an additional repulsion due Pauli’s
exclusion principle. Hence, the triplet potential VT (r) is shallower than the singlet
one VS (r).
The exact inclusion of the interatomic potential in the description of the gas would
be extremely complicated. However, the gases we are dealing with are ultradilute:
The range of the interatomic potential r0 (on the order of the van der Waals lengthr0 ∼ (µC6 /~2 ) ∼ 50 a0 for 6 Li) is much smaller than the interparticle distance n−1/3 ∼5 000−10 000 a0 . Also, our samples are ultracold, that is, typical collisions occur at low
relative momenta k ¿ 1/r0 with an associated de Broglie wavelength λ = 2π/k À r0 .
The scattering particles thus never explore the fine details of the short-range scattering
potential. The entire collision process can thus be described by a single quantity, thescattering length.
What follows is a quick reminder on scattering theory including the definition of
the scattering length.
The scattering amplitude
The Schrödinger equation for the reduced one-particle problem in the center-of-massframe of the colliding atoms (with reduced mass m/2 and distance vector r) is(∇2 + k 2 )Ψ(r) = v(r)Ψ(r) with k 2 =mE~2and v(r) =mV (r)~2(2.41)

Far away from the scattering potential, the wave function Ψ(r) will be given by thesum of the incident plane wave eik·r and an outgoing scattered wave, which satisfies
4
For distances on the order of or larger than the characteristic wavelength of radiation of theatom, λ À r0 , retardation effects change the potential to a −1/r7 law.

38(∇2 − k 2 )Ψout (r) = 0:

eikr
(2.42)rf (k, k0 ) is the scattering amplitude for scattering an incident plane wave with wave
vector k into the direction k0 (energy conservation implies k 0 = k). The probability
per unit time for the scattered particle to traverse the surface element dS = r2 dΩ
is v|f |2 /r2 dS = v|f |2 dΩ (v is the incident - and final - velocity). Dividing by theΨ(r) = eik·r + f (k, k0 )incident particle flux we arrive at the differential scattering cross sectiondσ = |f |2 dΩ(2.43)

Since we deal with a central potential, the scattered wave must be axially symmetricwith respect to the incident wave vector k, that we choose to lie in z-direction (noazimuthal φ-dependence). We can thus expand Ψ(r) into the basis of free sphericalwaves with angular momentum corresponding to quantum numbers l and zero zprojection [145]:
Ψ(r) =∞X

Pl (cos θ)Rkl (r)(2.44)

l=0

The radial functions Rkl (r) verify a radial Schrödinger equation containing the potential V (r) and the centrifugal potential ~2 l(l + 1)/mr2 that depends on the angularmomentum quantum number l. Far way from the potential, the Rkl (r) must behavelikeπ1(2.45)
Rkl (r) ∝ sin(r − l + δl )r
2
The incident plane wave can also be expanded like 2.44, but here δl = 0. The entire
effect of the potential V (r) on the incoming wave is thus to add a phase shift δl toeach spherical wave with angular momentum l. This quite remarkable result allows
us to write the scattering amplitude entirely in terms of the δl :∞f (θ) =

1 X
(2l + 1)(e2iδl − 1)Pl (cos θ)
2ik l=0(2.46)The total cross section is given by∞

4π X(2l + 1) sin2 δl
σ= 2
k l=0

(2.47)For our ultracold collisions, we are interested in describing the scattering process at39

low momenta k ¿ 1/r0 , where r0 is the range of the interatomic potential. That is, theparticles do not have enough energy to resolve the fine details of the potential (theirde Broglie wavelength 2π
being much larger than the size of these fine details). Usingkthe radial Schrödinger equation one can show that for low momenta δl ∼ k 2l+1 and1thus each partial wave fl = 2ik(e2iδl − 1) appearing in 2.46 behaves like fl ≈ δkl ∼ k 2l .
In the absence of resonance phenomena for l 6= 0, s-wave scattering l = 0 is dominant
over all other partial waves:f ≈ f0 =1 2iδ0(e− 1)2ik(2.48)We already know that at low momenta, δ0 ∼ k. It is thus useful to define the scattering
lengthtan δ0a = lim(2.49)
k¿1/r0kwith which we can rewrite f as
f =−a
1 + ika(2.50)In the limit ka ¿ 1, f becomes independent on momentum and equals −a. For
ka À 1, the scattering amplitude is limited by f = − ik1 and the cross section σ = 4π.
k2This is the so-called unitarity limit.
Consequence of quantum statistics
If the colliding particles are identical, there are two scattering processes with thesame outcome: direct and exchange scattering. In the case of bosons, the two-particle
wave function needs to be symmetric and we have to add the corresponding scattering
amplitudes. For fermions, the wave function needs to be antisymmetric, so we have tosubtract the amplitudes. Partial waves of even angular momentum quantum number
are symmetric under exchange, those with odd l are antisymmetric. It follows thatthe total scattering amplitude for bosons only contains (twice) the even terms in
the partial wave expansion, whereas for fermions it only contains (twice) the odd
terms. The total s-wave scattering cross section for two identical bosons is σ = 8πa2 ,while for two identical fermions the s-wave scattering cross section is identically zero.
This means that a polarized Fermi gas at ultracold temperatures is essentially noninteracting.

40

Pseudo-potentials
If the de Broglie wavelength 2πof the colliding particles is much larger than thekfine details of the interatomic potential, 1/k À r0 , we can ”cheat” a bit and modify
the potential in such a way that it is much easier to manipulate in the calculations,but still reproduces the correct s-wave scattering. An obvious candidate for such a”pseudo-potential” is a delta-potential δ(r). However, there is a subtlety involved
which we will address in the following. The goal is to find an expression for the2v(r)scattering amplitude f (k, k0 ) in terms of the potential V (r) = ~ m, so that we cantry out different pseudo-potentials, always ensuring that f → −a in the s-wave limit.
For this, let us go back to the Schrödinger equation Eq. 2.41. If we knew the solution
to the following equation:(∇2 + k 2 )G(r) = δ(r)(2.51)

we could write an integral equation for the wavefunction Ψ(r) as follows:Z
ik·r

Ψ(r) = ed3 r0 G(r − r0 )v(r0 )Ψ(r0 )+(2.52)

This can be simply checked by inserting this implicit solution for Ψ into Eq. 2.41.
G(r) can be easily obtained from the Fourier transform of Eq. 2.51, defining G(p) =R 3 −ip·r
d re
G(r):
(−p2 + k 2 )G(p) = 1(2.53)The solution for G(r) isZ
G+ (r) =

d3 p
eip·r
1 eikr=−(2π)3 k 2 − p2 + iη
4π r

(2.54)

where we have chosen (by adding the infinitesimal constant iη, with η > 0 in the
denominator) the solution that corresponds to an outgoing spherical wave, which iswhat we need. G+ (r) is the Green’s function of the scattering problem. Far awayfrom the origin, |r − r0 | ∼ r − r0 · u, with the unit vector u = r/r, andΨk,+ (r) ≈ e

ik·r

eikr
−
4πr

Z
0

0

d3 r0 e−ik ·r v(r0 )Ψk,+ (r0 )(2.55)

Now using the definition of the scattering amplitude in Eq. 2.42, this gives us aformula for f (k, k0 ):1
f (k, k ) = −
4π
0

Z
0

0

d3 r0 e−ik ·r v(r0 )Ψk,+ (r0 )41(2.56)

The fourier transform of the product v(r0 )Ψk,+ (r0 ) equals the convolution of the fouriertransforms of both factors, v(q) and Ψk (q):1
f (k, k ) = −
4π

Z

0

d3 q
v(q)Ψk (k0 − q)(2π)3(2.57)On the other hand, Ψk (q) satisfies the integral equation Eq. 2.52 written in momentum space:1Ψk (p) = (2π) δ(p − k) + 2
k − p2 + iη
4πf (k, k0 )3= (2π) δ(p − k) − 2
k − p2 + iη3Multiplying by v(q − p), integrating

R

d3 p
(2π)3

Z

d3 q
v(q)Ψk (k0 − q)3(2π)(2.58)(2.59)

and relabelling yields

Z−4πf (p, k) = v(p − k) − 4π

d3 q v(p − q)f (q, k)(2π)3 k 2 − q 2 + iη

(2.60)

This is an integral equation for f in terms of the potential v, useful to perform
perturbation expansion. If we only keep the first order in v, we obtain the scattering
amplitude in Born approximation:f (p, k) = −1v(p − k)4π(2.61)

which is simply given by the fourier transform of the potential, evaluated at thetransferred momentum vector p − k. For a delta-potential V (r) = V0 δ(r), we
obtain as f → −a4π~2 aV0 =(2.62)
m
However, already the second order term in the expansion of 2.60 would not converge,
R d3 q 1
as it involves the integral (2π)
3 q 2 . The reason is of course that the δ-potential does
not fall off at large momenta (ultra-violet divergence). Any physical potential does
fall off at some large momentum, so this is not a ”real” problem. We can formallywork with the fourier transform V0 of the delta-potential in the following, but if weare to calculate physical quantities, we should replace it in favor of the observableand meaningful quantity a using the formal prescription
1
m
m
=
− 2
2
V0
4π~ a ~
42Z

d3 q 1(2π)3 q 2

(2.63)

(for this, use the limit k = p = 0 in Eq. 2.60 and replace f → −a). We will always
find that the diverging integral is exactly balanced by another diverging integral in
the final expressions, so this is a ”sane” procedure [74, 115].

2.2.2

The interacting Bose gas

We are now in the position to formulate the problem of an interacting Bose gas. The
second-quantized hamiltonian reads
Z
Ĥ =

1
~ 2 ∇2+ Vext (r))Ψ̂(r) +d rΨ̂ (r)(−
2m
2
3

†

Z
d3 r d3 r0 Ψ̂† (r)Ψ̂† (r0 )V (r, r0 )Ψ̂(r0 )Ψ̂(r)(2.64)

where Vext (r) is the external trapping potential of Eq. 2.25 and V (r0 , r) = V (r − r0 )is the interatomic potential, for which we can choose the Pseudo-potential V (r) =gδ(r) of the preceding section. As long as we work in real space, we can safely set
2ag = 4π~. We will limit our discussion to the case of repulsive interactions, relevantmto our experiment. Let us assume that even in the interacting system, there still
is a macroscopic occupation of a single-particle state ψ(r). That is we set Ψ̂(r) =ψ(r) + δ Ψ̂(r),withDE ψ(r) being the thermal average of Ψ̂(r) in the many-body state:ψ(r) = Ψ̂ (r) , and δ Ψ̂(r) represents the fluctuations around this ”mean-field”.
Neglecting these fluctuations leads to the time-dependent Gross-Pitaevskii equationfor ψ(r, t):∂i~ ψ(r, t) =
∂t

µ 2 2
¶
~∇
2
−
+ Vext (r) + g |ψ(r, t)| ψ(r, t)
2m(2.65)

Its validity is limited to weakly interacting gases, for which na3 ¿ 1. In typical experiments on alkali BECs, this condition is very well fulfilled. For a sodium BEC withn ≈ 1014 cm−3 and a = 3.3 nm, we have na3 ≈ 4 · 10−6 . However, for the molecularcondensates produced in our experiment, the scattering length can be tuned at will,
so this condition can be violated. We will see in section 2.3 how to approximately
describe this regime.
The ground state wave function can be obtained from Eq. 2.65 by setting ψ(r, t) =−iµt/~
e
ψ(r):
µ 2 2
¶
~∇
2
−
+ Vext (r) + g |ψ(r)| ψ(r) = µψ(r)(2.66)
2m
In the Thomas-Fermi approximation, it is assumed that the condensate wave function
is only slowly varying in space such that one can neglect the kinetic energy term.
This is valid as long as interactions ∼ gn dominate over the zero-point motion of
the particles ∼ ~ω. Already for weakly interacting alkali gases, this condition is very
43

well fulfilled (not too close to the edge of the condensate), with typical interactionenergies in our experiment of gn ∼ 150 nK and ~ωr ≈ 4 nK. In this approximation
we obtain the condensate density nc (r) = |ψ(r)|2 :µnc (r) = max¶µ − V (r),0g(2.67)

The chemical potential µ is constrained by the normalization condition on nc (r):R
N = d3 r nc (r). For the harmonic trapping potential of Eq. 2.25, we obtain (comparewith the expression for a Fermi gas, Eq. 2.39)
Ãnc (r) = nc0 1 −

X x2
i

!

i
Ri2

with the central density nc0 = µ/g, the Thomas-Fermi radii Ri =1
µ = ~ω̄2µ

15N aāh.o.

(2.68)
q

2µ
mωi2

and

¶2/5
(2.69)

where āh.o. = (ax ay az )1/3 is the geometric mean of the harmonic oscillator lengths.
Interactions thus have a major effect on the shape of the Bose-Einstein condensate.
Instead of a gaussian, the density profile is now parabolic. The size of the condensate
is no longer given by the harmonic oscillator length but by the much larger Thomas´1/5
³
´1/5
³
15N a
a
.
The
parameter∼ 6 for a typicalFermi radius Rx,y,z = ax,y,z 15Nāh.o.
āh.o.
sodium BEC. However, such weakly interacting condensates are still considerably
smaller in size than a thermal cloud, as long as T > µ/kB , and more dense: At T = 0,
the density of a typical sodium BEC is an order of magnitude higher than the densityof a thermal cloud at TC . We conclude that the condensate in weakly interacting,trapped Bose gases can still be detected as a dense, central feature emerging in the
midst of a large thermal cloud. In the case of strong interactions, when µ becomescomparable to (kB times) TC , this direct signature will be considerably weaker (seechapter 5).
At finite temperatures, the question arises whether interactions in the thermal
cloud, and between the thermal cloud and the condensate have to be taken into2
account. In alkali BECs, gnth ∼ ~ma λ31 ∼ kB T (nth a3 )1/3 ¿ kB T , so one can neglectdB
interactions in the thermal cloud. The condensate is also practically unaffected by
the thermal cloud. In turn, however, the condensate has quite a strong effect on
the thermal atoms, as gnc = µ can be on the order of kB T . The thermal cloud44

thus experiences the ”mexican hat” potential Vext (r) + 2gnc (r), it is repelled by thecondensate (the factor of two coming from the exchange term, see section 2.2.1). In
more strongly interacting gases, a self-consistent mean-field approximation could be
carried out by including the self-interaction of the thermal cloud and its effect on
the condensate. For very strong interactions, the mutual repulsion can lead to an
essentially complete separation of the thermal cloud and the condensate. Imbalanced
Fermi mixtures allow to clearly demonstrate such a dramatic repulsion, see chapter 7.
Bogoliubov transformation for the interacting Bose gas
We will now turn to another point of view of the interacting Bose gas at zero temperature, given by Bogoliubov [32]. It allows the identification of the elementary
excitations of the system, thereby giving a natural explanation for superfluidity. It
also introduces the concept of quantum depletion, about which we will have more to
say in the following chapters. Lastly, it will be interesting to compare this description
to the BCS case in the following section.
The model hamiltonian considered by Bogoliubov describes a dilute Bose gas
confined in a box of volume Ω:Ĥ =X²k a†k ak +k

V0 X † †a a am ak+l−m
2Ω k,l,m k l(2.70)

2 2

where ²k = ~2mk is the free particle spectrum and the notation V0 for the Fouriertransform of the pseudo-potential reminds us that we have to correctly renormalize
the interaction according to Eq. 2.63 when calculating physical properties.
As we did before, we assume a macroscopic occupation of the ground state k = 0
and can thus replacepa0 , a†0 → N0
(2.71)
with N0 the ground state occupation number. It is supposed that N0 ≈ N , the total
number of particles. With that assumption, and keeping only terms of order N or N 2 ,the hamiltonian becomes quadratic in the operators ak , a†k . Bogoliubov showed that
this hamiltonian can be solved exactly by introducing a canonical transformation of
the operators ak , a†k :†ak = uk αk − vk α−ka†k = uk αk† − vk α−k(2.72)hi
The new operators αk , αk† also obey the bosonic commutation relations αk , αk† 0 =45

Excitation energy Ek / µ

5
4
3
2
1
0
0

1
2
Momentum p / m c

3

Figure 2-1: Excitation spectrum of the weakly interacting Bose gas. For low momenta
p < mc, excitations are sound waves with linear dispersion Ek = c~k (dashed line).
At large momentum transfer p À mc, excitations are free-particle like (dotted line),with energy offset µ.
hiδkk0 and [αk , αk0 ] = αk† , αk† 0 = 0. For a judicious choice of the real parameters uk
and vk the hamiltonian can be transformed into´
1X ³ ††
Ek αk αk + α−k α−k
Ĥ =
2 k

(2.73)

The correct choice is uk = cosh φk , vk = sinh φk , with tanh 2φk = ²kµ+µ and µ = ngfrom the preceding section. From the commutation relations for αk , αk† immediately
follows that αk† αk has eigenvalues 0, 1, 2, . . . and thus E0 represents the ground stateenergy and Ek is the excitation spectrum of the hamiltonian. The ground state isdefined by αk |Oi = 0 for all k 6= 0 and can be thought of as the ”quasi-particle
vacuum”. One finds for the average energy per particle, with the particle density
n = N/Ω,Ã!
r
E0
2π~2 an
128 na3=1+(2.74)Nm
15
π
The first term in this expression is simply 12 µ, which we could have obtained from thepreceding perturbation treatment. The second term is the Lee-Yang correction [148],
which is nonanalytic in the interaction parameter na3 and thus cannot be obtained
from perturbation theory.
The excitation spectrum Ek is given by (see Fig. 2-1)£
¤1/2
Ek ≡ (²k + µ)2 − µ246(2.75)

which has the two limiting cases
( p
Ek ≈

µ
~k,
m

²k + µ,k→0k→∞(2.76)The long-wavelength excitations are thus sound waves with a speed of sound c =pµ/m. At large momenta, the excitations correspond to free particles excited above
the condensate mean-field µ.
Quantum depletion
Even in the ground state, the particles are not all condensed in the same single-particle†
state k = 0. Indeed, ak |Oi = −vk α−k|Oi 6= 0 and thus the ground state contains
q
3
N −N0
8
particles in excited k-states. This number of excited particles is N = 3 naπ .
For a typical sodium BEC, it is less than 1%. Again, however, for our molecularcondensates this number can in principle be large. Of course, once na3 is no longer
small compared to 1, the approximation of weak interactions is no longer allowed.
Let us point out that the particles populating nonzero k-states are not to be confusedwith thermally excited particles. They are in fact perfectly correlated admixtures tothe many-body wave function. One can show that these admixtures occur in pairs of
opposite momenta (k, −k), which one could probe by noise correlation experiments.
If the range of available k-states is restricted, for example by imposing a weak latticepotential that singles out certain k-directions, the quantum depleted part of the manybody wavefunction should lead to a distinct structure in the interference pattern of
expanding condensates5 [96, 254].
Superfluidity in weakly interacting Bose gases
With the aid of the excitation spectrum we are now able to argue that weakly interacting Bose gases must show superfluidity. The well-known Landau criterion states
that, given a dispersion relation Ek of the medium, an object moving at a velocity
v smaller than vc = mink Ek /~k cannot scatter from the medium. For free particles(hence also for the Bose-Einstein condensate in an ideal gas), Ek ∼ k 2 and thus vc = 0.
However, for the weakly interacting Bose gas, we find vc = c > 0, and thus for particle
velocities smaller than v it is a frictionless superfluid. Indeed, superfluidity of weakly
5

This structure should be distinguished from the Bragg peaks due to atoms in the macroscopicallyoccupied single-particle state and from the incoherent (spherically symmetric gaussian) backgrounddue to atoms localized in single lattice sites

47

interacting Bose gases has been studied for example by creating a single vortex in a
two-component BEC [164, 11], through impurity scattering [58, 179] and as a mostspectacular demonstration via the creation of vortex lattices [160, 2, 120, 87]. The
origin of quantized vortices will be discussed in chapter 6.

2.2.3

Fermi gases with attractive interaction: Cooper instability

The non-interacting Fermi gas did not show any phase transition down to zero temperature. One might assume that this qualitative fact should not change as interactions
are introduced. This is true in the case of repulsive interactions. For attractive interactions, the situation is, however, dramatically different. By taking advantage ofthe attraction, the fermions regroup into pairs, which can then form a condensate.
The idea of pairing might be natural, as tightly bound pairs of fermions canbe regarded as point-like bosons, which should condense according to the preceding
section. However, for weak attractive interaction – as is the case for the electronelectron interaction in metals – it is not evident that a paired state exists. Indeed,
we will see in the following that in three dimensions there is no bound state for two
isolated particles and arbitrarily weak interaction.
Two-body bound states in 1D, 2D and 3D
Localizing a quantum-mechanical particle down to a certain range R leads to anincreased momentum uncertainty of p ∼ ~/R at a kinetic energy cost of about ER =p2 /2m = ~2 /mR2 . Clearly, a shallow potential well of size R and depth V with
V /ER ≡ ² ¿ 1 cannot confine the particle within its borders. But we can search forpa bound state at energy EB ¿ ER of much larger size rB = 1/k ≡ ~2 /2mEB À R.
• In 1D, the bound state wavefunction far away from the well necessarily behaves
like e±kx for negative (positive) x (see Fig. 2-2a). As we traverse the well,
the wavefunction has to change its slope by 2k over a range R. This costs
kinetic energy ~2 ψ 00 (x)/2mψ̄ ≈ ~2 k/mR that has to be provided by the potential
energy −V (here, ψ̄ is the average value of ψ(x) in the well). We deduce thatk ≈ mRV /~2 = ²/R, the size of the bound state rB ≈ R/² is indeed much larger
than the size of the well, and the bound state energy EB ≈ −ER ²2 /2 depends
quadratically on the weak attraction −V . Importantly, we can always find abound state even for arbitrarily weak (purely) attractive potentials.
48

1D

2D

ψ(x)

3D

ψ(r)
+kx

-kx

0
V(x)
-ER

V1D

R
rB ~ R/ε-log kr

-kr

e

e

ψ(r)
u(r)
-kr

e

e

x

0

V2D

-ER

r

0

-ER
rB ~ R e1/εr
Rr
V3D > ERrB ~ R ER/(V3D-Vc)

Figure 2-2: Bound state wavefunctions in 1D, 2D and 3D for a potential well of size R
and depth V . In 1D and 2D, bound states exist for arbitrarily shallow wells. In termsof the small parameter ² = V /ER with ER = ~2 /mR2 , the size of the bound state
in 1D is R/². In 2D, the bound state is exponentially large, of size Re−1/² . In 3D,
due to the steep slope in u(r) = rΨ(r), bound states can only exist for well depths
V3D larger than a certain threshold Vc ≈ ER . The size of the bound state diverges as
RER /(V3D − Vc ) for V3D > Vc .
• In 2D and for a spherically symmetric well, the Schrödinger equation for the radial wavefunction ψ(r) outside the well reads 1r ∂r (r∂r ψ) = k 2 ψ. The solution is
the modified Bessel function which vanishes like e−kr as r À 1/k (see Fig. 2-2b).
For R ¿ r ¿ 1/k, we can neglect the small bound state energy EB ∝ k 2 compared to the kinetic energy and have ∂r (rψ 0 ) = 0 or ψ(r) ≈ log(kr)/ log(kR),
where 1/k is the natural scale of evolution for ψ(r) and we have normalized ψto be of order 1. Note that in 2D, it is not the change in the slope ψ 0 of the
wavefunction which costs kinetic energy, but the change in rψ 0 . Inside the well,
we can assume ψ(r) to be practically constant as V ¿ ER . Thus, rψ 0 changes
from ≈ 1/ log kR (outside) to ≈ 0 (inside) over a distance R. The corresponding~2
kinetic energy cost is 2mr∂r (rψ 0 ) ≈ ~2 /2mR2 log(kR) = ER /2 log(kR), which
has to be provided by the potential energy −V . We deduce k ≈ R1 e−cER /2V andEB ≈ ER e−cER /V with a numerical constant c on the order of 1. The particle isextremely weakly bound, with its bound state energy depending exponentiallyon the shallow potential −V . Accordingly, the size of the bound state is exponentially large, rB ≈ RecER /2V . Nevertheless, we can always find this weakly
bound state, for arbitrarily small attraction6 .
6Mathematically,
the condition for the potential V (r) to allow a bound state in 2D requires theR∞integral 0 rV (r)dr to be finite [145].

49

V

1D
¿ ER

ψ(r > R)e−r/rBrB

R EVR

Re−cER /2V

R
R VE−Vc−V 2 /ER−ER e−cER /V−(V − Vc )2 /ER

2~
EB = − mr
2

B

2D
¿ ER
½
− log r/rB , R ¿ r ¿ rB
K0 ( rrB ) =e−r/rB ,r À rB

3D> Vc ≈ ER
e−r/rBr

Table 2.1: Bound states in 1D, 2D and 3D for a potential well of size R and depth
V . ψ(r > R) is the wave function outside the well, rB is the size of the bound state,
and EB its energy (ER = ~2 /mR2 ).
• In 3D and for a spherically symmetric well, the Schrödinger equation for the
wavefunction ψ transforms into an effective one-dimensional problem for the
wavefunction u = rψ (see Fig. 2-2c). We might now be tempted to think that
there must always be a bound state in 3D, as we already found this to be thecase in 1D. However, the boundary condition on u(r) is now to vanish linearlyat r = 0, in order for ψ(0) to be finite. Outside the potential well, we still
have u ∝ e−kr for a bound state. Inside the well the wavefunction must fall
off to zero at r = 0 and necessarily has to change its slope from −k outside to∼ 1/R inside the well over a distance R. This costs the large kinetic energy~2 u00 (r)/2mū ≈ ~2 /mR2 = ER (with ū the average value of u(r) in the well).
If the well depth V is smaller than a critical depth Vc on the order of ER , theparticle cannot be bound. At V = Vc , the first bound state enters at E = 0.
As k = 0, u is then constant outside the well. If the potential depth is further
increased by a small amount ∆V ¿ Vc , u again falls off like e−kr for r > R. Thisrequires an additional change in slope by k over the distance R, provided by∆V . So we find analogously to the 1D case k ∼ mR∆V /~2 . Hence, the boundstate energy E ≈ −∆V 2 /ER is quadratic in the ”detuning” ∆V = (V − VC ),and the size of the bound state diverges as rB ≈ RER /(V − VC ). We will findexactly the same behavior for a weakly bound state when discussing Feshbachresonances in chapter 3.
The analysis holds for quite general shapes V (r) of the potential well (in theformulas, we only need to replace V by its average over the well - if it exists -,R∞R∞1
V (r)dr in 1D, R12 −∞ rV (r)dr in 2D etc.). Table 2.1 summarizes the different
R −∞cases.
Applying these results to the equivalent problem of two interacting particles colliding in their center-of-mass frame, we see that in 1D and 2D, two isolated particles
50

~2
mΩ
1
= Ω1
|V0 |

R

1D
q
ρ(²)

ρn (²)
d² 2²+|E|²<ER
2 2

E = − ~ mk

2D

1

~2

1

π
q

2m²

2π
³

m
4~2 |E|

m
2
− 4~
2 V0

m
log
4π~2

2ER +|E|
|E|
2

− 4π~

−2ER e m|V0 |

´

1
2π 2

3D
q

2m²
~2

p ´
π
2E
−
|E|
R
~2
2
³
´2
0c
− π82 ER |V0|V|−V= −~2 /ma2
0|
1
2π 2

¡ m ¢3/2³√

Table 2.2: Link between the density of states and the existence of a bound state
for arbitrarily weak interaction. To compare with table 2.1 note that V0 ∼ V Rn .
22V0c = √2π( ~ )3/2 is the threshold interaction for the 3D case. The formula for
2ER m
the 3D bound state energy as a function of scattering length a follows from therenormalization procedure outlined in section 2.2.1.
can bind for an arbitrarily weak purely attractive interaction. Hence in 1D and 2D,
pairing of fermions and their subsequent condensation can in principle be understoodalready at the two-particle level. Indeed, one can show that the existence of a twobody bound state in 1D and 2D is a necessary and sufficient condition for a BCS
instability [201]. In 3D, however, there is a threshold interaction below which two
isolated particles are unbound. We conclude that if pairing and condensation occur
for arbitrarily weak interactions in 3D, then this must entirely be due to many-bodyeffects.
Density of states
What physical quantity decides whether there are bound states or not? To answer
this question, we formulate the problem of two interacting particles of mass m in
momentum space. This allows a particularly transparent treatment for all three cases
at once, and identifies the density of states in the different dimensions as the decisive
factor for the existence of bound states.
2 2

Searching for a shallow bound state of energy E = − ~ mk (m/2 is the reducedmass), we start by writing the Schrödinger equation for the relative wavefunction~2(∇2 − k 2 )ψ = V ψ in (n-dimensional) momentum space:mm1
ψk (q) = − 2 2~ q + k2Z

dn q 0V (q − q0 )ψk (q0 )n(2π)(2.77)For a short-range potential of range R ¿ 1/k, V (q) is practically constant, V (q) ≈V0 , for all relevant q, and falls off to zero on a large q-scale of ≈ 1/R. For example,

51

for a potential well of depth V and size R, we have V0 ∼ V Rn . Thus,mV01ψk (q) = − 2 2~ q + k2Z

dn q 0ψk (q0 )
1 (2π)n0
q <R(2.78)

We integrate once more over q, applying the same cut-off 1/R, and can then divide byR
dn q
the common factor q< 1 (2π)
n ψk (q). We obtain the following equation for the boundR
state energy E:
1
m
−
= 2
V0
~

Z

dn q
1
1
=
n
2
2
1 (2π) q + k
Ω
q< R

Z
d²
²<ER

ρn (²)
2² + |E|(2.79)

with the density of states in n dimensions ρn (²), the energy cut-off ER = ~2 /mR2
and the volume Ω of the system (note that V0 has units of energy times volume).
The question on the existence of bound states for arbitrarily weak interaction has
now been reformulated: As |V0 | → 0, the left hand side of Eq. 2.79 diverges. If
this equation is to have a solution for any small |V0 |, the right hand side needs todiverge as well as the bound state energy |E| → 0. The question is thus whether
the integral over the density of states in Eq. 2.79 can take on any large value - orwhether it is bound from above. Table 2.2 presents the different cases in 1D, 2D,p
3D. In 1D, the integral diverges as 1/ |E|, so one can always find a bound statesolution. The binding energy depends quadratically on the interaction, as we hadfound before. In 2D, where the density of states ρ2D is constant, the integral still
diverges logarithmically as |E| → 0, so that again one can find a solution |E| for any
small |V0 |. The binding energy now depends exponentially on the interaction and
ρ2D :
− 2Ω
(2.80)
E2D = −2ER e ρ2D |V0 |However, in 3D the integral is finite for vanishing |E|, and one needs to overcome aminimum threshold interaction in order to bind the two particles.
These results might give us a hint why there might be a paired state for two
fermions immersed in a medium, even for arbitrarily weak interactions: It must be
that, somehow, the free 3D density of states for the fermions is altered due to thepresence of the surrounding atoms. This is exactly what happens, as will be discussed
in the next section.

52

Figure 2-3: Cooper problem: Two particles scattering on top of a Fermi sea. a)
Particles with equal and opposite momenta can scatter as shown. The Fermi sea(grey shaded) blocks possible final momentum states. b) Here, the total momentumof the two particles q is nonzero. Possible scattering states k + q/2 and −k + q/2(or, equivalently, k − q/2, see construction) must lie outside the Fermi sea. Only inthe small shaded region (a circle in 3D) can both particles be close to their respective
Fermi surface. In contrast, pairs at rest in a) can take advantage of the entire Fermi
surface.
Pairing of fermions - The Cooper problem
Consider now two weakly interacting spin 1/2 fermions not in vacuum, but on topof a (non-interacting) filled Fermi sea, the Cooper problem [66]. Momentum states
below the Fermi surface are not available to the two scattering particles due to Pauliblocking (Fig. 2-3a). For weak interactions, the particles’ momenta are essentiallyconfined to a narrow shell above the Fermi surface. We thus realize that the problem
is effectively 2D, and we should find a bound state for the two-particle system forarbitrarily weak attractive interaction. The effective ”2D” density of states should beρ3D (EF ), the density of states at the Fermi surface.
In principle, the two fermions could form a pair at any finite momentum. However,considering the discussion in the previous section, the largest binding energy can be
expected for the pairs with the largest ”2D” density of scattering states. For zeromomentum pairs, the entire Fermi surface is available for scattering, as we can see
from Fig. 2-3a. If the pairs have finite center-of-mass momentum q, the number ofcontributing states is strongly reduced, as they are confined to a circle (see Fig. 2-3b).
Consequently, pairs at rest experience the strongest binding. In the following we willcalculate this energy.
We can write down the Schrödinger equation for the two interacting particles as
before, but now we need to search for a small binding energy EB = E − 2EF < 0 on

53

top of the large Fermi energy 2EF of the two particles. The equation for EB is
1
1
− =
V0
Ω

Z
d²
EF <²<EF +ER

ρ3D (²)2(² − EF ) + |EB |(2.81)The effect of Pauli blocking of momentum states below the Fermi surface is explicitly
included by only integrating over energies ² > EF .
In superconductors, the natural cut-off ER is given by the Debye frequency ωD ,ER = ~ωD , corresponding to the highest frequency at which ions in the crystal lattice
can respond to a bypassing electron. Since we have ~ωD ¿ EF , we can approximate
ρ3D (²) ≈ ρ3D (EF ) (a constant just like in two dimensions!) and find:
EB = −2~ωD e−2Ω/ρ3D (EF )|V0 |(2.82)In the case of an interacting Fermi gas, we should replace 1/V0 by the physically
relevant scattering length a < 0 using the prescription in Eq. 2.63. The equation forthe bound state becomes
m
1
−
=
2
4π~ a
Ω

Z EF +EREF

ρ3D (²)1d²−
2(² − EF ) + |EB | Ω

Z EF +ERd²0ρ3D (²)2²(2.83)The right hand expression is now finite as we let the cut-off ER → ∞, the result being(one assumes |EB | ¿ EF )m
ρ3D (EF )−=4π~2 a
2Ω

µ

µ
− log

|EB |
8EF

¶

¶
−2

(2.84)

pF
Inserting ρ3D (EF ) = Ωmk
with
the
Fermi
wave
vectork=
2mEF /~2 , one arrives
F
2
2
2π ~
at
8
(2.85)EB = − 2 EF e−π/kF |a|e
The binding energies Eqs. 2.82 and 2.85 can be compared with the result for thebound state of two particles in 2D, Eq. 2.80. The role of the constant density ofstates ρ2D is here played by the 3D density of states at the Fermi surface, ρ3D (EF ).
As this is largest for zero-momentum pairing (see Fig. 2-3), Cooper pairs at rest havethe largest binding energy.
The result is remarkable: Two weakly interacting fermions on top of a Fermi sea
form a bound state due to Pauli blocking. However, in this artificial problem weneglected the interactions between particles in the Fermi sea. As we ”switch on” theinteractions for all particles from top to the bottom of the Fermi sea, the preceding54

discussion shows that the gas will reorder itself into a completely new, paired state.
The Fermi sea is thus unstable towards pairing. The full many-body descriptionof such a paired state, including the necessary anti-symmetrization of the full wave
function, was achieved by Bardeen, Cooper and Schrieffer (BCS) in 1957 [23]. As wewill see in the next section, the self-consistent inclusion of all fermion pairs leads tomore available momentum space for pairing. The effective density of states is thentwice as large, giving a superfluid gap ∆ that is exponentially larger than |EB | from
Eq. 2.85:
8(2.86)∆ = 2 EF e−π/2kF |a|e

2.3

The BEC-BCS crossover

The preceding section has shown that in the many-body problem we can expect
pairing for an arbitrarily weak interaction between fermions. The ground state of
the gas will be a condensate of Cooper pairs as described by BCS theory. For strong
interactions, a true two-body bound state between fermions exists even in vacuum.
Here, we expect a Bose-Einstein condensate of these tightly bound fermion pairs ormolecules to be the ground state of the system. It was realized by Leggett [149],
building upon work by Eagles [85], that the crossover from the BCS- to the BECregime is smooth. This might surprise, as we know from before that two-body physics
shows a threshold behavior at a critical interaction strength, below which there is nobound state for two particles. In the medium, however, we simply cross over from aregime of tightly bound pairs to a regime where the pairs are of much larger size than
the interparticle spacing. Closely following the work of [149, 172, 74], we will describe
the BEC-BCS crossover in a simple ”one-channel” model, where the scattering lengtha is the parameter which ”tunes” the interaction. For positive a > 0, there is a
two-body bound state available at EB = −~2 /ma2 (see table 2.2), while for a < 0,corresponding to attractive interaction, pairing is purely a many-body effect. In either
case, for s-wave scattering the orbital part of the pair wavefunction ϕ(r1 , r2 ) will besymmetric under exchange of the paired particles’ coordinates and will only dependon their distance |r1 − r2 |. We will attempt to find a many-body wavefunction
Ψ (r1 , . . . , rN ) = Aϕ(|r1 − r2 |)χ12 . . . ϕ(|rN −1 − rN |)χN −1,N

(2.87)

that describes a condensate of such fermion pairs, with the operator A denoting
the correct antisymmetrization of all fermion coordinates, and the spin function
55

χij =↑ (i) ↓ (j).
In second quantization notation we write|ΨiN =Z Y

d3 ri ϕ(r1 −r2 )Ψ†↑ (r1 )Ψ†↓ (r2 ) . . . ϕ(rN −1 −rN )Ψ†↑ (rN −1 )Ψ†↓ (rN ) |0i (2.88)i

P † ik·r
where the fields Ψ†σ (r) =. With the Fourier transform ϕ(r1 − r2 ) =k ckσ eP
ik·r1 −ik·r2e
we can introduce the pair creation operatork ϕk eb† =X

ϕk c†k↑ c†−k↓(2.89)

N/2(2.90)kand write|ΨiN = b†|0i

In this way, |ΨiN looks very suggestive, reminding of a condensate of bosonic particles.
However, it is important to realize that the operators b† do not obey the Bose-Einstein
commutation relations and thus do not describe bosons. In fact, we can calculatehi£ † †¤P† †††b , b − = kk0 ϕk ϕk0 ck↑ c−k↓ , ck0 ↑ c−k0 ↓=0(2.91)−
P[b, b]− = kk0 ϕk ϕk0 [c−k↓ ck↑ , c−k0 ↓ ck0 ↑ ]− = 0(2.92)hiX£ †¤Pb, b − = kk0 ϕk ϕk0 c−k↓ ck↑ , c†k0 ↑ c†−k0 ↓ϕ2k (1 − nk↑ − nk↓ ) (2.93)=−k

The first two relations are fine, but the third one spoils Bose-Einstein statistics forthe b-particles. Only in the limit where the pairs are tightly bound do the b-particles
correspond to point-like bosons: Here, the pairs occupy a wide region in momentum
space and the occupation nk of any momentum state k is very small (see section 2.3.2£¤P
below). Thus, b, b† − ≈ k ϕ2k = 1.
Working with the N -particle state |ΨiN is inconvenient, as one would face acomplicated combinatoric problem in manipulating the sum over all the c†k ’s (as one
chooses a certain k for the first fermion the choices for the second depend on thisk, etc.). Rather, one passes over to the grand canonical formalism, not fixing thenumber of atoms but the chemical potential µ, and defines the many-body state:
N |Ψi ==PQ

1
N (N/2)! |ΨiN=XN

ϕk c†k↑ c†−k↓

ke

|0i =

Y
k

56

1
†
N/2
b†
|0i = eb |0i(N/2)!

(1 + ϕk c†k↑ c†−k↓ ) |0i(2.94)

where the last equation follows from c†2
k = 0. If we choose the constant N =Q
2 1/2, then |Ψi becomes a properly normalized statek (1 + ϕk )|ΨBCS i =YQ

(uk + vk c†k↑ c†−k↓ ) |0i

1
k uk =(2.95)

kwith vk = uk ϕk and |uk |2 + |vk |2 = 1. This is the BCS variational wavefunction, first
introduced by BCS to describe condensation of Cooper pairs in momentum space.
From the above derivation, however, it is clear that its applicability encompasses theentire regime of pairing, all the way towards the limit of small molecules.

2.3.1

Gap and number equation

The variational parameters Duk and vkEare derived in the standard way by minimizing
the free energy E − µN = Ĥ − µN̂ . The many-body hamiltonian for the systemisX †V0 X †²k ckσ ckσ +Ĥ =ck+ q ↑ c†−k+ q ↓ ck0 + 2q ↓ ck0 − 2q ↑(2.96)
22Ω0k,σk,k ,qThe approach taken by BCS consists in finding the wave function |Ψi that describes a condensate of pairs at rest. As we have seen in section 2.2.3, such Cooper
pairs have the largest binding energy. Neglecting interactions between pairs at finite
momentum, we only keep the term for q = 0. The free energy becomes
D
E
X
V0 X
2ξk vk2 +Ĥ − µN̂=
uk v k uk 0 v k 0
Ω
0
k
k,k

(2.97)

with ξk = ²k − µ
Minimizing E − µN leads tovk2
u2k
with Ek

µ
¶
1
ξk
=
1−
2
Ek
µ
¶
1
ξk
=
1+
2
Ek
q
=
ξk2 + ∆2

combined with the gap equation ∆ ≡ − VΩ0
or

1
− =
V0

Z

P

V0
k hck↑ c−k↓ i = − Ωd3 k 1(2π)3 2Ek
57

(2.98)P

V0
k u k vk = − Ω

P

∆
k 2Ek

(2.99)

(compare with the analogous equation in free space Eq. 2.79 and for the simplified
Cooper problem Eq. 2.81) and the number equation for the total particle density
n = N/ΩZ
n=2

d3 k 2vk(2π)3(2.100)which have to be solved simultaneously to yield the two unknowns µ and ∆. We will
once more replace V0 by the scattering length a using prescription Eq. 2.63, so thatthe gap equation becomes (compare Eq. 2.83)m−=
4π~2 a

Z

d3 k
(2π)3

µ

1
1
−
2Ek 2²k¶(2.101)

where the integral is now well-defined. The equations can be rewritten in dimensionless form with the Fermi energy EF = ~2 kF2 /2m and wave vector kF = (3π 2 n)1/3
r

³µ´
∆
I1
EF
∆
µ
¶3/2 ³ ´
3 ∆
µ
1 =
I2
2 EF
∆


Z ∞
1
1
with I1 (z) =dx x2  q− 2x0(x2 − z)2 + 1


Z ∞
2
x −z

and I2 (z) =dx x2 1 − q
202(x − z) + 1
1

2
−
=
kF a
π

(2.102)(2.103)(2.104)(2.105)This givesµ¶1/3 ³ ´
2
2
µ
−
=
I1
kF a
π 3I2 (µ/∆)
∆
Ã
!2/3
∆
2
¡µ¢
=
EF
3I2 ∆
1

(2.106)(2.107)

The first equation can be inverted to obtain µ/∆ as a function of the interaction
parameter 1/kF a, which can then be inserted in the second equation to yield the gap∆. The result for µ and ∆ as a function of 1/kF a is shown in Fig. 2-4. Marini,
Pistolesi and Strinati have shown that one can even obtain analytic expressions forthe above in terms of complete elliptic integrals [163].

58

1
1.5

-1
1.0
-2

∆/EF

µ/EF

0

0.5

-3
-4
2

1

0
1/kFa

-1

0.0
-2

Figure 2-4: Chemical potential (dotted line) and gap (straight line, red) in the BECBCS crossover as a function of the interaction parameter 1/kF a. The BCS-limit ofnegative 1/kF a is to the right on the graph. The resonance where 1/kF a = 0 is shownby the dashed line.
BCS limit
In the BCS-limit of weak attractive interaction, kF a → 0− , we have7µ ≈ EF8∆ ≈ 2 e−π/2kF |a|e(2.108)(2.109)The first equation tells us that adding a spin up and spin down particle to thesystem costs a Fermi energy per particle 8 : In the weakly interacting BCS limit Pauliblocking still dominates over interactions, and hence the particles can only be addedat the Fermi surface. The second equation is the classic result of BCS theory forthe superfluid gap9 . Compared to the bound state energy for a single Cooper pair
on top of a non-interacting Fermi sea, Eq. 2.85, the gap is exponentially larger (afactor eπ/2kF |a| ), as the entire collection of particles now takes part in the pairing10 .
7This√follows by substituting ξ = x2 − z in the integrals and taking the limit z → ∞. One has
I1 (z) ≈ z (log(8z) − 2) and I2 (z) = 23 z 3/2 .
8
BCS theory implicitly assumes an equal number of spin up and spin down atoms. The chemical
potential is thus the energy cost for adding a spin up atom if at the same time a spin down particle
is added.
9
The present mean-field treatment does not include density fluctuations, which modify the prefactor in the expression for the gap ∆ [98].
10
In the self-consistent BCS solution, not only the momentum states above the Fermi surface
contribute to pairing, but also those below it, in a symmetric shell around the Fermi momentum. In
the Cooper problem the states below the Fermi surface were excluded, reducing the effective densityof states by a factor of two. The pairing energy depends exponentially on the density of states,
which explains the difference between the binding energy in the Cooper problem and the gap energyof BCS theory.

59However, the gap is still exponentially small compared to the Fermi energy: Cooper
pairing is fragile.
To give a sense of scale, Fermi energies in our dilute lithium gas are on the orderof a µK, corresponding to 1/kF ∼ 4 000 a0 . A ”typical” scattering length for lithium,in the absence of scattering resonances, will be about 50 − 100 a0 (on the order of thevan der Waals-range). The gap is thus vanishingly small, ∆/kB ≈ 10−30 . . . 10−60 K.
With that number in mind, it seems simply hopeless to achieve superfluidity in Fermi
gases. However, Feshbach resonances allow tuning of the scattering length, bringing
the gas into the strongly interacting regime where kF |a| > 1 (see chapter 3). In
this case, there is a good chance to reach superfluidity: The above mean-field theory
predicts ∆ = 0.22EF or ∆/kB ≈ 200 nK for kF |a| = 1. Such temperatures are now
routinely achieved in experiments on ultracold gases.
BEC limit
In the BEC limit of tightly bound pairs, for kF a → 0+ , one finds11~2
π~2 an
+
2
2mar
m
16 EF
√
∆≈
3π kF aµ=−(2.110)(2.111)

The first term in the expression for the chemical potential is the binding energy
per fermion in a tightly bound molecule (see table 2.2). Naturally, this energy is only
available if we add two fermions of opposite spin at the same time to the system (wehave assumed an equal mixture of spin-up and spin-down fermions when writing down
the wave-function in Eq. 2.87). Adding only one spin-up fermion to the system does
not give or cost any energy to first approximation: The tightly bound fermions occupy
a large region in momentum space, hence the Fermi gas is not degenerate and Pauliblocking not relevant anymore12 . The second term is a mean-field contribution describing the repulsive interaction between molecules in the gas. Indeed, a condensate
of molecules of mass mM = 2m, density nM = n/2 and a molecule-molecule scattering2
M nM
length aM will have a chemical potential µM = 4π~maM(see section 2.2.2). Since µM
is twice the chemical potential for each fermion, we obtain from the above expression
11

This result
from the expansion of the integrals for z < 0 and |z| → ∞. One finds
p follows1π
and I2 (z) = π8 √1 .
I1 (z) = − π2 |z| − 32
|z|3/2|z|

12

To the next order of approximation, the single fermion feels a repulsive mean-field interaction
from the gas of molecules, so it costs an energy to add this particle. This is discussed in detail inchapter 7.

60

1
-2
-0.5

0.8
0

nk

0.6
0.5
0.4
1

0.2

0.5

1

1.5

2

k/kF

Figure 2-5: Occupation nk of momentum states k in the BEC-BCS crossover. The
numbers give the interaction parameter 1/kF a.
the molecule-molecule scattering length aM = 2a. However, this result is not exact.
Petrov, Shlyapnikov and Salomon [193] have performed an exact calculation for theinteraction between four fermions and shown that aM = 0.6a. The present mean-fieldapproach neglects correlations between different pairs, or between one fermion and apair. If those are included, the correct few-body physics is recovered [195, 123].

2.3.2

Evolution from BCS to BEC

Our variational approach smoothly interpolates between the two known regimes of a
BCS-type superfluid and a BEC of molecules. The transition, which occurs approximately between 1/kF a = −1 and +1, appears fully continuous. The occupation of
momentum states nk = vk2 evolves smoothly from the step-function Θ(kF − k) of adegenerate Fermi gas, broadened over a width ∆ ¿ EF due to pairing, to n times thesquare of the molecular wave-function ϕk ∝ 1+k12 a2 (see Fig. 2-5). It is also interesting
to follow the evolution of the ”Cooper pair” wavefunction13 in k-space, where it isgiven by hΨBCS | c†k↑ c†−k↓ |ΨBCS i = uk vk , and in real space, where it isZ
φ(r1 − r2 ) =hΨBCS | Ψ†↑ (r1 )Ψ†↓ (r2 ) |ΨBCS i =

1
=
2

Z

d3 k
∆
p
eik·(r1 −r2 )
3
2
2
(2π)
ξk + ∆

13d3 k
uk vk eik·(r1 −r2 ) (2.112)
3(2π)(2.113)Note that this definition is not equal to the Fourier transform of the pair wavefunction ϕ(r)introduced in Eq. 2.87, which would be vk /uk . The definition given here is the two-point correlation
function. Both definitions for the Cooper pair wavefunction show a sharp feature, either a peakor an edge at the Fermi surface, of width ∼ δk, thus giving similar behavior for the real space
wavefunction.

61

0

1

2 3
k / kF4µ = 0.59 EF∆ = 0.69 EF

1.2
0.8
0.4
0.0

5 2

0

1

2 3
k / kF4µ = 0.95 EF∆ = 0.21 EF

4
3
2
1
0

5 2

5
0

1

2 3
k / kF4µ = 0.997 EF∆ = 0.046 EF

20
15
10
5
0
0

5

1

2 3
k / kF

4

5

0

1

1

4

1/kFa = -2
φ(k) [a.u.]

3

1/kFa = -1
φ(k) [a.u.]φ(k) [a.u.]rφ(r) / ∆ [arb. units]

2

1/kFa = 0

µ = - 0.8 EF∆ = 1.33 EF

0.5
0.4
0.3
0.2
0.1

φ(k) [a.u.]

1/kFa = 1
3

0

0
0
0

1

2

3
kFr

4

5 0

4

8
kFr

12

0

10

20

30 0

20

kFr

40

60

k Fr

Figure 2-6: Evolution of the spatial pair wavefunction φ(r) in the BEC-BCS crossover.
The inset shows the Fourier transform φ(k), showing clearly that in the BCS-limit,
momentum states around the Fermi surface make the dominant contribution to the
wavefunction. In the crossover, the entire Fermi sphere takes part in the pairing.
In the BEC-limit, φ(k) broadens as the pairs become more
and more tightly bound.
R ∞ sin(r√ξ+µ)φ(r) was obtained via numerical integration of −µ dξ √ 2 2 (here, ~ = 1 = m), anξ +∆expression that follows from Eq. 2.113.
In the BCS limit, the fermions taking the most advantage from pairing resideδ² ≈ ~v∆F , where vF is theat the Fermi surface k = kF , in a region of width δk ∼ ∂k∂²velocity of fermions at the Fermi surface. We thus expect the spatial wavefunctionof Cooper pairs to have a strong modulation at the inverse wave vector 1/kF , and an
overall extent of the inverse width of the pairing region, ∼ 1/δk ∼ ~v∆F À 1/kF . This
is indeed the result of Eq. 2.113, which gives (setting r = |r1 − r2 |) [23]kF ∆φ(r) = 2
sin(kF r) K0
π r ~vF

µ

¶

r

r→∞

πξBCS

∼

sin (kF r) e−r/(πξBCS )(2.114)

where K0 (kr) is the modified Bessel function that falls off as e−kr at infinity. The
envelope function is of the same type as what we have found for a two-body boundstate (see table 2.1). The characteristic size of q
the Cooper pair, or the two-particle2

hφ(r)|r |φ(r)i, and this gives indeed
correlation length ξ0 , can be defined as ξ0 =hφ(r)|φ(r)iξ0 ∼ 1/δk,~vF
ξ0 ≈ ξBCS ≡À 1/kF in the BCS-limit(2.115)π∆
1
In the BEC limit, uk vk ∝ 1+(ka)2 , and soe−|r1 −r2 |/aφ(r1 − r2 ) ∼|r1 − r2 |62

(2.116)

2

kF ξ0

10
6
4
2

1
6
4
2

4

2
1/kFa

0

-2

Figure 2-7: From tightly
bound molecules to long-range Cooper pairs. Evolution ofq2

hφ(r)|r |φ(r)ithe pair size ξ0 =as a function of the interaction parameter 1/kF a.
hφ(r)|φ(r)i
On resonance (dashed line), the pair size is on the order of the inverse wave vector,
ξ0 (0) ∼ k1F , about a third of the interparticle spacing.

which is simply the wavefunction of a molecule of size ∼ a (see table 2.1). The twoparticle correlation length14 is thus ξ0 ∼ a. Fig. 2-6 summarizes the evolution of the
2 |φ(r)ipair wavefunction, and Fig. 2-7 shows the pair size given by [23, 197] ξ02 = hφ(r)|rhφ(r)|φ(r)i
as a function of the interaction parameter 1/kF a.

2.3.3

Single-particle and collective excitations — Landau criterion

The BCS-state |Ψi describes a collection of pairs b†k |0i = uk |0i + vk c†k↑ c†−k↓ |0i. A
single fermion that we add in state k ↑, say, cannot find a pairing partner and requiresa kinetic energy ξk . The new state does not profit from pairing in (k ↑, −k ↓),P
as −k ↓ is empty, and we loose the pairing energy 2ξk vk2 − 2 VΩ0 uk vk k0 uk0 vk0 =2= ξk − Ek (see Eq. 2.98). The total cost for adding one fermion isξk (1 − Eξkk ) − ∆Ekthus simply ξk −(ξk −Ek ) = Ek . In the same way, one calculates the cost for removing
a fermion from the BCS-state, which is again Ek .
Fig. 2-8 shows the single-particle excitation energy Ek for different interaction
strengths in the BEC-BCS crossover. For µ > 0, the minimum energy required toexcite a particle out of the condensate occurs for ξk = µ and is ∆, which gives ∆14This length scale should be distinguished from the coherence length ξphase that is associatedwith spatial fluctuations of the order parameter. The two length scales coincide in the BCS-limit,
but differ in the BEC-limit, where ξphase is given by the healing length ∝ √1na . See [197] for a
detailed discussion.

63

1/kFa = 1

Ek / EF

3

1/kFa = 0.553µ = - 0.8 EF∆ = 1.33 EF

1/kFa = 0

µ = 0 EF∆ = 1.05 EF

1/kFa = -1

µ = 0.59 EF∆ = 0.69 EF

1/kFa = -2

µ = 0.95 EF∆ = 0.21 EFµ = 0.997 EF∆ = 0.046 EF

2
1
0
0

1
k / kF

2 0

1
k / kF

2 0

1
k / kF

2 0

1
k / kF

2 0

1
k / kF

2

Figure 2-8: Evolution of the single-particle excitation spectrum in the BEC-BCS
crossover.
On the BEC-side, for µ < 0, the minimum required energy to add a particleq2
is |µ| + ∆2 and occurs at k = 0. This qualitatively changes at 1/kF a = 0.553 wherepµ = 0. For µ > 0, the minimum energy is ∆ and occurs at k = 2mµ/~2 .
the name of the superfluid gap. Possibly the most visual demonstration of this gap,
which prevents single fermions to enter the superfluid, is the phenomenon of phaseseparation in imbalanced Fermi
q mixtures (see chapter 7). However, for µ < 0 theminimum energy becomes |µ|2 + ∆2 and occurs for k = 0. Deep in the BECregime, this corresponds to ~2 /ma2 , which is simply the energy needed to break amolecule. If we want to make a distinction between the BEC- and the BCS-limit,that seem to be so smoothly connected, the natural point would be the interaction forwhich µ = 0, as here the character of single-particle excitations changes from havingk > 0 to having k = 0. In the variational solution, this occurs for 1/kF a = 0.553.
We may now ask for the critical velocity of the superfluid in the BEC-BCS
crossover. Given the single-particle excitation spectrum above, Landau’s criterion
for the critical velocity would predict
Ekvc,BCS = min
=
k ~k

r³
p

´
µ2 + ∆2 − µ /m

∆
→0+
µ

→

∆
~kF

(2.117)

An object that is dragged through the superfluid at this velocity will break a fermion
pair. In contrast, in the BEC-limit of a condensate of tightly bound molecules wewould expect density fluctuations, sound waves, to provide the mode of smallest
pp
critical velocity, at the speed of sound cs = µM /mM = π~2 an/m2 . This Bogoliubov sound mode of the BEC-limit finds its analog in the BCS-regime, where it√is called the Bogoliubov-Anderson mode, propagating at the speed of sound vF / 3,with vF = ~kF /m the Fermi velocity15 . The connection is smooth, as expected and
15qThis speed of sound can be calculated using the hydrodynamic equation c =64∂P∂ρ , ρ = m n

0.7

cS (Bog.-And.)vc/vF, cs/vF

0.6
0.5
0.4

Sound waves

Pair breakingcS (Bog.)

0.3

vC

0.2
0.1
0.0
4

2
1/kFa

0

-2

Figure 2-9: Critical velocity vc in the BEC-BCS crossover. The relevant excitationsin the BEC-regime
correspondto Bogoliubov (Bog.) sound waves with speed of
pµ
√
vF
√
sound cs = m = 3π kF a. This sound mode eventually becomes the BogoliubovvF
Anderson (Bog.-And.) mode in the BCS-regime, with cs = √. The evolution is3
smooth [74, 176, 27, 64], but only the limiting cases are shown here. In the BCSregime the excitations with the lowest critical velocity are single-particle excitations
that break a Cooper pair. Here, vc ≈ ~k∆F . After [64].
found by [74, 176, 27, 64]16 .
The critical velocity for the superfluid is given by the smaller of the two velocities,
the speed of sound cs and the critical velocity for pair breaking vc,BCS . For the
BEC-side, it is shown in [64] that for small momenta k ¿ 1/a which do not resolve
the composite
q nature of the molecules, the expression for the Bogoliubov-dispersion~2 k2
Ek,BEC = ( 2m
+ µM )2 − µ2M remains valid even well into the crossover region.
M
This allows us to determine the speed of sound in an approximate way in Fig. 2-9.
Notable in this figure is the sharp peak in the critical velocity around resonance. It
is here that the superfluid is the most stable [219, 64]. Importantly, we observe thatthe nature of the dominant low-lying excitations changes from rather benign (lowk’s) sound waves in the BEC-limit to the more violent pair breaking in the BCS-limit(large k). An experimental consequence is that the superfluid is very fragile in the
BCS-regime in response to stirring (see chapter 6).
2~
and the pressure of a normal Fermi gas P = 23 VE = 25 EF n = 25 2m(3π 2 )2/3 n5/3 . Thus, the soundmode is already present in the normal Fermi gas, the main effect of pairing being to push low-lying
single-particle excitations up in energy, which would otherwise provide damping. While this damping
would also vanish in a non-interacting gas at zero temperature, this sound mode cannot propagate,
as collisions are absent and the gas can no longer maintain local equilibrium.
16
Note that in superconductors, density fluctuations can only occur at the plasma frequency, as
they are charged superfluids.

65

2.3.4

Finite temperatures

To study the system at finite temperature, we have to include fluctuations aroundthe BCS variational ground state. This is done most elegantly using the procedure by
Bogoliubov [33] and Valatin [249]. The goal is to introduce new quasi-particles thatapproximately diagonalize the original many-body hamiltonian, just like what wasdone for the interacting Bose gas in section 2.2.2. For this, new fermionic operators
are introduced:
γk↑ = uk ck↑ − vk c†−k↓
†
γ−k↓

=

(2.118)uk c†−k↓ + vk ck↑(2.119)with the requirement that the BCS state presents the new ”vacuum” for these quasiparticles: γk↑ |Ψi = 0 = γ−k↓ |Ψi. If uk and vk are chosen as before, the hamiltonian
becomesXX††
Ek (γk↑γk↑ + γk↓
γk↓ )(2.120)(ξk − Ek ) +Ĥ − µN̂ =kk

The first term is the free energy E − µN in the BCS state. The second gives the†energy of a collection of quasi-particles. γk↑
simply creates the state we considered
above, where a fermion is added in state k ↑, the paired state −k ↓ being empty. As
we have seen, the energy for this excitation is Ek , as is correctly accounted for in the
diagonalized hamiltonian.

Gap equation at finite temperature
The gap equation is modified at finite temperature. With the definitions above one
finds
³
D
E D
E´
†
†
hck↑ c−k↓ i = uk vk 1 − γk↑γk↑ − γk↓γk↓(2.121)DE†
As the quasi-particles are fermions, they follow the Fermi-Dirac distribution γk↑
γk↑ =
P
1
. The equation for the gap ∆ = − VΩ0 k hck↑ c−k↓ i thus becomes (replacing V0
1+eβEkas above by the scattering length a)m−=
4π~2 a

Z

d3 k
(2π)3

µ

1
tanh
2Ek

66

µ

βEk
2

¶

1
−
2²k

¶
(2.122)

Temperature of pair creation
We are interested in determining the temperature T ∗ = 1/β ∗ at which the gap
vanishes. In the BCS-limit, this procedure gives the critical temperature for thenormal-to-superfluid transition. Setting ∆ = 0 in the gap equation, one needs tosolve [172, 83, 74]m−=
4π~2 a

Z

d3 k
(2π)3

µ

1
tanh
2ξk

µ

β ∗ ξk
2

¶

1
−
2²k

¶
(2.123)This has to be solved simultaneously with the constraint on the total number ofatoms. Above the temperature T ∗ , we have a normal Fermi gas with a Fermi-Dirac
distribution, so the number equation becomes
Z
n=2

d3 k
1
3
(2π) 1 + eβ∗ξk(2.124)In the BCS-limit, we can expect µ À kB T ∗ and thus find µ ≈ EF . Inserted in the
gap equation, this gives the expected critical temperature for BCS superfluidity∗
TBCS= TC,BCS =eγ 8 −π/2kF |a| eγ
e
= ∆0
π e2
π

(2.125)

with Euler’s constant γ, and eγ ≈ 1.78. Here, we distinguish ∆0 , the value of thesuperfluid gap at zero temperature, from the temperature-dependent gap ∆(T ). One
can show that ∆0 − √2π∆0 kB T e−∆0 /kB T ,qq∆(T ) ≈
8π 2kT
1 − TTC ,7ζ(3) B C

T ¿ TCT C − T ¿ TC

(2.126)In the BEC-limit, the chemical potential µ = −Eb /2 = −~2 /2ma2 is again given
by half the molecular binding energy as before, and the temperature T ∗ is found to
be
∗
≈
TBEC

³
2

|Eb |
b|
ln |E
EF

´3/2

(2.127)This is not the critical temperature for the superfluid transition but simply the temperature around which pairs start to form. The logarithmic factor has its origin in
the entropy of the molecule-free fermion-mixture, which favors unbound fermions andlowers kB T ∗ below the binding energy Eb . There is no phase transition at T ∗ .

67TC / TF, T* / TF

0.4

0.2

0
-1

0

1

1/kFa

Figure 2-10: Superfluid transition temperature TC and pair creation temperature T ∗(dashed line) in the BEC-BCS crossover. In the BEC regime, TC corresponds to the
BEC transition temperature for a gas of molecules. In the BCS regime, the criticaltemperature depends exponentially on the interaction strength, drastically reducing
TC . Figure extracted from [74].
Critical temperature
Determining TC , the temperature at which long-range order is established, is more
involved. In the deep BEC-regime, the critical temperature is simply given by the
non-interacting value for the BEC transition of a gas of molecules at density nM = n/2
and mass mM = 2m,
2π~2TC,BEC =mMµ

nM
ζ( 32 )

¶2/3

π~2
=
m

µ

n
2ζ( 23 )¶2/3= 0.21EF(2.128)This result holds for weakly interacting gases, with only a small correction due tointeractions. On the BCS-side, the critical temperature should smoothly connect tothe BCS result given above. Fig. 2-10 shows the behavior of TC as a function of
the interaction strength. Whether there is a local maximum of TC somewhere in the
crossover region is not fully settled [115].
”Preformed” pairs
In the region between TC and T ∗ , we will already find bound pairs in the gas, which
are not yet condensed. In the BCS-limit, where T∗ → TC , condensation occurs at
the same time as pairing, which we see now is not true for stronger interactions.
68On the deep molecular side, it is of course not surprising to find thermal moleculesin the gas above TC . However, the qualitative picture of thermal pairs still holds
in the entire crossover region from −1 < 1/kF a < 1. These uncondensed pairs are
sometimes called ”preformed” (pairing occurs before condensation) and are detected
as a ”pseudo-gap” in RF spectroscopy experiments [59].

2.3.5

Definition of the condensate wavefunction

As Fritz London proposed in 1938, superfluidity should be a quantum mechanicalphenomenon occurring over macroscopic distances. A ”wavefunction” ψ(r) should be
ascribed to the superfluid, whose ”stiffness” against perturbations would result in thepeculiar properties of the superfluid.
In the case of weakly interacting Bose-Einstein condensates, we can readily identify
ψ(r) with the condensate wavefunction shared by all the bosons in the ground stateof the system. Its squared magnitude |ψ(r)|2 ≡ n(r) is equal to the density of thecondensed gas. Thus, taking an absorption image of a weakly interacting BoseEinstein condensate directly reveals the magnitude of the wavefunction. This has
led to the spectacular demonstration of coherence between two Bose condensates [13]and the observation of vortex lattices [160, 2, 120, 87] by straight-forward absorption
imaging (to name only a few examples).
For fermionic superfluids, it is the center-of-mass wavefunction of the condensed
fermion pairs which takes on the role of ψ(r). We can define it as the thermal averageof the pair creation operator Ψ†↑ (r) Ψ†↓ (r):DE
ψ(r) = N Ψ†↑ (r) Ψ†↓ (r)(2.129)where N is a constant we can conveniently choose.
One realizes an immediate difficulty in measuring this wavefunction: If the pairs
are not tightly bound, like on resonance or in the BCS-regime, an image of the atomic
density will not reveal the center-of-mass wavefunction of the
D pairs. OneE would rather
observe the distribution of, say, the spin-up atoms n↑ (r) = Ψ†↑ (r)Ψ↑ (r) , which turns
out to practically not change as one enters the superfluid regime17 . This presents a
more profound difficulty than the mere absence of a bimodal density profile of thegas in the presence of a thermal cloud. It shows that in the BCS-regime, even at zerotemperature, we cannot easily detect interference, vortices or other effects related to17

We have experimentally observed slight changes in the density profile on resonance, see chapter 5.
Note also that for unequal mixtures, there is a dramatic signature, see chapter 7.

69

the condensate’s coherence.
As one can readily checkD with the
E toolsPfrom previous sections, in a uniform BCS
P
† †
N
∆
state one has ψ = Ω k ck↑ c−k↓ = NΩ k 2E=N∆, where we have used the BCSVkgap equation. We see that ψ is simply proportional to the superfluid gap ∆, whichis zero in the normal and finite in the superfluid state. As such, ψ(r) naturally lends
itself as the order parameter for the superfluid phase transition, an angle of thoughtthat was pioneered by Landau.
BEC-limit: Gross-Pitaevskii equation
How should we choose the prefactor N ? In the BEC-limit of tightly bound molecules, ψ(r) should represent the wavefunction of the molecular condensate. As such,its squared magnitude should give the density of condensed molecules
D
EnM , equal tohalf the total density n at zero temperature: |ψ|2 = n/2. As Ψ†↑ Ψ†↓ already hasunits of density, the prefactor N must be the square root of a volume.
q We found in
16 √EFsection 2.3.1 that in the BEC-limit and at zero temperature, ∆ = 3π
. UsingkF a√
V = 4π~2 a/m it can be seen that one should choose N = 2π a3/2 . The resulting condensate wavefunction ψ(r) solves the Gross-Pitaevskii-equation we know fromatomic Bose-Einstein condensates:
µ 2 2
¶
~∇
2
−
+ Vext (r) + g |ψ(r)| ψ(r) = µM ψ(r)(2.130)2mMwith g = 4π~2 aM /mM , the molecular scattering length aM , the mass mM = 2m andthe molecular chemical potential µM .
In complete analogy with atomic BEC, we directly know that (in the ThomasFermi regime, neglecting the kinetic energy) the condensate density will be givenbyµ2

nM (r) = |ψ(r)| = max¶µM − Vext (r),0g(2.131)

which results in the famous inverted parabola in the case of harmonic trapping. Notethat for experiments in optical traps, the polarizability of molecules is twice that ofsingle atoms, so that Vext for molecules is twice as large as the potential experienced
by atoms.
We thus expect a direct signature of condensation in the density profile of themolecular gas. This was therefore our starting point for studying fermionic superfluids, as will be described in chapter 5.

70

BCS-limit: Ginzburg-Landau equation
In the limit of tightly bound molecules, the fermionic degrees of freedom are not relevant for the description of the condensate. The momenta of the fermions forming
the molecules are spread out over the large range of 1/a, the inverse size of the molecule, which is in this limit much larger than the Fermi momentum kF . The fermionsthemselves thus form a non-degenerate gas. In the BCS-limit, the fermionic degrees
of freedom are dominant, with each spin species forming its respective Fermi sea. We
can then no longer expect that there exists a single equation for the center-of-mass
wavefunction of pairs ψ(r), which can correctly describe the physics of the fermion
pair condensate. The Cooper pairs are spread out in space and overlap strongly, sothat a priori, non-local equations should replace the local Gross-Pitaevskii equation we
had found for the condensate of molecules. Indeed, the Gorkov formulation based on
Green’s functions allows to write BCS theory in Dthe form of two coupledequations for
E
®
­ †
†
†
the normal Ψσ (r, t) Ψσ0 (r0 , t0 ) and anomalous Ψ↑ (r, t) Ψ↓ (r0 , t0 ) Green’s functions
and their complex conjugates (see for example [90]). Still, in the limit where the gap∆(r) is small, and close to the critical temperature TC , one can find a single equation
for the center-of-mass wavefunction of the pairs, the Ginzburg-Landau equation. It
was introduced in 1950 as a purely phenomenological equation for the condensate
wave function ψ(r) with parameters that needed to be fixed by experiments18 :
µ 2 2
¶
~∇
2
−
+ b(r) |ψ(r)| ψ(r) = −a(r) ψ(r)2m∗

(2.132)

Closely resembling the Gross-Pitaevskii equation above, this equation describes
the motion of ”superfermions” of mass m∗ . The condensate wavefunction is normalized to give the density of these ”superfermions”, |ψ(r)|2 = n∗s . In a uniform system,
the two solutions of Eq. 2.132 are ψ = 0 (the normal state) or |ψ|2 = −a/b. The orderparameter ψ should vanish at TC , so a natural choice is to set a(T ) = (T − TC )a0 .
This implies a ”superfermion” density that vanishes linearly close to TC . The simplest choice for b is a constant. For a non-uniform system, we see by inspection thatEq. 2.132 defines a natural length scale over which the order parameter varies, the18

We omit the effect of a magnetic field for the case of electron pairs, which could be included byintroducing a vector potential A. Rotation of fermion pairs in a neutral fermionic superfluid could
be included as well, as it is formally equivalent to the motion of superconducting electron pairs in amagnetic field: One simply replaces eA/c by mΩ × r, where Ω is the rotation frequency.

71

Ginzburg-Landau coherence lengthsξGL (T ) =s~22m∗ |a(T )|=~22m∗ (TC − T ) |a0 |(2.133)This length scale becomes very large close to the critical temperature, and in particular it can be large compared to the BCS-coherence length ξBCS = ~vF /π∆0 , defined
above via the zero-temperature gap ∆0 . Spatial variations of the order parameterthus occur at a length scale much larger than the size of a Cooper pair. This is inessence the reason why, even in the BCS-regime of long-range fermion pairs, the condensate can be described by a local equation [236]. Remarkably, the Ginzburg-Landau
equation, introduced on phenomenological grounds, was later derived by Gorkov from
BCS theory. The ”superfermions” turned out to be fermion pairs, so that a natural
normalization of the wavefunction is provided by the density of pairs, |ψ(r)|2 = ns /2,
where ns is the total superfluid density (at zero temperature, we necessarily havens = n, the total fermion density). This normalization fixes the phenomenological³´
2 (k T )2
T
B C
1
−
and
parameters, m∗ = 2m as expected for fermion pairs, a = − 6π7ζ(3)ETCF³´|a|/b = n 1 − TTC . Comparing with the equation for the superfluid gap close to
TC , Eq. 2.126, and using the result for the critical temperature TC = ∆0 eγ /π in the
BCS-limit, we arrive atµT|ψ| = n 1 −
TC
2

¶
=

7ζ(3) ∆2
∆2
n
≈
0.33
n
8e2γ ∆20∆20(2.134)

The numerical prefactor is not correct at zero temperature, where one should expect|ψ|2 = n/2, as here, the superfluid density must equal the total density. This should
not be of concern, however, since T = 0 lies outside the applicability of the GinzburgLandau equation. The approximate expression for the gap close to TC in Eq. 2.126
extrapolates to an incorrect value (≈ 1.7∆0 ) at T = 0.
At least close to TC , Eq. 2.134 finally fixes the normalization parameter N in
Eq. 2.129 for the BCS-regime,
r
N =

r
√
1
7ζ(3)
n
56ζ(3)
EF 3/2
p
V
=
|a|
8e2γ
∆0
3e2γ
kF |a| ∆0(2.135)

It is satisfying to noteq that, using the limiting expression for the gap ∆0 in the

16 √EF
BEC-regime, ∆0 =, one recovers, up to a numerical constant, the same
3π kF anormalization constant N ∝ a3/2 we had found before to arrive at the Gross-Pitaevskii

72equation. This suggests that the regime of applicability of the Ginzburg-Landau
equation increases from the far BCS-side, where it is TC − T ¿ TC , to the far BECside, where it encompasses all temperatures from T = 0 up to TC . In the intermediate
regime of strong interactions (kF |a| À 1), the Ginzburg-Landau approach is expected
to still provide a reasonable, qualitative description of the condensate [74, 236].

73

Chapter 3
Feshbach resonances
Typical scattering lengths in alkali atoms are on the order of the van der Waalsrange r0 ≈ 50 − 100 a0 . Common interparticle spacings in our ultradilute gases
are n−1/3 ∼ 10 000 a0 . We have seen in the preceding chapter that it is simply
hopeless to achieve superfluidity in Fermi gases with such a small interaction strength
kF |a| ∼ 0.02, as the critical temperature depends exponentially on this parameter.
Clearly, one needs a way to enhance the interatomic interactions by some sort of
scattering resonance.
Early on, 6 Li was considered a very promising candidate to achieve fermionicsuperfluidity [237], as its triplet scattering length was found to be unusually large andnegative, about −2 000 a0 [3]. As we will see, this is due to an almost bound statein the interatomic potential of 6 Li, causing low-energy collisions to be indeed almost
resonant. While this seems already too good to be true, 6 Li offers in addition a ”magicknob” to tune the interaction strength at will, an unusually wide Feshbach resonance,
predicted in 1997 [124]. These resonances occur as a bound state in the interatomic
potential is tuned into resonance with the energy of two colliding atoms. This tuning
is possible via an applied magnetic field if the magnetic moment of the bound state
differs from that of the two unbound atoms. Feshbach resonances have not only
allowed to reach the strongly interacting regime where kF |a| > 1 [174, 106, 36], butalso to create ultracold molecules from samples of ultracold atoms [205, 69, 239, 128], a
new way of doing ”quantum chemistry”. Of great importance here was the realizationthat the gas close to resonance is stable [78, 174, 69, 239, 128, 202], a property thatis directly linked to the Pauli principle [193].
In the following, I will describe scattering resonances in general and the phenomenon of Feshbach resonances in particular.

74

3.1

Scattering resonances

Let us consider the archetypical example of a square-well potential of size R and
depth V (see Fig. 3-1). We have already seen in the preceding chapter that in order
have a 3D bound state for two particles of mass m, the potential well needs to be
2 2

π ~deeper than a minimum depth Vc . For a square-well potential [145], Vc = 4mR
2 . Letus discuss the relevant situations.

• If no bound state is present, for V < Vc , two colliding particles will feel attraction, corresponding to a negative scattering length a < 0. That is, ther>Rradial wavefunction for their relative motion u(r) ∼ sin(k(r − a)) appears tobe ”pulled in”, to originate from negative values of the distance r between theatoms (see upper left panel in Fig. 3-1). The closer V gets to the critical Vc ,the more negative the scattering length becomes. It is helpful to think of an”almost” bound state that exists just above the energy of the colliding atoms(the ”threshold”). Second-order perturbation theory then shows that this ”virtual” state will repel the incoming state, lowering the energy of the colliding
particles and thus leading to their attraction. The closer the virtual state lies
to threshold, the stronger the attraction.
• If the well is just a bit deeper than the critical Vc , a true bound state is available
below zero energy. This state pushes the incoming state up in energy and thus
leads to repulsion of the colliding atoms. The closer the bound state lies withrespect to the incoming state, the stronger the repulsion. The wavefunction
u(r) appears to be ”pushed out” from the origin, to originate from a > 0. We
see that necessarily, whenever a new bound state enters the scattering potential,
the wavefunction u(r) acquires a new node.
• Right at V = Vc , a true bound state forms at zero energy. Atoms colliding atvery low energies are thus in resonance with this state. The scattering length
diverges, causing the wavefunction u(r) to look ”flat” over distances r < 1/k. In
fact, over those distances u(r) cannot be distinguished from the equally ”flat”
wave-function of the very large bound state (see section 2.2.3), the overlap
between the two is at its maximum, as we expect for resonant coupling. Likein optics, this situation corresponds to maximum ”absorption” of the incoming
wave by the potential well, with an absorption cross section given by σ = 4π.
k2This reminds us of the resonant absorption cross section of light by atoms, which

75

4
3

a/R

2
1
0
-1
-2
0

EB / ER

-20

-40

-60

0

2

4

6
(V/ER)

1/2

8

10

12

Figure 3-1: Scattering resonances for the square well potential. Shown is the scattering length a and the bound state energies EB as a function of the square well depth
V . Close to a resonance (say at Vc ), the energy of the associated bound state depends
quadratically on V − Vc . The four panels above illustrate the evolution of the wavefunction (red) at selected points (blue circles). The zero-crossing of its tangent (bluedashed line) gives the scattering length. It is negative for a shallow well (left panel),corresponding to attraction. As a bound state enters the well at a critical depth(green), the scattering length diverges and reenters from +∞. The wavefunction now
has a new node at a.

76

is σ = 6πwith k the photon wave vector1 (see section 2.2.1).
k2
It is not difficult to show that the scattering length a is given by [145]µtan κRa=R 1−
κR

¶
(3.1)

q

where κ = mV
. Away from any resonances, the scattering length is always close~2
to the ”background” scattering length R. However, on top of this background value,
scattering resonances occur when κR = (2n + 1) π2 with integer n. This coincides
exactly with the appearance of a new bound state in the potential (see Fig. 3-1). The
scattering length is positive if the nearest state is truly bound and negative when it
is still ”virtual” or almost bound. Close to a resonance (say at Vc = ~2 kc2 /m), thescattering length diverges as
a≈

1
2~2
≈
kR(k − kc )mR(V − Vc )(3.2)

and for a > 0 the bound state energy is related to a by
EB = −

1 (V − Vc )2
~2
=− 2
4
ER
ma

(3.3)

We found this general behavior for weakly bound states already in chapter 2.2.3: The
binding energy depends quadratically on the ”detuning”. The beauty of the Feshbach resonance mechanism is that this detuning is indeed an externally controllable
parameter.

3.2

Feshbach resonances

We now turn to realistic interaction potentials between alkali atoms. Here, the interaction actually depends on the internal structure of the two colliding atoms, namely
on the relative spin orientation of their valence electrons, singlet or triplet. In Fig. 3-2
for example, the atoms enter in a triplet configuration. If there was no coupling between the singlet VS and the triplet potential VT , the atoms would simply scatter off
each other in VT (r), acquiring some certain, fixed phase shift. However, the hyperfine
1

The factor ”6” is related to the photon being a spin-1 particle, such that atom-photon collisions
always involve transfer of angular momentum. The lowest-energy collisions can thus be thought ofas being p-wave collisions.

77Energy

EnergyBound State∆µ B

Interatomic distance

B0

Magnetic Field

Figure 3-2: Origin of Feshbach resonances. Atoms entering (for example) in the triplet
potential are coupled to a singlet bound molecular state. By tuning the external
magnetic field, this bound state can be brought into resonance with the incoming
state (at B0 in the right graph).
interaction Vhf is not diagonal in the total electronic spin S = s1 + s2 of the two atoms
and thus provides a coupling between singlet and triplet potentials:
Vhf = ahf (s1 · i1 + s2 · i2 )
ahf
ahf
=
S (i1 + i2 ) +(s1 − s2 ) (i1 − i2 )22
= Vhf+ + Vhf−

(3.4)

with the hyperfine constant ahf and the nuclear spins i1,2 of the two atoms. Given
the coupling Vhf− , the atoms can explore the singlet potential as well. It is in ourcase a ”closed channel”, meaning that singlet continuum states are not available asfinal scattering states by energy conservation. A Feshbach resonance occurs when thestate in which the atoms collide (the ”incoming” state) is in resonance with a boundstate in this potential. The energy difference between the incoming and the Feshbachbound state can be tuned via an applied magnetic field, due to their different magnetic
moments (see Fig. 3-2).

3.2.1

A simple model

Let us consider a simple model of a Feshbach resonance, in which there is only one
bound state of importance |mi in the closed channel, the others being too far detuned
in energy (see Fig. 3-3). The continuum of plane waves of relative momentum k78

Two-particle energy

>

|k

0

>

|m

>

|ϕ

0

Detuning δδ0Figure 3-3: Simple model for a Feshbach resonance. The dashed lines show theuncoupled states: The closed channel molecular state |mi and the scattering states|ki of the continuum. The uncoupled resonance position lies at zero detuning, δ = 0.
The solid lines show the coupled states: The state |ϕi connects the molecular state |miat δ ¿ 0 to the lowest state of the continuum above resonance. At positive detuning,the molecular state is ”dissolved” in the continuum, merely causing an upshift of allcontinuum states as ϕ becomes the new lowest continuum state. In this illustration,
the continuum is discretized in equidistant energy levels. In the continuum limit(Fig. 3-4), the dressed molecular energy reaches zero at a finite, shifted resonanceposition δ0 .

79

between the two particles in the incoming channel will be denoted as |ki. In theabsence of coupling, these are eigenstates of the free hamiltonian
H0 |ki = 2²k |kiwith ²k =~2 k 22m

H0 |mi = δ |mi(3.5)

where δ, the bound state energy of the ”bare” molecular state, is the parameter we
can experimentally control. We will only consider interactions between |mi and the|ki’s, neglecting scattering that occurs exclusively in the incoming channel. First, weare interested in seeing how the molecular state is altered due to the coupling to thecontinuum |ki. Thus, we want to solve
H |ϕi = E |ϕiwith |ϕi = α |mi +X

ck |ki(3.6)

kfor E < 0 where H = H0 + V and the only non-zero matrix elements of V are√hm| V |ki = gk / Ω and their complex conjugates (we will take gk to be real). Ω isthe volume of the system and introduced in this definition for later convenience. We
quickly findgk
(E − 2²k ) ck = √ αΩ
1 X gk2 α
1 X
gk ck =(E − δ) α = √Ω k E − 2²kΩ k
1 X gk2
and thusE−δ =Ω k E − 2²k(3.7)

As we only consider low-energy s-wave scattering, one can take the gk ’s to be constant
g0 up to some cut-off ER and zero beyond. The right hand side of Eq. 3.7 side is thenidentical to Eq. 2.79, and we can read off the solution from table 2.2 (3D case):Zp
g02 ERρ(²)|E| + δ == δ0 − 2E0 |E|d²Ω 02² + |E|Z
2
X
g02 ER
4p
1
gk
1
=
E0 ER
with
δ0 ≡
d² ρ(²) =Ω k 2²k
Ω 0
2²
π
µ 2³
¶2
g0 m ´3/2
andE0 ≡2π 2~280

where we have assumed ER À |E|. As illustrated in Fig. 3-3, for positive detuning
δ > δ0 the original molecular state is ”dissolved” in the continuum. For δ − δ0 < 0,
we find a true bound state at
E = −E0 + δ − δ0 +p

(E0 − δ + δ0 )2 − (δ − δ0 )2(3.8)

This ”dressed” bound state energy is shown in Fig. 3-4. Far away from the resonancep√
region, for δ0 − δ À E0 , we have E ≈ −( δ0 − δ − E0 /2)2 − E20 , which describes,
apart from energy shifts, the original bound state. In the other limit, the bound statecrosses the energy of the incoming state at a shifted position δ = δ0 . Close to thispoint, the energy E ≈ − 21 (δ − δ0 )2 /E0 depends quadratically on the detuning δ − δ0 ,as expected.
Scattering length
To find the scattering length, it is convenient to project the Schrödinger equation
H |ψi = E |ψi onto the two relevant subspaces, the space of the scattering states |kiof the incoming channel and the molecular state of the closed channel, |mi. The goal
is to find out how the coupling to the molecular state affects scattering in k-space.
P
For this, we introduce the projectors Pk = k |ki hk| and Pm = |mi hm|. The weightof all other states being vanishingly small, we have Pk + Pm ≈ 1. The projected
wavefunctions are |ψk i = Pk |ψi and |ψm i = Pm |ψi. The free hamiltonian H0 acts
exclusively within each subspace, so it commutes with the projectors. We define H0k =Pk H0 Pk and H0m = Pm H0 Pm . In contrast, the interaction V couples k-space to thePmolecular state, and its only non-zero projections are Vkm = Pk V Pm = k √gkΩ |ki hm|
P
and Vmk = Pm V Pk = k √gkΩ |mi hk|. We can now easily calculate the projection of
H |ψi onto k-space: Pk H |ψi = Pk (H0 + V ) |ψi = H0k |ψk i + Pk V (Pk + Pm ) |ψi =H0k |ψk i + Vkm |ψm i. In the same way we get Pm H |ψi = H0m |ψm i + Vmk |ψk i. The
Schrödinger equation then reads(E − H0k ) |ψk i = Vkm |ψm i(E − H0m ) |ψm i = Vmk |ψk i(3.9)

We can formally solve the problem in the closed channel2 , |ψm i = (E − H0m +iη)−1 Vmk |ψk i and insert this into the equation for |ψk i:
2

The small positive infinitesimal part iη ensures that the scattered wave will correspond to an
outgoing wave.

81

(E − H0k ) |ψk i = Vkm (E − H0m + iη)−1 Vmk |ψk i =Vkm Vmk|ψk i
E − δ + iη

(3.10)

P
where we have used H0m Vmk = H0m |mi k √gkΩ hk| = δ Vmk .
The scattering problem is now entirely formulated in the open channel. The
molecular state causes an effective interaction that corresponds to two atoms colliding, spending some (infinitesimal) time in the molecular state and exiting again
as two unbound atoms. From equations 2.60 and 2.63 we can read off the scattering amplitude for zero-energy collisions and hence the scattering length. We only
need to insert the Fourier transform of the effective interaction at E = 0, which isg2

km VmkV0 = Ω limE,k,k0 →0 hk| VE−δ+iη|k 0 i = − δ0 :

δ − δ0
m
δ
1X 1
=
−
=
−
+
2
4π~2 ag0 Ω k 2²k
g02
or

a =

m
g02
4π~2 δ0 − δ(3.11)The scattering length, shown in Fig. 3-4, diverges at the shifted resonance positionδ = δ0 . Not surprisingly, we recover E = −~2 /ma2 for the bound state energy close
to resonance for positive a > 0, as it should be.
In the experiment, the Feshbach resonance occurs for a certain magnetic field
B0 . With the magnetic moment difference ∆µ between the incoming state and theclosed (uncoupled) molecular state, we have δ − δ0 = ∆µ(B − B0 ) (taking ∆µ to beconstant). Including the background scattering length abg for collisions that occur
entirely in the open channel, the scattering length can be written in its usual formµ
a = abg

3.2.2

∆B
1−
B − B0¶with ∆B =m
g02
4π~2 ∆µ abg(3.12)

Feshbach resonances in 6 Li

We will now discuss the case of 6 Li relevant for our experiments. Fig. 3-5 shows
the s-wave scattering length for collisions between the two lowest hyperfine states,|F, mi = |1/2, 1/2i and |1/2, −1/2i of 6 Li. It shows a broad Feshbach resonance centered around B0 = 834.15 G. The resonance is approximately described by Eq. 3.12
with abg = −1 405 a0 , ∆B = 300 G [24]. These values are highly unusual when comparing scattering lengths and Feshbach resonances in other alkali atoms. Background
82

0.5

Scattering length [a.u.]

Energy [a.u.]

10

5

0.0

0

-5

δ0

-10

-0.5
0.0

0.5

1.0

1.5

δ / δ0Figure 3-4: Bare, uncoupled molecular state (dashed line), coupled, bound molecularstate (solid line) and scattering length (dotted line) close to a Feshbach resonance.
The shaded area represents the continuum of scattering states, starting at the collisionthreshold at E = 0. Interaction between the molecular state and the continuum shifts
the position of the resonance by δ0 from the crossing of the uncoupled molecular statewith threshold. Note the quadratic behavior of the bound state energy with detuning(δ − δ0 ) close to resonance.

5

20

Energy [MHz]

0

0

-10

Scattering length [a0]

10

3

-5

-20x10543

650
834
Magnetic Field [G]Figure 3-5: Feshbach resonances in 6 Li between the two lowest hyperfine states|F, mi = |1/2, 1/2i and |1/2, −1/2i. A wide Feshbach resonance occurs at 834.15
G. The resonance position is shifted by an unusually large amount of ∼ 300 G fromthe crossing of the uncoupled molecular state at 543 G. A second, narrow Feshbachresonance occurs right at 543 G, shifted by less than 200 mG.

83

0

-500
1

Evs

+

X Σg , v = 38

0

I=0
-1000

Energy [MHz]Energy [MHz]

1

-1

I=2

∆B
0

543.25Magnetic Field [G]834.15Figure 3-6: Bound state energies for 6 Li2 in a magnetic field. The weakest boundstate of the singlet potential, X 1 Σ+
g , v = 38, splits into two hyperfine components
with total nuclear spin I = 0 and I = 2. The state with I = 0 is almost not coupled
to the triplet scattering continuum, causing the narrow resonance at 543.2 G. In
turn, the state I = 2 is very strongly coupled and leads to the broad resonance at
834 G, a shift of ∆B ≈ 300 G. The strong coupling is caused by a virtual state of thetriplet potential (dashed line in the inset), which is only Evs ≈ h · 300 kHz away fromthreshold (E = 0). It is this virtual state that is responsible for the large backgroundscattering length for triplet collisions.
scattering lengths are typically on the order of ±100 a0 or less, which is roughly therange of the van der Waals-potential. Typical widths of other observed Feshbach resonances are at least two, rather three orders of magnitude smaller than ∆B. Clearly,
the broad Feshbach resonance in 6 Li is a special case.
The unusually large background scattering length of 6 Li that decreases even to−2 100 a0 at high fields, hints at a resonance phenomenon even away from the wide
Feshbach resonance. Indeed, it turns out that there is an almost bound, virtualstate in the triplet potential of 6 Li very close to the collision threshold. From thelarge background scattering length we can estimate that the virtual state must lie
approximately Evs ≈ ~2 /ma2bg ≈ h · 300 kHz above threshold.
It turns out that the wide Feshbach resonance at 834 G is caused by an interplay
between the molecular state in the singlet potential and the open channel resonancedue to the virtual state. Such a situation has been analyzed in [162] and applied to thecase of 6 Li in [248]. As a crude approximation, we can argue that the molecular state
couples to the scattering continuum exclusively via the virtual bound state. Fromthe equation for the bound state energy, Eq. 3.7, we can estimate a resonance shift of84

2
δ ≈ gvs/Evs due to the coupling to the virtual state. Even for a coupling gvs that is
not unusually large we expect the resonance position to be shifted considerably (andhence the resonance to be wide) if the virtual state is close to threshold. Indeed, fromthe resonance width ∆B ≈ 300 G we can estimate
gvs ≈

p

δ Evs =p∆µ ∆B Evs ≈ h · 15 MHz(3.13)

where we have used ∆µ ≈ 2µB on resonance, where the incoming state is essentiallya pure triplet, while the molecular state is a singlet. The hyperfine constant for 6 Li is
ahf ≈ h · 150 MHz, which sets the scale for the interaction Vhf− . A typical coupling gk
R
is thus ahf times the overlap integral d3 rϕb (r)ψk (r) of the bound state wavefunction
and the scattering states3 . We see indeed that gvs is not particularly large, and thatthe cause for the wide Feshbach resonance at 834 G is the virtual bound state4 .

3.2.3

Broad versus narrow Feshbach resonances

We have seen how it is possible to tune the scattering length in atomic gas collisions
via Feshbach resonances. To what extent can the many-body description of chapter 2
be carried over to this case?
Let us recall that the derivation of the BEC-BCS crossover made use of a contact
potential V = V0 δ(r), which was adjusted in such a way as to reproduce the correct
scattering length a. However, the use of such a contact potential implies the relation
EB = −~2
ma2(3.14)

for the energy of the bound molecular state5 . This signifies that the character ofthe molecular state is entirely described by the scattering length, a property of the3

To estimate the overlap integral, we assume the weakly bound singlet state to have a characteristic size aB ∼ 50 a0 , the van der Waals-range of the potential. Over this small scale, the virtual√state’s radial wavefunction uvs = rψvs is approximately constant and equal to ∼ 1/ abg (for norp
Rmalization). The integral drub (r)uvs (r) is thus on the order of aB /abg ≈ 0.1 − 0.2. With thisreduction we obtain the correct order of magnitude for g.
4
The resonance at 543 G has a narrow width, because the molecular state causing it has avanishing matrix element (Clebsch-Gordan coefficient) with the virtual state.
5This can be seen by using a square well potential V (r) = −V for r < R and zero outside. The
relation between the scattering length, the Fourier transform of V (r), Ve (q) ≈ Ve (0) = const (forq ¿ 1/R) and R is given by Eq. 2.63:
X 1
m
1
=−
+
4π~2 a
2²k
Ve
k<1/R

85

(3.15)

scattering states in the open channel.
For any real situation, this relation holds only close to resonance. We see from thesimple model system of section 3.2.1 that the bound state energy starts to deviate
from the above relation once |EB | ≈ E0 . E0 is the energy scale associated with thestrength of the Feshbach coupling between the bare molecular state and the scatteringcontinuum. Close to resonance, for |EB | ¿ E0 , the ”dressed” molecular state|ϕi = α |mi +X

ck |ki(3.18)kwill be almost
q entirely described by the scattering states of the open channel |ki (one2|EB |
E0

finds α2 =¿ 1). However, as |EB | starts to grow beyond E0 , the dressedmolecular state resembles more and more the original ”bare” state |mi, and α ≈ 1.
The crucial question for the many-body description is now: How does the Fermi
energy EF compare to the coupling scale E0 ? The BEC-BCS crossover occurs for−1 < 1/kF a < 1. If we first reach the weakly interacting regime kF a ¿ 1 (equivalently EF ¿ ~2 /ma2 ) before ~2 /ma2 ≈ E0 , we can safely use the single channel
description for the crossover. The ”dressed” molecular state is almost completely
dissolved in the open channel continuum throughout the crossover and the detailsof the original molecular state |mi do not play a role (case of a ”broad” Feshbachresonance). However, if E0 is comparable to EF , then the molecular state affects the
many-body physics and it needs to be included in the description of the gas (case ofa ”narrow” Feshbach resonance) [227].
Using Eq. 3.12 and the definition for E0 , Eq. 3.8, we can relate the coupling energy
scale E0 to experimentally observable parameters. One has
E0 =1 (∆µ∆B)2
2 ~2 /ma2bg(3.19)

The fraction of the dressed molecular wave function that is in the deeply bound state
The equation for the bound state energy E < 0 was derived in Eq. 2.79 and isX
1
1=
2²k + |E|Ve

(3.16)

¶
µ
¶
X µ 1
m
1
1 ³ m ´3/2 p
ER=−=|E| arctan
4π~2 a2²k
2²k + |E|
2π 2 ~22 |E|(3.17)

k<1/R

We then havek<1/R

where we have used the cut-off energy scale ER = ~2 /mR2 . For ER À |E| we obtain Eq. 3.14.

86

|mi is

r
2

α =

2EB
=2
E0

r

EF 1
E0 kF a(3.20)For the resonance used in the experiments by D. Jin on 40 K, E0 /kB ≈ 1 mK, whichshould be compared to a typical Fermi energy of EF /kB = 1µK. This resonance isthus broad. Nevertheless, at kF a = 1 the fraction of the wavefunction in the closed
channel molecule is α2 ≈ 6%. This might possibly explain the shorter lifetime of thegas of molecules 40 K2 close to resonance [202] as compared to the case in 6 Li2 [35].
For the wide Feshbach resonance in 6 Li, one has E0 /kB ≈ 50 K, which makes it
an unusually broad resonance. However, as pointed out above, the simple model ofsection 3.2.1 does not apply to the case of 6 Li, and one has to take into account the
influence of the virtual state in the open channel. If this is done, one finds that theenergy scale equivalent to E0 is6 rather ≈ ∆µ∆B = kB · 40 mK, which is still verybroad. The simple relation EB = −~2 /ma2 holds still to better than 3% at a magnetic
field of 600 G, 234 G away from resonance, while many-body effects (kF a > 1) start
to become important only above 750 G. Indeed, the closed channel contribution to
the dressed molecular state has been measured in the group of R. Hulet [188] to beless than 1% at magnetic fields beyond 600 G and less than 10−3 throughout theentire strongly interacting regime beyond kF a ≈ 1.
The Feshbach resonance in 6 Li at 543 G, in turn, has E0 /kB ≈ 1 µK and is thus
narrow. In the case of a narrow resonance, the many-body physics is qualitatively
different from the BEC-BCS crossover picture. For a detuning δ − δ0 < 0, all fermion
pairs are still tightly bound in the closed channel molecular state, where they form acondensate. For δ − δ0 < 2EF , the molecular condensate coexists with a BCS-type
fermionic superfluid. Here, the molecular state (unstable in vacuum above threshold)is stabilized by Pauli blocking, as the outgoing momentum states are occupied byfermions in the BCS-state. Only for δ − δ0 > 2EF is the molecular state truly
emptied and we are left with a BCS-type superfluid. However, since the resonanceisqE0
narrow, the interactions for δ −δ0 > 2EF À E0 will be very small, kF |a| < EF ¿ 1,
rendering the observation of such a state extremely difficult.

6

At a detuning δ − δ0 = E0 , the bound state energy in the simple Feshbach model is about one
half times ~2 /ma2 . We can thus use the detuning where |EB | = 21 ~2 /ma2 as the definition of thecoupling energy scale.

87

Chapter 4
Cooling fermions - The apparatus
Cooling a gas of atoms into the nanokelvin regime appears to be (almost) commonplace: Eight years after the discovery of gaseous Bose-Einstein condensates in 1995,
BECs were reliably produced in about 50 laboratories around the world, a number has
steadily grown since. At this pace, one would expect the number ultracold fermion
experiments to exceed 50: The first degenerate Fermi gas of atoms was created in1998 [76]. However, by now only thirteen laboratories in the world are studying ultracold fermions. This can hardly be explained by lack of interest in fermions, as they
are the building blocks of matter and Fermi statistics governs the structure of nuclei,
atoms, and the behavior of metals. The explanation lies in experimental complexity. Of the two corner stones for achieving ultracold temperatures, laser cooling andevaporative cooling, the latter fails to work for a sample of spin-polarized fermions.
In evaporative cooling [118, 73], the most energetic particles are forcibly removed
from the trap, while the remaining atoms rethermalize to a lower temperature. Ina spin-polarized cloud of fermionic atoms, however, s-wave (”head-on”) collisions
are forbidden by the Pauli principle, and p-wave collisions will be frozen out at low
temperatures1 . After removal of energetic particles, the sample would not thermalize.
The way out of this dilemma is to use a coolant: This can be either a second
spin state of the same fermionic atom [76, 100, 130] or another atomic species [247,214, 112, 207, 97, 137, 225, 184, 19]. In such a mixture, s-wave collisions between
different atoms or spin states are allowed and sympathetic cooling of fermions can
1
We can estimate the temperature for which p-wave collisions can occur, using the van-der-Waals
potential −C6 /r6 . The kinetic energy of the colliding particles with angular momentum ~ must
exceed the centrifugal barrier: E > ~2 /µr02 − C6 /r06 = 2~2 /3µr02 , where µ = m/2 is the reduced mass¡
¢1/4
is the position of the barrier. For 6 Li, r0 = 70 a0 and thus kB T ≈ 8 mK. At
and r0 = 3µC6 /~2
the temperatures achieved by laser-cooling (∼ 500 µK in our magneto-optical trap for 6 Li), p-wave
collisions are frozen out.

88

proceed. In our experiment, fermionic 6 Li is cooled by a cloud of bosonic 23 Na atoms
in a magnetic trap, after an initial stage of laser cooling.
The choice of 6 Li as the fermion species is simply explained by the relatively highnatural abundance of about 7% percent, allowing the use of an atomic beam source,
and the ease of laser cooling of alkali atoms. The only other stable fermionic alkali
atom, 40 K, has a natural abundance of only 0.01%. The common conclusion is that itneeds to be enriched in an expensive process, precluding its use in a high-flux atomic
beam oven2 . Our group’s expertise in Bose-Einstein condensation of sodium atoms in 2001 there existed three working sodium BEC-machines - made the choice of thecoolant a natural one. With 23 Na, the largest alkali condensates (containing up to
100 million atoms) have been produced. The large number of available sodium atoms
presents an ideal refrigerator with large heat capacity. However, at the time theproject was started, the collision properties between sodium and lithium atoms werecompletely unknown. It was thus a gamble whether sympathetic cooling would befeasible at all. Nature was kind: The approach led to the production of large cloudsof about 50 million degenerate fermions [111] and to condensates containing about 10
million fermion pairs, a number that is a factor of 50-100 larger than that produced
in other experiments. This strong atomic signal certainly helped in the discovery ofsmall features such as vortices (chapter 6) or faint kinks (chapters 5 and 7) in thedensity profile of fermionic superfluids.

4.1

Experimental setup

Table 4.1 summarizes the strategy followed to cool a gas of 6 Li fermions from temperatures of 700 K in an oven down to 50 nK deep in the superfluid regime. It involves aseries of different techniques which all have to work reliably to produce a consistentsample in every experimental run. From beginning to end, the sequence lasts about
40 s, the longest stage being the ∼ 15 − 20 s of RF evaporation of sodium in themagnetic trap.
The techniques of laser cooling and evaporative cooling in a magnetic trap, leading
to Bose-Einstein condensation, are nicely described in the Nobel lectures of 1997 (laser
cooling, S. Chu [60], W. Phillips [194] and C. Cohen-Tannoudji [63]) and 2001 (BoseEinstein condensation, E. Cornell, C. Wieman [67] and W. Ketterle [136]). Detailed
descriptions of a sodium BEC-apparatus can be found in PhD-theses [232, 57] of
2

An atomic beam source for non-enriched potassium has been demonstrated very recently in the
Ketterle group [261].

89

StageTemperature

Two-species oven
Laser cooling(Zeeman slower & MOT)
Sympathetic cooling(Magnetic trap)
Evaporative cooling(Optical trap)

6

Li Atom number

Density

T /TF

720 K

3 × 1011 atomss1014 cm−3

108

1 mK

∼ 3 × 1081010 cm−3

104

1 µK

5 × 107

5 × 1012 cm−3

0.3

50 nK

2 × 107

5 × 1012 cm−3

0.05

Table 4.1: Different stages during the preparation of a superfluid Fermi gas. Typicalnumbers for the temperature, atom number, density and degeneracy factor T /TF aregiven. Up to the last stage, the gas is spin-polarized. In the optical trap, a spinmixture of the two lowest hyperfine states of 6 Li is evaporatively cooled close to a
Feshbach resonance.
former members of the Ketterle group. The addition of fermionic 6 Li to an existing
sodium BEC apparatus3 required designing a double-species oven, described in detailin the PhD thesis of Cladiu Stan [233], as well as providing laser light for laser-coolingof 6 Li. Details on the latter can be found in my thesis written for the French Magistèrede Physique [264], as well as in Zoran Hadzibabic’s PhD thesis [110].
The experimental setup is sketched in Fig. 4-1. Hot (∼450 degrees C) 6 Li and 23 Na
atoms escape from the oven chamber, traverse a Zeeman slower and are captured ina double-species magneto-optical trap (MOT). We obtain about 1010 sodium atoms
and 3 · 108 lithium atoms stored at temperatures of about 1 mK and densities of∼ 1010 cm−3 for 6 Li and ∼ 1011 cm−3 for 23 Na. The phase space densities of the clouds
are about 10−7 . The temperature is limited by the natural line width of the opticaltransition (Doppler limit, see e.g. [194]). Hence, to advance further, the trapping lightneeds to be switched off and the atoms are transferred into the conservative potentialof a Ioffe-Pritchard DC magnetic trap. It is crucial in our scheme to prepare the two
species in their respective ”stretched hyperfine states”, that is, both the nuclear andelectronic spins of each atom need to be aligned with the magnetic field. This ensuresthat interatomic collisions (which to a good approximation preserve the total spin
projection on the magnetic field axis) cannot transfer atoms into other spin states(so-called spin-exchange collisions) - a process that could potentially release a large
amount of energy and heat the sample.
3

It is the same apparatus that produced the first sodium Bose-Einstein condensates in 1995.

90

Figure 4-1: Experimental setup. a) The source of atoms, a double species oven for
Li and 23 Na. b) Main chamber, showing the setup for the MOT and slowing beams
and one of the two possible imaging directions.

6

Optical Pumping into the Stretched State Fig. 4-2 provides a closer look at
the hyperfine states of 23 Na and 6 Li. All alkali atoms have a single unpaired valence
electron, so S = 1/2. As 23 Na has a nuclear spin I = 3/2, we have (2S +1)(2I +1) = 8
hyperfine states available. For 6 Li with I = 1, there are 6 hyperfine states. Only socalled ”low-field seeking” states can be magnetically trapped4 , leaving us in principlewith three options for sodium (not counting the weakly trapped |2, 0i state) andthree for lithium. Inspection of possible (that is, energetically favored) spin-exchange
collisions eliminates all combinations but two [110]:|2, 2iNa + |3/2, 3/2iLi(stretched states)|1, −1iNa + |1/2, −1/2iLi
As we see from Fig. 4-2, the lithium state |1/2, −1/2i is only low-field seeking
below 27 G. The lithium cloud would need to be colder than about 300µK in order
not to spill out from the trap. Clearly, the robust solution is to prepare both atomic
species in their respective stretched state5 .
4Maxwell’s equations only allow magnetic field minima in free space. Thus, only states thatminimize their energy at low field can be trapped.
5
Historically, the stretched state combination was not the first to be tried [112], since sodium had
initially only been condensed in the |1, −1i state, and inelastic losses in the |2, 2i state were known

91

23Na

6Li

|6>
|5>
|4>

400
2000

F = 3/2

F=2
1000

mS = +1/2

0mS = -1/2

Energy [MHz]Energy [MHz]mS = +1/2

200

0mS = -1/2

-200

-1000

F = 1/2

F=1
-2000

|3>

-400

|2>
|1>

-3000
0

400

800

1200

0

Magnetic Field [G]

50

100

150

200

250

300

Magnetic Field [G]Figure 4-2: Hyperfine states of 23 Na and 6 Li. Energies are relative to the situation
without hyperfine interaction. The sodium hyperfine splitting is aNa
hf = 1.77 GHz, forLilithium it is ahf = 228 MHz. Good quantum numbers at low field are the total (nuclear
plus electronic) spin F and its z-projection, mF . At high field, it is the electronic
and nuclear spin projections mS and mI . The states relevant for the experimentare marked with thick lines. Evaporative cooling of sodium and sympathetic coolingof lithium are performed in the upper, stretched states. In the final stage of theexperiment, the strongly interacting Fermi mixture is formed around the Feshbachresonance at 834 G (not shown in the graph) using the two lowest hyperfine states oflithium, labelled |1i and |2i.

We achieve spin-polarization of lithium and sodium by optical pumping for ∼ 500µs with σ + polarized light right after switching off the MOT trapping light. For 6 Li,
both F (hyperfine-changing) and mF (Zeeman-changing) pumping is used, while for23

Na we only employ F -pumping6 . Next, the Ioffe-Pritchard magnetic trap is switchedon. A ”purification” stage ensures that only stretched states remain in the trap: Withthe 23 Na F -pumping light (resonantly exciting F = 1 atoms) still illuminating thecloud, the bias field of the trap is increased to 80 G. At this field, all hyperfine states
are resolved by RF spectroscopy at our initial temperatures of about 1 mK. A 0.6 s
to be more severe. Only later was condensation achieved also in the stretched state [111], greatly
facilitating cooling of lithium.
6
Zeeman or mF -pumping for 23 Na can improve the sodium numbers by ∼30%, which is usually
not worth the additional effort.

92

Figure 4-3: Magnetic trap for sympathetic cooling of 6 Li with 23 Na. The trap consistsof a ”curvature coil” that produces an axially confining potential. Its offset magnetic
field is cancelled to about 0.5 G by the ”anti-bias coil”. Radial confinement is provided
by the gradient coils which are wound in ”cloverleaf” configuration. They replace thefour Ioffe bars in a standard Ioffe-Pritchard trap. The anti-bias coils are used at alater stage to access the wide Feshbach resonance between states |1i and |2i at 834
G.

microwave sweep near the hyperfine splitting of 23 Na at 1.77 GHz transfers possible|2, 1iNa and |2, 0iNa 23 Na-atoms into the untrapped |1, 0iNa or |1, 1iNa state. Here, theatoms get ”a second chance” to be transferred into the stretched state via absorptionof the F -pumping light. If they again end up in the ”wrong” mF -state, they are
finally expelled from the trap by another 1.2 s microwave sweep, this time with thepumping light switched off. The final transfer efficiency from the MOT into stretchedstates in the magnetic trap is about 30-40% for sodium, and about 40-50% for 6 Li.

4.2

Sympathetic cooling of 6Li by 23Na

Once the atoms are loaded into the magnetic trap (see Fig. 4-3), 23 Na is evaporatively
cooled by forcibly removing energetic atoms with a microwave ”knife” tuned to the|2, 2i → |1, 1i hyperfine transition above 1.77 GHz. The lithium atoms are insensitive
to this frequency and simply cool via thermalization with sodium atoms. After 15 s ofevaporative cooling, the trap is adiabatically decompressed over 1 s in order to reduce
the effect of inelastic collisions in the increasingly dense sodium cloud. Without thelithium cooling load, another 5 s of evaporative cooling result in a Bose-Einstein93

condensate of 107 sodium atoms. For experiments with lithium, sodium is completely
evaporated over 5 s, leaving a degenerate cloud of about N = 5 · 107 spin-polarized
6

Li fermions at a temperature of T ≈ 0.3TF , where kB TF = ~ω(6N )1/3 is the Fermi
energy, and ω is the geometric mean of the trap frequencies.
If less lithium atoms are initially loaded into the magnetic trap, the cooling load
for sodium is smaller, resulting in a lower final temperature of the fermionic cloud,with T ≈ 0.05TF [111]. However, it is not preferable to minimize temperature at thecost of atom numbers at this stage, as further manipulations of the cloud necessarily
increase entropy. There are two main reasons for this increase in entropy:
1. The next stage of the experiment will require the transfer of the spin-polarized
cloud from the magnetic into an optical trap. As the two traps are never
perfectly ”mode-matched” (e.g. they do not have equal trapping frequencies in
the case of harmonic trapping), the non-interacting Fermi gas, initially preparedin equilibrium, is necessarily brought into a non-equilibrium distribution. Oncethe gas is allowed to thermalize (see below), this will result in an increased
entropy.
2. The experiment proceeds by creating a two-state Fermi mixture from theinitially spin-polarized cloud. This is achieved in two steps: First, starting withall atoms in the initial state |1i, a radiofrequency (RF) Landau-Zener sweep
transfers all atoms into a coherent superposition of the initial state and the statethat we want to populate, |2i. This coherent manipulation obviously preserves
entropy. However, as atoms travel along different paths in the trap, they pick up
different phases in the inhomogeneous magnetic field and decohere. Even if we
had started with a zero-temperature Fermi sea of |1i-atoms, we are now left with
two Fermi clouds of |1i- and |2i-atoms in which every second momentum stateis unoccupied within the original Fermi sphere. Clearly, the final temperature
will not be zero but rather a good fraction of the Fermi temperature.
We see that it does not pay off to reach very low temperatures at the cost of atom
number. Further evaporative cooling after loading the optical trap and creating thetwo-state mixture is needed. This will actually be more efficient than sympathetic
cooling by sodium atoms, due to the strong interactions between the two fermionicspin states. The true figure of merit of the sympathetic cooling stage is thus thenumber of atoms that can be loaded into the finite volume of the optical trap.

94

4.3

Preparation of an interacting Fermi mixture
in an optical trap

At the end of the sympathetic cooling stage, we are left with a degenerate Fermi
gas of atoms polarized in the stretched hyperfine state of 6 Li, |F, mi = |3/2, 3/2i.
As p-wave collisions are frozen out and s-wave collisions are forbidden by the Pauliprinciple, such a non-interacting gas represents indeed one of the most ”ideal” gases
imaginable. Evidently, in order to study interacting Fermions, a second spin stateneeds to be populated to again allow for s-wave collisions. Unfortunately, all mixturesof two magnetically trappable hyperfine states in 6 Li are unstable with respect to spinchanging collisions7 . However, the lowest two hyperfine states of 6 Li, |1i ≡ |1/2, 1/2i
and |2i ≡ |1/2, −1/2i are stable. Furthermore, as we have seen in chapter 3, theirinteractions are tunable over a wide range in the vicinity of a Feshbach resonance.
Both states |1i and |2i are high-field seeking at the relevant magnetic fields aroundthe Feshbach resonance. We are thus forced to use an optical trap, which can holdatoms regardless of their spin.

4.3.1

Magnetic plus optical trapping

As atoms are polarizable particles, they are attracted by a static or slowly oscillating
electric field (slow on the timescale of electronic transitions). A laser beam detuned
to the red of the atomic transition (= slowly oscillating) thus traps the atoms at
its intensity maximum. Our optical dipole trap (ODT) consists of a single laser
beam 8 at a wavelength of λ = 1064 nm, a maximum power of 4 W and a waist(1/e2 -radius) of about9 w = 125 µm. The maximum trap depth is about kB × 10 µK,
just about enough to catch the degenerate Fermi cloud in the magnetic trap with
Fermi energy kB × 3 µK. We take great care to ensure a radially symmetric trapping
potential, needed for the study of rotating quantum gases described in chapter 6.
The maximum radial trapping frequency is 300 Hz. The axial trapping provided by
the laser beam is very weak, due to the laser beam’s large waist and accordingly
slow beam divergence (Rayleigh range z0 = πw2 /λ = 4.6 cm!). If the optical trap
was the only potential acting on the atoms, the aspect ratio of the trapped cloud7|3/2, 3/2i + |3/2, 1/2i → |3/2, 3/2i + |1/2, 1/2i + 11mK; |3/2, 3/2i + |1/2, −1/2i → |3/2, 1/2i +|1/2, 1/2i, and |1/2, 1/2i is untrapped; |3/2, 1/2i + |1/2, −1/2i → |3/2, −1/2i + |1/2, 1/2i, bothstates are untrapped.
8
Yb fiber laser, 20 W, IPG Photonics model PYL-20-LP
9
Waist and maximum power have changed many times over the years. I present here the mostrecent setup, used to detect vortices in fermionic superfluids (see chapter 6).

95

√would be 2z0 /w ≈ 500 and the axial trapping frequency only 0.5 Hz. To ensure amore manageable geometry and faster dynamics, we clearly need additional magnetic
confinement in the axial direction. This is conveniently provided by the ”curvaturecoils” already used in the magnetic trapping stage. They create a magnetic saddle
point potential with typical axial trapping frequency of νz = 23 Hz. The radial anti√
trapping with (imaginary) frequency νr = i 23/ 2 Hz is negligible compared to thestrong radial confinement provided by the ODT.

4.3.2

Spin transfer

At the end of sympathetic cooling in the magnetic trap, the spin-polarized Fermi gas
is loaded into the optical trap by ramping up the laser power over 300 ms. The radialconfinement of the magnetic trap can now be removed, which is done by reducing
the current in the ”gradient coils” (see Fig. 4-3) to zero10 over 1 s. The atoms,
all polarized in the stretched state |F, mi = |3/2, 3/2i, experience now the radialconfinement of the optical trap plus the axial magnetic curvature. The transfer into
state |1i ≡ |1/2, 1/2i can be simply achieved by a RF sweep across the single-photon|3/2, 3/2i → |1/2, 1/2i hyperfine transition at 228 MHz. However, state |1i atoms
are high-field seekers and thus experience an anti-trapping axial curvature after the
spin flip. By quickly reversing the sign of the axial magnetic bias field the atoms aretrapped again (see Fig. 4-4). To minimize heating of the cloud due to expansion in
the transient anti-trap, the field curvature is reduced before the spin transfer, andthe magnetic field reversal takes place after less than half a trapping period. A moreelegant solution to this spin flip problem would be to use a more tightly focussed
ODT (which we avoid for other reasons, see chapter 6) or a second optical dipole trapthat provides axial confinement regardless of the atomic spin. However, subsequentevaporative cooling of the interacting |1i-|2i mixture (see below) will efficiently remove
the produced heat.
At this stage, the magnetic field is now increased to values around the Feshbachresonance between state |1i and |2i (see chapter 3), located at B = 834 G. Here, the6
Li hyperfine states are in the Paschen-Back regime, that is, the electron and nuclear
10

This is a slightly delicate process, as one needs to ensure that the center of the magnetic trap
remains aligned with the optical trap at all times during the current ramp. In the radial direction,
this trap center is defined by the zero-point of the radial magnetic fields. To this one must add
the influence of gravity, leading to sag. The position of the optical trap can be different, due torequirements at later stages of the experiment (as in chapter 6). Additional bias fields in the radial
direction allow to overlap the magnetic trap with the optical trap, but the required field values
depend on the current in the cloverleaf coils. Hence, the bias fields need to be dynamically adjusted
as the gradient current is reduced.

96

Spin Flip
State:|3/2,3/2>

Field Flip
|1/2,1/2>|1/2,1/2>

Spin Flip
Trapping Potential [a.u.]

Magnetic Field B [a.u.]|3/2,3/2>

1

B0

B0

0

B1

|1/2,1/2>

1

0
-1

-1

Field Flip
|1/2,1/2>

0

1 -1

0

1 -1

0

1

Axial Position [a.u.]
-1

0

1 -1

0

1 -1

0

1

Axial Position [a.u.]

Figure 4-4: Hyperfine transfer of the cloud in a magnetic field curvature. The atoms
are initially trapped in state |3/2, 3/2i. After the spin transfer into state |1/2, 1/2i,
the atoms are no longer trapped. Quickly reversing the sign of the magnetic field
retraps the atoms.

spins are decoupled. Both state |1i and |2i have their electronic spin aligned withthe applied magnetic field and thus have practically identical magnetic moments of
1 Bohr magneton. They only differ by the projection of the nuclear spin, mI = +1
for state |2i, and mI = 0 for state |1i. An RF Landau-Zener sweep at about 80 MHz
can easily transfer atoms in state |1i into state |2i. The final population in the two
spin states can be controlled by the sweep rate: The Landau-Zener probability forthe transfer |1i → |2i is
P|1i→|2i = 1 − exp(−2π

Ω2R
),
ω̇

where Ω2R is the square of the Rabi frequency, proportional to the power of the RF
wave, and ω̇ is the frequency sweep rate. For a fully adiabatic transfer, Ω2R À ω̇ andall atoms are transferred into state |2i. A non-adiabatic transfer can produce any
final spin mixture.
One should note that the RF sweep is a coherent process that does not directly
convert atoms into another spin state (see [266], reprinted in appendix A). Rather,each atom is brought into a coherent superposition state ∼ a |1i + b |2i. A noninteracting, degenerate Fermi gas of atoms in state |1iwill still be a spin-polarizedFermi gas after the sweep, but now all atoms are polarized in the superposition state.
Clearly, we require a decoherence mechanism to truly populate the two distinct spinstates. This mechanism is found in the inhomogeneous magnetic field experienced by
the atoms, which scrambles their relative phase as they follow different paths in the97trap. By recording the onset of interactions in our cloud, we have measured [106]the typical timescale for this decoherence to be tens of milliseconds for our magnetic
field curvatures. This is on the order of the axial trapping period, which is a sensible
result.
Once the fully decohered spin-mixture is created, efficient evaporative cooling of
the gas can begin. We typically lower the trap depth U over several seconds down
to about 1 µK to enforce evaporative cooling. At this point, U becomes comparable
to the Fermi energy of the cloud, and further reduction in trap depth leads to strong
losses as atoms spill out from the trap. We reach temperatures of T /TF ≈ 0.05 in a
Fermi mixture containing a total of about 107 fermion pairs.

98Chapter 5
Bose-Einstein condensation offermion pairs
The hallmark of Bose-Einstein condensation in atomic gases was the sudden appearance of a dense central core in the midst of a large thermal cloud [12, 72]. This
direct signature in the density distribution derives from a clear separation of energy
scales in weakly interacting gases. The condensate’s repulsive mean-field µ ∝ na ismuch smaller than the critical temperature (times kB ) at which condensation occurs,
TC ∝ n2/3 : The gas parameter na3 is much less than 1 (about 4 × 10−6 for our sodium
condensates). In a harmonic trap, the different energy scales directly translate into√√
the different sizes of a thermal cloud, Rth ∝ T , and of a condensate RC ∝ µ.
However, in the case where µ becomes comparable to kB TC , the condensate is nolonger clearly separated from the thermal cloud, hindering the direct observation ofcondensation.
This is the situation we encounter in our Fermi mixture close to the Feshbachresonance. In not too strongly interacting gases of molecules, it was indeed possible toobserve the characteristic bimodal density distribution known from atomic BEC [102,271, 26, 36, 188] (section 5.1). The fact that molecules could be created [205] and were
found to be stable and long-lived close to a Feshbach resonance [69, 128, 239, 202]
was crucial for this observation. It finds its explanation in the fermionic nature of themolecules’ constituents [193].
As the interactions between molecules are increased by moving closer to the Feshbach resonance, the size of the molecular condensate grows and the bimodal feature
close to TC becomes invisible1 . Now interactions dominate the appearance of boththe normal and the condensed cloud. At the same time - this is the essence of the1

This is almost true. See section 5.3 below and section 7.4.

99BEC-BCS crossover - the composite nature of the molecules starts to play a role. The
constituent fermions, no longer spread out in k-space due to the molecular binding,start experiencing Pauli pressure from the presence of others. The cloud size increases
even further until finally, on the BCS-side of the resonance, the size is completely dominated by the single-particle energy, the Fermi energy EF . This is evidently the same
size the system would have in the normal state.
For this reason it seems no longer possible, in the strongly interacting regime, todetect the superfluid phase transition in the density profile of the gas. To the rescue
comes the fact that in atomic Fermi gases, the interaction strength can be tuned atwill, on timescales short compared to the growth time of the condensates. This allows
to effectively ”map” the condensate wavefunction in the strongly interacting regime
onto a molecular condensate wavefunction far away from resonance. If this is done in
the moment of releasing the gas from the trap, the final image of a molecular cloud
will again reveal a bimodal density distribution: The condensate surrounded by athermal cloud.
This method was employed by the JILA group [203] in a Fermi mixture of 40 K,
and subsequently in our group for 6 Li [272] to infer condensation of fermion pairs on
the BCS-side of the Feshbach resonance.
In the following I will first introduce the methods that have allowed the formationof molecules, and present the observation of one of the first molecular Bose-Einsteincondensates. Next, I will describe the technique that has allowed to infer condensationof fermion pairs on the BCS-side of the Feshbach resonance, where pairing is purely amany-body effect. Finally, I will present unpublished data showing a direct signatureof the condensate in the density profile of the gas on resonance.

5.1

Bose-Einstein condensation of molecules

5.1.1

Feshbach resonances

Even before one could enter the realm of many-body physics, Feshbach resonances
have opened a new avenue in the physics of atoms and molecules: They have allowed
the production of ultracold molecules starting with ultracold atoms. The first observation of a Feshbach resonance in ultracold atoms [126, 235] showed strong losses in
the atomic signal that were attributed [250, 168, 259] to the formation of ultracold,
highly vibrationally excited molecules. It was predicted that these molecules, formedout of two bosonic atoms, would undergo fast vibrational relaxation into more tightly
100bound molecules. Still, in experiments on 85 Rb, the presence of the molecules, as
short-lived as they were (lifetime ∼ 100 µs) could be detected via coherent beats between the free atomic and the bound molecular state [81]. Studies of the decay offermionic gases close to a Feshbach resonance [78, 36, 202] held a peculiar surprise:
The maximum atom loss was not centered on resonance, but was shifted towards
regions where the Feshbach molecular state was already quite deeply bound. The
gas close to resonance was stable [78, 174, 36, 204], in stark contrast to the bosoniccase. This molecular state could be reversibly populated via a magnetic field sweep
across resonance [205], at a conversion efficiency exceeding 90% [69, 121]. Most importantly, it was found to be long-lived [69, 128, 239], with lifetimes between about
100 ms (for 40 K) and several 10 s (for 6 Li). This is to be compared to the conversion
efficiencies of only 5%-10% and lifetimes on the order of 5 ms observed in bosonicgases [117, 255, 84].
It is this long lifetime which in the case of 6 Li has allowed to simply evaporatively
cool the gas close to the Feshbach resonance to create molecules [128], populating themolecular state via collisions, and not via a magnetic field sweep.

Methods for making molecules

Two-particle energy

5.1.2

kBT

EB

Magnetic Field

Figure 5-1: Creating molecules via three-body collisions. A molecular state is coupled
to the continuum (see Fig. 3-3). As the gas is cooled on the molecular side, themolecular state is populated vie three-body collisions. If the binding energy is notmuch larger than kB times the temperature, the energy carried away by the third body
does not substantially heat the sample. For fermionic particles, further vibrational
decay into small molecules is strongly suppressed due to Pauli blocking.
There are essentially two ways to create molecules out of unbound atoms close to a
Feshbach resonance. Three-body collisions and coherent two-body adiabatic transfer.
101

Two-particle energy

Magnetic Field

Figure 5-2: Creating molecules via magnetic field ramps. A magnetic field sweep cantransfer unbound atoms adiabatically into the molecular state, much like a two-level
Landau-Zener transition.

In the three-body process, two of the colliding particles form a molecule, the third
carrying away the leftover energy and momentum. In a typical situation, molecularstates are deeply bound compared to the temperature of the gas (times kB ). The
released energy in such a collision thus usually heats up the cloud, leading to trap
loss (hence the name ”bad collisions”). However, in the case of a very weakly boundmolecular state close to a Feshbach resonance, the binding energy can be on theorder of the temperature, and molecules can efficiently form without severe heating
and trap loss (Fig. 5-1). Subsequently, leftover atoms can be efficiently evaporated
from the optical trap. Weakly bound molecules have twice the atomic polarizability,
hence the optical trap is twice as deep for molecules than for single atoms.
The molecules are only stable if the decay into more deeply bound molecular states
is suppressed in some ways. It turns out that for fermions there is such a suppression
mechanism: The Pauli principle. Decay into a deeply bound state requires threefermions to be very close to each other in the final state of this process. With only two
spin states available, this is strongly forbidden [193]. For bosons, the decay process isactually enhanced close to a Feshbach resonance, which results in molecular lifetimesof only a few ms [117, 255, 84].
The second process is reversible, it does not generate heat and makes use of thetunability of the molecular state: Starting with unbound atoms in the continuum,
one can adiabatically sweep the magnetic field across resonance and form a boundmolecule (Fig. 5-2). It is very helpful to think of this sweep in terms of a two-level
Landau-Zener sweep through an avoided crossing. For a coupling matrix element V

102

between two ”bare” states, |ai and |bi, and an energy sweep rate Ė, one finds [168]|V |2P|ai→|bi = 1 − e−c ~Ė(5.1)

for the probability P|ai→|bi to make a transition from |ai to |bi as the bare stateenergies are swept through resonance. Here c is a numerical constant on the order of1.
In the case of Feshbach resonances, the two ”levels” are the molecular state andpthe energy of two unbound atoms. V is the coupling matrix element V = N/Ω g0 =√N hϕb | Vhf− |Ψk i that we have encountered in chapter 3. The number N of atom pairsthat appears in |V |2 accounts for the fact that each spin up atom has N chances toform a molecule with a spin down atom per volume Ω. If we take our simple Feshbach
2model of chapter 3, we can replace g02 = 4π~
abg ∆µ∆B. The bare state energies aremtuned via the magnetic field, so Ė = µB Ḃ. We have thennPatoms→molecules ' 1 − e−A Ḃ
with A = c(5.2)

4π~abg ∆B. That is, the higher the density and the slower the magnetic fieldmramp across resonance, the more efficient the production of molecules [141].
The simple two-state picture is obviously flawed, as the state of two unboundatoms belongs to a continuum. Also, we neglect here the fact that two atoms need toencounter each other in real space before the transition to the molecular state can takeplace. In a ”hot” atomic cloud, the production efficiency for adiabatic sweeps willthus be less than 100%. In a semi-classical picture, we can expect that the efficiency
will be maximum if the two atoms belong to the same phase space cell. In a zerotemperature cloud, one would expect 100% efficiency of the sweep, as was discussed
in [121].
It should be noted that this coherent way of producing molecules generally does
not lead to a thermal equilibrium state if the sweep is too fast. The molecular cloudneeds to thermalize via collisions before it can settle into equilibrium (where, forexample, parts of it might be Bose condensed).
Distinguishing atoms from molecules
An amazing property of the molecular state close to resonance is its long range. A
most important consequence is that we can directly image the molecules using thesame laser light that is absorbed by unbound atoms. This is possible as long as the103Imaging Atoms
at zero Field

Imaging Atoms + Molecules
close to resonance

Energy [MHz]

0

-500

-1000

0

834.15Magnetic Field [G]Figure 5-3: Imaging atoms and molecules. At zero field, only unbound atoms areimaged. Tightly bound molecules do not absorb the imaging light resonant withunbound atoms. Close to resonance, the molecules are so weakly bound that theirconstituent atoms absorb light just like unbound atoms.

molecular size is much larger than the wavelength (divided by 2π).
The easiest way to distinguish atoms from molecules is to take an image at low
field, where the tightly bound molecules are detuned far from resonance (and their line
strength is diluted over the many excited molecular states), see Fig. 5-3. Alternatively,
one can dissociate the molecules into an unoccupied state using an RF pulse [205] andsubsequently image that state with a resonant laser beam. Yet another diagnostic
tool is the use of a Stern-Gerlach field gradient [128] to separate molecules from atoms
in a regime where their magnetic moments already differ substantially.

5.1.3

Formation of molecules

In a first experiment to locate the resonance, one can prepare a Fermi mixture on
the ”BCS”-side of the Feshbach resonance, where no two-body molecular bound stateis available to atom pairs. As the magnetic field is swept across resonance, molecules will form and, accordingly, the signal from unbound atoms taken with low-fieldimaging will diminish (Fig. 5-4). This is a standard technique to locate Feshbachresonances [205, 117, 234]. From Fig. 5-4 we determine a value of B0 = 838 ± 27 G
104

b)

1000

2.0

800

6

Atom number [10 ]Magnetic Field [G]

a)

600
400
200

1.5
1.0
0.5
0.0

0
-4

0
4
time [ms]

750

8

800

850

900

950

Magnetic Field [G]Figure 5-4: Molecule formation by magnetic field sweep across the Feshbach resonance. a) Experimental procedure. A Fermi mixture prepared on the BCS-side of
the Feshbach resonance is swept across resonance (shown as the dashed line) to form
molecules. The gas is released from the trap at the end-point of the ramp at time
t = 0 ms. Zero-field imaging is used to be sensitive to only the leftover atoms. b)
Atomic signal vs end-point of the magnetic field sweep. The line is a fit to an errorfunction, whose center is determined to be 838 ± 27 G, with an uncertainty given by
the 10%-90% width (54 G).

for the position of the resonance.
Seeing the atomic signal disappear can easily mislead one to believe that theatoms are simply lost from the trap. However, the process is reversible: Ramping
back across resonance will dissociate the molecules (if done non-adiabatically, thisprocess can heat the cloud in the trap [171]).
In fact, the dissociation method gives a more accurate determination of the location of the Feshbach resonance [203, 272]. To avoid effects due to the high density inthe trap (i.e. many-body physics), the molecular cloud is expanded to a 1000 times
lower density, about 1010 cm−3 . Then the magnetic field is ramped to a value Btest .
If Btest lies above the Feshbach resonance, the molecules will dissociate into unboundatoms, which can subsequently be detected at low field.
The very sharp onset of the atomic signal at Btest = 821±1 G is striking (see Fig. 55) and suggests this magnetic field value as the position of the Feshbach resonance.
However, using molecular RF spectroscopy on a bound-bound transition between two
different Feshbach molecular states, the group of R. Grimm in collaboration with
the NIST theory group was able to infer the location of the Feshbach resonance 2 at
2

A reason for our lower value could be few-body collisions, even at 1000 times lower density:
At 821 G, a = 30 000 a0 and kF a is still on the order of 1. Another systematic shift could be thefinite ramp speed employed, that might dissociate the extremely weakly bound molecules at 821 G

105

Magnetic Field [G]

950
900
850
800
750
time [ms]

700
0

10

20

30

Figure 5-5: Locating the Feshbach resonance by molecule dissociation. The experimental procedure is shown in the inset. A molecular cloud is prepared on the BEC-sideof the Feshbach resonance, at 780 G, and released from the trap at t = 0 ms. Aftersome expansion, the field is ramped to a test value around resonance (shown as thedashed line), held constant and is finally brought to zero field, where only unboundatoms are imaged. a) The atomic signal as a function of the test field shows a sharp
threshold behavior at 821 ± 1 G, where the uncertainty is the statistical error of athreshold fit, shown in b).

834.1 ± 1.5 G [24].
From Fig. 5-5 we can directly see that before dissociation, more than 99% of thegas exists in form of molecules. The reason is that this molecular cloud was formed
via the three-body process, by simply cooling the gas at the fixed field of 780 G(the BEC-side of the resonance). The lifetime of the weakly bound molecules is solong, and the binding energy is so small that losses and heating are negligible, and,after evaporation of leftover unbound atoms, essentially all particles are bound intomolecules.
Coupling strength
Fig. 5-6 demonstrates the extremely strong coupling strength of the 6 Li Feshbachresonance. In this experiment, a Fermi mixture is released from the trap at B =840 G, slightly above the Feshbach resonance. After a varying expansion time, themagnetic field is switched off to zero field (at an initial slew rate of Ḃ = 30 µs/G),
where the remaining atomic signal is detected. At the high density of the trap, almost(binding energy 30 nK!). We searched for a dependence of the resonance location on these effects
and estimated that they could contribute an up-shift of the position by 3 ± 3 G each. This is barely
within the error margin of the RF spectroscopy result.

106

a)

b)

Atomic signal [10 ]

2.0

0.8

1.0

0.5

0.5

Molecule fraction

1.5

6

2.0

Atomic Signal [10 ]

6

1.5

1.0

780

0.6
0.4
0.2

0.0

0.0

1.0

800 820 840
Magnetic field [G]

860

0
1
2
3
4
Expansion time before magnetic field ramp [ms]

0.0

0

1
2
3
4
5
12
-3
Density per spin state [10 cm ]Figure 5-6: Revival of the atomic signal during expansion and strength of Feshbach
coupling. a) The magnetic field is switched off after varying expansion times for a
cloud released at 840 G. The field ramp creates molecules more efficiently at the highdensities of the trap than at low densities after long expansion. The inset shows theatomic signal for immediate switch off (triangles pointing down) and for a switch off
after 4 ms expansion (triangles pointing up). In b), all of the atomic signal loss isinterpreted as molecular conversion and plotted as a function of density. The density
was calibrated by imaging the cloud at high field for varying expansion times. All fits
are for the simple Landau-Zener-model described in the text.

the entire atomic signal has vanished, the loss being almost 95%. However, as thedensity decreases during expansion, the atomic signal revives, indicating that theatoms were not truly ”lost” but only ”hidden” in the molecular state. In Fig. 5-6b) we tentatively interpret all the atomic signal loss as conversion into the weaklybound molecular state3 . This conversion efficiency is plotted versus the density per
spin state at the given expansion time. Using the simple fit function given by theG
Landau-Zener formula Eq. 5.2, we find A ≈ 24 1012 cm−3 µs with a relative error of 50%
due to the uncertainty in the atom number.
The prediction is Eq. 5.2 with c = 2π, and using the lithium parameters we findA=G
8π 2 ~abg ∆B= 19 12 −3m
10 cm µs

(5.3)

The agreement is quite good but might be fortuitous, as the simple formula does
not take temperature and other effects into account (see [121] for a discussion of
more complete models and comparison with experiments). Also, here we have notdirectly observed the molecular state, but only the loss in atomic signal, which could
3

Other loss-channels like unobserved molecular states cannot be excluded here.

107

be enhanced due to collisions after the ramp, leading to the formation of more deeply
bound molecules. Nevertheless, while the maximum conversion efficiency might besubject to these systematics, the shape of the density dependence is likely less affected,
especially at low densities where collisions after the ramp should be negligible.
The strong coupling to the molecular state does not allow us to be fast comparedto the two-body time scale at the density of the trapped gas. For a typical n =5 × 1012 cm−3 , in order to convert less than 10% of the atoms into molecules, wewould have to ramp at a rate of 1200 G / µs, about 40 times faster than the limit set
by our coil inductance.
On the other hand, this efficient molecule conversion will be very useful in section 5.2, where the state of the fermion pair condensate above resonance is ”frozen in”
by the rapid ramp into a condensate of molecules. That is, the ramp can be faster orcomparable to the many-body time scale but still slow enough for efficient molecule
formation.
For comparison, for the Feshbach resonance at B = 202 G used in experiments
G
G
with 40 K we have A = 0.9 1014 cm−3 µs (close to the measured value 0.7 1014 cm−3 µs [205]),
resulting in efficient conversion into molecules at about 2000 times slower ramp rates.

5.1.4

Observation of molecular Bose-Einstein condensates

After the successful creation of ultracold molecules out of ultracold atoms via Feshbach resonances in gases of fermions [205, 69, 128, 239] and bosons [81, 117, 255, 84]
the intriguing question was whether thermal equilibrium could be reached and coldenough temperatures realized to observe the onset of Bose-Einstein condensation ingases of molecules.
In the case of 6 Li the long lifetime of molecules [36] (see Fig. 5-7) enables us toevaporate the Fermi mixture at a fixed magnetic field, just as if one were to coola cloud of bosonic atoms towards BEC. As the mixture is cooled by ramping down
the trapping laser power, molecules form as the temperature becomes comparable
to the binding energy. Accordingly, the atomic signal observed in zero-field imaging
vanishes: We can see this in Fig. 5-5 for fields below resonance, where essentially no
atomic signal is measured. What renders evaporation very efficient is the fact thatmolecules, having twice the polarizability of an atom, experience twice the trap depth
for unpaired atoms. The potentially energetic atoms produced during a moleculeforming three-body collision are efficiently removed from the trap.
To observe the molecules directly, one can either use high-field imaging (see Fig. 5-

108

Magnetic Field [G]
760

780

800

810
20

0.6

0.4

10

0.2

Lifetime [s]Remaining fraction after 9.9 s

740

5
0.0

0
3

4

5

6

7

8

9

2

1
4

Scattering length [10 a0]

Figure 5-7: Lifetime of molecules in partially condensed clouds. The cloud with initially about 1 × 106 molecules was held for 9.9 s (initial density about 5 × 1012 cm−3 ,slightly varying with the interaction strength). Shown is the remaining fraction as
a function of scattering length. The lifetime 1/Γ is calculated under the simplifying assumption of a pure exponential decay e−Γt . The line is a fit with a powerlaw for Γ = ca−p , giving p = −0.9. The clouds were partially condensed (up to
80% condensate fraction at the largest scattering length), and all measurements weredone in the strongly interacting regime where a > 1/kF , so the expression for therelaxation rate differs from the prediction for weakly interacting, thermal molecules(Γ = ca−2.55 ) [193].

3) or one can dissociate them as in Fig. 5-5 by crossing the resonance shortly before
taking an absorption image at zero field. Below a certain temperature, one observes
the striking onset of a bimodal density distribution, the hallmark of Bose-Einsteincondensation (see Figs. 5-8 and 5-9). In contrast to weakly interacting Bose gases,
the condensate peak is not much smaller than the thermal cloud, indicating a large
mean-field energy of the BEC, comparable to kB times the condensation temperature.
As we move closer to the Feshbach resonance, the size of the condensate grows to
be almost that of a degenerate Fermi gas. Figure 5-10 illustrates this fact for an
essentially pure condensate at 780 G.
We thus start entering the situation outlined in the introduction: All energy scales
approach each other. The molecules are no longer tightly bound and far from each
1/3
other, but rather their average distance nM from each other becomes comparable tothe molecular size in free space, given approximately by the scattering length. One
has already entered the strongly interacting regime of the BEC-BCS crossover.

109

Figure 5-8: Observation of Bose-Einstein condensation of molecules. As the trap
depth is lowered (from left to the right image), the characteristic bimodal density
distribution appears in the molecular cloud.

Molecule Number Per Unit Length [ 1/mm ]

4 mW

10 mW

11 mW

14 mW

19 mW

6

1x10

38 mW

0
0.0

0.5

1.0

Position [ mm ]Figure 5-9: Bimodal density distribution emerging in a cloud of molecules. Shown
are radially integrated profiles of absorption images such as those in Fig. 5-8, as afunction of final laser power. The dashed lines are fits to the thermal clouds.

110Radial density [a.u.]

0.4

0.2

0.0
-1.0

-0.5

0.0
0.5
Radius [mm]1.0 -1.0

-0.5

0.0
0.5
Radius [mm]1.0

Figure 5-10: Comparison between a molecular BEC (a) and a degenerate Fermi sea(b). The condensate containing Nm = 6 × 106 molecules is in the strongly interacting
regime at a magnetic field of 780 G (1/kF a = 0.6). Its size is almost as large (factor∼ 0.7) as a non-interacting Fermi sea containing Nm atoms, indicated by the whitecircle. Image and profile b) show an essentially spin-polarized Fermi sea (minority
component of < 2% not shown) containing N = 8 × 106 atoms at the same field. The
images were taken after 12 ms expansion with the probe light aligned with the longaxis of the cigar-shaped clouds.

111

5.2

Condensation of fermion pairs close to a
Feshbach resonanceIn contrast to the case of molecular condensates, the superfluid in the strongly interacting regime does not appear to spatially separate from the normal state. Thisis the main ”dilemma” with strongly interacting Fermi gases, dramatically shown inrecent high-resolution images (see Fig. 5-11). A second difficulty with Fermion paircondensates on the BCS-side is the fragility of the pairs: During expansion, the gas
becomes more dilute, and the pair binding energy can decrease below (kB times) the
local temperature. Thus, the pairs can break in time of flight (see chapter 6 for astudy of this effect using vortices). Clearly, a new detection method is needed to infercondensation on the BCS-side of the resonance.

5.2.1

The rapid ramp technique

Such a method was introduced by the JILA group [203], and later adapted to 6 Li
by our group [272]. The idea behind the rapid ramp technique is to ”rescue” thefragile fermion pairs by sweeping the magnetic field towards the BEC-side of theresonance, thereby transforming them into stable molecules (see Fig. 5-12). This isdone in the moment the trap is switched off for expansion. If each fermion pair istransferred into a tightly bound molecule, the momentum information of the originalpair is preserved. Time-of-flight analysis of the resulting molecules should thus allow
to infer the momentum distribution of pairs in the gas above resonance.
This technique has enabled us to demonstrate fermion pair condensation in theentire BEC-BCS crossover. Sample images and profiles of the resulting molecular
clouds are shown in Fig. 5-13. The drastically reduced interaction results in a clear
separation of the condensate from the ”thermal” or uncondensed part of the cloud4 .
The fact that this can be seen even with low-resolution imaging (compare the profilesin Fig. 5-13 from February 2004 with those in Fig. 5-11 obtained two years later)
demonstrates the power of the method. Note that the rapid ramp converts atoms intomolecules with higher than 90% efficiency for all fields in the BEC-BCS crossover,4

At zero field, the scattering length between molecules should be on the order of the singlet
scattering length of lithium atoms, which is about 40 a0 . The exact value is not known. In fact,
the residual mean-field interaction at zero field is so low that the condensate practically does not
expand if the rapid ramp is performed immediately after switching off the trap. For this reason, it
is sometimes beneficial to let the cloud expand by some amount before the rapid ramp is performed.
This converts some of the interaction energy in the cloud into kinetic energy, which lets one ”choose”
the final expanded size of the molecular condensate.

112

b

c

Optical density [a.u.]

a

-1.0

-0.5

0.0
0.5
Radius [mm]1.0 -1.0

-0.5

0.0
0.5
Radius [mm]1.0 -1.0

-0.5

0.0
0.5
Radius [mm]1.0

Figure 5-11: The dilemma with strongly interacting Fermi gases: The superfluid
appears not to leave a trace in the density profile of the gas. Shown are high-resolution
images (from February 2006) of spin up atoms in a resonantly interacting, equal
mixture of spin up and spin down for different temperatures. The lower graphs show
aximuthally averaged radial profiles (noise level well below 1% of the maximum opticaldensity). All three clouds are very well fit using a finite-temperature Thomas-Fermiedistribution (with fugacity eµ/kB T , central density n0 and mean square radius hr2 i as
free parameters, see Eq. 2.28). However, the empirical temperatures of Te/TF = 0.22(a), 0.13 (b) and 0.075 (c) determined from the profiles’ wings indicate that at least
clouds b and c should be in the superfluid regime (which is indeed the case, seechapter 6). Trap parameters νr = 162 Hz, νz = 22.8 Hz, 10 ms time of flight,
expansion factor 13.9, atom numbers N per spin state were 10.2 (a), 9.5 (b) and 7.5
×106 .

113

Figure 5-12: Rapid ramp to the molecular side to observe pair condensation. Immediately after switching off the trapping beam, the magnetic field is ramped to zero
field. This converts long-range pairs into stable, tightly bound molecules. Momentum
information of the original pairs is thus preserved.

see the discussion in section 5.1.3 above and Fig. 5-6.

5.2.2

Condensate fraction in the BEC-BCS crossover

The condensate fraction was determined by fitting a bimodal distribution to the
profiles like those in Fig. 5-13, a parabola for the central dense part and a gaussian
for the thermal background. To our surprise, we found very large condensate fractions
throughout the entire BEC-BCS crossover, with a peak of 80% at B ≈ 820 G, close
to, but still on the BEC-side of the resonance (determined via RF spectroscopy andtheory [24] to be B0 = 834 G)5 (see Fig. 5-14). It is intriguing to note that aroundthe same field, a minimum in the damping of collective excitations was found [25].
The high condensate fraction might be a hint that the pairs in the strongly interacting regime on the BCS-side of the resonance are still smaller than the interparticle
spacing, and not larger as one would expect from long-range Cooper pairs. In thiscase, it is intuitive that each atom of a given pair can still form a molecule with its
original partner during the rapid ramp. If the pairs were much larger than the interparticle spacing, molecules might rather form out of uncorrelated atoms, resulting ina thermal cloud after the ramp. In accord with this argument, BEC-BCS crossovertheory predicts that the pairsize ξ will be smaller than the interparticle spacing n−1/35Interestingly it is at this magnetic field where the sharp onset of the dissociation threshold wasfound (see discussion of Fig. 5-5 above).

114

BEC-sideb)Resonancec)

BCS-Side

Radial density [a.u.]a)-300 -200 -100
0100
Position [µm]

200

300

-300 -200 -100
0100
Position [µm]

200

300

-300 -200 -100
0100
Position [µm]

200

300

Figure 5-13: Fermion pair condensates. Axial density of the atomic cloud after the
rapid ramp to zero field (in < 100 µs), further expansion (for 10 ms), and dissociationof the resulting molecules by ramping back across resonance. The initial field B0 , thenumber of fermion pairs N , the condensate fraction and the interaction parameter
1/kF a where a) 745 G, 700 000, 47%, 1.2; b) 835 G, 1.4 × 106 , 81%, 0.0 (resonance);c) 912 G, 1 × 106 , 49%, -0.5.

up to kF a = −1 (see chapter 2), which is indeed on the BCS-side of the resonance.
So far no experiment on Fermi gases has shown condensation or superfluidity in a
regime where kF |a| (a < 0) is significantly less than 1 and hence where pairing istruly long-range.
Let us also note that the high condensate fraction is in stark contrast to the
maximum fraction of about 14% found in experiments with 40 K [203]. The cause forthis discrepancy might be connected with the shorter lifetime of the Fermi mixture
in 40 K close to resonance, on the order of 100 ms [202]. This lifetime might not havebeen sufficient to reach full thermal equilibrium of the gas in all three dimensions6 .
In our experiments, the condensates were found to be very long-lived. For a holdtime of 10 s, the condensate fraction on resonance was observed to still be close to
its initial value7 . In fact, these lifetimes can very favorably compare to those foundfor atomic Bose-Einstein condensates.
6

The axial trapping period in [203] was about 200 ms, comparable to the lifetime of atoms in theregime where condensation has been reported [202].
7
On the BEC-side, it decayed more rapidly due to the increasing rate of vibrational relaxation of
the molecules away from resonance. The decay in the condensate fraction on the BCS-side is likely
caused by atom loss without change in temperature.

115

Condensate Fraction

0.8

0.4

0.0
600

1000

800

Magnetic Field [G]Figure 5-14: Condensate fraction in the BEC-BCS crossover as a function of themagnetic field before the rapid ramp. The symbols correspond to different holdtimes, 2 ms (crosses), 100 ms (squares) and 10 s (circles).

5.2.3

Discussion of the condensate fraction

Fig. 5-15 shows a ”phase diagram” for the condensate fraction as a function of temperature. Several theoretical works [79, 20, 51] have confirmed the general behaviorof the ”critical temperature” of the observed condensation phenomenon in 40 K and in
6
Li. Also, the fact that high condensate fractions were found in our experiment could
be explained within the BEC-BCS crossover model [191, 209, 181] and by Quantum
Monte-Carlo simulations [18].
How does the condensate fraction change in the BEC-BCS crossover? On themolecular side of the resonance, even in the absence of losses due to vibrational
relaxation, one would expect quantum depletion
q to reduce the condensate fraction.
The Bogoliubov theory of chapter 2 predicts 38

nM a3Mπ

for the condensate depletionin a molecular gas of density nM and scattering length aM . At kF a = 1 (at about 750
G in our experiment) this would give 9% (we assume aM = 0.6a [193]). Evidently,
close to resonance this first-order expression diverges. The quantum depletion on
resonance will be a universal number.
On the BCS-side of the resonance, it is a priori not evident how a ”condensate
fraction” should be defined. If we ask for the superfluid density, this must of course
equal the total density of the gas at zero temperature. This was in fact the way wenormalized the order parameter ψ(r) on the BCS-side in section 2.3.5. If the question
concerns the fraction of paired fermions, this, too, would be given by 100% at T = 0.
116

Figure 5-15: ”Phase diagram” for the condensate fraction as a function of magnetic
field and temperature. Condensates are obtained in the entire BEC-BCS crossover.
The highest condensate fraction and highest onset temperature are obtained on the
BEC-side close to resonance. As a model-independent measure of temperature, thecondensate fraction at 822 G (see arrow) is used as the vertical axis. The Feshbachresonance lies close to this point, at 834 G.

However, we have seen in section 2.3 that we should not regard the BCS-state asa condensate of bosonic pairs: Anti-symmetrization of the many-body wavefunction
spoils this picture. The definition of ”condensate fraction” that is usually employed
is a measure of coherence in the gas, the long-range order due to condensation. The
quantitythat expresses
this coherence is the ”Cooper pair wavefunction” φ(r1 − r2 ) =DE††Ψ↑ (r1 )Ψ↓ (r2 ) . The number of ”condensed pairs” N0 is the norm of φ(r1 −r2 ). Withthe tools of chapter 2 and following [209], we can calculate the condensate fraction
within the variational BCS Ansatz:ZN0 =d3 r1 d3 r2 |φ(r1 − r2 )|2(5.4)

Calculating this quantity for the BCS state one observes that the density of condensed
pairs isZ
N0
1X 2 2
∆2
1
d3 k
n0 =
=
(5.5)
uk v k =ΩΩ k4(2π)3 ξk2 + ∆2
Quite satisfactorily, the integral can be obtained analytically and we find with [209]
s
3/2

m
∆3/2
n0 =
3
8π~

µ
+
∆

117

r
1+

µ2
∆2

(5.6)

µ and ∆ were calculated in chapter 2. In the BEC-limit, with the help of Eq. 2.111,
this becomes the density of molecules or half the total atomic density n, as expected,
n0 = n/2(5.7)

that is, all molecules occupy the same single-particle ground state. In the BCS-regime,
where the gap is exponentially small, one finds [209]
n0 =

3π ∆
mkF
∆=
n
2
8π~
16 EF

(5.8)

Also this result is natural: Loosely speaking, the pairs that truly contribute to coherence in the system reside mainly close to the Fermi surface, in an energy width ofsize ∆, so their density is simply ∼ n∆/EF .
On resonance, µ/∆ and ∆/EF are universal numbers, for which the variational
BCS approach provides the approximations 0.85 and 0.69, respectively. One finds a
condensate fraction of
2N0= 0.70(5.9)
NHence, the variational BCS-Ansatz provides a hint that the condensate fraction on
resonance can indeed be large. Other studies have come to the same conclusion. For
example, a Monte-Carlo study gives 2N0 /N ≈ 0.60 [18].
It is not evident that the rapid ramp method of the experiment directly measuresthis condensate fraction. Our experiments on the growth time of fermion pair condensates [270] and on the in-situ detection of condensation (see section 5.3 and chapter 7)
and theoretical modelling of the ramp [79, 191, 10, 260] have shown that the observed
bimodal density distributions provide a qualitative signature of a condensate on the
BCS-side. It is an open question whether the obtained condensate fractions can provide a good quantitative measure of the coherent part on the BCS-side. Fig. 5-16
compares the variational BCS prediction to the data obtained via the rapid ramp.
The very close agreement is fortuitous, but it is encouraging to see the correct trend
on the BCS-side.

5.2.4

Discussion of the time scales involved

One objection to the rapid ramp method might be that the condensates could beformed during the ramp, on the molecular side of the Feshbach resonance. To ruleout this possibility, we need to address the question on what time scales condensate

118

Condensate fraction

1.0
0.8
0.6
0.4
0.2

2

1

0

-1

-2

1 / kFa

Figure 5-16: Condensate fraction as a function of the interaction strength in the
BEC-BCS crossover. The circles show the 100 ms data of Fig. 5-14. The interaction
strength is calculated using the known scattering length as a function of magnetic
field and the experimental value 1/kF = 2 000 a0 . The curve shows the variational
BCS prediction for the condensate fraction. The good agreement on the BCS-side
should not be taken as proof that the rapid ramp truly measures the condensate
fraction. On the BEC-side, heating due to vibrational relaxation leads to fast decay
on the condensate. Figure adapted from [209] using Eq. 5.6.

formation takes place. Table 5.1 lists the different time scales of importance in our
experiment.
Time scaleTwo-body physics
Magnetic field ramp (timescale on which the molecular state changes with respect to threshold)Fermi energy
Time required to leave resonance
Evolution of the gap at kF |a| = 2
Gap at kF |a| = 2
Inverse collision rate at unitarity and T /TF = 0.1
Growth time of a pair condensate at kF |a| = 2
Radial trapping period

Formula
√
~/g
2πn
0
√
g0 2πn/∆µḂ

Value
20 ns
80 ns

~/EF
∆B/Ḃ
˙
∆/∆
~/∆
≈ 0.23 ~EF /(kB T )2 [95]≈ ~EF /∆2 [21]
2π/ωr

3 µs
10 µs
10 µs
15 µs
70 µs
75 µs
2 ms

Table 5.1: Time scales involved in the rapid ramp technique
The fastest timescale is that relevant to two-body physics, the Feshbach couplingstrength. At the high densities of our trapped samples, we cannot beat this evenwith our fastest magnetic field ramps (that is, switching off the power supply). As
discussed in section 5.1.3, for our ramp speeds the conversion into molecules is always
better than 90% for in-trap densities. The next fastest time scale is that set by the
Fermi energy, which in the unitarity regime on resonance would set the timescale
119for collisions in the normal Fermi gas, were it not for Pauli blocking (and factors of
2π etc.). The time it takes to leave the resonance region in our gas is a bit longer
than the Fermi time scale, but might be smaller than the inverse collision rate. This
would mean that the original momentum distribution of fermion pairs is truly ”frozen
in” during the ramp. The momentum distribution of the final molecules would thenreflect that on the BCS-side.
The ramp is non-adiabatic on the time scale of the gap, which is forced to evolve˙
faster than it can respond to the change in interaction strength, ∆/∆& ∆/~. Finally,
there is the relaxation time scale of the gas in response to a change in the particle
distribution. In a normal Fermi gas of N particles at temperatures T ¿ TF , relaxation
occurs via collisions of particles close to the Fermi surface, of number N T /TF . Pauliblocking reduces the available final states for collisions by another factor of T /TF ,giving a relaxation time τR ≈ ~EF /(kB T )2 . In general, if the Fermi surface is smeared
over an energy width ∆E, the relaxation time is τ ≈ ~EF /∆E 2 . In the case of a
(BCS-type) superfluid, ∆E = ∆, and the relaxation time thus scales as τR = ~EF /∆2 .
We were able to show that fermion pair condensates grow indeed on a timescale long compared to the ramp time, comparable to the expected relaxation timescale [270]. This lends validity to the rapid ramp method.
This technique actually poses a difficult problem for many-body theory. For a
Bose-Einstein condensate, problems that involve time variations can be treated withthe time-dependent Gross-Pitaevskii equation (Eq. 2.65), which is applicable also
at zero temperature. The single characteristic timescale involved is the mean-fieldtimescale τµ = ~/µ. For the BCS-case, the pendant to the Gross-Pitaevskii equationis the time-dependent Ginzburg-Landau equation for the order parameter ∆(r, t) (seesection 2.3.5), which is unfortunately only valid in a region of size (TC /EF )4 close to
the critical temperature, and other approaches are needed. It turns out that the
gap introduces the time scale of coherent evolution of the wavefunction, while therelaxation time ~EF /∆2 gives the time scale for incoherent redistributions of the
many-body state.
If the ramp is faster than all many-body time scales, the BCS-state needs to”project” itself onto the eigenstates on the BEC-side [79, 191, 10, 260], a processsimilar to an orthogonality catastrophe [260]. As the BCS-state can either decay
into condensed or excited molecules, it is natural to expect the typical image of acondensate embedded in a thermal cloud after expansion.
In summary, our experiments show that one can indeed infer the existence of afermion pair condensate on the BCS-side using the rapid ramp technique. It provides
120

us with a qualitative signature of condensation. The quantitative comparison between
the observed condensate fractions and what is usually defined as a condensate fractionin BCS theory is, however, far from trivial [79, 191, 10, 51].

5.3

Direct observation of the phase transition in aresonantly interacting, equal mixture of fermions

5.3.1

The normal and the superfluid gas on resonanceLet us now return to the dilemma with strongly interacting Fermi gases we haveseen in Fig. 5-11: The superfluid does not appear to leave a detectable trace in thedensity profiles of the gas on resonance. We can understand this qualitatively fromthe principle of universality: As the scattering length diverges, the only length scaleof relevance is the interparticle spacing, n−1/3 . Accordingly, the only energy scale~2
available is the Fermi energy, ²F = 2m
(3π 2 n)2/3 . The chemical potential µ of the gas
can thus only be ²F times a universal constant:µ = ξs ²F(5.10)In a trap, the local chemical potential varies like µ(r) = µ − V (r), so we immediately
deduce that for harmonic trapping the shape of the cloud, at least at zero temperature,must be given byµn(r) = n0

r2
1− 2
RTF

¶3/2
(5.11)q2µ3/2
with n0 = 3π1 2 ( 2mµ
)
and
the
Thomas-Fermi
radiusR=(we consider a
TF
ξs ~2
mω 2
spherical trap without loss of generality). This shape is identical to the one we expect
for a normal Fermi gas (see chapter 2). The only difference is the universal constantξs appearing in µ and n0 . This parameter can indeed be measured for example bydetermining the cloud size [174, 26, 139, 187], or the release energy of the gas [37]. It
is not difficult to check that
µ=

p

ξs EF

(5.12)

RTF = ξs1/4 RF(5.13)

121

with the usual expression for the Fermi energy of an equal mixture in a harmonic
trap, EF = ~ω(3N )1/3 ≡ 21 mω 2 RF2 and the total number of atoms, N .
The above consideration holds strictly only at zero temperature, where the gas we
describe is the superfluid (hence the subscript s). The crucial point is now that above
the critical temperature TC , the normal state can have a different universal parameterξn describing its zero temperature energy and size (which are, of course, never attained
due to the transition to the superfluid state). It should then, in principle, be possible
to directly observe the superfluid phase transition by a faint change in the density
profiles, as an inner superfluid core emerges with a different equation of state (thatis, if not accidentally ξn = ξs ). It turns out that the superfluid constant ξs has beenmeasured [174, 26, 139, 187] and calculated [43, 17, 44, 39] to be aboutξs ≈ 0.45(5.14)

while the universal constant for the normal state has been obtained in [38, 39] to beaboutξn ≈ 0.59(5.15)

Hence the characteristic size of a normal and a superfluid gas in a trap are expected
to differ.

5.3.2

Anomalous features in the density profiles of expanded
clouds at unitarity

To test this hypothesis, we have studied the density profiles of the unitary gas onresonance after expansion. To a very good approximation, the trapping potential was
cylindrically symmetric (see chapter 6). This allowed us to obtain high-resolution
profiles via azimuthal averaging. Temperature was varied by changing the final trap
depth used during evaporation. After evaporative cooling, the trap was recompressed
to a fixed trap depth (U = 2.6 µK) before expansion. As an independent check for thesuperfluid phase transition, the condensate fraction was determined using the rapid
ramp method, described in the previous section.
Sample profiles are shown in Fig. 5-17. The goal is to detect a deviation from theshape of a single-component fermionic cloud. We thus fit an unconstrained finitetemperature Thomas-Fermi function to the profiles. The relevant information is now
contained in the residuals of such a fitting procedure. The quality of the fit, givenby the parameter χ2 , is shown in Fig. 5-18 as a function of the trap depth. At tem122

OD]
-2

Residual

0

[10

2

Optical Density [a.u.]

a)

b) 2
0

-2

-2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
0.0

0.0
0.0

0.4

0.8

1.2

0.0

Radius [R F]

0.4

0.8

1.2

Radius [R F]

Figure 5-17: Density profiles of an equal Fermi mixture on resonance. The temperature in a) was T /TF ≈ 0.15, whereas in b) it was T /TF ≈ 0.09. Temperatures were
determined from the thermal molecular cloud after the rapid ramp, and might not
be quantitatively accurate. Both gas clouds contained a condensate after the rapid
ramp to the BEC-side. The condensate fraction was: a) 7%. b) 60%.

peratures above 200 nK, the cloud is well fit by the Thomas-Fermi profile. However,as we lower the trap depth during evaporation below a certain value, the fits become
suddenly worse than at high temperatures. Small ”kinks” or changes in slope appear
in the profiles. This could indicate that now (at least) two ”species” in the gas compete for space. At our lowest temperatures, we see the quality of the fits to improveagain, indicating that now a large fraction of the gas is in the superfluid state. The
size of the cloud is then RTF = 0.83RF , which gives ξs ≈ 0.47, which is in accord withother experiments and theory [174, 26, 139, 187, 43, 17, 44, 39].
One should note that we never observe the fit residuals to deviate by more than
2% from the Thomas-Fermi shape. This explains why this effect has so far been too
small to be observable. Note that the conventional way of measuring an effective
temperature is to fit a Thomas-Fermi profile to the entire cloud. The small features we observe will bias the fits and thus systematically affect such temperature
measurements. For example, the Thomas-Fermi fit to Fig. 5-17b would yield an unrealistically low temperature of T /TF < 0.01. A better way to determine temperature
would probably be to only fit the profile’s wings, where the gas should be normal.
The region to be included in the fit could be defined by requiring the quality indicatorχ2 to be as low as that found for normal, ”hot” clouds above the phase transition.
In Fig. 5-17b), such a procedure yields T̃ /TF ≈ 0.11. This compares quite well tothe value T /TF ≈ 0.09 obtained from the thermal distribution of molecular clouds
observed after a rapid ramp to the BEC-side. However, the fact that the normal state123T [nK]

200
100

n0

1

0.001

100

2

χ [a.u.]

150

50

0.0

0.5

1.0
1.5
2.0
Trap depth [a.u.]

2.5

Figure 5-18: χ2 of the Thomas-Fermi fit as a function of the final evaporation trap
depth. χ2 shows a strong dependence on temperature, as expected from theory [122,190, 119, 230]. The condensate starts to form already around a trap depth of U ∼ 2.2,
and the fit-residuals in Fig. 5-19 indicate the same transition. The quality of the
global fit does not appear to be sensitive to small condensates. The temperature wasdetermined from the thermal wings of expanding molecular clouds after the rapid
ramp. The Fermi temperature decreased slowly from 1.5 µK for U = 2.6 to 1.4 µK at
U = 0.4, and dropped quickly due to atom spilling below U = 0.2. All measurements
were done after recompression into a deeper trap with U = 2.0.

on resonance is still strongly interacting limits its use as a reliable thermometer.
A convenient way to graph fit residuals as a function of temperature is by meansof a ”density” plot of gray shades, with white and black corresponding to positiveor negative deviations of the measured profile from the fit. This is shown in Fig. 519. Also included in this figure is the information on the density profiles and theircurvature. While the profiles themselves do not appear to change with temperature(trap depth) on the scale of the plot, we observe an intriguing structure appearing
in the residuals at an evaporation depth of about U = 2 µK. The curvature of thedensity profiles shows a similar qualitative behavior.
As a strong indication that the observed feature indeed stems from the superfluid,124

Figure 5-19: Density profiles, their curvature and their fit-residuals on resonance. a)
Density profiles on resonance as a function of trap depth. There is no sign of a phasetransition at this resolution. b) After the rapid ramp to the BEC-side and expansion,a condensate is clearly visible below a certain trap depth. c) The curvature of thedensity profiles on resonance in a) carries a signature of the condensate. No fieldramp is required. d) The fit residuals for a finite temperature Thomas-Fermi fit. Also
here, the condensate’s imprint in the density profile is clearly visible. To obtain thecurvature, the noisy central region of ±0.1RF in each profile was replaced by a fit.

we also include a density plot of the profiles obtained with the rapid ramp method.
As explained in the previous section, this allows to clearly separate the condensate
and thermal cloud in expansion. The condensate fraction is included in Fig. 5-18, andshows, as one would guess from inspection of Fig. 5-19, that the condensate appears
around U = 2.2 (in a.u.), a trap depth of 4.2 µK in real units. We conclude that asmall condensate does not leave a strong signature in the gas cloud, unlike the case of
weakly interacting Bose gases. Only when the condensate has grown to an appreciable
size (about 20% in our data) does it significantly deform the density profiles.
To conclude, we have shown that the dilemma stated in the beginning of thischapter does not truly exist on resonance. The gas does show features in the density
profiles upon condensation. Such features have been predicted by several authors [122,190, 119, 230], but up to now they were too small to be seen after column integration.

125

Chapter 6
High-temperature superfluidity:
Observation of vortices in astrongly interacting Fermi gas
V o r t e x is king, having kicked out Zeus.
Aristophanes, Clouds 828 (423 B.C.)

All things happen by virtue of necessity, t h e v o r t e x being the causeof the creation of all things, and this he calls necessity.
Diogenes Laertius, Lives of eminent Philosophers, IX 45(Democritus, 460-357 B.C.)

Superfluidity is the hallmark property of interacting quantum fluids, whether it occurs
in superconductors, in liquid helium or inside a neutron star. It is intimately, although
not directly related to the presence of a condensate. It might be evident from analogy
with atomic BEC that the molecular condensates of the preceding chapter are indeed
superfluid. But how can one demonstrate superfluidity for the strongly interacting
Fermi gas on resonance?
Quantized vortices in superfluids and superconductors
One of the most striking properties of superfluids is their response to rotation. In
contrast to a normal fluid, which rotates just like a rigid body, a superfluid can carry126

angular momentum only in the form of quantized vortices. Their mutual repulsion
results in a regular distribution of the vortex lines, the Abrikosov lattice [4]. In superconductors it is an applied magnetic field that attempts to set the charged superfluidof electron pairs in rotation. The superconductor responds by either completely expelling the magnetic field (the Meissner effect) or by allowing quantized magnetic flux
lines, vortices, to penetrate it. Quantized magnetic flux was measured by Deaver andFairbanks [75] and Doll and Näbauer in 1961 [80] by moving a thin superconducting
cylinder of tin toward and away from a conducting coil and measuring the electromotive force induced in the coil as a function of applied field. Entire Abrikosov lattices ofmagnetic flux lines were observed by using ferromagnetic particles that were trappedat the lines’ end-points (Träuble and Essmann [246], Sarma [212], independently in1967). Preceding these experiments in superconductors, quantized circulation in superfluid helium-4 [252] was observed by Vinen in 1958 by measuring the frequencyof circular motion of a thin wire placed at the center of the rotating superfluid. The
direct observation of vortex lattices in superfluid 4 He was achieved in 1979 by Yarmchuk, Gordon and Packard [257] by imaging ions trapped in the core of the vortex
lines. Using the method of the vibrating wire, the presence of quantized circulation
was confirmed for the fermionic superfluid 3 He in 1990 by Davis, Close, Zieve andPackard [71]. Our work presents the first direct imaging of vortices in neutral fermionic superfluids. It is interesting to add to this list that glitches in the frequency of
pulsars, fast rotating neutron stars, have been attributed to the spontaneous decayof vortex lines leaving the neutron pair superfluid [9, 82].
Superfluidity in dilute gases
The observation of superfluidity in gaseous Bose-Einstein condensates had been thesubject of many experimental efforts. One class of experiments performed at MIT
consisted in observing the energy dissipated in the cloud by a moving obstacle [200,58, 179]. Another class of experiments addressed the change of the moment of inertia
as the gas turns superfluid. This could be evidenced by the observation of the scissors
mode for a Bose-Einstein condensate [161], the oscillation of a dilute gas generated by
the sudden rotation of the confining trap [108]. Finally, the observation of vortices inweakly interacting BECs provided the most spectacular demonstration of superfluidity. In a two-component condensate, the JILA group created a single vortex using aphase-imprinting technique, in which one component was stationary while the otherwas rotating around it [164, 11]. In an adaption of the ”rotating bucket” idea, vortex
lattices in single-component BECs were created [160, 2, 120, 87] by driving surface
127excitations [178].
In Fermi gases, the direct observation of superfluidity was hindered by the factthat the normal cloud above TC is still strongly interacting and obeys (collisional)hydrodynamic equations of motion, of the same form as that expected for superfluidhydrodynamics. The observation of an inversion of the aspect ratio of the cloud
during expansion [174], a hallmark for Bose-Einstein condensation in weakly interacting Bose gases, indicated strong interactions in the gas but could not provide
proof for superfluidity [105, 36, 204]. Condensation on both sides of the Feshbachresonance [102, 129, 271, 203, 272] showed the presence of very low-energetic fermion
pairs, but did not directly address questions of superfluidity or phase coherence. Evidence for superfluidity was provided via measurements of collective excitations of thegas [138, 25], and via the observation of pairing in the strongly interacting regime [59].
A signature of a phase transition was not observed in these experiments, in the formof an abrupt change of the physical behavior of the gas as a function of temperature.
It is a remarkable fact that the BEC-BCS crossover supports bound fermion pairseven above TC , in an uncondensed state. Superfluid pairs could not be distinguishedfrom uncondensed pairs, and theoretical modelling was needed to infer the presenceof a superfluid.
In the experiments I will describe in the following, superfluidity was directly
demonstrated in ultracold Fermi gases through the observation of vortices. At thesame time, these results established phase coherence in gases of molecules and offermionic atom pairs. Vortices have now become a tool to study the limits of superfluidity close to the phase transition (section 6.3.3) or in imbalanced Fermi mixtures(chapter 7).
In the following, I will present the experimental techniques used to set stronglyinteracting gases into rotation (section 6.2), the observation of vortex lattices in section 6.3 along with measurements on the damping time, and the phenomenon of pairbreaking in expanding superfluids (section 6.3.3).

6.1

Origin of the vortex phenomenon

Superfluids are described by a macroscopic wavefunction ψ(r) (see chapters 2 and 5).
It is this quantity which is zero in the normal state and non-zero in the superfluid
state, so it qualifies as the order parameter of the superfluid phase transition. As a128

wavefunction, it is a complex quantity, with a magnitude and phase φψ(r) = |ψ(r)| eiφ(r)(6.1)

The velocity of the superfluid is simply the gradient of its phase,
v=

~
∇φ
m∗

(6.2)

where m∗ is the mass of the bosonic entities forming the superfluid. In the caseof fermionic superfluids, we have m∗ = 2m, where m is the fermion mass. If we
integrate Eq. 6.2 around a closed loop inside the superfluid, we immediately arrive at
the Onsager-Feynman quantization condition [180, 91, 92],Iv · dl = n

h
m∗

(6.3)

with integer n. If the superfluid wavefunction has no nodal lines and the loop fullylies in a simply connected region of space, we must have n = 0. However, Eq. 6.3 canbe fulfilled with n 6= 0 if the wavefunction contains a vortex, that is, a flow field that
depends on the vortex core distance r like v ∼ 1/r. At the location of the vortex, thewavefunction identically vanishes, it has a nodal line. This is the way a superfluid cancarry angular momentum. In case of cylindrical symmetry, the angular momentum
per boson or fermion pair is quantized in units of ~. Note that the vortices are an
equilibrium property of the superfluid at given angular momentum. This is in marked
contrast to classical vortices, which can only be found in non-equilibrium situations(dynamical equilibrium). Here, vorticity must decay whenever the viscosity is nonzero. Also, classical vortices do not obey any quantization rule and can occur withany circulation.
It turns out that for quantized vortices, only the case of n = ±1 is of relevance,since more highly ”charged” vortices are energetically not favorable and quickly decay
into singly charged vortices [222].
How can quantized vortices nucleate? Imagine that we move a stick or spoon
through the condensate. We know that the condensate will not respond to the spoon if
its motion is slower than a certain critical velocity. Another fact is that vortices cannot
suddenly appear within the condensate, as the angular momentum contained within aclosed loop inside the condensate cannot abruptly jump. Rather, the nodal lines haveto enter the condensate from its surface, where the condensate’s wavefunction is zero.
This surface can also be the surface of the stirrer, if it fully expels the condensate.
129Figure 6-1: Fate of a quadrupole oscillation in a rotating atomic Bose-Einstein condensate. The images show a sodium condensate in the magnetic trap after stirring
slightly above the quadrupole resonance (at 52 Hz, trapping frequencies ν⊥ = 73 Hz
and νz = 18 Hz) and equilibrating for a certain time t (time given in ms). First, thecondensate rotates in the form of a perfect quadrupolar collective excitation. Afterabout 100 ms, density depletions looking like vortex cores start to appear at the edgesof the condensate. Between 500 ms and 1 s, some of these penetrate inside the condensate as vortex lines, which arrange themselves into an ordered lattice after about
1-2 s.

We conclude that the spoon has to excite surface excitations, moving faster thanthe local critical velocity vc for such excitations [14]. What surface excitations are
efficiently created depends on the shape of the stirrer [70], or, in the case of a rotating
container, the roughness of the container walls. Accordingly, the necessary criticalangular velocity Ωc to nucleate vortices will likewise depend on the stirrer’s shape.
Note that Ωc is typically much higher than the thermodynamic critical angular velocityΩth . The latter is the angular velocity at which, in the rotating frame, the groundstate of the condensate contains a single vortex. But simply rotating the condensate
at Ωth will not lead to this ground state. If the condensate contains N bosons, N
units of angular momentum have to be provided to form the vortex. This presents
an energy barrier between the metastable state without vortex and the ground statewith vortex. Driving a surface excitation provides the necessary coupling mechanism
to ”pump” angular momentum into the condensate, which can subsequently relax
into a state containing vortices.

6.2

Experimental techniques

In the case of weakly interacting Bose gases in magnetic traps, the techniques forsetting the cloud in rotation are well established [178, 160, 2, 113, 120]. In [113, 120],the initially axially symmetric magnetic potential is deformed into an ellipse in the130radial plane, which is then set in rotation. In [160], an asymmetric optical dipole
potential is superimposed onto a cylindrically symmetric magnetic trap, again resulting in an elliptically deformed potential. In these cases, the role of the ”roughcontainer walls” needed to nucleate vortices is played by the smooth elliptical deformation. In contrast to the case of a sharp object or ”spoon” stirring in the superfluid,
this smooth potential can only excite a specific surface excitation of the condensate,a rotating quadrupole mode. This collective excitation carries angular momentum
m = ±2 (the third component of angular momentum) and can only be excited at a√
certain angular frequency1 ΩQ = ω⊥ / 2, where ω⊥ is the radial trapping frequency2 .
Below this threshold angular frequency, the rotating elliptical potential cannot excite
the condensate. Above this threshold frequency, the condensate’s quadrupolar modeis excited, and will eventually decay (via a dynamical instability) into vortices [159](see Fig. 6-1).
In the MIT experiments [178, 2, 1], two small, focussed laser beams were symmetrically rotated around the cloud. Vortices could be created efficiently over a large
range of stirring frequencies [2, 199]. The small beams presented a sharp obstacle tothe superfluid, most likely creating vortices locally at their surface [199], corresponding to high angular momentum excitations at low critical angular velocities. This is
the strategy followed in our experiment on rotating Fermi gases.

6.2.1

Roadblocks

However, in contrast to the case of weakly interacting Bose gases in magnetic traps,
the rotation of Fermi gases and the observation of vortices is hindered by several
issues, some of them technical and some of them due to the physics of Fermi gases.
Technical difficulties:
1. Strongly interacting fermions in 6 Li require an optical trap formed by a laser
beam. In contrast to magnetic traps, which can be designed to be very roundin the transverse plane, laser beams typically suffer from aberrations, such asa non-uniform beam profile with potentially high spatial frequency noise, and without special care - are generally not round.
√
While a collective excitation carrying angular momentum
m has an energy ~ω⊥ m, it m-fold√symmetric and is thus excited at a frequency Ω = ω⊥ / m, see [240].
q
1

2

In the presence of an elliptic deformation, one needs to replace ω⊥ by (ωx2 + ωy2 )/2, where ωx,y
are the trapping frequencies in the direction of the long and short axis of the ellipse.

1312. The experiments have to be carried out at high magnetic fields3 . Stray gradientscan distort the optical trapping potential. Also, the expansion from the opticaltrap, crucial for the magnification of the vortex cores, can be strongly affectedby the curvature of the magnetic potential.
3. Gravity is usually not considered a problem in magnetic traps, since it merely
shifts the center of the (to a very good approximation) harmonic trapping potential. However, optical traps provide a gaussian potential for the atoms.
Gravitational sag can thus distort the trapping potential.
The technical difficulties can be overcome one by one using an atomic BEC of
sodium atoms as the ”testbed”. However, the following problems are specific to
Fermi gases:
Difficulties arising from the physics of Fermi gases:
1. Far on the molecular side of the Feshbach resonance, in the regime where kF a issmall, the situation should be fully analogous to the case of atomic condensates.
This is thus the natural starting point for observing vortices. However, too far
away from resonance, the molecules (which are in the highest vibrational state of
the interatomic potential!) can undergo rapid vibrational relaxation via threebody collisions, leading to heating and trap loss. The lifetime of the gas needs tobe longer than the vortex nucleation and equilibration time (typically 1 s). This
limits the smallest values of kF a one can study (kF a & 3 in our experiment).
2. On the other hand, closer to resonance, quantum depletion becomes important:
The density of atoms in one spin state is no longer given by the square of thesuperfluid wavefunction |ψ(r)|2 as was the case in the deep molecular regime.
The contribution of the coherent part of the gas to the atomic density actually
becomes exponentially small in the BCS regime (for definitions of the condensate
wavefunction see sections 2.3.5 and 5.2.3). Loosely speaking, only the Cooper
pairs, of weight N ∆/EF , contribute to the condensate wavefunction ψ(r). Even
though ψ(r) still has to identically vanish inside the vortex core, the densityn(r) does not anymore, and the vortex contrast is greatly reduced. This will
3

”High” refers to the scale of atomic physics experiments, that is, the scale set by the hyperfine
interaction, ranging from about 100 G to several 1000 G. In condensed matter, magnetic fields
around tens of Tesla are considered ”high”, as here µB B becomes comparable to the gap energy andkB times the critical temperature.

132

set a technical limit on how deep into the strongly interacting regime one canobserve vortices as density depletions.
3. The vortex core size in the molecular Bose-Einstein condensate is given by
the healing length ξ ∝ k1F √k1F a . As kF a is increased moving into the stronglyinteracting regime, the vortex core shrinks. The core size will of course neverdecrease below the value at unitarity, where it has to be on the order of 1/kF ,the only length scale available when a → ±∞. Still, one might be concerned
whether this length scale can be resolved.
4. It is well-known from studies on atomic BECs [1] that vortices are strongly
damped by friction with the normal gas as the temperature approaches TC . As
we move to the BCS-side of the resonance, TC is strongly reduced (see chapter 2)
and we might expect vortices and rotation to be heavily damped.
5. On the BCS-side, the lowest lying excitations in the superfluid are those which
break a pair. However, as experience with sodium condensates in magnetic
traps suggests (see Fig. 6-2), one needs to strongly perturb the cloud in orderto nucleate many vortices. In the case of a gas in the BCS-regime, such a strong
perturbation is likely to simply destroy the superfluid via pair-breaking.
We will first describe the solutions to the technical problems, using an atomic
sodium BEC as the stepping stone.

6.2.2”Sanding off the bumps”

Optical Trap
The experimental setup used for rotating Fermi gases is shown in 6-3. The atom trap
is formed by the combination of an optical dipole trap (waist w), confining the atoms
radially, and a magnetic saddle point potential that provides axial confinement (andweak radial anti-confinement). We favored this geometry out of several reasons:• The large waist for the optical trap allows for a smooth beam profile, suppressing
high spatial frequency noise that is usually introduced when focussing light down
to sizes not much larger than the wavelength.
• We aim for a small aspect ratio of the cigar-shaped trap (about νz /νr = 5), sothat the stirring beam can rotate the entire cloud at once. For very elongatedclouds, a tightly focussed stirrer will excite the sample only locally, leading toinefficient transfer of angular momentum into the cloud.
133

Figure 6-2: Vortex nucleation for violent stirring in an atomic BEC. The upper row
shows expansion images of sodium condensates after 500 ms of stirring at the quadrupole frequency, for different laser powers of the stirring beam. The lower row shows
the resulting BEC after 300 ms of equilibration time. This suggests that the condensate has to be severely excited to generate many vortices. From left to right, the laser
power was increased for each subsequent image by a factor of two.

The aspect ratio of purely optical traps at wavelength λ (= 1064 nm in our experi√ment) is given by 2πw/λ. We see that it is not possible to fulfill both requirementswith purely optical (near-infrared) traps. In the combined optical and magnetic trap,
the trap frequencies are essentially decoupled, νr being given by the optical trap andνz fixed by the magnetic field curvature.
Another important requirement is that the cloud is allowed to cool after stirring.
Without dissipation, the excited cloud would never form an ordered vortex lattice. In
experiments using magnetic traps, this could be achieved by leaving an ”RF-knife”
on during equilibration, which removed energetic atoms from the trap. Optical traps
have a finite trap depth ”built in”. Cooling of the cloud will be efficient if the trap
depth U is not much higher than the Fermi energy EF . This constraint fixes thewaist, once the aspect ratio a = νr /νz and the axial trapping frequency νz is given:
The relation between U and the waist w is
1
1
U = mωr2 w2 = mωz2 a2 w2
4
4

134

(6.4)

a)

b)
10 W IR1064 nm

40 MHzAOM

Dichroic
MOT

beam
dump

Clean-up
Fiber
4 W IRPBS

XY-AOM

Dichroic
Imaging
671 nm

500 mW
532 nmFigure 6-3: Experimental setup for the observation of vortices in a Fermi gas. a)
Sketch of the geometry. The atomic cloud (in red) is trapped in a weakly focussed
optical dipole trap (pink). The coils (blue) provide the high magnetic offset field toaccess the Feshbach resonance as well as the axial confinement (additional curvaturecoils not shown). Two blue-detuned laser beams (green) rotate symmetrically aroundthe cloud. An absorption image of the expanded cloud shows the vortices. b) Opticalsetup for the vortex experiment. The laser beam forming the dipole trap is spatiallyfiltered using an optical fiber tolerating high laser power. The stirring beam (green)
passes through two crossed AOMs that deflect it in the transverse (XY) plane. Thesebeams are overlapped with the imaging light by dichroic mirrors. The light for themagneto-optical trap (MOT) is overlapped on a polarizing beam splitter cube (PBS).

The Fermi energy per spin state for a total number of atoms N is given by (using theharmonic approximation for the radially gaussian potential):
EF = ~(ωr2 ωz )1/3 (3N )1/3 = ~ωz a2/3 (3N )1/3

(6.5)

Requiring U & EF results inrw&2~ −2/3a(3N )1/6
mωz

(6.6)

If we want to trap N = 1×107 atoms with an aspect ratio a = 5 and an axial trapping
frequency νz = 20 Hz (a typical value), we need the waist to be larger than 100 µm.
Note that this requirement is quite stringent: Changing ωz is limited: Increasing the135Magnetic Potential TOP

Magnetic Potential SIDE
1

0

-1

1

x [mm]

x [mm]z [mm]

1

Mag.+ Grav. Potential SIDE

0

-1
-1

0
y [mm]

1

g

0

-1
-1

0
y [mm]

1-1

0
y [mm]

1

Figure 6-4: Trapping potential for the study of rotating Fermi gases. The magnetic
field produces a saddle potential (left), confining in the axial and anti-confining in
the radial direction. The optical trap needs to be aligned with the confining axis inorder to avoid stray gradients. In the vertical direction, gravity shifts the center ofthe potential seen by the atoms (middle + right). The optical trap thus needs to be
below the symmetry axis of the magnetic fields.

current in the curvature coils by a factor of two (which increases power dissipation
by four) only reduces the required waist by 20%. Allowing for an aspect ratio of 10
would only give another reduction by 35%, at the expense of making uniform stirring
across the longer cigar more difficult.
We choose w = 120 µm. The large waist ensures that there is negligible variationof the optical trap along the axial direction of the cloud: The Rayleigh range of thebeam is z0 = πw2 /λ > 4 cm while a typical axial cloud size is 1 mm. The opticaltrap is thus quite perfectly a cylindrically symmetric potential. The symmetry in theradial plane is ensured using a high-power optical fiber to spatially filter and shapethe beam profile. Care is taken not to deteriorate the quality of the gaussian beam’sroundness when passing through several lenses after the fiber exit.
The maximum power in the laser beam is 4 W, which limits the trap depth toabout 10 µK. This is still deep enough to load about 3 × 107 degenerate fermions
from the magnetic trap after the sympathetic cooling stage with sodium (The Fermi
temperature in the combined magnetic and optical trap during this loading is 5 µK,
and the degenerate cloud at T /TF ≈ 0.3 is not much larger than a zero-temperature
Fermi sea).

136

Magnetic Field profile
The offset magnetic field tunes the interatomic interactions close to the Feshbachresonance. We would like this to be an independent parameter, so one requires twocoils in Helmholtz configuration that create a very flat magnetic field along the axial
dimension. Our ”Feshbach” coils provide a residual magnetic field curvature that
corresponds to an axial trapping frequency of 11.0 Hz at 834 G (resonance). While√
this frequency depends only as BFB on the offset field BFB due to these coils, we canrender the dependence negligible by adding an additional magnetic field curvature via
independent sets of ”curvature” coils. These coils also produce an offset field thatis directed opposite to the field from the Feshbach coils, yielding a total offset fieldB0 < BFB . The final curvature corresponds to νz = 22.8 Hz axial confinement, andvaries only by 1 Hz from B0 = 700 G to B0 = 1000 G. The optical trap providing
radial confinement with νr in the range of 50 to 300 Hz, the aspect ratio of the cloud
can be varied between about 2 and 12.
The combined potential of Feshbach and curvature coils creates a magnetic saddle
potential V (x, y, z) = 12 mωz2 (z 2 − 12 x2 − 12 y 2 ). It is then necessary to align the opticaltrapping beam judiciously on top of the saddle, aligned with its confining (z-)axis(see Fig. 6-4). Otherwise the cloud would experience stray gradients, distorting theround trapping potential provided by the laser beam.
Gravity
Along the vertical (x-)axis, the combined potential of gravity+magnetic fields is− 14 mωz2 x2 − mgx. Thus, gravity shifts the saddle potential by an amount 2g/ωz2 ≈1 mm. The ”sweet spot” in the radial plane to which the ODT needs to be aligned isthus not the center of the magnetic field coils, but about 1 mm below it (see Fig. 6-4).
In this position no gradients act on the atoms. If the optical trap is round in theradial plane, the combined potential experienced by the atoms is round as well.
The alignment procedure of the ODT is shown in Fig. 6-5. At the end of evaporation of the lithium condensate, the trap depth is reduced in about 30 ms to a very
shallow depth which is not sufficient to hold the atoms if they are not in the ”sweetspot”. After 10 ms of expansion from the optical trap one clearly observes in which
direction the atoms spill out, and one can counteract by moving the optical trap.

137

Figure 6-5: Alignment of the optical trap to achieve a radially symmetric potential.
In the left image, the trap is still far from the ”sweet spot”. In the right image, straygradients are almost completely cancelled. The absorption images are of a lithium
pair condensate. The field of view for each image is 1.1 × 1.3 mm.

Expansion
While the trap is now round and the cloud can be set in rotation, one still needs toobserve the vortex cores. In the trap, the vortex size is on the order of the healing
length (for an atomic or a molecular BEC) or the inverse wavevector (for a stronglyinteracting Fermi gas), about 200 nm. It is essentially impossible to detect themin-situ using absorption imaging with larger optical wavelength (589 nm for 23 Na,
671 nm for 6 Li). Fortunately, vortices survive the expansion of the condensate, which
we can thus use as a ”magnifying glass”. However, only in simple geometries is the
expansion a faithful magnification. Complications arise due to the expansion into asaddle potential.
Let us discuss the case of an atomic or molecular condensate containing a single
vortex, initially confined in our cigar-shaped trap and with axial and radial ThomasFermi radii Ra and Rr . It is well-known that the condensate preserves its parabolic
density profile during expansion, and its radii scale according to a simple scaling
law [48, 131, 167, 47]. For very long cigars, Ra À Rr , the condensate’s mean-field
mostly escapes in the tight radial dimension and the expansion is effectively 2D. Forpfree space expansion, Rr (t) = Rr (0) 1 + ωr2 t2 , while for expansion into our radial
anti-confining potential,
v
!2
u
µ
µ
¶ Ã√
¶
u
ωz
ωz
2ωr
2
2
t
Rr (t) = Rr (0) cosh √ t +sinh √ t
ωz
2
2

138

(6.7)

Figure 6-6: Decrease of vortex visibility for a sodium condensate expanding from theoptical trap into a magnetic saddle potential. Top and bottom row show axially andradially integrated optical densities, respectively. The saddle potential is confining in
the axial and anti-confining in the radial direction. As the condensate expands radially, it collapses in the axial dimension, a direct consequence of hydrodynamic flow.
The vortex cores shrink and collapse onto themselves, thereby filling in completely
and forming ring-like structures (see text for details). For the images, the magnetic
field curvature (νz = 26 Hz) was switched off after, from left to right, 0, 2, 3, 4, 5, 7
and 10 ms. The total time of flight was constant at 35 ms.
√
where ωz / 2 is the rate of radial anti-trapping4 .
How does the vortex core size change during expansion? There are two regimes one
can simply understand, the initial hydrodynamic expansion and the ballistic expansion at long times of flight. In the first part of the expansion, the mean-field µ ∝ nachanges so slowly that the condensate wavefunction can still react to the change indensity: Adjustments on the healing length scale ξ - about the size of a vortex inequilibrium - can occur at a rate ~/mξ 2 = µ/~. As long as the rate of change of µ
- essentially the rate of change of Rr - is smaller than µ/~, the vortex core canq stillpadjust in size to the local mean-field. It thus grows as ξ ∝ 1/ n(t)a ∝ Rr (t) Raa(t) .
If Ra does not vary appreciably, ξ/Rr (t) will remain constant during the expansion,
the vortex core grows just as the size of the condensate, and the magnification isfaithful.
Once the rate of change of µ(t) becomes comparable to µ(t)/~, the condensate can4

In the case of hydrodynamic expansion of a gas with arbitrary equation of state ² ∝ nγ , thescaling factors br,z (t) for the radial (r) and axial (z) direction do not have an analytic expression,
but can be easily calculated as the solution to the differential equations [167]
b̈z

=

b̈r

=

ωz2
− ωz2 bz
bγ+1
b2γ
z
r
ωr2
1 2
2γ+1 γ + 2 ωz br
br
bz

(6.8)
(6.9)

Here we assume that the axial confinement is still on during expansion, only the radial confinementfrom the optical trap being switched off.

139

no longer adiabatically adapt to the lowering density. The characteristic expansion
rate being ωr , this occurs when µ(t) ≈ ~ωr . For much longer expansion times, weare in the limit of ballistic expansion. Here, each particle escapes outward with thegiven velocity (in free space) or, in the case of the saddle potential, with a radialacceleration proportional to its distance from the origin. This simply rescales theradial dimension, and thus stretches the vortex core and the cloud size by the same
factor. Again the magnification is faithful.
However, in our experiment we are not in the quasi-2D regime where Ra À Rr .
The saddle potential ”squishes” the cloud in the axial dimension, as the decreasing
mean-field no longer stabilizes the condensate’s axial size. According to the above
estimate, the vortex cores will shrink in comparison to the cloud size by a factorp∝ Ra (t). We can see the effect on a sodium condensate in our optical trap in
Fig. 6-6, where the axial curvature was left on for longer and longer times.
To work around this problem, we quickly reduce the magnetic field curvatureduring expansion by ramping down the curvature coils (in about 1 ms). As thisincreases the overall offset field (see above), the current in the Feshbach coils isdecreased accordingly, so as to leave the offset field B0 - and the interaction parameterof the Fermi mixture - constant. We can speed up the radial expansion even further
in comparison to the axial evolution by actively ”squishing” the cloud about 3 ms
before release. This is done by simply ramping up the power in the optical trapping
beam by a factor of 4. Not only does this increase the radial trapping frequency, but it
also excites a ”breathing” mode in the condensate. The result is that the condensate
expands almost twice as fast as without these steps.
Overcoming all technical difficulties, the resulting trapping potential can now
favorably compare with our magnetic traps. The residual ellipticity of the transverse
potential is estimated to be less than5 2%. Large vortex lattices containing about
120 vortices can be created in sodium Bose-Einstein condensates (see Fig. 6-7). The
vortex lifetime is about 4 s, which is at least half the lifetime of vortices in ourmagnetic trap.

5

Of course, this cannot compare with the almost perfect roundness of a (magnetic) TOP trap,with residual ellipticity of less than 0.1% [113].

140

Figure 6-7: Vortex lattice in a Bose-Einstein condensate of sodium atoms in themagnetic trap (left) and the optical trap (right image). The optical trap (highest
number obtained ∼ 120 vortices) can favorably compare with the magnetic trap
(highest number in our experiment ∼ 150 vortices).

141

Figure 6-8: Vortex lattice in a 6 Li2 molecular condensate. In a), stirring, equilibrationand imaging of the vortex lattice all took place at a fixed field, 766 G (1/kF a). Imagea) shows the very first clear signature we observed. The vortex core depletion isbarely 10%. b) A Fourier filter applied to a) clearly shows the Abrikosov vortex
lattice. c) The improved scheme of ”squishing and release” (see text), as well as asudden reduction of the interaction strength led to an improved vortex contrast.

6.3

Observation of vortices in a Fermi gas

The search for vortices in Fermi gases started on the molecular side of the resonance,
to stay close to the analogous case of an atomic BEC. It was not clear whether there
would be a window in which 1. the molecular condensate lives long enough to nucleate
vortices and 2. the interactions are weak enough such that vortices show up as clear
density depletions.
It turned out that at a field of 766 G (1/kF a = 1.3), we were successful: After
stirring the cloud for 800 ms and letting the cloud equilibrate in 400 ms, we observed
a vortex lattice in the density profile (Fig. 6-8). This established superfluidity formolecular condensates.
With this as the starting point, different methods could be tried to improve thevortex contrast. What turned out to be crucial for good contrast was - as expected- the reduction of the interaction strength. In the moment the vortex lattice isreleased from the trap, the magnetic field is lowered to fields around 700-740 G(1/kF a ≈ 2 − 4 initially, further growing during expansion). If the condensate still√
has time to react to this change in scattering length, the vortex size ξ ∝ Rr (t)/ awill increase relative to the condensate’s radius (the expression for ξ is valid in the
BEC-limit, and assumes radial expansion, see previous section). Also, at the newinteraction strength, quantum depletion of the condensate wavefunction is reduced.

142

Figure 6-9: Observation of vortices in a strongly interacting Fermi gas. This establishes superfluidity and phase coherence in gases of fermions.

The vortices are effectively ”emptied” by weakening the interactions6 .
This technique proved to work on both sides of the Feshbach resonance. Stirring,equilibration and initial expansion could be performed at magnetic fields between 740
G (1/kF a = 2) and 860 G (1/kF a = −0.35), before switching to the BEC-side during
expansion for imaging. The observation of ordered vortex lattices above the Feshbachresonance at 834 G, on the BCS-side of the resonance, establishes superfluidity andphase coherence of fermionic gases at interaction strengths where there is no two-bodybound state available for pairing.
One might argue that the gas could have been in the normal state above resonance,
while the vortex lattice might have formed during the 10 ms of expansion at theimaging field, on the BEC-side of the resonance. This is not possible. We showedthat the vortex lattice needs many hundreds of milliseconds to form in the stirred
cloud. This is the same time scale found for the lattice formation time in atomic
BECs [159, 1]. It is also in agreement with a theoretical study of vortex formation
6

The ramp should not move too far into the weakly interacting regime: The condensate would
simply not expand anymore as practically all the repulsive mean-field has been taken out of thecloud. On the other hand, letting the condensate expand for too long before the ramp should be
avoided as well. At reduced density the ramp would not allow the condensate to adjust to the newinteraction strength and modify the vortex size. The condition is again (see previous section) thatthe ”reaction time” of the condensate wavefunction, µ/~, evaluated at the final field, should be fasterthan the rate of change of µ, that is, the rate of change of Rr (t).

143

in strongly interacting Fermi gases [245]. When a thermal cloud is slowly cooled
through the transition temperature [113], the condensate first forms without a vortex.
As the condensate grows, vortices are nucleated at the surface and then enter thecondensate [14]. When a thermal cloud is suddenly cooled, a condensate with phasefluctuations will form [16, 224] which can arrange themselves into a vortex tangle.
In either case, one would expect a crystallization time of at least several hundred
milliseconds before a regular vortex lattice would emerge. It simply needs several
axial trapping periods for the vortex tangle to stretch out. It should be noted that
this time scale was found to be independent of temperature [1] and seems to represent
an intrinsic time scale of superfluid hydrodynamics.
Even if these timescales were not known, it is not possible to establish a regularvortex lattice with long-range order in a gas that expands at the speed of sound of thetrapped gas. Opposing edges of the expanding cloud simply cannot ”communicate”
fast enough with each other.
The regularity of the lattice proves that all vortices have the same vorticity. Fromtheir number, the size of the cloud and the quantum of circulation h/2m for eachvortex, we can estimate the rotational frequency of the lattice. For an optimized
stirring procedure, we find that it is close to the stirring frequency. This excludes aquantum of circulation of h/m or doubly charged vortices.

6.3.1

Vortex number and lifetime

The number of vortices that could directly be created on the BCS-side was rather
low in the first experiments, as the stirring seems to have had an adverse effect on
the stability of the pairs. This corresponds to the expectation that the gas is morerobust on the BEC-side, where the lowest excitations are sound waves, while on the
BCS-side it is pair breaking. To optimize the vortex number on the BCS-side, firsta large vortex lattice was produced close to resonance, at 812 G, before ramping themagnetic field beyond the Feshbach resonance. In this way, large numbers of vortices
could be obtained in the entire BEC-BCS crossover (see Fig. 6-9).
The vortex lifetime is on the same order of what was found for atomic BECs.
This displays the high degree of metastability of vortices in superfluids. The longest
lifetime is found close to, but not right on resonance. In fact, there is a peculiar dip
in the lifetime on resonance, which we speculate to be caused by pair breaking. The
idea is the following: Pairs far outside, in the low-density regions of the cloud, andclose to resonance, have very small binding energies. For example, the (two-body)

144

1.5

1 / k Fa
0
-0.4

2.0

1.0

0

-0.5

-0.9

-1.0

80
0.01
0.02
0.04

4
2

10

0.1
0.2
0.4

4
2

1

1
2
4

4
2

730

780

834
880
930
Magnetic Field [G]Vortex Lifetime [s]Vortex Decay Rate [1/s]

100

b)
-0.7

Number of Vortices

0.6

a)

60

40

20

0

980

700

750

800 834
900
Magnetic Field [G]950

Figure 6-10: 1/e-Lifetime (a) and number (b) of vortices in the BEC-BCS crossover.
Vortices are long-lived across the entire BEC-BCS crossover. A narrow dip in the
lifetime on resonance is presumably due to breaking of pairs whose binding energies
are comparable to h× the rotation frequency (several tens of Hz, corresponding to
the binding energy of free molecules a few G away from resonance). Open symbols
are data from April 2005. Optimization of the system using a deeper trap resulted in
improved characteristics on the BCS-side, shown by the full symbols (data from June
2005). In b), the triangles give the number obtained by stirring and equilibrating
both at the given field. Stirring at 812 G and subsequently ramping to the final field
for equilibration resulted in the data shown as circles.

molecular binding energy at 830 G, 4 G away from resonance, is only kB × 3 nK or
h × 60 Hz. This is on the order of ~× the rotation frequency Ω of the lattice. If there
are small ”bumps” in the trapping potential, the molecules rotating around the trap
are excited at their binding energy and thus tend to break. The effect could thus
be an interesting coupling of the external degrees of freedom with the internal ones.
Why do clouds in the BCS-regime not suffer from this pair breaking as well? Here,the binding energy in the low-density regions is even smaller if not identically zero.
We speculate that this is because only the pairs that have a binding energy ∼ ~Ω arein danger. Such pairs reside in a certain shell within the condensate. If this is not theoutermost paired shell, the surrounding superfluid ”protects” the endangered pairs
from breaking.
This scenario might also explain why stirring the cloud on the BCS-side creates
heating and creates vortices only inefficiently. As the cloud is spun up, all frequencies
from Ω = 0 Hz up to the final value will occur, thus destroying one shell of pairs after
the other.

145

a)

b)

Number of vortices

25
20
15
10
5
0
0

5

10
15
Time spent in bucket [s]20

10 s in rotating bucket

Figure 6-11: Rotating bucket for superfluid Fermi gases. a) Vortex number vs timespent in the rotating trap. After a certain nucleation time the number of vortices
saturates and stays constant. The final vortex number depends on the power of thegreen beam, indicating slippage due to residual friction with the ”container walls”.
The lower and upper curve correspond to lower and higher green beam power. b)
Absorption image of an optimized vortex lattice, containing about 75 vortices, after
10 s hold time in the rotating bucket. The magnetic field for all data was 812 G,
corresponding to 1/kF a ≈ 0.2.

The dip in the vortex lifetime on resonance might thus be a spurious effect due
to residual trap anisotropy. Its presence is a curiosity, since the critical velocity ofthe superfluid should be highest, and thus the superfluid the most stable close toresonance [219, 64].

6.3.2

A rotating bucket

All the experiments described before first set the cloud in rotation using the stirring
beam and then let the gas equilibrate in the stationary trap to settle into a vortex lattice. In the stationary trap, the configuration is of course only metastable. Granted, if
the trap was perfectly round, there would be no way for angular momentum to escapeand the vortex lattice would persist indefinitely. In this case, the vortex configurationis the ground state of the equivalent system with a given angular momentum imposed(mathematically speaking, it is the ground state for the hamiltonian in the rotating
frame H − ΩLz , where H is the hamiltonian in the absence of rotation and Lz plays
the role of a lagrangian multiplier: the angular momentum imposed on the systemthat fixes the rotation frequency Ω). At zero temperature, it would not even matter
if the trap was not round, as the superfluid does not experience friction with the146container walls. The only way the superfluid can spin down is via friction with thethermal cloud [263, 88]. The thermal cloud’s rotation, in turn, is slowed down quickly
by the container walls [107].
However, in the original ”rotating bucket” experiments on helium-4 [257], it was
actually possible to continue rotating the trap practically indefinitely (for as long as
the refrigerator helium supply lasted). In the case of atomic condensates, the rotating
anisotropy or the ”spoon” used to stir up the cloud usually needs to be switched off
before an ordered vortex lattice can form7 . This is possibly because the spoon is notmoving smoothly enough to allow equilibration into a vortex lattice. Before theirobservation of ordered vortex arrays, the experiments on helium-4 initially sufferedfrom similar perturbations [257].
In contrast to previous experiments, we have found that it is possible to keep
the stirring beams on during formation of the vortex lattice. The experiment was
performed at 812 G in a trap with radial trapping frequency νR = 90 Hz. The twostirring laser beams (power in each beam ≈ 100 µW, waist w = 16 µm) created only
a weak potential of about 20 nK each on the cloud (mean-field µ ≈ 400 nK). They
were rotated around the cloud at a frequency of 70 Hz. For imaging, the atoms werereleased from the combined trap, the confining optical potential plus the repulsive
stirring beam. We found that it was possible to stabilize a vortex lattice containing
19 vortices for 20 s (see Fig. 6-11). The maximum time studied was only limited by
the memory of the computer controlling the experiment.
The final vortex number depended on the laser power or the ”stiffness” of thestirrer. Increasing the power in the stirring laser by 60 % allowed a larger equilibrium
vortex number, 29. This suggests that the stirrer is fighting a residual trap anisotropy.
The ”slippage” is worse for a weaker drive, and causes the condensate to rotate at alower speed.
It is interesting to note that vortices become visible in the condensate only after
a considerable delay of 1 − 2 s. In contrast to the more severe perturbations usedin the previous studies (stirring potential ∼ 100 nK), where the total stirring time
was about 500 ms to 1 s, the more gentle but continuous stirring is less efficientin nucleating vortices. The system is stuck longer in the metastable state without
vorticity. Again, with the stronger stirrer, the time needed to nucleate the firstvortices was considerably reduced. That indicates that the barrier to the energeticallymore favorable state containing vortices could now be overcome more easily.
7

In Paris, 11 vortices ordered in a lattice were observed in expansion after ramping down thestirring beam in only 8 ms [160, 54].

147

Figure 6-12: Observation of pair breaking in a superfluid Fermi gas. Shown areabsorption images for different expansion times on the BCS-side of the Feshbachresonance at 910 G (0,1,2,3,3.5,4, and 4.5 ms) and at 960 G (0, 0.5, 1, 1.5, 2, 2.5, and3 ms), before ramping to the BEC-side for further expansion and imaging. The total
time-of-flight was constant. The vortices served as markers for the superfluid parts ofthe cloud. Superfluidity survived during expansion for several ms, and was gradually
lost from the low-density edges of the cloud towards its center. The field of view of
each image is 1.2 mm × 1.2 mm.

At later stages of the experiment, we were able to stabilize 75 vortices for 10 s ina deeper trap with νr = 120 Hz.

6.3.3

Pair breaking

A true qualitative distinction between the BEC- and the BCS-limit in fermionic superfluids is the fragility of the pairs on the BCS-side. Here, pairing is purely amany-body effect. In contrast to the case of Bose-Einstein condensates, the simple
expansion from the trap can destroy the superfluid (at finite temperatures): The now
isolated pairs will break. All information about the center-of-mass wavefunction of
the pairs ψ(r) and hence the vortices, will be gradually lost. In our experiments, therapid ramp to the BEC-side ensures that the pairs survive as stable molecules.
It is now natural to delay the ”rescuing” ramp to the BEC-side and observe
when exactly this breakdown of superfluidity occurs, using the presence or absence ofvortices as markers for superfluidity [216]. We thus let the gas expand on the BCSside for a certain time tBCS , then ramp down to the BEC-side for further expansion
and imaging. The total expansion time is kept constant.
It is found that superfluid flow initially persists during the expansion. Then,
vortices start to disappear first at the edges of the cloud, then, for longer BCS148

Figure 6-13: (Color online) Peak interaction strength kF a during superfluid expansion. Starting at a peak kF a in the optical trap, vortices survive up to an interaction
strength kF a ≈ −0.8, almost independent of the magnetic field (scattering length a).
Filled circles correspond to partially superfluid, open circles to normal clouds. The
observed number of vortices is color coded.

expansion tBCS , further inwards until the last vortex disappears at the cloud’s center(see Fig. 6-12). The time tBCS for which the last vortex disappears, increases the closer
we are to resonance, that is, the larger the interaction strength and the stronger thefermion pairs are bound.
In fact, by varying the magnetic field and thus the scattering length, we find thatthe last vortex always disappears at about the same value of the interaction parameterkF a ≈ −0.8 (see Fig. 6-13). Here one enters the regime of weak interactions, where
the pair size is large, the critical temperature for superfluidity is small and hence thesuperfluid state is fragile.
The simplest explanation for our observation is naturally that we cross the phasetransition line during expansion. While T /TF is an adiabatic constant for the expansion, T /TC is not, as TC /TF depends exponentially on the density. As the density
decreases, the critical temperature in the outer regions of the cloud eventually drops
below T , superfluidity is lost starting from the edges inwards.
We can estimate the critical interaction strength for this breakdown to occur.
At our coldest temperatures, T /TF = 0.05 (see chapter 7). The formula for TCγ ¡ ¢7/3
due to Gorkov and Melik-Barkhudarov [98] gives TC = eπ 2e
TF e−π/2kF |a| =0.28 TF e−π/2kF |a| . This formula should be valid in the BCS-regime where kF |a| . 1.
We can now equate
1=

T
T TF
TF
=
≈ 0.05= 0.18 eπ/2kF |a|
TC
TF TC
TC
149

(6.10)

Figure 6-14: Loss of vortex contrast on resonance, at B = 834 G. Shown are absorption images at fixed total time-of-flight, but for different expansion times on resonance(2, 2.5, 3, 3.5, 4,5, and 6 ms) before the magnetic field ramp to the BEC-side for further expansion. The vortex contrast decreases uniformly across the cloud from 15%(for 2 ms resonant expansion) to about 3% (for 5 ms). The field of view of each imageis 1.2 mm × 1.2 mm.

The result gives the critical interaction strength kF |a| = 0.9. This is close to theobserved value.
However, we can think about other scenarios that might explain the observed
disappearance of vortices: In the BCS-regime around kF |a| = 1, the pairs are large
and fragile. The expansion itself might be too much of a perturbation so that they
break apart. This dynamic pair breaking will occur if the rate of change in the binding
energy ∆ is larger than the rate at which the pairs can respond to the reduced density,˙∆/~. For our experimental parameters, we find ~∆∆2 ≤ 0.4, so the pair binding energy
still changes more slowly than the characteristic response time. However, the rates are
not too different, and this dynamic pair breaking mechanism might not be negligible.
Note that in the images, even for short ”BCS-expansion” times, we see a clear
boundary between the coherent part containing vortices and a part of the fluid which
does not contain vortices and appears normal. For these short expansion times, thedynamic pair breaking mechanism is ruled out and the most plausible scenario is that
we are simply seeing the normal-to-superfluid transition in the gas cloud: TC = T
defines a certain density in the cloud beyond which the gas is necessarily normal.
On resonance, TC ∝ TF , and one would not expect to loose superfluidity during
expansion. Indeed, here we observe that vortices survive expansion even in the lowdensity edges of the cloud (see Fig. 6-14). However, the longer the ramp to the
BEC-side is delayed, the lower the vortex contrast in the final cloud, and the smaller
the apparent size of the vortex cores. We believe that this is due to the low density of
the cloud after long expansion on resonance. Then the gas is so dilute that, after theramp, the vortex cores cannot adjust anymore to the larger equilibrium size and theincreased contrast they should have on the BEC-side. It is expected that expansion
on resonance simply scales the dimensions of the cloud (at least for purely radial
150expansion) [47]. Hence we can speculate that, the longer we delay the ramp to the
BEC-side, the closer the final vortex contrast and vortex size will correspond to thesituation on resonance. Here, vortices are expected to be small, of size ξ ∼ 1/kF , andto show a reduced contrast [41].
To conclude, we observe superfluid flow in an expanded Fermi gas down to densities
of about 1011 cm−3 . At these densities, the average distance between two atoms is 2
µm! The average distance between neutrons in a neutron star is a few fm, corresponding to densities of 1038 cm−3 . This nicely illustrates how general the phenomenon offermionic superfluidity is.

151Chapter 7
Fermionic superfluidity withimbalanced spin populations
In previous chapters, we have exclusively studied an equal spin-mixture of fermionic
atoms. Here, pairing can be complete and the entire gas can become superfluid at zerotemperature. What, however, happens if we deliberately imbalance the populationsin the two spin states? In this case, not every spin up (majority) fermion can find aspin down (minority) partner. Immediately, several questions arise: Can the gas still
be superfluid? If so, are the excess fermions tolerated inside the gas of pairs or are
they expelled from the superfluid? If superfluidity breaks down, what is the natureof such a strongly interacting, imbalanced normal mixture?
We studied these questions in a strongly interacting Fermi mixture by varying
the spin imbalance, temperature and interaction strength. Superfluidity was foundto be robust against population imbalance in the strongly interacting regime. We
could show that, below a certain temperature, the superfluid state requires equal
spin densities, and phase separates from the partially polarized normal state. At acritical population imbalance, we observed the final breakdown of the superfluid state,
the Clogston or Pauli limit of superfluidity. Studying imbalanced Fermi mixtures
enabled us to directly observe the superfluid transition in situ, without any magnetic
field ramps into the molecular regime. The non-interacting, fully polarized part of thecloud served as a reliable thermometer, solving, for the imbalanced case, the problem
of measuring temperature in a strongly interacting gas.
In contrast to the smooth BEC-BCS crossover physics in an equal Fermi mixture
with varying interaction strength, the imbalanced mixtures provide us with a richphase diagram, including zero temperature phase transitions, which challenges present
many-body theories.
152

7.1

Historic overview

The question of imbalanced superfluidity has a 50 year old history. Bardeen, Cooper
and Schrieffer described the superconducting state as a condensate of pairs of electrons
carrying opposite spin. However, an applied magnetic field tends to align the electron
spins, thereby putting a ”stress” on the fully paired superfluid state. Following thisthought, Clogston [61] derived in 1962 an upper limit for the critical magnetic field of
a superconductor: Once the energy gain for aligning the electron spins is larger thanthe energy gain from pairing opposite spins, Cooper pairs must break and the systemshould turn normal. This is the Pauli or Clogston limit of superconductivity. In 1964,
Fulde and Ferrell [94] and independently Larkin and Ovchinnikov [146] found that not
all the pairs necessarily break at once, but that there exists a novel superfluid state
which tolerates a certain amount of broken pairs by allowing the remaining Cooper
pairs to have a common non-zero momentum Q (FFLO or LOFF state). Thus, theorder parameter (the gap) is not a constant, but it oscillates at a wavelength ∼ 1/Q,
where the typical value of Q is given by the Fermi momentum difference between
the two spin states. This state of the imbalanced superconductor will thus breaktranslational and rotational symmetry by picking out a certain direction Q. Forty
years later, the debate about the ground state of imbalanced superfluidity is still ongoing. The topic had been revived from several directions: From condensed matter,through studies of exotic superconductors [46, 198, 30] that are essentially Paulilimited, and from nuclear physics and the study of the quark-gluon plasma, where
similar questions arose in the study of superfluid pairing of quarks at unequal Fermi
energies [7], and from atomic physics, were the advent of ultracold atomic Fermi gases
presented a new opportunity to study imbalanced superfluidity.
Preceding the experimental studies, Liu and Wilczek [151] proposed the breachedpair or Sarma state [211] as an alternative to the FFLO state. Another suggestion,that did not require breaking of translational, only rotational symmetry, was a statewith a deformed fermi surface [218]. Combescot and Mora [65] studied the FFLOstate in Fermi gases in the BCS regime. Its direct detection by imaging the gas was
proposed by Mizushima et al. [169] (see also [49]). Bedaque et al. [28] noted that it isfavorable for a weakly interacting imbalanced Fermi mixture to phase separate into asuperfluid part at equal densities and a normal part at unequal densities. In the regimeof strong interactions, a Monte-Carlo study of the problem of imbalanced superfluidity
was performed by Carlson and Reddy [44]. They found that the imbalanced Fermi
gas on resonance was intriguingly close, within their error margins, to be a polarized153superfluid, where unpaired excess fermions were tolerated by the Cooper pairs. The
implications of universality on resonance were discussed by Cohen [62]. Pao et al. [185]performed a stability analysis of the superfluid state across a Feshbach resonance atfixed densities, and found that homogeneous superfluidity was unstable on the BCSside and in the strongly interacting regime.
The subject has received much attention since our first observation of superfluidity
with imbalanced spin populations, reported in June 2005 [135]. We determined thephase diagram of imbalanced Fermi gases, and found the Clogston limit of superfluidity [268]. Hints of phase separation were seen in our work and at Rice University [187].
Imbalanced Fermi mixtures allowed us to observe the superfluid phase transition directly, without the necessity of field ramps [269]. Finally, using a novel imaging technique, we were able to demonstrate phase separation in strongly interacting Fermi
mixtures [223].
Motivated by these findings, phase diagrams for imbalanced mixtures close to a
Feshbach resonance were obtained1 [185, 221, 229, 256, 152, 157, 103, 50, 127, 186],
and the shape of imbalanced Fermi gases in atom traps was studied [196, 140, 226,
258, 53, 114, 55, 220]. Insight into the equation of state on resonance was gained by
exploiting the principle of universality [52, 40] and via Monte-Carlo calculations [153].
The properties of the vortex state in imbalanced fermionic superfluid was discussed
in [242], showing that particles of the majority spin species will be trapped inside thevortex cores.
Naturally, there are still many open questions: The role of temperature, finite size
effects or the nature of the strongly interacting normal state. The search is still ongoing for a signature of the FFLO-state or other proposed new forms of superfluidity.
We can expect that the next months and years will bring us exciting new experimental
results and theoretical insight.

7.2

The Clogston limit of superfluidity

7.2.1

The Clogston limit of superconductors

Naturally, it is not a simple task to magnetize a superconductor because of the Meissner effect [166]: Magnetic fields are actively expelled from charged superfluids. They
can only enter as magnetic flux lines or vortices (and only in so-called Type II super1

This list is far from complete. Over 80 articles have been written on the topic over the past ninemonths. I refer the reader to the cited publications and references therein.

154

conductors). Once the energy cost per volume for expelling the field, B 2 /2µ0 , exceeds
the difference in the free energy density F between the normal and the superconducting state, the superconductor turns into a normal metal. If the critical field is Bc , we
have
B2
FN = FS + c
(7.1)
2µ0
where the subscripts N and S stand for the normal and the superconducting state,
respectively. The free energy density difference FN −FS is given by the pairing energy∆ of ρ(EF )∆ electron pairs at the Fermi surface per unit volume (ρ(EF ) is the densityof states per spin state at the Fermi energy),
FN − F S =1
ρ(EF )∆22(7.2)

which relates the critical magnetic field Bc to the superconducting gap,
Bc =p

µ0 ρ(EF )∆(7.3)

To get a sense of scale, for free electrons, ρ(EF ) = 34 EnF , with n ≈ 1−10·1022 cm−3 thedensity of electrons and EF ≈ kB × 50 000 − 100 000 K, and assuming ∆ = 1.76 kB TC
from BCS theory (see Eq. 2.125 in chapter 2), we get
Bc ≈ 50 G

TC
K

(7.4)

This is a very small field compared to the natural scale set by the critical temperatureand the electron’s magnetic moment, BTC ∼ kB TC /µB = 15 000 G TKC . However, atthe surface of a superconductor, the magnetic field does not suddenly drop to zero,but only over a certain penetration depth. For very thin superconductors or certain
special materials, the magnetic field can enter the superconductor, increasing thecritical magnetic field substantially. It was Clogston’s insight that under the extreme
condition that there is no Meissner effect, there will still be a critical magnetic fieldthat destroys superconductivity à la BCS.
The reason is the following: The superconducting state is fully paired, that is,there must be an equal number of spin up and spin down electrons present. In fact,by changing a spin down into a spin up electron one gains an energy 2µB B, but
this would necessitate breaking the initial pair (cost ∆), and in addition there isno pairing partner for the new spin up electron (cost another ∆) (for adding andremoving single particles to the BCS state see section 2.3.3). Thus, for µB B < ∆ this155

cannot occur. However, the normal state is free to adjust the number of spin up vsspin down electrons, thereby reducing its energy.
E

µ

}

0

2 µBB

Figure 7-1: Imbalanced chemical potentials for electrons in the normal state under
the influence of a magnetic field.

Let us calculate this reduction of free energy density in the normal state in thepresence of a magnetic field. The field changes the energies of spin-up electrons
according to ²k↑ = ²k − µB B, which we can incorporate into the chemical potentialµ↑ = EF + µB B of the up-electrons2 (see Fig. 7-1). The energy density of a Fermi gas
at chemical potential µ↑ is ² = 53 µ↑ n↑ , with the density n↑ = 6π1 2 kF3 ↑ and the Fermi
wave vector kF ↑ related to µ↑ via µ↑ =

2
~2 kF
↑
.
2m

Hence, the free energy density of the”two-species mixture” of spin-up and spin-down electrons is given by33µ↑ n↑ + µ↓ n↓ − µ↑ n↑ − µ↓ n↓
5
5
³
´
4
5/2
5/2
= − ρ(1) µ↑ + µ↓
15

(7.5)

FN (B) =

(7.6)

3/2

with ρ(1) = √m2π2 ~3 . Inserting the chemical potentials given as a function of EF andB, we finally have4
FN (B) = − ρ(EF ) EF2
15

(µ

µB B
1+
EF

≈ FN (0) − ρ(EF ) µ2B B 2
2

¶5/2

µ

µB B
+ 1−
EF

¶5/2 )

valid for µB B ¿ EF

(7.7)
(7.8)

The sign-convention makes sure the spin-up electrons will be the majority species. This means,the electron spin of spin-up electrons points into the direction opposite the magnetic field.

156

N
The magnetization of the normal state is M = − ∂F
= ρ(EF )µ2B B, proportional to∂Bthe applied field.

If the reduction in FN is larger than the energy gain from forming a fully paired superconducting state, Eq. 7.2, the metal favors the normal state. The critical magnetic
field found by Clogston is thus
1 ∆BClogston = √2 µB= 18 500 G

(7.9)
TC
K

(7.10)The expression holds for weak interactions, where ∆ ¿ EF . This is evidently very
well satisfied by conventional superconductors, but it will be interesting to see whether
such a critical field might still exist in the strongly interacting regime.

7.2.2

Translation to dilute Fermi gases in atom traps

In dilute Fermi gases, the chemical potential difference δµ ≡ 2h = µ↑ − µ↓ is notdirectly controlled, as it was the case for superconductors in a magnetic field whereh = µB B. What can be freely chosen in our case are the numbers of spin-up andspin-down fermions, i.e. the populations of the two hyperfine states |1i and |2i.
Furthermore, the experiments take place in the presence of a trapping potential
V (r). The chemical potentials of the two species are determined by the global requirement that they must correspond to the state that has the given particle numbers N↑
and N↓ in the two spin states. Let us say the system finds that µ0↑ and µ0↓ are thecorrect (global) chemical potentials. Locally, the chemical potentials at point r in thetrap will thus be3µ↑ (r) = µ0↑ − V (r)µ↓ (r) = µ0↓ − V (r)µ (r)+µ (r)(7.11)

While the average chemical potential µ(r) = ↑ 2 ↓difference chemical potential is constant throughout,varies across the trap, theδµ(r) = 2h = µ0↑ − µ0↓(7.12)

Thus, effectively, the trapping potential tunes the ratio h/µ over a certain range. An
3

We assume here that the system can be described by local quantities, that is, we assume thatthe local density approximation holds (see section 7.5).

157

a)
µ(r)

Energy

V(r)
∆(r)
√2

µ↑

δµ
}
µ

µ

↑

n↑(r) , n↓(r)b)

R↑

ntot(r), δn(r)

R

↑

RSF

c)

0

r

Position in trap

Figure 7-2: A weakly interacting, imbalanced Fermi mixture with superfluid component in a harmonic trap. a) Trapping
potential V (r),√√ local, average chemical potentialµ(r) and Clogston limit ∆(r)/ 2. Where ∆(r)/ 2 > δµ/2, the gas is in the superfluid phase. b) shows the densities of spin up and spin down atoms, c) the total andthe difference density distribution. For this illustration, the peak gap was chosen tobe ∆(0) = 0.25µ(0).

158imbalanced Fermi mixture in a trap can thus explore several local phases. Let us forexample consider the Clogston limit above. It occurs when the ”field” h satisfies
1
h = √ ∆(r)2

for ∆ ¿ µ(7.13)

where the superfluid gap ∆(r) is to be calculated as usual using the average chemical¡ ¢7/3potential µ(r), that is in the weakly interacting regime4 ∆(r) = 2eµ(r) e−π/2kF (r)|a| ,with the Fermi wave vector kF (r) given by µ(r) = ~2 kF (r)2 /2m. The situation
is sketched in Fig. 7-2: The Clogston criterion defines a critical radius RSF where√∆(RSF ) = 2 h = √12 δµ. For radii r < RSF , the gas is in the BCS-type superfluid state and thus fully paired. For radii exceeding RSF , the gas is in thenormal state. Here we can further distinguish the region where both spin up andspin down particles are present, RSF < r < R↓ and the region where only spin
up atoms are present, R↓ < r < R↑ , the case of a fully polarized sample. Theseradii are - in this weakly interacting
case - simply given by the usual relationsq
2µ↑,↓
V (R↑,↓ ) = µ↑,↓ , that is R↑,↓ =, for harmonic trapping. The densities aremω 2
22mµ↑,↓ 3/2) (1 − Rr2 )3/2 (for RSF < r < R↑,↓ ), whereas for the
~2
↑,↓
2
2
3/2
superfluid state at r < RSF one has nSF (r) = 6π1 2 ( 2mµ(1 − Rr 2 )3/2 · (1 + O( ∆2 )2 )) with~µq

given by n↑,↓ (r) = 6π1 2 (
R=

2µ
mω 2

given by the average chemical potential and a small enhancement of the2
density (O( ∆) due to pairing [220].
µ2

We will see in the experiments that the important qualitative features, a central
superfluid region of equal densities, a mixed normal region and a fully polarized region
further outside, remain valid also in the strongly interacting regime where kF |a| À 1.

7.2.3

Phase separation in the BCS-regime

Fig. 7-2 shows that the densities of the two spin components are expected to jumpfrom their respective values in the normal state to a common value in the superfluid
state. This discontinuity nicely illustrates that the Clogston limit is a first-order transition: As h/µ is tuned starting from zero (either directly via a magnetic field for a
superconductor, or implicitly via the varying trapping potential), the density difference is first zero, as one is still in the BCS superconducting phase. At the Clogston
limit h/µ = √12 ∆the system suddenly turns normal and the density difference jumpsµ
to a finite value. In the language of thermodynamics, the free energy density, con4

This is the BCS-gap when density fluctuations are taken into account [98]. Note that in the
weakly interacting regime, µ does not depend on the interaction strength and is equal to the average
Fermi energy of a non-interacting Fermi mixture.

159

sidered a function of the superfluid gap ∆, has developed a new global minimum at∆ = 0, the normal state.
We thus see that in the presence of a trapping potential, which tunes h/µ, we
expect to observe phase separation between an inner superfluid core of equal densities
and a normal state outside. However, phase separation would also occur in the caseof an imbalanced Fermi mixture confined in a large box potential. In this case,the global densities n↑ and n↓ are fixed. We can consider three possibilities: 1. A
uniform superfluid state with imbalanced densities. 2. A uniform normal state withimbalanced densities. 3. A phase separated state containing bubbles of a superfluid
equal mixture, surrounded by a mixed normal state.
A uniform BCS state with imbalanced densities is not possible, as we have alreadyseen in the case of superconductors. Polarization of the BCS state could only happen
if the ”Zeeman field”
δµh=>∆(7.14)
2
Before h could ever equal the gap, however, the system already turns normal at√
h = ∆/ 2, the Clogston limit. The BCS state thus does not tolerate imbalanced
densities.
One might consider other possible states than the fully paired BCS-state, forwhich, for increasing mismatch between the two Fermi surfaces of the homogeneous
superfluid, pairing still occurs, but becomes less efficient with decreasing Fermi surface
overlap, and the gap decreases until superfluidity finally disappears. One candidate is
the Sarma (or breached pair) state [211, 151]. For fixed ”magnetic field” h, this stateis a maximum of the thermodynamic potential, hence unstable, but it was argued thatit might be stable in the case of fixed densities. For the BCS-case it turns out that thephase separated state containing superfluid bubbles of equal densities is energeticallyfavored compared to a uniform normal state of unequal densities and the Sarma phase,as was first pointed out by Bedaque, Caldas and Rupak [28]. In a certain parameter
regime, calculations indicate that the FFLO-state, where Cooper pairs have finite
momentum, should be the ground state of the system [94, 146, 169, 221, 157].

7.2.4

Critical number imbalance

In the case of weak interactions (∆ ¿ µ) and harmonic trapping, it is easy to derivea critical population imbalance for which there will be no superfluid region. The

160

condition is simply RSF = 0 or1
h = √ ∆(0)2
µ ¶7/32h1⇒= √e−π/2kF |a|µ2 e(7.15)(7.16)

where, as before, µ is the average Fermi energy of a non-interacting, equal Fermi
mixture, µ = EF . The numbers of particles in the normal state are given by the¡ µ↑,↓ ¢3
usual relation for a harmonic trap (see Eq. 2.40 in chapter 2), N↑,↓ = 61 ~ω
. We
thus find the critical population imbalance for weak interactions(1 + µh )3 − (1 − µh )3N↑ − N↓h
3 ∆(0)
δc ≡=≈3 = √hh33N↑ + N↓µ(1 + µ ) + (1 − µ )2 µ(7.17)

Population imbalance δ

1.0
0.8
0.6

Superfluid

Normal

0.4
0.2
0.0

2

1

0
1/kF a

-1

-2

N −N

Figure 7-3: Critical population imbalance δ = N↑↑ +N↓↓ for BCS-type superfluidity ina harmonic trap. The data-points give the experimental result [268]. The crossmarks the point on the BEC-side of the resonance where superfluidity occurs at any
imbalance smaller than 1. Using the Gorkov expression for the gap, this point lies at
1/kF a = 0.7, outside the applicability of our derivation. However, the point should
lie at 1/kF a > 0, probably around 1/kF a ≈ 1, where the chemical potential µ changes
from positive to negative values and BCS-type physics is replaced by condensation ofsmall molecules.

Fig. 7-3 shows the resulting phase diagram, along with the experimental results described in the next section. The calculation assumes the expression for h/µ, Eq. 7.16,
to still hold in the strongly interacting regime, which is discussed below. In the161weakly interacting regime, where h ¿ µ, we see that δc is proportional to the gap
and thus depends exponentially on the interaction strength 1/kF |a|. This presents apart from the exponentially low critical temperature TC - another reason why in the
beginnings it was not clear whether experiments with fermionic atoms were able toobserve superfluidity. It appeared that the two spin populations had to be essentiallyperfectly matched: For a Fermi energy of EF = 1 µK (1/kF ≈ 3800 a0 ) and a scattering length of a = −500 a0 , still ten times larger than ”typical” scattering lengths inalkali atoms, the critical population imbalance δc is already smaller than one out of
105 atoms! If experiments do not control the number difference to better than 1%,
they cannot explore superfluidity at interaction strengths smaller than kF |a| ≈ 0.35.
Obviously, the use of Feshbach resonances solved this problem.
Using several independent techniques described in the following sections, we measured the critical number imbalance as a function of interaction strength. The datafor one type of measurement (via the condensate fraction) is shown in Fig. 7-3. It issatisfying to see that the experimental data closely follows the expected trend for thecritical number imbalance.
Resonance
Note that in the above calculation of the critical number imbalance we have assumed
that the normal state is non-interacting. Also, the BCS-approach neglects Cooper
pairs at finite momenta as well as interactions between different Cooper pairs. In
addition, the simple formula Eq. 7.2 for the free energy density difference between
the normal and the superfluid state will be altered for large kF |a|. For these reasons,the results cannot be trusted in the strongly interacting regime. Nevertheless, it isreasonable that the qualitative behavior remains unchanged all the way to (1/kF a =0), in the same way the qualitative behavior of the BEC-BCS crossover model appears
to capture the essential physics of an equal mixture. Moreover, the expression for the
gap used in Eq. 7.16 above, due to Gorkov and Melik-Barkhudarov [98], gives for the
resonance case ∆ = (2/e)7/3 EF = 0.49EF , which agrees surprisingly well with theresult of Monte-Carlo calculations, ∆ = 0.50EF [44]. With all caveats in mind, wegive the critical population imbalance on resonance, using the expression for the gap
due to Gorkov and Melik-Barkhudarov
δc = 80%

at 1/kF a = 0(7.18)We have measured this critical imbalance on resonance (see below) and find δcexp =162

74(5)%. A recent Monte-Carlo study predicts δc = 77% [153]. Using the variational
BCS-approach, for which ∆ = 0.69EF on resonance, yields δc = 92%. However,
one should not expect the variational Ansatz to predict any parameter on resonancein a quantitative way. Fluctuations, such as those included by Gorkov and MelikBarkhudarov in the BCS-regime, seem to play an important role. This example gives
hope that the study of imbalanced Fermi mixtures will substantially improve ourunderstanding of strongly interacting fermions.
Both experiment and recent theory obtain δc < 1 on resonance. By continuity,
this implies that there is a region on the BEC-side of the BEC-BCS crossover wherecondensation cannot occur, even at zero temperature, in the presence of large spin
imbalance. This is surprising, since for a > 0, there is a molecular state available for
two atoms in vacuum, and one would expect these bosonic molecules to condense atzero temperature (essentially, bosons have no other choice at zero T ). A simple way
out of this dilemma is of course that the very presence of the large Fermi sea of spin
up atoms prohibits the formation of the molecular state. Essentially, the (two-body)molecular state requires a width k ∼ 1/a in k-space. Once a ≈ 1/kF in the stronglyinteracting regime close to resonance, the required momenta are Pauli-blocked by thepresence of spin-up atoms occupying k-space from k = 0 to k = kF ↑ > 1/a. Hence,
the molecular state is not available anymore.

7.3

Fermionic superfluidity with spin imbalance

In this experimental section, we will demonstrate superfluidity for imbalanced mixtures. The first experiment is a study of rotating fermionic mixtures. Vortices directly
demonstrate that superfluidity persists in trapped, imbalanced Fermi mixtures. The
second experiment determines the condensate fraction using the rapid ramp methodof chapter 5. Hereby we could show the robustness of the superfluid in the stronglyinteracting regime and directly demonstrate the Clogston limit of superfluidity. In
the first paragraph I describe how the population imbalance can be varied in our
experiment.

7.3.1

Creation of imbalanced Fermi mixtures

The experiments start at maximum optical trap depth with a fully polarized sample
in the lowest hyperfine state of lithium, |1i, and at a magnetic field above resonance(typically at 875 G). The energies of states |1i and |2i are split by about h × 76 MHz.
163

A radiofrequency sweep then transfers a certain fraction from state |1i into state2|2i according to the Landau-Zener law P = 1 − e−2πΩR /ω̇ , where ~ΩR is the Rabifrequency of the transition, tunable via the power of the radiofrequency drive, and ω̇
is the sweep rate at which the transition is crossed. We typically keep ~ΩR fixed andvary ω̇. After the sweep, all fermions are still polarized in a coherent superposition
state of |1i and |2i and need several tens of milliseconds to decohere, as discussed inchapter 4 and in [106]. As the sample has decohered, evaporative cooling can proceed.
Starting with an initial temperature of about T /TF = 0.3, we reach T /TF = 0.05.
EKin [nK]

Population imbalance

Atom number [x 106]

Atom number [x 106]

175
150
20
16
-1.0

-0.5

0.0
0.5
Population imbalance

1.0

20
15
10
5

0.5
0.0
-0.5
0.0

0.2
0.4
0.6
0.8
Landau-Zener probability

1.0

Figure 7-4: Creation of imbalanced Fermi mixtures. A faster-than-adiabatic Landau
Zener RF sweep is used to transfer a fraction of atoms from state |1i into state |2i.
For a fixed final trap depth at the end of evaporation, the temperature varies only
slightly with imbalance.

During evaporation, mostly majority atoms escape from the trap, as their Fermi
energy is closer to the trap depth. Thus, evaporation will always reduce the imbalancein the cloud. Fig. 7-4 displays the final atom number as a function of the LandauZener probability. The dependence of the total atom number and temperature on the
population imbalance is fairly small. It could be further minimized by adjusting thetrap depth at the end of evaporation as a function of imbalance.

164

Figure 7-5: Superfluidity in an imbalanced fermion mixture. For the 812 G data, thepopulation imbalance was (from left to right) 100%, 90%, 80%, 62%, 28%, 18%, 10%
and 0%. For the 853 G data, the mismatch was 100%, 74%, 58%, 48%, 32%, 16%,
7% and 0%.

7.3.2

Rotating imbalanced mixtures: Vortices

After a strongly interacting, imbalanced spin mixture has been created, the cloud is
set in rotation, equilibrates and is then imaged as described in chapter 6. Fig. 7-5
shows the resulting images, and Fig. 7-6 the number of vortices as a function of imbalance and interaction strength. Vortices are observed for a wide range of imbalances.
Clearly, superfluidity in the strongly interacting Fermi gas is not constrained to a narrow region around the perfectly balanced spin-mixture, but instead superfluid flow
is observed for large population asymmetries. Gradually, for decreasing interaction
strength 1/kF a, the superfluid window closes around the optimal situation of an equal
mixture. This is what we expect from the discussion of the Clogston limit above.

165

Number of Vortices

40
30
20
10
0
-0.8

-0.4
0.0
0.4
Population Imbalance

0.8

Figure 7-6: Number of vortices obtained in an imbalanced Fermi mixture. The curves,from top to bottom, correspond to decreasing interaction strength 1/kF a = 0.2 (BECside), -0.15, -0.3, -0.4, and -0.5 (BCS-side of the Feshbach resonance).

166

Figure 7-7: Fermion pair condensates and the Clogston limit of superfluidity in an
imbalanced Fermi mixture. For high population imbalance (δ > δc ), the gas is anormal mixture (left). At δ . δc , a condensate emerges in the clouds’ center, which
grows as the imbalance is further reduced (right). The images are obtained using the
rapid ramp method, starting from resonance.

7.3.3

Condensate fraction

Vortices are a direct indicator for superfluid flow. However, close to the limit ofsuperfluidity, the large normal cloud should quickly damp the rotation of the superfluid [107, 1, 265]. Therefore, the detection of vortices can only provide a lower
bound for the size of the superfluid window. The methods of chapter 5 can be used
to directly detect whether or not a condensate was present in the imbalanced clouds.
A rapid ramp to the BEC-side, directly after release from the optical trap, stabilizes
the fermion pairs and transforms the initial pair condensate on the BCS-side into a
molecular condensate (see Fig. 7-7). Using this method has allowed to map out theregime of pair condensation as a function of population imbalance and demonstrate
the Clogston limit of superfluidity.
Fig. 7-8a) shows the condensate fraction as a function of population imbalancefor varying interaction strength. This directly displays the strong dependence ofthe superfluid window on 1/kF a. At a given interaction strength, there is a sharp
threshold behavior at a certain critical imbalance, below which condensates are found.
The critical imbalance determined from such data is shown in Fig. 7-3. On the BCSside, it should closely follow the value of the superfluid gap ∆. The qualitative
agreement with this expectation is excellent.
Fig. 7-8b) contrasts this quite dramatic dependence of the critical imbalance on the167

Varying interaction strength
0.4

Condensate Fractionb)
B, 1/kFa
820 G, 0.11
833 G, 0 (Res.)880 G, – 0.27
920 G, – 0.44

0.3
0.2
0.1
0.0
-1.0

-0.5
0.0
0.5
Population imbalance

Varying temperature
0.4

Condensate Fraction

a)

0.3

Ekin
150 nK
175 nK
195 nK
210 nK
250 nK

0.2
0.1
0.0
-1.0

1.0

833 G
1/kFa = 0

-0.5
0.0
0.5
Population Imbalance

1.0

Figure 7-8: Superfluid window of imbalanced Fermi mixtures. The condensate fraction is shown as a function of population imbalance for a) varying interaction strength
at the lowest temperature, and b) for varying temperature on resonance.

interaction strength with its weak dependence on temperature. While increasing thetemperature reduces the condensate fraction substantially, it only affects the criticalimbalance close to TC . This is reminiscent of the weak dependence of the superfluid
gap on temperature5 .
From Fig. 7-8a) we recognize a dramatic consequence of population imbalance.
There is no longer a smooth crossover from the BEC- to the BCS-limit in the imbalanced case, in contrast to the case of equal mixtures studied in previous chapters.
Rather, a phase transition occurs as the interaction strength is reduced below a critical value that depends on the interaction strength. In principle, it could well be that
more than one phase transition occurs around the critical interaction, from a BCStype superfluid to several forms of FFLO-type states, until finally the normal stateis reached. If such intermediate superfluid states exist, they leave - at our present
resolution - no distinct feature in the density profiles of the gas.

5

BCS-gap ∆(T ) ≈ 1.74 ∆0√ For the −∆2π∆0 kB T e 0 /kB T for T ¿ TC [89].

p

1 − T /TC for TC − T ¿ TC and ∆(T ) ≈ ∆0 −

168

7.4

Direct observation of the phase transition, imbalanced case

The study of the condensate fraction has allowed us to establish the superfluid-tonormal phase diagram of Fig. 7-3. One of the most amazing results of the analysis
that led to this phase diagram is clearly the prediction that the superfluid appears tophase separate from the normal cloud (see Fig. 7-2). This suggests that in striking
contrast to the case of an equal mixture, one should be able to directly observe thesuperfluid phase transition in the density profile of the gas, without the need of poorly
understood magnetic field ramps.

7.4.1

On resonance: Universality

We have indeed found this idea to be correct. Fig. 7-9 shows density profiles of anunequal mixture on resonance at an imbalance of about 60%. As the sample wascooled below a certain temperature, a central dense core appeared in the minority
cloud. Even at our coldest temperatures, this core was observed only below a criticalimbalance δc = 74(5)%, the same value that we have found before using the rapid
ramp technique. It is thus natural to identify the central core with the fermion paircondensate.
A different imaging technique, a variant of phase-contrast imaging (explained
below in section 7.5) was used to directly image the density difference as a function oftemperature. According to our expectation of Fig. 7-2c), the density difference should
rapidly drop to zero when crossing the boundary between the normal mixture andthe superfluid core. Indeed, as we cool down the sample, we observe the appearanceof a striking central depletion in the difference profiles, see Fig. 7-10. Naturally, thisdepletion emerges at the same temperature at which we observe the central dense corein the minority cloud (Fig. 7-9). In section 7.5 we will study whether the depletionis complete, that is, whether the central densities are indeed equal in the superfluid
regime.
Apart from providing us with a direct method to detect the condensate, imbalanced mixtures also enable us to measure temperatures. A substance can only
function as a reliable thermometer if it does not strongly interact with the sample tobe measured. This explains the difficulty of assigning temperatures to the stronglyinteracting equal mixtures of chapter 5. In the case of imbalanced mixtures, atoms
in the outer wings of the majority cloud are not in contact with minority atoms (see
169

a

b

|1

c

>

>

Optical density [a.u.]

|2

-1.0

-0.5

0.0
Radius [mm]0.5

1.0 -1.0-0.5

0.0
Radius [mm]0.5

1.0 -1.0

-0.5

0.0

0.5

1.0

Radius [mm]Figure 7-9: Direct observation of the superfluid phase transition in a resonantlyinteracting, imbalanced Fermi mixture. In contrast to the case of equal populations,the fermion pair condensate in an unequal mixture is clearly visible in the density
profiles. This should be compared to the density profiles of an equal mixture, Fig. 511. The upper row shows majority clouds, the lower row minority clouds, for an
imbalance of δ = 60%. All images were obtained after 11 ms of expansion. The
dashed line is a fit to the wings of the minority cloud to a Thomas-Fermi profile,
clearly missing the central feature. Temperature was varied by lowering the trapping
power. It can be obtained from the ballistically expanding wings of the majority
cloud. We have T /TF = 0.14 (a), 0.09 (b) and 0.06 (c). Here, kB TF is the Fermi
temperature of an equal mixture containing the same total atom number.

170

Figure 7-10: Direct observation of the superfluid phase transition via in-situ phase
contrast imaging. At a critical temperature, a central depletion appears in the difference density, corresponding to the appearance of the condensate as observed inFig. 7-9.

Fig. 7-9). They expand ballistically from our trap and can thus be used as a reliable
thermometer. This has allowed us to measure the critical temperature for a given
population imbalance (see Fig. 7-11). Future experiments can obtain the criticaltemperature as a function of imbalance and might thus - by extrapolation - lead to areliable value even for the case of an equal mixture.
A mixed, unbalanced region on resonanceFig. 7-9 shows that there exists a mixed region outside the condensate, where minority
and majority spins coexist. This is a necessary consequence of the existence of thecritical imbalance δc on resonance. Starting with a normal mixture at δc , the gas will
not suddenly fully condense for slightly lower imbalance. This normal mixture aboveδc (or beyond the radius of the condensate at RSF ) is an intriguing state of matter: It isstrongly (resonantly!) interacting, yet normal (as far as we know from the observationof condensates and vortices). Moreover, we find that the minority spin state in thismixed region is very well fit with a Thomas-Fermi profile of a non-interacting Fermi
gas. Only the cloud radius shows that this is indeed a strongly interacting gas, as
it is found to be smaller than that of an equivalent non-interacting cloud (we findR↓ /RTF↓ ≈ 0.75 at δ = δc ). This might indicate that the strongly interacting Fermi
gas behaves like a collection of non-interacting fermions with an effective mass. This171δ [%]

65
55
10
6
4
2

2

χ [arb. units]

2

1
0.08 0.10 0.12 0.14 0.16 0.18
T / TF

Figure 7-11: Critical temperature for an imbalanced Fermi mixture on resonance. The
quality of a Thomas-Fermi fit, expressed as its χ2 -value, is plotted as a function of
the reduced temperature T /TF as determined from the majority cloud’s ballistically
expanding wings. A threshold fit (black line) gives the critical temperature for δ =59% as TC /TF = 0.12.

is the essence of Landau’s Fermi liquid theory and the reason why other than pairing
interactions between fermions can be neglected in BCS theory [215]. Here, the onlyrole of interactions is to ”dress” the ”bare” fermions with an effective mass, the result
being a reformulation of the interacting system in terms of weakly interacting quasiparticles. It is an important goal of future experiments to understand whether thisidea still works in the strongly interacting regime on resonance.
Here, we will give an elegant argument independently found by F. Chevy [52] andBulgac and Forbes [40] that shows how the different radii of the superfluid, the mixed
region and the fully polarized cloud constrain the conditions for being in either ofthese three phases.
At unitarity, there are only two energy scales available, the chemical potentialsµµ↑ and µ↓ for the two species. Their ratio η = µ↑↓ determines in which of the possible
states the Fermi mixture is in. Clearly, for η = 1 we are in the superfluid state. As
we reduce η, we will, according to our results, leave the superfluid state and enter
the normal mixed state, say at a critical ηa . Then, upon further reduction of η, thesystem will eventually switch into the fully polarized state, say at ηb . The following
considerations allow us to constrain these parameters.
Bounds for ηa and ηb
We start with the same analysis as for the Clogston limit in the weakly interacting
case: We need to minimize the free energy density F of the two-species mixture, thatis, find the state with maximum pressure P = −F. In the fully polarized state, the172pressure is (see Eq. 7.6)

4
5/2
ρ(1) µ↑(7.19)
15
Due to universality on resonance, the chemical potential µ of a fully paired superfluid~2state must be proportional to the Fermi energy, µ = ξEF = ξ 2m
(6π 2 n)2/3 . From∂²µ = ∂nit follows that the energy density in the superfluid state has the same formPN =as that of a normal, two-component Fermi gas: ²(µ↑ , µ↓ ) = 53 µ↑ n↑ + 35 µ↓ n↓ = 56 µ n,
µ +µwith n = n↑ = n↓ and6 µ = ↑ 2 ↓ . The pressure PS ≡ µ↑ n↑ + µ↓ n↓ − ²(µ↑ , µ↓ ) = 54 µn
in the superfluid state is then
PS =

4
1
ρ(1)
(µ↑ + µ↓ )5/2
15(2ξ)3/2(7.20)

If there was no intermediate mixed phase, we would have a phase transition fromthe superfluid to the fully polarized normal state when their pressures become equal:
PN = PS

(7.21)⇒ µ↓ = ((2ξ)3/5 − 1) µ↑
The universal parameter ξ has been measured in experiments on equal mixtures[174, 26, 37, 139, 187], most simply by determining the cloud size, and calculated forexample via Monte-Carlo calculations [43, 44, 17]. Our cloud size measurements yieldξ = 0.47 (see section 5.3), in close agreement with other determinations. This meansthat the superfluid-to-fully polarized normal transition would occur at ηc = −0.04.
Since we observe the intermediate phase before the system becomes fully polarized,
we must have ηa > ηc .
An upper bound for ηb , the transition from the mixed to the fully polarized region,
can be obtained as follows. At ηb , we essentially have the situation of only one minority
atom experiencing the interaction with the cloud of N↑ majority atoms. In the BCSregime, this interaction would simply be the (attractive) mean-field-term 4π~2 an↑ /m.
On resonance it will be a universal number times the only energy scale left in thissituation, µ↑ . We naturally expect this interaction to be still attractive, corresponding
to some sort of ”binding” of the minority particle to the majority Fermi sea. Two
independent variational approaches by Chevy [52] and Bulgac and Forbes [40] result
6

Note that this only depends on the average chemical potential µ, because the superfluid is notsensitive to the difference δµ: The magnitization M = −∂²/∂δµ = 0 is zero in the BCS state.

173

Figure 7-12: Constrained radii of the superfluid (Rα ), the mixed (Rβ ) and the fullypolarized phase (R1 ) on resonance. The departure for larger minority clouds is likely
due to the fact that for small population imbalance the superfluid did not have avery distinct boundary, which increased the uncertainty in determining Rα . Figure
extracted from [52].

in the upper bounds for the ”binding” energy µb↓ :(µb↓ ≤−0.44 µb↑ , (Chevy)−0.50 µb↑ , (Bulgac and Forbes)(7.22)

Therefore we have ηb ≤ −0.50.
Constraint from the measured cloud radii
In the case of a harmonic trap, the chemical potentials vary as given in Eq. 7.11. The
transition from the superfluid to the mixed normal region will occur when2µ0↓ − 12 mω 2 RSF= ηa2µ0↑ − 12 mω 2 RSF(7.23)

defining the radius of the superfluid RSF . The transition from the mixed region to thefully polarized normal one occurs at a radius R↓ , where the minority cloud vanishes:µ0↓ − 12 mω 2 R↓2= ηbµ0↑ − 12 mω 2 R↓2(7.24)

Finally, one can note that the chemical potential µ0↑ of the majority spin state issimply given by the majority cloud radius, µ0↑ = 21 mω 2 R↑2 . These equations allow us
174to eliminate both µ0↑ and µ0↓ and we obtain an equation for the radius of the superfluid
core [52, 40]:
s
RSF
=
R↑

(R↓ /R↑ )2 − q
1−q(7.25)a −ηbwhere q = η1−ηis the value of (R↓ /R↑ )2 at which the superfluid vanishes.
b
The comparison of Eq. 7.25 to our data [269] is reproduced from [52] in Fig. 7-

12. Our data results in q = 0.32, from which Chevy deduces the following stringent
bounds on ηa and ηb :−0.04 < ηa < 0.02

(7.26)−0.55 < ηb < −0.50(7.27)The parameter ηa has a meaningful interpretation: It is the ratio of the density discontinuities in the majority and minority clouds at the superfluid-to-normaltransition7 ,
∆n↑
= −ηa
(7.28)∆n↓
While the minority cloud clearly shows a sharp change in its density at the phasetransition, the majority profiles do not feature a dramatic jump (see Fig. 7-9). We
were able to see faint traces of the condensate in the majority density (see supplemental material in [269], reprinted in appendix I), but the smallness of the effect certainly
speaks for a small value of ηa .
One should remark that the variational BCS approach (not including the correction due to Gorkov-Melikbarkhudarov) would predict a considerably smaller intermediate phase. The critical density imbalance beyond which no superfluid region
persists in a uniform system would be ≈ 93%, corresponding to ηa ≈ 0.1, as one cancalculate using the results for the Clogston limit above. The energy needed to add aminority atom to a majority Fermi sea would be identically zero, ηb = 0, as interactions between atoms that do not form a Cooper pair are neglected. In contrast, theupper bounds found by Chevy and Bulgac and Forbes show that the minority atomis ”bound” by an energy that is at least one half of the majority cloud’s chemical
potential. This nicely confirms the intuitive picture that the large spin up Fermi sea
presents an attractive ”mean-field” potential to a single minority atom.
7

5/2

Due to universality, the pressure in both phases can be written as P = f (η)µ↑ . The densities
µ

5/2

∂P
are given by n↑ = ∂µ
= f 0 (η)(− µ2↓ ) µ↑
↑
↑

5/2

∂P
+ 52 µP↑ and n↓ = ∂µ
= f 0 (η)( µ1↑ ) µ↑ . The pressure and↓µ

the chemical potentials are continuous across the transition, so ∆n↑ = − µ↓↑ ∆n↓ .

175

Concluding this section, we see how the simple measurement of cloud radii inthe strongly interacting regime has allowed to extract viable information about theequation of state of the system. The general equation of state for the two-species
mixture will read²(µ↑ , µ↓ ) = f (η)EF(7.29)
with an unknown universal function f (η) that, hopefully, future studies will be able
to determine.

7.4.2

On the BEC side: strongly interacting Bose-Fermi mixture

On the BEC-side, at 1/kF a = 0.5, we find several features that are qualitatively
different from the resonance case.
1. At the lowest temperatures, the minority atoms are now essentially all part ofthe condensate. That is, there is practically no mixed region.
2. The majority spin species now also shows a dramatic signature of condensation.
A closer look on the resonance profiles reveals, however, that this latter distinction is
only quantitative: There is indeed a very faint trace of the condensate in the majority
distribution (see supplementary information in [269], reprinted in Appendix I).
Fig. 7-14 displays the condensate fraction as a function of the reduced temperature T /TC , where T is determined as before from the non-interacting wings of the
majority component, and TC is the critical temperature of a non-interacting cloudof N↓ molecules. We see that interactions strongly reduce the critical temperaturefrom the non-interacting value. The downshift by a factor of two is in accord with amean-field model of condensates in the strongly interacting regime [35]. The cloud ofexcess atoms will act to further reduce the critical temperature.
Mean-field model
The situation on the BEC-side of the resonance resembles that of a strongly interacting mixture of bosons (the molecules) and fermions (the unpaired atoms of the
majority species). Indeed, Pieri and Strinati [196] have shown that on the far BECside of the Feshbach resonance, where kF a . 1, this view is indeed correct. Whatemerges is a mean-field description in which the condensed bosons repel the unpaired

176

a

b

c

>

|1

>

Optical density [a.u.]

|2

-1.0

-0.5
0.0
0.5
Radius [mm]

1.0

-1.0

-0.5

0.0
0.5
Radius [mm]

1.0

-0.5

0.0
Radius [mm]0.5

Figure 7-13: Phase transition in a strongly interacting atom-molecule mixture, at
B = 780 G or 1/kF a ≈ 0.5. Unlike on resonance, all minority particles are part of thecondensate at low temperatures: there is essentially no mixed normal region. The
reduced temperatures were T /TF = 0.2 (a), T /TF = 0.12 (b) and T /TF ≤ 0.05 (c).
The noise in the center is an artefact of azimuthal averaging: the number of averaged
pixels evidently increases with distance from the center.

177

Imbalance
1/6N↓ a / ah.o.

1.0
0.6
0.2
1.2
1.0
0.8

Condensate Fraction

1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.1

0.2
0.3
0.4
TMajority / TC, Minority

0.5

0.6

Figure 7-14: Condensate fraction of the molecular condensate (at B = 780 G) as afunction of temperature. Strong interactions reduce the condensation temperature
by a factor of two compared to the non-interacting value. The critical temperature is
1/6
T /TF = 0.18 for an imbalance of δ = 75%. N↓ a/ah.o. is the gas parameter giving theinteraction strength. In a weakly interacting atomic BEC, this number is typically on
the order of 10−2 . The condensate fraction was determined using a simple bimodal
fit, a gaussian plus a parabola (solid circles) or a fit that assumes full expulsion of
the thermal cloud by the condensate (open circles).

fermions. For increasing interaction strength, the fermions are essentially fully expelled from the condensate. This form of phase separation has also been theoreticallystudied in ”true” dilute Bose-Fermi mixtures, both in the trapped system [170] andin the bulk [253], and it certainly reminds of the phase separation between 3 He and4
He mixtures [99] (only 6% of 3 He are soluble in 4 He at low temperatures).
We can quite easily ”guess” the mean-field equations for this mixture, and refer the reader to the full derivation given in [196]. To start, we need to know therelevant scattering lengths for the system. The molecule-molecule scattering length
was calculated by Petrov, Shlyapnikov and Salomon [193] by solving the four-bodyproblem exactly. The result is aBB = 0.6a, where a is the atom-atom scattering
length. Thus, the molecules repel each other, as we had found in chapter 2 within the
BCS variational approach. This, however, predicted the larger value8 aBB = 2a. The
8
The reason is that the BCS variational approach neglects interactions between pairs, and threebody interactions between a fermion and a molecule.

178fermion-boson scattering length aBF = 1.18a was calculated almost half a century
ago [228] but has so far not been verified experimentally.
At low temperature, all minority fermions will be paired with majority ones,
and the resulting molecules will practically all be condensed. We thus have only
two contributions, the molecular condensate ψ(r) and the density of excess fermionsδn(r). The excess atoms experience the molecules as an additional external potential∝ aBF |ψ(r)|2 , so we have (see Eq. 2.28 in chapter 2)µ¶3π~2 aBF21(µ↑ −Vext (r)−|ψ(r)|)/kTBm
δn(r) = − 3 PolyLog 3/2, −eλdB(7.30)

where µ↑ is the chemical potential of the majority atoms. The molecules, in turn, feelthe mean-field repulsion from the excess fermions, in addition to their own repulsive
mean-field. The Gross-Pitaevskii equation for ψ(r) thus reads
µ
¶
~2 ∇2
3π~2 aBF
4π~2 aBB−
ψ(r) + 2Vext (r) +δn(r) ψ(r) +|ψ(r)|2 ψ(r) = µM ψ(r)2mM
mmM
(7.31)
with the mass mM = 2m of molecules. The factor 2 in front of Vext simply signifies
that each molecule experiences twice the potential felt by a single atom. The chemical
potential for molecules µM in an equal mixture is µM = 2µ + EB with EB = ~2 /ma2
the molecular binding energy (see chapter 2). In the case of unequal mixtures, chemical equilibrium in the atom-molecule mixture requires µM = µ↑ + µ↓ + EB . The setof equations is closed by introducing the total density n(r) = 2 |ψ(r)|2 + δn(r) andrequiringZ
N↑ + N↓ =d3 r n(r)(7.32)

d3 r δn(r)(7.33)

ZN↑ − N↓ =with the given number N↑ and N↓ of majority and minority atoms.
We can proceed further using the Thomas-Fermi approximation, neglecting the
kinetic term in the Gross-Pitaevskii-equation. Also, we now work at zero temperature.
The equations for ψ(r) and δn(r) becomeµ2|ψ(r)| =δn(r) =m2π~2 aBB1
6π 2¡ 2m ¢3/2~2¶
3π~2 aBFmax µM − 2Vext (r) −δn(r), 0mµ¶3/2
3π~2 aBF2
max µ↑ − Vext (r) −|ψ(r)| , 0m
179(7.34)(7.35)

Energy [µK]

2 a)

2

b)

2

µ↑
1

1

0

0

1

-3

cm ]

Veff (r)

0
10

20

20

nC(r)

12

Density [10

c)

5

10

10

δn(r)
0

0

0
100

0
Radius [µm]

100

100

0
Radius [µm]

100

100

0
Radius [µm]100

Figure 7-15: Effective potential and density of excess fermions in a Bose-Fermi
mixture. The upper row shows the self-consistent effective potential Veff (r) =Vext (r) + gBF |ψ(r)|2 experienced by excess fermions in the presence of a molecular condensate with wavefunction ψ(r). The dotted line gives the chemical potentialµ↑ of the majority spin species. The lower row shows the density of excess fermionsδn(r) and the condensate density nC (r) = |ψ(r)|2 . The population imbalance δ wasa) 20%, b) 70% and c) 99%, and the interaction strength 1/kF a = 2. Calculations
are for a spherical trap with trapping frequency ω = 2π 100 Hz and a total number of
107 fermions.

These equations allow us to study several cases (see Fig. 7-15).
Small imbalance
If the excess density is small, we can neglect the influenceof the few excess atoms on the molecular condensate. The excess atoms, in turn,experience an effective ”Mexican hat potential”, the trapping potential plus the meanfield interaction from the condensate (see Fig. 7-15a):
Veff = Vext (r) +gBFmax(µM − 2Vext (r), 0)gBB(7.36)

with the couplings gBB = 2π~2 aBB /m and gBF = 3π~2 aBF /m. The inverted potentialfrom the condensate mean-field on the right always dominates the ”bare” potential in
the center, as 2gBF /gBB = 3aBF /aBB ≈ 6 > 1. This shows that the excess atoms will,
for small imbalance, always reside in the wings of the condensate, never in the center.
As the imbalance or the strength of the interactions are increased, the mean-field
repulsion from the excess atoms starts to affect the condensate. Still, the effective
potential retains its qualitative shape (see Fig. 7-15). Thus we arrive at the fact that180

in the trapped case, there will always be an extended region in the phase diagram
where an ”unmagnetized” superfluid at equal spin densities is spatially separated froma mixed region.
A ”magnetized superfluid” As the population imbalance is increased, or theinteraction strength decreased, more and more excess atoms enter the condensate
from the edge inwards. At the critical point where the chemical potential of majority
atoms becomes large enough to overcome the condensate’s mean-field repulsion at
BF
the trap center, µ↑ = ggBB
µM , excess particles can penetrate all the way into thecondensate (see Fig. 7-15b). The superfluid phase is ”magnetized” throughout. In
the language of the BCS-state, now the cost of having unpaired atoms inside thesuperfluid is outweighed by the gain in ”Zeeman energy” 2h = µ↑ − µ↓ of changing
a spin down (minority) into a spin up (majority) particle. Recall from chapter 2pthat the role of the excitation gap on the BEC-side is played by µ2 + ∆2 . Thepfact that polarization of the superfluid is favorable means h >µ2 + ∆2 . This
is indeed possible on the far BEC-side, as there is no Clogston limit anymore (seep
BF
µM in the far BEC-limit (with theFig. 7-3). One can note that µ2 + ∆2 → |µ| + ggBB
variational BCS-Ansatz, that leads however to the ”wrong” values of gBF and gBB ),and we recover the above criterion for µ↑ .
Strong interactions: complete phase separation For strong interactions, on
the other hand, the condensate expels essentially all excess fermions. Phase separation
between the superfluid and the normal region is thus complete and occurs for all
imbalances. The situation now closely resembles our expectations for the BCS-side.
Fig. 7-16 summarizes these findings by plotting the central density of excess fermions δn(0) as a function of imbalance and the interaction parameter 1/kF a. The
equations Eqs. 7.34 and 7.35 were solved by iterative insertion of one density profile
into the equation for the other, and adjusting at each step the value of the chemical
potentials µM and µ↑ to converge to the required particle number.
As pointed out by [196], the details of the density profiles - the maxima in thedensity of excess fermions, the critical imbalance / interaction where phase separation
starts, etc. - depend on the value of the atom-molecule scattering length. Accurate
measurements of the density profiles will allow to verify the 50 year old prediction
aBF = 1.18.

181

cm ]

-3

10
0
0

50
100
Radius [ µm]

0

50
100
Radius [ µm]

0

50
100
Radius [ µm]-3Central excess density δn(0) [cm ]Density [10

12

20

100
12

60
3
40

2
20

1
0

0
5

4

3
1/kFa

2

1

Population Imbalance δ [%]

80
4x10

Figure 7-16: Phase diagram for the strongly interacting Bose-Fermi mixture in aharmonic trap at zero temperature. Shown is the density of excess fermions in thecenter of the trap. Excess fermions can enter the condensate either for large imbalanceor fixed (non-zero) imbalance and weak interactions, above the critical line shown inblack. Near a critical interaction around 1/kF a = 1, excess fermions are expelledfrom the condensate’s center at all imbalances. The profiles on top show the casefor δ = 80% at varying interaction parameters, 1/kF a = 3 (left), 2 (middle) and 1(right). Other parameters as in 7-15.

7.5

Observation of phase separation

In previous sections we have seen that both on the BCS, and on the BEC-side of theresonance, we would expect to observe phase separation in the sample. We define aphase separated state by the presence of a region in space where the spin densities
have equal densities, and which is surrounded by a shell at unequal densities. Hints ofphase separation were found by subtracting the column density profiles in Fig. 7-9 andfinding a dip in the center of the difference distribution. Similar central depletions
have been found in [187]. However, to distinguish a phase-separated state from amerely strongly interacting normal state that would show similar depletions [267], itis necessary to reconstruct the 3D density difference. In principle this can be achievedby subtracting the absorption pictures of the two spin species, and inverting theresulting profiles [267]. But forming the difference between two large and fluctuating182

quantities leads to large fluctuations in the result. A method was needed to directly
image the density difference. Phase contrast imaging provides such a method, as is
discussed in the following.

7.5.1

Experimental method: Phase-contrast imaging

Figure 7-17: Phase contrast imaging for the observation of phase separation in an
imbalanced Fermi mixture. a) The probe beam is tuned right in between the two
resonance frequencies for atoms in states |1i and |2i. The resulting phase shift isthus proportional to the difference in column densities. b) Phase contrast image ofatoms in state |1i (left) and |2i (right) and that of an equal mixture of the two states(middle).
In order to directly measure the density difference distribution, we employ a variant of phase contrast imaging (see Fig. 7-17 and [232]). Light that is off-resonantwith an atomic transition (detuning |δ| much larger than the linewidth Γ) experiR
ences a phase shift φ(x, y) ∝ 1δ dz n(x, y, z) when passing through the atomic cloudwith density n(x, y, z). Atoms in states |1i and |2i have imaging resonances that areabout 80 MHz apart. Thus, if a laser is detuned right in between those two imaging
resonances, the phase shift due to atoms in state |1i will have the opposite sign ofthat due to atoms in |2i. The total phase shift will be proportional to the differenceof the column densities.
The phase information can be transformed into an intensity information by astandard homodyning technique. If Ei is electric field of the collimated incoming light,
and Ef is the field after passing through the atoms, then Es = Ef − Ei = Ei ( eiφ − 1)is the scattered light from the atoms (we neglect absorption in our description as|δ| À Γ). The scattered light is imaged onto a CCD camera, where it interferes withthe unscattered light Ei . This would give an intensity propotional to |Ei + Es |2 ∝|1 + iφ|2 ∝ 1 + φ2 /2 for small phase shifts φ. The dependence is quadratic and thus
183

weak. To increase the sensitivity to φ, the collimated light that misses the atoms
is phase shifted by π/2 using a phase spot in the focus of a lens. The scattered,
phase-shifted light from the atoms in the object plane will miss the phase spot. The¯¯2intensity on the camera is now proportional to ¯eiπ/2 + iφ¯ ∝ |1 + φ|2 ∝ 1 + 2φ. The
signal is thus linear in φ and thus sensitive to small phase shifts.

7.5.2

In-situ profiles of the column density difference

Figure 7-18: In situ imaging of an imbalanced Fermi mixture for various population
imbalances. The profiles directly measure the column density difference of the two
spin states. Below an imbalance of δ ≤ 75%, a distinctive central depletion is observed
and a shell structure emerges.
This new experimental tool has allowed us to directly probe the density difference
in imbalanced Fermi mixtures in-situ (in the trap) (see Fig. 7-18). We observed how
the depleted core region diminished for increasing imbalance and eventually vanishedcompletely around an imbalance δ ≈ 75%.
The 3D reconstruction of the density difference finally revealed that at low temperatures, the central depletion in the density difference actually results from a central
core at equal densities, surrounded by a shell at unequal densities (see Fig. 7-19).
The reconstruction is performed using an inverse Abel transformation. The only assumption entering this analysis is that of cylindrical symmetry of our trap, which weknow from chapter 6 to be very well fulfilled. This allows us to check two further
assumptions that are often made, namely the local density approximation (LDA) andthe approximation of harmonic trapping [226, 125].
In local density approximation, the densities n↑,↓ (r, z) and their difference canonly depend on the value of the local potential V (r, z) = 21 mωr2 r2 + 12 mωz2 z 2 . That is,184

Figure 7-19: Reconstruction of 3D density profiles. Double integration results in a flat
difference density profile, as expected from LDA when integrating over a hollow shell.
The 3D reconstruction of the difference density profile shows an extended region ofequal densities surrounded by a shell with unequal densities.

they are functions of λ2 r2 + z 2 only, where λ = ωr /ωz is the aspect ratio of the trap.
The border of the region of equal densities must likewise be defined by a certain value
2of the potential V , hence constant λ2 r2 + z 2 ≡ Rcore. Let us now calculate the doubly
integrated difference profileZ∆nax (z) =πdx dy ∆n(λ (x +y )+z ) = 2
λ
2

2

2

2

Z ∞
0

π
d(ρ ) ∆n(ρ +z ) = 2
λ
2

2

2

Z ∞
du ∆n(u)z2(7.37)

2
Now if |z| < Rcore , then ∆n(u) is zero from u = z 2 up to u = Rcore. The doubly
integrated density is thus constant:π∆nax (|z| < Rcore ) = 2
λZ ∞du ∆n(u)(7.38)

2
Rcore

Indeed, when performing the double integration, we observed such a characteristic ”flat top” distribution, indicating that LDA and harmonic trapping are good
approximations.

7.5.3

Two regimes of phase separation

For the range of interaction parameters studied in the experiment [223] (−0.4 ≤1/kF a ≤ 0.6), the observation of a central core with equal densities was correlatedwith the existence of a condensate after the rapid ramp to the BEC-side (see Fig. 7185

Figure 7-20: Correlating the condensate fraction with the observation of phase separation. Whenever a condensate is found after the rapid ramp, the in-situ densities of
the two species are equal.

20). This shows that the physics on both sides of (but still close to) the Feshbachresonance is qualitatively the same, which does not surprise given the experience withequal mixtures.
However, there is an important qualitative difference between the BEC- and the
BCS-regimes. On the BCS-side, it is always energetically favorable to place a new
majority atom not inside the superfluid, but on top of the Fermi sea of excess atoms
in the phase separated normal state. To reverse this situation, the ”Zeeman” energy
h would have to overcome the pairing gap ∆, but the Clogston limit occurs already√before this can happen, at h = ∆/ 2. In the far-BEC-regime, on the other hand,
we are dealing with a Bose-Fermi mixture of tightly bound molecules and unpaired
atoms. Excess fermions can enter the molecular condensate without destroying it.
While for small imbalance, the central region of the trap will still contain a superfluid
at equal spin densities, at a certain imbalance excess atoms start to fill in the central
region. This means that there will be a ”magnetized superfluid” on the BEC-side of
the imbalance phase diagram (see Fig. 7-21).

186

Figure 7-21: Phase diagram for trapped, strongly interacting, imbalanced Fermi mixtures. This shows the combined diagrams of Figs. 7-3 and 7-16, along with the dataof [268] (empty circles) and [223] (full circles). The normal state density differencein the BCS-regime was calculated for a non-interacting mixture. This central density
difference should sharply rise from zero on the BCS-side as the system turns normal.
In contrast, on the BEC-side the density difference increases smoothly from zero,starting at some critical imbalance. The superfluid is not destroyed, and one obtainsa ”magnetized superfluid”.

7.6

OutlookThe study of imbalanced Fermi mixtures has revealed a host of interesting effects.
Their quantitative explanation presents a challenge to many-body theories, especiallyin the strongly interacting regime. Varying the spin imbalance led to new insight into
the strongly interacting normal state above the critical imbalance. The critical polarization is itself an important number. It provides a measure for the stability of thesuperfluid state at equal densities, when compared to the strongly interacting normal
state. On resonance, it is a universal number, of the same importance as the gap
or the chemical potential of an equal mixture in units of the Fermi energy. In the
BCS-limit, the critical polarization is linearly related to the pairing gap, with a universal coefficient (see Eq. 7.17). Future experiments can study the superfluid-normal
mixture, as well as the mixed normal state in the trap for example by measuring theircollective excitations, the damping of out-of-phase oscillations (”second sound”), theexcitation spectrum via RF spectroscopy and the behavior under rotation. An exciting experiment would be to observe majority atoms trapped inside vortex cores. In
principle, this should be possible using phase contrast imaging of a rotating mixture.

187

New forms of superfluidity? In all of the above, we only considered the normal state and the BCS state as possible ”candidates” for the ground state of theimbalanced system. However, the BCS derivation does not include the possibility offorming Cooper pairs at finite center-of-mass momentum. In the balanced case, thiscan well be justified: As a system (without currents) turns superfluid, the growthrate of the zero-momentum pair population is largest [215]. By ”mode-competition”
these zero-momentum pairs completely dominate other possible populations. However, in the imbalanced case, Fulde, Ferrell [94] and Larkin and Ovchinnikov [146]have shown that there is indeed a state of lower energy than both the normal andthe BCS-type superconducting state, which involves pairing of fermions at finite momenta. This superfluid state is expected to approximately occur between (slightly
below) the Clogston limit h = √12 ∆ = 0.707 ∆ and hFFLO = 0.754 ∆. While thisstate is favorable compared to the normal and BCS state, there is a host of otherpossible proposals for the ground state, involving more than one common Cooper
pair momentum or states with deformed Fermi surfaces [46]. Unfortunately, the parameter space for these exotic states of superfluidity appears to be very small [221](but see [169] for a more optimistic view). The trapping potential tunes h/µ quickly
over the interesting range, and the FFLO-state might be obscured by surface effects
at the superfluid-to-normal phase boundary.
Nevertheless, the study of imbalanced Fermi mixtures will necessarily improve ourunderstanding of fermionic superfluidity in the strongly interacting regime, and will
help us to decide on the features a full theory of Fermi gases must have.

188

Chapter 8
Conclusion
Not even three years ago the first Bose-Einstein condensates of molecules had been
observed. The rapid developments, the quick succession of major experimental breakthroughs and new theoretical insights were and still are truly breathtaking. The amazing pace in the field of fermionic superfluids in atomic gases has its origins evidently
in the (comparatively) long experience with the bosonic counterparts, Bose-Einsteincondensates of atoms. While four and a half years had passed between the discoveryof the first atomic BEC and the first observation of a vortex lattice in a condensate,
it only took a third of that time for fermionic superfluids. By now the field has
matured as experiments are in a position to address open questions of many-body
physics. My colleagues and I have been fortunate to be able to contribute to thisexciting field of physics. To conclude this thesis, let me shortly summarize what has
been accomplished by our group, what questions still await an answer and in what
directions we can go from here.
From the moment it became clear that molecules, formed out of two fermionic
atoms, would not undergo fast vibrational decay close to a Feshbach resonance, thestrategy in the search for fermionic superfluidity was evident: One should first produce a stable molecular condensate before entering the unknown realm of strong
interactions. For this task, 6 Li turned out to be a ”magic” substance, allowing
molecular lifetimes of several tens of seconds close to resonance. Molecular BEC
could thus be achieved by evaporating the gas at fixed interaction strength, muchin the same way bosonic atoms are condensed. These condensates allow to explore
the regime of strong interactions, where beyond mean-field corrections apply. Sucheffects are difficult to observe in atomic condensates, where quantum depletion is
marginal (∼ 1%) (at least in the bulk). One might worry that the fermionic nature of
the molecular constituents might dominate beyond mean-field corrections expected
189for point-like bosons. However, a diagrammatic expansion in the BEC-limit of thecrossover shows [195], and Monte-Carlo calculations confirm [17], that the only effectof the fermionic constituents in this regime is to modify the scattering length formolecules into aM = 0.6a. The study of strongly interacting molecules could thus
confirm decade old predictions, for example the Lee-Yang correction for the equationof state for point-bosons [148]. Preliminary data on collective oscillations from theInnsbruck group indicates these corrections to be present and to have the expectedvalue.
The next step in achieving fermionic superfluidity was to change the sign of thescattering length from repulsive to attractive and search for condensation of fermion
pairs. In this regime binding is purely a many-body effect, as the molecular state
has disappeared. The binding can be understood as arising from Pauli-blocking of
possible scattering states, as in conventional Cooper pairing. However, the pairs in
our experiments are likely never larger than the interparticle spacing, as kF |a| & 1
in all cases studied so far. This might explain why the transfer into tightly boundmolecules is so efficient. This transfer allows to detect condensates on the attractiveside of the Feshbach resonance, by effectively ”mapping” the many-body wavefunction onto a thermally excited state containing a molecular condensate. We could
confirm that the transfer occurred on a timescale short compared to the growth timeof pair condensates, which gave an important piece of evidence that the fermion paircondensates were indeed formed before the transfer. This work also demonstrates thegreat potential given by the Feshbach resonance mechanism to study time-dependentand out-of-equilibrium situations. These are theoretically challenging problems, as
the most widespread tool, the time-dependent Ginzburg-Landau equation, only works
if the order parameter ∆ varies so slowly that the system is always close to equilibrium [260].
The final goal of the experiments was evidently to prove superfluidity and phasecoherence in atomic Fermi mixtures. Both these goals were simultaneously achievedby observing ordered vortex lattices in a rotating Fermi gas. These Abrikosov lattices
are well-known from Type-II superconductors placed in a magnetic field. They have
not yet been observed in Helium-3, the only other neutral fermionic superfluid. A
future direction for the work on rotating Fermi gases can be the observation of a
critical rotation frequency [251, 262], which is analogous to the critical magnetic field
Hc2 in Type-II superconductors. Another interesting topic is the non-trivial density
profile of a vortex on the BCS-side of the resonance [143, 109, 116, 173, 158, 219].
√
On the BEC-side, the only microscopic scale is the healing length ξ ∝ 1/ na which
190

then necessarily sets the size of the vortex core. On the BCS-side, however, there are
two length scales at zero temperature, the Fermi wavelength ∝ 1/kF and the Pippard
or BCS coherence length ξ0 = ~vF /π∆ À 1/kF . The vortex profile should show
variation on the scale of ξ0 and oscillations on the scale of the Fermi wavelength. To
complicate things further, the vortex core size should also depend on temperature(the Kramer and Pesch effect) [143, 109, 116]. Furthermore, the density depletiondue to the vortex core is strongly reduced when tuning the interaction towards the
BCS-side [41, 134, 158, 219, 56]. One can relate this effect to the presence of quasiparticle bound states inside the vortex core [45, 219, 56]. Preparing imbalanced Fermi
mixtures might allow the direct observation of these bound states as locations where
the majority atoms reside preferentially [242].
In all of the above examples, we could be guided either by intuition borrowed
from atomic BEC or from BCS superconductors. The last chapter of this thesis has
described arguably the first experiment where truly new questions were addressed,that cannot be studied in this way in superconductors or other materials, namely
the study of fermionic superfluidity with imbalanced spin populations. Superfluidity
survived in a certain window of imbalance, a result that one could have guessed on
the molecular side of the resonance, but that is less trivial on the BCS-side. Quite
intriguing was the observation of direct signatures of the phase transition in thedensity profile, without any magnetic field ramps. A remarkable result was that ina resonantly interacting gas, superfluidity can be destroyed by a chemical potential
mismatch. The nature of this strongly interacting ”normal” state (as it appears)
remains to be understood. Can it be described as a collection of quasi-particles
with effective mass? A recent Monte-Carlo study indicates this to be the case [153].
However, Monte-Carlo methods have their limits when dealing with unequal Fermi
mixtures because of the famous ”sign-problem” in requiring the antisymmetry of the
many-body wavefunction. Experiments might be able to determine the full equationof state for arbitrary density imbalance. An important question is whether a fermionicsuperfluid in three dimensions can exist (at zero temperature) at unequal densities.
On the BEC-side, such a ”magnetized” superfluid should clearly be observable. On
the BCS-side, calculations indicate the FFLO-state to be the ground state of thesystem, in a certain window of parameters (see for example [65, 169, 221, 157]). The
question is whether this parameter regime translates into a large enough region inreal space for the novel state to be observable in experiments. As there is a multitude
of candidate FFLO-like states with crystal-like distribution of Cooper pair momenta,
it is not at all clear how such a state could be unambiguously observed.
191

Finally, let me attempt to give a brief outlook as to where atomic physics canhelp to answer burning questions in condensed matter, nuclear and astrophysics. As
both bosonic and fermionic superfluids are now available, these gases can be applied
to form model systems that simulate real materials. One of the most challenging
and important problems is that of High-TC superconductors, specifically the question whether d-wave Cooper pairs are contained in the Hubbard model, one of themost promising candidate hamiltonians for High-TC materials. Fermionic atoms withrepulsive interaction in optical lattices could potentially realize the Hubbard model.
One of the first signatures to look for will be anti-ferromagnetic ordering of spin-up
and spin-down atoms, a robust normal state one obtains when all lattice-sites are
singly occupied. Experiments are being set up to study this important topic.
An important future subject will be the study of strongly interacting quantum
mixtures of different atomic species. For example, a mixture of 6 Li and 40 K could
be used to observe fermionic superfluidity of unequal particles, with unequal masses.
This is yet another way to imbalance the chemical potentials of the two species,
qualitatively different from the population imbalance studied in the preceding chapter.
It will be intriguing to see how nature copes with this mass mismatch. One can go
further and dream about studying superfluids with three different atoms or spin states,
which would be a (simplified) analog of quark (”Color”) superfluidity, where different
quarks with different Fermi momenta form superfluid pairs.
Furthermore, the use of interspecies Feshbach resonances, like the ones discoveredby our group [234], might provide a new road towards ultracold, heteronuclear groundstate molecules. Such molecules would have a large electric dipole moment, very
useful for a variety of applications. These range from measurements of fundamental
constants [142] to the use of dipolar molecules as q-bits in a quantum computer [77].
For many-body physics, ultracold dipolar molecules would open the door to the studyof quantum gases with strong anisotropic interactions [210].
About hundred years have passed since the first formation of a superfluid in 1908,
and the discovery of superconductivity in 1911. After the discovery of Bose-Einsteincondensation in dilute gases in 1995, atomic physics has been in fast-forward mode forten years to ”catch up” with condensed matter physics. We appear to finally be in theposition to start solving open problems, using atoms, the building blocks of matter,as model systems for real materials. With ever more experimental and theoreticalgroups joining in for the excitement, we can look forward to new discoveries andinsights in the years ahead. The last four years were certainly a thrill for me.

192

Appendix A
Spectroscopic Insensitivity to Cold
Collisions in a Two-State Mixtureof FermionsThis appendix contains a reprint of Ref. [266]: Martin W. Zwierlein, Zoran Hadzibabic, Subhadeep Gupta, and Wolfgang Ketterle, Spectroscopic Insensitivity to Cold
Collisions in a Two-State Mixture of Fermions, Phys. Rev. Lett. 91, 250404 (2003).

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Spectroscopic Insensitivity to Cold Collisions in a Two-State Mixture of FermionsMartin W. Zwierlein, Zoran Hadzibabic, Subhadeep Gupta, and Wolfgang Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT,
Cambridge, Massachusetts 02139, USA(Received 24 June 2003; published 19 December 2003)
We have experimentally demonstrated the absence of spectroscopic resonance shifts in a mixture of
two interacting Fermi gases. This result is linked to observations in an ultracold gas of thermal bosons.
There, the measured resonance shift due to interstate collisions is independent of the coherence in thesystem, and twice that expected from the equilibrium energy splitting between the two internal states ina fully decohered cloud. We give a simple theoretical explanation of these observations, which elucidates
the effect of coherent radiation on an incoherent mixture of atoms.
DOI: 10.1103/PhysRevLett.91.250404

PACS numbers: 03.75.Ss, 05.30.Jp, 32.30.Bv, 34.20.Cf
87 Rb bosons to measure g2 in the interstate collisional

The coherence properties of light and matter are intimately connected with the quantum statistics of theconstituent particles. One quantitative measure of thecoherence in a system is the two-particle correlation
function at zero distance, g2 , which measures the probability that two particles are simultaneously detected.
Intensity fluctuations in the incoherent light emitted bya light bulb lead to photon ‘‘bunching,’’ making thisprobability twice higher than in the coherent light of alaser. Identical fermions on the other hand exhibit ‘‘antibunching,’’ making such a probability zero.
Interactions in ultracold atomic gases crucially dependon the value of g2 [1]. The reason is that s-wave scattering relies on particles overlapping in real space. Theinteraction energy in a many-body system is determinedby coherent collisions, for which the outgoing and theincoming two-particle states are identical. Under thisconstraint, the two colliding particles can at most do
two things — either preserve their momenta, or exchange
them. We can thus distinguish four cases: (i) Two identicalbosons in a thermal gas can collide in both ways, corresponding to g2  2. (ii) Two atoms in a Bose-Einsteincondensate (BEC) have the same momenta and cannot
undergo the exchange interaction. Here g2  1. (iii) Two
distinguishable particles, fermions or bosons, also cannot
exchange their momenta because that would make theoutgoing state different from the incoming one. Again,g2  1. (iv) Two identical fermions cannot collide at all,so g2  0. In all cases, the mean-field energy of a particle with mass m is given by g2 4h 2 =man, where a isthe s-wave scattering length, and n is the density of atoms
it interacts with.
Mean field energies and therefore g2 can be measured
spectroscopically. In experiments on ultracold hydrogen,mean-field shifts of the 1S-2S two-photon transition wereused to prove the existence of a BEC [2]. However,quantitative interpretation of the shifts led to a vivid
theoretical discussion about the coherence related ‘‘factors of 2’’ [3–6]. More recently, Harber et al. performedRamsey spectroscopy in a two-component, thermal gas ofshift [7]. Their measurements yielded g2  2, independent of the degree of coherence between the two states.
The spectroscopic results thus seemed to correspond tothe case of all particles being in an identical coherent
superposition of the two internal states, even though thebinary mixture was partially decohered and should have
had a mean-field energy corresponding to 1 < g2 < 2.
The authors commented on this mystery [8]: ‘‘it is apleasure to note that a two-level system can still yieldsurprises, 75 years after the advent of quantum mechanics.’’ The mystery can be formally resolved using a quantum Boltzmann equation [9–13].
Here, we experimentally address the relation between
coherence and spectroscopic measurements in a binary
mixture of ultracold fermions. We demonstrate that shiftsof spectroscopic lines are absent even in a fully decoheredbinary mixture, in which the particles are distinguishable, and the many-body mean-field energy in the system
has developed. We theoretically show that this is a directconsequence of the coherent nature of the radiofrequency(rf) excitation, which, in general, leads to a final statewith g2 different from the initial state.
Our calculation intuitively explains both our results forfermions, and the results for bosons of Ref. [7].
In a recent paper [14], we demonstrated the absence ofmean-field ‘‘clock shifts’’ in a coherent two-state superposition of 6 Li fermions. In this case, rf spectroscopy was
performed on a gas prepared purely in one internal state.
Since an rf pulse acts as a rotation in the two-state Hilbert
space, all the atoms stayed in an identical (superposition)state and could not interact. As long as the fermionic
atoms were indistinguishable, g2  0, and the resonance
was thus found to be unperturbed at 0  E12 =h, where
E12 is the energy difference between the internal states j1i
and j2i.
However, once decoherence sets in, for example due toinhomogeneous magnetic fields across the cloud, the spatial overlap between atoms in different states grows andmean-field energy density builds up:

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0031-9007=03=91(25)=250404(4)$20.00

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VOLUME 91, N UMBER 25
E int r  g2 V12 n1 r n2 r;

PHYSICA L R EVIEW LET T ERS
V12 

4h 2a ; (1)m 12

where n1 and n2 are the local densities of particles instates j1i and j2i, and a12 is the interstate s-wave scattering length. Here decoherence means that off-diagonal
matrix elements of the density matrix have vanishedlocally. As a result, everywhere in the sample, atoms
are no longer in one pure state, but occupy two orthogonalstates, and s-wave collisions are no longer suppressed by
the Pauli principle. In a fully decohered cloud, we have abinary mixture of two distinct species of atoms, with amean-field energy density E int  V12 n1 n2 . This interaction changes the equilibrium energy level of atoms instate j1i (j2i) according to 1;2  V12 n2;1 . The difference in equilibrium mean-field energy of the two statesis thenEint  2  1  V12 n1  n2 :(2)Population of State |2 After rf-Pulse [%]This suggests [7,8,14] that in a decohering sample, theresonant frequency for population transfer between thetwo states gradually changes from 12  0 to 12 
1
0  h Eint . Here, we show both experimentally andtheoretically that this conclusion is wrong, and that the
spectroscopic resonance frequency 12 is always the unperturbed frequency 0 .
Our experimental setup was described in [14,15].
About 107 fermionic 6 Li atoms were confined in an
optical dipole trap at a temperature of 35 K. The twolevel system under consideration is formed by the twolowest ground state hyperfine levels, j1i and j2i, corresponding to jF; mF i  j1=2; 1=2i and j1=2; 1=2i in the
low field basis, respectively. A dc magnetic field of B 80

320 G was applied to the sample in order to tune theinterstate scattering length a12 to a large value of 300a0 , where a0 is the Bohr radius [14].
We created a superposition of atoms in states j1i and j2i
using a nonadiabatic rf sweep around the energy splittingof 74 MHz. As the sample decohered, efficient evaporative cooling set in, confirming a large elastic scattering
length. After 1 sec, we were left with a fully decohered
mixture at a mean density n  5 1013 cm3 . The rateof the rf sweep was adjusted so that after decoherence andcooling, 80% of the atoms were in state j2i. The meanfield interaction should thus have increased the energy
splitting of the two levels by h  2  1 
V12 n1  n2  h 10 kHz. Our experiments involving
a third state [14] have confirmed the presence of suchenergy shifts, and prove that full decoherence has beenreached.
Rabi spectroscopy in the interacting binary mixture
was performed by applying 200 s rf pulses of different
frequencies, and recording the final populations in thetwo states by simultaneous absorption imaging (Fig. 1).
In order to eliminate the systematic uncertainty in thevalue of 0 , we performed a second experiment with the
population ratios of states j1i and j2i reversed. According
to Eq. (2), one would expect an opposite shift of theresonance.
Within our precision, no interaction shift of the resonance frequency was observed. Comparing the expecteddifference in mean-field shifts for the two experiments,2  20 kHz, with the measured line separation of
34146 Hz, we arrive at an apparent value for g2 
0:0027. This demonstrates the universal absence of aresonance shift in a very cold two-level Fermi gas, independent of the coherence in the system.
Evidently, rf spectroscopy does not measure the expected difference in thermodynamic chemical potentials
for the two states. Experiments with thermal bosons have
posed a similar puzzle [7]. Here we explain that this is a

60

40

20
-5
0
5
Frequency - 74,159.672 kHz

FIG. 1. Absence of mean-field shift of an rf transition in abinary Fermi system. The resonance curves were measured forfully decohered 80=20% two-state mixtures of fermions. The
measured frequency difference between the two lines is 34
146 Hz, even though Eq. (2) would predict a splitting of
20 kHz.

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FIG. 2. Bloch sphere representation of rf transitions. (a) An rf
pulse rotates a pure state A into B. The superposition state
decoheres into a ‘‘ring’’ distribution, represented by its average, C. (b) A second rf pulse transforms the fully decoheredstate C into a partially coherent state D. The final state E isreached only after further decoherence. (c) Transfers A ! B
and C ! D are coherent and reversible. B ! C and D ! E are
irreversible.

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VOLUME 91, N UMBER 25

direct consequence of the coherent nature of the rf
excitation.
In Fig. 2, the average properties of the many-body state
at a specific point r in the trap are described by the threecoordinates of the local spin-1=2 Bloch vector mr 
mz re^ z  m? r. In the following, we omit the label r.
mz  n2  n1 =2 represents the population differencein the two states, whereas the transverse component m? isa measure of the coherence in the system. The length of
the Bloch vector measures the purity of the average stateand hence the entropy of the system. Fully decoheredstatistical mixtures do not have off-diagonal matrix elements of the density matrix and are represented by vectors with m?  0, with state A being the special case of apure state. In Fig. 2(a), state B is created by applying an rf
pulse on a pure sample A. In this case, there is no interaction energy in the system during the rf pulse, and nofrequency shift is expected [14]. State C is formed
through subsequent decoherence of state B. States B andC have the same number of particles in j1i and j2i, but inC the mean-field has fully developed.
Our experiment is performed on a C-like state[Fig. 2(b)]. Here we explain why Eq. (2) still does notgive the correct resonance frequency for an infinitesimaltransfer of atoms between j1i and j2i. The key point is thateven though the sample is fully decohered, the applied rf
pulse reintroduces coherence into the system. According
to Eq. (6) below, this will change the value of g2 . Let us
consider two fully decohered states, C and E. Equation (2)correctly gives the energy of the transformation C ! E.
However, these two states have different entropies, as
indicated by Bloch vectors of different lengths. An rf
pulse is a unitary transformation of the system, andmust preserve entropy. The true effect of the rf pulse isthus to change the relative populations of j1i and j2i bytilting the Bloch vector away from the z axis, into state D.
It is the energy of this transformation, C ! D, that needs
to be calculated in order to find the correct resonant rf
frequency.
In the case of fermions with short-range (delta function) interactions, we can prove very generally that theresonance frequency will always be 0 , by showing thatthe interaction Hamiltonian is invariant under rotations
on the Bloch sphere. The interstate s-wave interaction atpoint r is described by the second-quantized Hamiltonian
density
Hint r  V12 1y r 2y r 2 r 1 r:

(3)

Under a general rotation, described by polar angles ; ,ythe field operators 1;2transform according to: i=2 yyi=2 y1  sin e2;1;  cos e22(4) i=2 y i=2 y
y
e
e


sin

cos
:
1
2
2;
2
2
Using the standard fermionic anticommutation relations
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PHYSICA L R EVIEW LET T ERS( 1

2  2

1;

1

;
Hint
 V12

1  0, etc.), it is easy to show that
y
1;

y
2;

2;

1;  Hint :(5)

We therefore see that an rf-induced rotation on the Bloch
sphere commutes with the interaction Hamiltonian, andhence does not change the energy of the many-body state.
It is then obvious that the resonant frequency will always
be 0 , independent of the coherence of the system.
We now present a more general calculation of themean-field frequency shifts, which holds for both fermions and bosons. To reduce complexity and concentrate on
the only controversial case of interstate interactions, weconsider a fictitious boson with no intrastate interactions(a11  a22  0). The (local) mean-field expectation
value of the Hamiltonian density in Eq. (3) is [16]E int r  hHint i  V12 n1 n2  n12 n21 ) g2  1  n12 n21
;
n1 n2

(6)

where n1  h 1y 1 i and n2  h 2y 2 i are the local
densities in the two states, we have introduced ‘‘coherences’’ n12  h 1y 2 i and n21  h 2y 1 i, and   1 forbosons/fermions. In a fully coherent sample n12 n21 
n1 n2 and g2  1  . As decoherence sets in, g2 increases (decreases) from 0 (2) to 1 for fermions (bosons).
For the most general case of a partially decohered sample,
we can rewrite Eq. (6) in terms of the (local) Blochvector, using n1;2  n2  mz , n12  mx  i my  n21 , andn12 n21  m2x  m2y  m2? , where n is the total particle
density. This gives
E int  V12n2
 V12 jmj2  1  V12 m2z :4(7)

Two samples with the same numbers of atoms in states
j1i and j2i, but different levels of coherence, have thesame mz , but different jm? j [e.g., states D and E in
Fig. 2(b)]. Again we see that two such samples indeed
have different interaction energies.
Now, let us evaluate the effect of coherence on theresonant rf frequency. A coherent rf excitation preserves
entropy (jmj  const), and the total density n. In an
infinitesimal tilt of the Bloch vector, the density of atoms
transferred from j1i to j2i is dn2  dn1  dmz .
Therefore, the change of interaction energy per transferred particle, and thus the shift in the resonant frequency  , comes out to be
1 @E int 
1


 
 1  V12 n1  n2 : (8)
h @m 
hzn;jmj

In analogy with a spinning top which precesses in
the gravitational field, the resonant frequency for an
infinitesimal tilt of the Bloch vector is also equal to
the frequency of its free precession. In the traditional
language of atomic physics, this analogy just reiterates
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PHYSICA L R EVIEW LET T ERSVOLUME 91, N UMBER 25
a)

b)

2

2

m

θ

Bdirect

Bdirect θ

θ

Bexch

Bexch
θ

Btot
Fermions

m

1

Btot
1

BosonsFIG. 3. Mean-field represented as effective magnetic field.
(a) Fermions: The exchange and direct interaction add up toform a magnetic field aligned with the average spin (V12 < 0 in
the drawing). The net torque vanishes and the Bloch vector m
precesses at the unperturbed frequency 0 . (b) Bosons: The
exchange interaction has opposite sign to that in fermions. It
exerts a torque on the average spin equal to the torque inducedby the direct interaction, as can be seen by comparing the twocross products with m. The Bloch vector thus precesses at 0
plus twice the frequency shift due to direct interaction.

that Rabi [14] and Ramsey [7] spectroscopy measure thesame characteristic frequency of the system. The striking
result is that in contrast to the interaction energy [Eqs. (6)and (7)], the precession of the Bloch vector, or equivalently the rf frequency shift [Eq. (8)], does not depend on
the level of coherence in the sample. Remarkably, the final
state may have a value of g2 different from the initialstate, such that the energy difference per transferred
particle is independent of the initial g2 . Equation (8)explains both our measurements with fermions, and theexperiment with thermal bosons of Ref. [7].
In order to further elucidate the role of coherencesin the precession of the Bloch vector, we employ theinterpretation of the mean-field energy as the interaction of the average spin with an effective magnetic
field [12,13]. Using Eq. (7), we obtain E int  const 
1
2 Beff  m [17] with
Beff  2V12 mz e^ z  m? :(9)In this picture, the precession of the spin due to interactions is driven by the torque Beff m. The magnetic fieldalong the z axis is induced by the direct interaction, andhas the same sign for fermions and bosons (Fig. 3). The
transverse magnetic field comes from the exchange interaction, and has opposite signs for fermions and bosons.
For fermions, Beff is parallel to m [Eq. (9)] and hence
does not cause any precession. Equivalently, the direct
and exchange interaction exert torques equal and opposite
to each other. For bosons, the two contributions add up toyield exactly twice the precession frequency given by the
direct interaction alone. During decoherence, the exerted
torque shrinks in proportion to the decaying transverse
spin. Therefore, the precession frequency remains constant, no matter how small the coherences are.
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In conclusion, we have demonstrated the absence of themean-field shift of rf transitions in a fully decohered,interacting binary mixture of fermions. This was explained by proving the invariance of the interaction energy under coherent Hilbert space rotations. This result isrelevant for the potential use of a fermionic atom supplying the frequency standard in an atomic or optical clock,since it implies a robust elimination of the systematic
errors due to density dependent frequency shifts.
Previously, the absence of such clock shifts was explainedby the absence of mean-field energy in a purely coherent
superposition state [14]. Now we have shown that there isno spectroscopic shift even after decoherence has led tomeasurable mean-field energies. Further, we have presented a simple theoretical framework for calculating
the precession frequency of the Bloch vector which describes an arbitrary spin state of either fermions or bosons. This resolves ‘‘the mystery of the Ramsey fringethat did not chirp’’ [8] with a simple and intuitive picture.
We thank Claudiu Stan and Christian Schunck forexperimental assistance, and Michele Saba for criticalreading of the manuscript. This work was supported by
the NSF, ONR, ARO, and NASA.

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[7] D. M. Harber et al., Phys. Rev. A 66, 053616 (2002).
[8] D. M. Harber et al. in Proceedings of the XVIIIInternational Conference on Atomic Physics, edited by
H. R. Sadeghpour, E. J. Heller, and D. E. Pritchard (World
Scientific, Cambridge, Massachusetts, 2003), pp. 3–10.
[9] M. Ö. Oktel and L. S. Levitov, Phys. Rev. Lett. 88, 230403(2002).
[10] J. E. Williams, T. Nikuni, and C.W. Clark, Phys. Rev.
Lett. 88, 230405 (2002).
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[16] A. L. Fetter and J. D. Walecka, Quantum Theory of ManyParticle Systems (Dover Publications, New York, 2003).
[17] The factor of 1=2 reflects the self-interaction.

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Appendix B
Observation of Bose-Einstein
Condensation of Molecules
This appendix contains a reprint of Ref. [271]: M.W. Zwierlein, C. A. Stan, C. H.
Schunck, S.M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, Observationof Bose-Einstein Condensation of Molecules, Phys. Rev. Lett. 91, 250401 (2003).

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Observation of Bose-Einstein Condensation of Molecules
M.W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle
Department of Physics, MIT–Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT,
Cambridge, Massachusetts 02139, USA(Received 27 November 2003; published 15 December 2003)
We have observed Bose-Einstein condensation of molecules. When a spin mixture of fermionic 6 Li
atoms was evaporatively cooled in an optical dipole trap near a Feshbach resonance, the atomic gas wasconverted into 6 Li2 molecules. Below 600 nK, a Bose-Einstein condensate of up to 900 000 molecules
was identified by the sudden onset of a bimodal density distribution. This condensate realizes the limitof tightly bound fermion pairs in the crossover between BCS superfluidity and Bose-Einsteincondensation.
DOI: 10.1103/PhysRevLett.91.250401

PACS numbers: 03.75.Ss, 05.30.Jp

Over the past few years, many different approaches
have been used to cool and trap molecules [1,2]. One
major goal has been the creation of molecular BoseEinstein condensates, which could lead to advances inmolecular spectroscopy, studies of collisions, and precision tests of fundamental symmetries.
Recently, a new technique for creating ultracold molecules led to major advances towards molecular BoseEinstein condensation (BEC). Molecules were produced
from ultracold atoms [3–9] near a Feshbach resonance[10], where a molecular state is resonant with the atomic
state and molecules can form without heat release. Thesemolecules are highly vibrationally excited and would
usually undergo fast decay. However, in the case of fermionic atoms the molecules showed very long lifetimes[7–9,11]. This has been attributed to Pauli suppression of
the vibrational quenching process, which couples a very
weakly bound molecular state to much more tightly
bound lower lying vibrational states [12]. We havenow been able to cool such molecules to Bose-Einsteincondensation.
This Bose-Einstein condensate represents one extremeof the crossover from Bose-Einstein condensation oftightly bound pairs (molecules) to BCS superfluidity of
Cooper pairs, where fermions form delocalized pairs inmomentum space [13].
In most of the recent experiments, molecules wereformed by sweeping an external magnetic field throughthe Feshbach resonance, adiabatically converting atoms
to molecules [3–6,8]. This atom-molecule coupling is acoherent two-body process [14].
In the case of 6 Li, experimental work indicated [7,9]and theoretical work predicted [15,16] that cooling theatoms at constant magnetic field would create an atommolecule mixture in thermal equilibrium. In this case, theatoms and molecules are coupled by three-body recombination [17]. For temperatures lower than the binding
energy of the molecular state, an almost pure molecular
gas should form, and at even lower temperatures, a molecular Bose-Einstein condensate. This work demonstrates that this surprisingly simple method to create

molecular condensates works. The success of this approach depends on a very favorable ratio of collisional
rates for formation and decay of molecules which may beunique to 6 Li.
The goal of molecular BEC was reached in several
steps. Using Feshbach resonances, atomic condensates
were put into an atom-molecule superposition state [2].
Pure molecular gases made of bosonic atoms were created close to [6] or clearly in [4] the quantum-degenerate
regime, but the effective heating time (of about 2 ms in
Ref. [4]) was too short to reach equilibrium. Earlier thismonth, while this work was in progress, two papers were
submitted. Reference [18] observed a quantum-degenerate gas of potassium molecules with an effective lifetimeof 5 to 10 ms, sufficiently long to reach equilibrium in
two dimensions and to form a quasi- or nonequilibrium
condensate [19]. Reference [20] provided indirect evidence for a long-lived condensate of lithium molecules[21]. Here we observe the formation of a condensate
by evaporative cooling of a molecular gas close toequilibrium.
The basic scheme of the experiment is similar to ourearlier work when we identified two Feshbach resonancesin lithium by monitoring the loss of trapped atoms due tothree-body recombination as a function of the external
magnetic field [22]. This process leads to ultracold molecules in the highest vibrational state below the continuum.
However, no attempt was made to detect these molecules
because until very recently [7,9] they were believed todecay rapidly.
Our experimental setup was described in Refs. [22,23].
After laser cooling and sympathetic cooling with sodium
atoms in a magnetic trap, 35  106 lithium atoms in the
jF; mF i  j3=2; 3=2i state were transferred into an optical trap formed by a single far detuned laser beam with up
to 7 Wof power at 1064 nm. The beam had a 20 m waist
and was aligned horizontally along the symmetry axis of
the magnetic trap. This generated a 650 K deep trapping
potential with 15 kHz radial and 175 Hz axial trapping
frequencies. They were determined with an accuracyof 10% by exciting dipolar oscillations with an atomic

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sodium condensate and scaling them to lithium atoms
using the ratios of polarizabilities and masses.
The 6 Li atoms were then transferred to the lowest energy state j1i, using an adiabatic frequency sweep aroundthe lithium hyperfine splitting of 228 MHz. dc magnetic
fields of up to 1025 G could be applied, a range encompassing the j1i  j2i Feshbach resonance [22,24,25],
where j2i denotes the second lowest hyperfine state of 6 Li.
Most of our experiments were performed at a magnetic
field of 770 G. This was below but still within the width of
the broad Feshbach resonance. Here the atomic scattering
length is positive corresponding to a stable weakly boundmolecular state. Using rf-induced transitions near
80 MHz, an equal mixture of states j1i and j2i wasprepared with a ratio of temperature T to Fermi temperature TF around 0.3. The sample was cooled in 350 ms by
ramping down the laser power of the optical trap totypically 103 of the maximum power resulting in a
calculated trap depth for unbound atoms of 650 nK. The
weakly bound molecules have twice the atomic polarizability. They experience the same trap frequencies andtwice the trap depth as the lithium atoms. Therefore, we
expect mainly atoms to be evaporated.
Atoms and molecules were detected by absorption
imaging after ballistic expansion times of 1 to 30 ms.
During the time of flight, the magnetic field was suddenly
switched off, and atoms in both states were imaged
simultaneously since the two optical transition frequencies are equal at zero field. Molecules were detected byfirst dissociating them by sweeping the magnetic field
across the Feshbach resonance up to 925 G and then byimaging the resulting atoms at zero field. With the
Feshbach sweep, molecules and residual atoms were imaged together. Without it, only the unbound atoms weredetected after switching off the magnetic field. We havefound that during the initial phase of the evaporative
cooling the atomic population dominated. A significant
molecule fraction formed around T  2 K, and in the
final phase of the cooling, no atoms could be discerned.
The absorption images and profiles shown in Figs. 1 and 2
therefore represent purely molecular column densities.
When the laser power of the optical dipole trap was
ramped down, the shrinking size of the cloud in absorption imaging signaled lower temperatures. Very abruptly,
the smooth distribution changed to a bimodal distribution—the well-known ‘‘smoking gun’’ of Bose-Einsteincondensation [26,27] (Figs. 1 and 2). Because of a slightasymmetry of our trapping potential, the centers of thecondensate and of the thermal cloud were shifted.
The phase transition could be identified by plotting theeffective area of the cloud vs laser power P [Fig. 3(a)]. At
the phase transition, there was an abrupt change in slope,
whereas the temperature changed smoothly. For a classical gas, the area depends only on temperature and trap
frequencies, which vary smoothly with P.
Quantitative information on temperature, total atom
number, and condensate fraction was obtained by fitting
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FIG. 1. Observation of Bose-Einstein condensation in a molecular gas. Shown are three single-shot absorption images
after 6 ms of ballistic expansion for progressively lower
temperatures (left to right). The appearance of a dark spot
marks the onset of BEC. The field of view for each image is
1:4  1:4 mm2 . The long axis of the optical dipole trap wasvertical in the image.

axial profiles (such as in Fig. 2) using a bimodal distribution: a Bose-Einstein distribution for the broad normalcomponent and a Thomas-Fermi distribution for the narrow (condensate) component. Condensates containing up
to 900 000 molecules and condensate fractions of up to
75% were obtained. The onset of BEC was observed at atemperature of 600 nK with 1:4  106 molecules. For an
ideal gas with this number of molecules, the predicted
BEC transition temperature TC  0:94 h! N 1=3 =kB is
650 nK, where ! denotes the geometric mean of the three
trapping frequencies. This agreement is fortuitous, given
the uncertainty in the trap frequencies at low power [28].
The cooldown is characterized in Fig. 3. Axial temperatures were obtained from the spatial extent of thethermal cloud (the size stayed almost constant during thetime of flight). The axial temperatures are in arbitrary

Molecule Number Per Unit Length [ 1/mm ]VOLUME 91, N UMBER 25

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2.4 mW

2.7 mW

3.3 mW

4.5 mW6

1x10

8.9 mW

0
0.0

0.5

1.0

Position [ mm ]FIG. 2. Bimodality in the density distribution of a moleculargas. Shown are radially averaged profiles through absorption
images such as those in Fig. 1, as a function of final laser power.
The dashed lines are fits to the thermal clouds.

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(b)
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500

Condensate Fraction

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(c)

0.5

0.0
0

5
10
Laser Power [mW]15

FIG. 3. Characterization of the phase transition. (a) The effective area is the integrated optical density divided by thepeak optical density of the absorption images. The ‘‘sudden
shrinking’’ of the area coincides with the appearance of abimodal density distribution and indicates the BEC phasetransition. (b) The radial (squares) and axial (triangles) temperatures varied smoothly during the evaporation. The axial
temperatures are in arbitrary units. (c) Condensate fraction.
Below the critical temperature, the condensate fraction increased from zero to up to 75%.

units [28]. Absolute radial temperatures were obtained
from the ballistic expansion. All of our temperature
measurements are consistent with equilibration in three
dimensions.
From the expansion of the largest condensates, we
determined the mean-field energy  to be 300 
100 nK. The peak density n of the condensate was obtained from the fit to be 7  1013 cm3 . The relation  4h2 an=m, where m is the molecular mass, implies amolecular scattering length of a  8 nm with an uncertainty of 60%. These uncertainties were estimated fromthe discrepancy of fits done at different times of flightwhich were not fully consistent. This might reflect asymmetries and anharmonicities of the trapping potential
which were not characterized.
It was predicted that the molecular scattering length a
is 0.6 times the atomic scattering length aa [12]. At ourmagnetic field of 770 G, the predicted value of aa lies
between 140 and 340 nm depending on the uncertain
position of the Feshbach resonance between 810 and
850 G [25]. The discrepancy between the predicted andobserved values of a needs further study.
The molecular gas decayed faster than extrapolated
from Refs. [7,9]. Just above Tc , the thermal cloud had apeak density of 1  1013 cm3 and an initial decay timeof about 1 s. Condensate numbers decayed to one third
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after a hold time of 30 ms. Those short lifetimes may
reflect leakage or heating in our optical dipole trap at low
laser power. In the present experiments, the laser power
was not stabilized. The lifetime of the thermal gas ismuch longer than estimated values of the axial trap
period of 100 ms and of the collision time of 2 ms, whichshould lead to full equilibrium. Depending on how thecondensate grew during the evaporative cooling, its lifetime may have been too short to develop phase coherencein the axial direction [19].
Using the experimentally determined scattering
length, we find that the molecular cloud at Tc has a ratioof mean-free path to radius close to 10 and should show
only negligible anisotropy during ballistic expansion[19]. Therefore, the onset of strongly anisotropic expansion is a distinguishing feature of the molecular condensate (Fig. 1).
The 6 Li2 molecules are extremely weakly bound. The
molecular binding energy depends on the atomic scattering length aa as h2 =ma2a [29]. For an assumed aa 
200 nm the binding energy is 2 K. The molecular state
which causes the Feshbach resonance is the X 1 g ;v 
38 state. This state is tightly bound, but near the Feshbachresonance it is strongly mixed with the state of the colliding atoms, and the molecular wave function is spread out
over an extension of order aa =2 [29].
Direct evidence for the large size of the molecules was
obtained by resonant imaging during ballistic expansion
at high magnetic fields. At 770 G, molecules could bedirectly imaged using light in resonance with the atoms at
the same field. The absorption was comparable to thezero-field absorption signal obtained after dissociating
the molecules. This shows that the molecular bond is so
weak that the absorption line is shifted from the atomic
line by less than a linewidth . The molecules are expected to absorb most strongly near the outer turning
point R. The excited state potential is split by the resonantvan der Waals interaction  h =R 3 , where  isthe resonant wavelength divided by 2 and  is
3=4; 3=2 for different excited molecular states. The
observed absorption signal implies a molecular size R
100 nm. It is this huge size compared with the muchsmaller size of the molecule in lower vibrational states
which, together with Fermi statistics, inhibits vibrational
relaxation and leads to the long lifetimes [12].
Condensates were observed after evaporative cooling at
various magnetic fields between 720 and 820 G. At thelower magnetic fields, the condensate expanded more
slowly, consistent with the predictions of a smaller repulsive mean-field energy.
In future work, we plan to use the molecular BEC as
the starting point for studying the BEC-BCS crossover[13]. By ramping up the magnetic field across the
Feshbach resonance, the molecules are dissociated into
fermionic atoms and the interaction between the atoms
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condensate and conserving entropy, a Fermi sea should
form with temperatures well within the range where BCS
type superfluidity has been predicted [30,31].
In conclusion, we have realized Bose-Einstein condensates of up to 900 000 molecules by evaporative coolingof a spin mixture of fermionic lithium atoms.
This work was supported by the NSF, ONR, ARO, andNASA. We thank A. Leanhardt for helpful comments.
S. Raupach is grateful to the Dr. Jürgen Ulderup foundation for financial support.
Note added.—In an optical trap with a slightly enlarged beam waist, we were recently able to hold molecular condensates for up to 400 ms (or three axial trapping
periods) which should result in 3D equilibration. The 1=edecay time was about 200 ms.

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Stoof, M. Houbiers, C. A. Sackett, and R. G. Hulet,
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Rev. Lett. 89, 130402 (2003).
[14] P. D. Drummond, K.V. Kheruntsyan, and H. He,
Phys. Rev. Lett. 81, 3055 (1998); P. Tommasini,
E. Timmermans, M. Hussein, and A. Kerman, e-printcond-mat/9804015; F. A. van Abeelen and B. J. Verhaar,Phys. Rev. Lett. 83, 1550 (1999); F. H. Mies, E. Tiesinga,
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C. Salomon, e-print cond-mat/0308384.
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field dependence of the loss rate consistent with meanfield effects. No observation of a phase transition ormeasurements of equilibration, temperature, or condensate fraction were reported.
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and C. Salomon, Phys. Rev. Lett. 91, 020402 (2003).
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done at full power. The axial frequency was affectedby small magnetic field curvatures which were negligible
at high laser power.
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Holland, Phys. Rev. A 66, 043604 (2002).

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Appendix C
Condensation of Pairs of FermionicAtoms near a Feshbach ResonanceThis appendix contains a reprint of Ref. [272]: M.W. Zwierlein, C. A. Stan, C. H.
Schunck, S.M. F. Raupach, A. J. Kerman, and W. Ketterle, Condensation of Pairs
of Fermionic Atoms near a Feshbach Resonance, Phys. Rev. Lett. 92, 120403 (2004).

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Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance
M.W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT,
Cambridge, Massachusetts 02139, USA(Received 1 March 2004; published 25 March 2004)
We have observed Bose-Einstein condensation of pairs of fermionic atoms in an ultracold 6 Li gas at
magnetic fields above a Feshbach resonance, where no stable 6 Li2 molecules would exist in vacuum. We
accurately determined the position of the resonance to be 822  3 G. Molecular Bose-Einsteincondensates were detected after a fast magnetic field ramp, which transferred pairs of atoms at close
distances into bound molecules. Condensate fractions as high as 80% were obtained. The large
condensate fractions are interpreted in terms of preexisting molecules which are quasistable even
above the two-body Feshbach resonance due to the presence of the degenerate Fermi gas.
DOI: 10.1103/PhysRevLett.92.120403

PACS numbers: 03.75.Ss, 05.30.Fk

Ultracold atomic gases have become a medium torealize novel phenomena in condensed matter physics
and test many-body theories in new regimes. The particle
densities are 108 times lower than in solids, but at temperatures in the nanokelvin range, which are now routinely achieved, interactions and correlations become
important. Of particular interest are pairing phenomena
in fermionic gases, which have direct analogies to superconductivity [1].
The interactions which drive the pairing in these gases
can be controlled using a Feshbach resonance [2], inwhich a molecular level is Zeeman tuned through zerobinding energy using an external magnetic field. This
provides an opportunity to experimentally probe what is
known as the BCS-BEC crossover; as the strength of theeffective attractive interaction between particles is increased a continuous transition from condensation ofdelocalized Cooper pairs to condensation of tightly
bound bosonic molecules is predicted [3–6]. Whereas in
the BCS limit the pairing is a strictly many-body effect[7], in the BEC limit a pair of fermions is bound even as
an isolated molecule. A novel form of high-temperature
superfluidity has been predicted to emerge in the crossover region [3–6]. Until recently, the observation of condensation phenomena in fermionic atomic gases was
restricted to the extreme BEC limit, where several groups
have observed Bose-Einstein condensation of diatomic
molecules [8–11].
An important step was recently reported, in whichcondensation of atomic 40 K fermion pairs was observedon the BCS side of a Feshbach resonance [12]. It wasargued that those pairs were not bound into molecules, but
merely moved together in a correlated fashion, similar to
Cooper pairs of electrons in a superconductor [13].
However, the exact nature of these pairs remained unclear.
In this Letter, we apply similar techniques to 6 Li atoms,
which have very different collisional properties [14], andobserve the pair condensation phenomenon above a
Feshbach resonance. In contrast to the previous work,where at most 15% of the atom pairs were condensed[12], condensate fractions of up to 80% were observed. We
argue that such a high condensate fraction is unlikely forpairs which are long range, but rather it indicates acondensate of short-range atom pairs which are essentially molecular in character even on the BCS side of theresonance.
A simple argument supports this possibility. In thebasic picture of a Feshbach resonance, a molecular state
above the dissociation threshold has a finite lifetime,
which becomes shorter as the energy of the state increases, as recently observed [15]. In the presence of the
Fermi sea, its lifetime will be increased due to Pauliblocking. The molecular level will be populated until its
energy becomes larger than twice the Fermi energy corresponding to the total number of atoms. The BCS-BEC
crossover is expected to occur at this point, and not at thelocation of the two-body Feshbach resonance [5,6].
The basic setup of our experiment was described in[10]. By sympathetic cooling of 6 Li atoms with 23 Na in amagnetic trap, a degenerate gas of about 3  107 6 Li
fermions at 0:3T=TF was created. After transfer into
an optical dipole trap (maximum power 9 W focused toan e2 radius of 25 m), an equal mixture of atoms in thelowest two hyperfine states j1i and j2i was prepared. The
sample was evaporatively cooled at a magnetic field of
770 G using an exponential ramp-down (time scale 
400 ms) of the optical trap to a final laser power of
15 mW. This created essentially pure Bose-Einstein condensates of up to 3  106 6 Li2 molecules. The observed
trap vibrational frequencies could be pdescribedby thefollowing
expression:


115
Hz
P
,


1:1 Hzradaxp
P 120B, where P is the optical power in mW, and B
is the magnetic field in kG. The latter dependence arises
from the residual axial curvature of the magnetic field.
Considerable improvements over our previous setup [10]led to an improved e1 condensate lifetime of 10 s at
770 G. Moreover, within the experimental uncertainty
in the total number of molecules (50%), mean-field

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measurements were consistent with a molecule-molecule
scattering length of 0:6a, where a is the atomic scattering
length [16,17].
Previously, the location of the 6 Li Feshbach resonancewas determined either by observing a peak in the inelastic loss [18] or the interaction energy of a j1i  j2i mixture [19]. A more accurate determination can be made bymapping out the onset of dissociation of the molecularstate [12,15]. After releasing an almost pure molecularsample from the trap at 770 G, the magnetic field waslinearly ramped up in 10 ms to a variable value. Duringthat time, the particle density dropped by a factor of 1000.
If the field crossed the resonance, molecules dissociatedinto atoms. These atoms were then imaged at zero field,
where the remaining molecules were not detected [10].
The Feshbach resonance appeared as a sharp onset in thenumber of detected atoms (Fig. 1). The speed of thedownward ramp to zero field had to be chosen carefully.
Fast ramps could dissociate very weakly bound molecules[20], such that the Feshbach resonance appeared systematically shifted to lower fields. For too slow a ramp-down,on the other hand, we found that even for clouds as diluteas 3  1010 cm3 molecules were recreated, lowering
the measured atomic fraction. However, when we varied
the ramp rate over more than 3 orders of magnitude, we
found a range of rates which gave identical thresholds at
822  3 G (Fig. 1).
To produce samples in the crossover region, we started
with an essentially pure Bose-Einstein condensate ofmolecules formed at 770 G. The laser power of the opticaltrap was increased in 500 ms from 15 to 25 mW in orderto accommodate larger Fermi clouds above the resonance.
In some experiments, we used a deeper trap with up to
150 mWof power; the additional compression was carried

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out after ramping in 500 ms to 900 G to avoid enhanced
losses on the BEC side of the resonance. Once the final
trap depth was reached, the magnetic field was ramped in
500 ms to values between 650 and 1025 G. The adiabaticity of this ramp was checked by ramping back to 770 G
and observing an identical density profile and condensate
fraction, similar to studies in Ref. [8]. At 1025 G, the total
peak density of the spin mixture in the deepest trap was
3  1013 cm3 , corresponding to a Fermi energy of
3:6 K and inverse Fermi wave vector k1
F ’ 2000a0 ,where a0 denotes the Bohr radius.
To probe the gas, we released it from the trap, and aftera variable delay d of usually 40 s, applied a rapidtransfer technique [12]: the magnetic field was switched
off exponentially to zero with an initial slew rate of
30 G=s, which adiabatically converted pairs of atoms
into deeply bound molecules at zero field [21]. As long as
no collisions or other dynamics occur during this ramp,
the velocity distribution of the resulting molecules thenconstitutes a probe of the atom pairs’ center-of-mass
motion before the measurement. After 3–6 ms time offlight at zero field, we dissociated the molecules with a
3 ms field pulse to 900 G and imaged the resulting atoms
after 2 ms at zero field [10,22]. We could also selectively
detect any remaining atoms by omitting the dissociation
pulse, and we observed that for d 500 s, less than
10% of the sample consisted of atoms, independent of theinitial magnetic field. At longer delay times, the atommolecule conversion became less efficient due to thedecreased density.
Typical absorption pictures of molecular clouds after
the rapid transfer ramp are shown in Fig. 2 for different
temperatures, clearly exhibiting a bimodal distribution.
This is evidence for condensation of pairs of 6 Li atoms on
the BCS side of the Feshbach resonance. The condensate
fractions were extracted from images like these, using a
Gaussian fit function for the ‘‘thermal’’ part and a

0.5

0.0
750

800
850
Magnetic Field [G]900FIG. 1. Determination of the Feshbach resonance position.
Shown is the onset of molecule dissociation when the magnetic
field was slowly raised and then ramped down to zero field witha variable rate: Using a switch-off of the power supply at an
initial rate of 30 G=s (crosses), a linear ramp to zero field of
100 G=ms (circles), a linear ramp for 16 ms at 12:5 G=ms,
followed by switch off (triangles). The identical threshold forthe two lowest ramp rates determines the resonance position to
be 822  3 G, marked by an arrow.

120403-2

FIG. 2 (color online). Emergence of a Bose-Einstein condensate of atom pairs as the temperature was lowered. Shown arecolumn densities (after 6 ms of time of flight) of the fermion
mixture after a rapid transfer ramp from 900 G for threedifferent initial temperatures T=TF  0:2, 0.1 and 0.05, together with their axially integrated radial density profiles.
The dashed line is a Gaussian fit to the thermal component.
Condensate fractions are 0.0, 0.1, and 0.6. Each cloud consistsof about 2  106 molecules. The field of view is 3  3 mm.

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Condensate Fraction

0.8

Condensate Fraction

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PHYSICA L R EVIEW LET T ERSVOLUME 92, N UMBER 12

0.4

0.4

0.0
0.0

700
600

800

800
900
Magnetic Field [G]

1000

1000

Magnetic Field [G]Thomas-Fermi profile for the ‘‘condensate.’’ Figures 3–5
show the observed condensate fraction as a function ofboth magnetic field and temperature. The striking features of these data are the high condensate fraction of
80% near resonance, and the persistence of large condensate fractions on the BCS side of the resonance all
the way to our maximum field of 1025 G. After 10 shold time, this value was still as high as 20%. Theseobservations were independent of whether the final magnetic field was approached starting with a Fermi sea or amolecular condensate. Note that for our peak densities,
the strongly interacting region of kF jaj > 1 extends from710 G onward.
There is experimental evidence that the observed paircondensates existed before the sweep and were not produced during the sweep by collisions. First, the observed
condensate fraction depended on the initial magnetic
field. Second, the condensate fraction did not change
when we varied the delay time d (between release of
the atoms from the trap and the magnetic field ramp) from0 to 200 s, although the density of the cloud changed bya factor of 4 [25]. However, we cannot rule out withcertainty that the momentum distribution of the pairs is
modified by collisions during the ramp [26]. At our highest densities, it takes about 4 s to take the molecules
created during the ramp out of the strongly interacting
region (kF jaj 1). A classical gas at the Fermi temperature would have a unitarity limited collision time comparable to the inverse of the Fermi energy divided by h,which is about 2 s. However, this may be affected by
120403-3

FIG. 4. Condensate fraction for different temperatures as afunction of magnetic field. The temperature of the molecular
cloud was varied by stopping the evaporative cooling earlier
and applying parametric heating before ramping to the final
magnetic field. Temperatures are parametrized by the molecular condensate fraction N0 =N at 820 G (open circles: 0.8; filled
circles: 0.58; open squares: 0.51; filled squares: 0.34; ‘‘ ’’:
0.21; triangles: 0.08; ‘‘’’: <0:01). The lowest temperature
was realized in the shallow trap (P 25 mW); the higher
temperatures required a deeper trap (P 150 mW).

Pauli blocking for the atoms and bosonic stimulation forthe molecules.
Assuming that collisions during the ramp can be neglected, it is still crucial to ask what exactly happens
during the rapid transfer ramp, and what kind of pairs
would likely be detected. A reasonable assumption is that
atoms form molecules preferentially with their nearest
neighbor, independent of the center-of-mass velocity ofthe pair. If, as our data show, a large fraction of the0
Condensate Fraction at 820 G

FIG. 3. Condensate fraction after the rapid transfer vs initial
magnetic field, for different hold times at that field in theshallow trap (P 25 mW). Crosses: 2 ms hold time, after
500 ms ramp to 1000 G and 4 ms ramp to the desired field;
squares and circles: 100 ms and 10 s hold time, after 500 ms
ramp from 770 G. The reduction of the condensate fraction forlong hold times far on the left side of the resonance is probably
due to the rapidly increasing inelastic losses for the more
tightly bound molecules [20,23]. The lower condensate fraction
at high field for long hold times is probably an effect of lower
density since the number of atoms had decayed by a factor of 4
without change in temperature.

0.2
0.1

0.2
0.3

T/TF

0.4
0.5

0.6
0.7

0.8

0.05
700

800
900
Magnetic Field [G]1000

FIG. 5 (color online). Temperature and magnetic field ranges
over which pair condensation was observed (using the samedata as in Fig. 4). The right axis shows the range in T=TF
(measured at 1025 G) which was covered. For high degeneracies, fitting T=TF was less reliable and we regard the condensate fraction as a superior ‘‘thermometer.’’ Note that for an
isentropic crossover from a BEC to a Fermi sea, T=TF isapproximately linearly related to the condensate fraction on
the BEC side [24]. For our maximum densities the region wherekF jaj 1 extends from about 710 G onward.

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VOLUME 92, N UMBER 12

PHYSICA L R EVIEW LET T ERS

detected molecules are in a zero-momentum state after
the fast transfer, this means that nearest neighbors had
opposite momenta. If the distance between the fermions
with opposite momenta making up each pair were comparable to or larger than the interatomic distance (as inlong-range Cooper pairs) one would not expect to findhigh condensate fractions; on the contrary, the transfer
into a tightly bound molecular state would randomly pick
one of the nearest neighbors, resulting in a thermal molecule. We regard our observed high condensate fractions asevidence for the existence of condensed atomic pairs
above the Feshbach resonance, which are smaller in size
than the interatomic distance and, therefore, molecular incharacter. Their stability may be affected by Pauli blocking and mean-field effects, but their binding should be a
two-body effect and not a many-body effect as in the caseof Cooper pairs.
In conclusion, we have observed 6 Li2 molecular BoseEinstein condensates after a fast downward magnetic
field ramp starting with equilibrium samples at fields
on either side of the broad 6 Li Feshbach resonance.
Since there are no truly bound molecular states above
the resonance, we tentatively interpret our results as a
Bose-Einstein condensate of pairs of atoms which aremolecular in character and stabilized by the existenceof the Fermi sea. This condensate would drain particles
from the Fermi sea and lead to a reduced atomic Fermi
energy roughly equal to half the energy of the molecularlevel [5,6]. Indeed, both in Ref. [8] and in the present
work, a reduction in the size of the cloud was observed as
the Feshbach resonance was approached from above,
which may be due to this effect. In agreement withtheoretical predictions [6] we have observed pair condensation in the regime where T=TF < 0:2 and kF jaj > 1(Fig. 5). The exact nature of the atom pairs remains tobe elucidated; they could be related to virtual states orscattering resonances in the continuum; they may turnout to be the tight-binding limit of Cooper pairs. It is also
possible that the pair condensate is a superposition stateof molecules and Cooper pairs [4 –6]. We regard thecharacterizing feature of the BCS-BEC crossover a qualitative change of the pairing phenomenon which has not
yet been observed.
This work was supported by NSF, ONR, ARO, andNASA. We thank Walter Hofstetter, Michele Saba, andZoran Hadzibabic for stimulating discussions. S. R. isgrateful to the Dr. Jürgen Ulderup foundation for financial
support.

[1] H. T. C. Stoof and M. Houbiers, in Bose-Einstein
Condensation in Atomic Gases, Proceedings of theInternational School of Physics Enrico Fermi, Course
CXL, edited by M. Inguscio, S. Stringari, and C. Wieman(IOS Press, Amsterdam, 1999), pp. 537–553.

120403-4

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26 MARCH 2004

[2] E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A
47, 4114 (1993); W. C. Stwalley, Phys. Rev. Lett. 37, 1628
(1976); S. Inouye et al., Nature (London) 392, 151(1998); P. Courteille et al., Phys. Rev. Lett. 81, 69 (1998).
[3] D. M. Eagles, Phys. Rev. 186, 456 (1969); A. J. Leggett, inModern Trends in the Theory of Condensed Matter.
Proceedings of the XVIth Karpacz Winter School ofTheoretical Physics, Karpacz, Poland, 1980 (SpringerVerlag, Berlin, Karpacz, Poland, 1980), pp. 13–27;
P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys.
59, 195 (1985).
[4] E. Timmermans et al., Phys. Lett. A 285, 228 (2001);
M. Holland et al., Phys. Rev. Lett. 87, 120406 (2001);
J. Stajic et al., e-print cond-mat/0309329; A. Perali et al.,e-print cond-mat/0311309; R. Combescot, e-print condmat/0310583; J. Kinnunen, M. Rodriguez, and P. Törmä,e-print cond-mat/0401543.
[5] Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402(2002).
[6] G. M. Falco and H. T. C. Stoof, e-print cond-mat/
0402579.
[7] M. Houbiers et al., Phys. Rev. A 56, 4864 (1997).
[8] S. Jochim et al., Science 302, 2101 (2003); M. Bartenstein et al., Phys. Rev. Lett. 92, 120401 (2004).
[9] M. Greiner, C. A. Regal, and D. S. Jin, Nature (London)
426, 537 (2003).
[10] M.W. Zwierlein et al., Phys. Rev. Lett. 91, 250401 (2003).
[11] T. Bourdel et al., e-print cond-mat/0403091.
[12] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92,
040403 (2004).
[13] NIST and University of Colorado at Boulder press release,
January 28, 2004; http://www.nist.gov/public_affairs/
releases/fermi_condensate.htm.
[14] M. Houbiers et al., Phys. Rev. A 57, R1497 (1998).
[15] T. Mukaiyama et al., e-print cond-mat/0311558 [Phys.
Rev. Lett.(to be published)].
[16] D. S. Petrov, C. Salomon, and G.V. Shlyapnikov, e-printcond-mat/0309010.
[17] The smaller value reported in our previous work [10] wasbased on an incorrect extrapolation of trap frequencies to
100 times lower power. The radial trap frequency wasstrongly influenced by multiple intensity maxima in the
optical focus.
[18] K. Dieckmann et al., Phys. Rev. Lett. 89, 203201 (2002).
[19] T. Bourdel et al., Phys. Rev. Lett. 91, 020402 (2003).
[20] J. Cubizolles et al., Phys. Rev. Lett. 91, 240401 (2003).
[21] According to Fig. 1, this fast ramp can dissociate weakly
bound molecules at low density, but at high density weobserved an atom-molecule conversion 90%.
[22] At this point the cloud was sufficiently dilute that the
final switch off of the field did not reconvert atoms intomolecules.
[23] S. Jochim et al., Phys. Rev. Lett. 91, 240402 (2003).
[24] L. D. Carr, G.V. Shlyapnikov, and Y. Castin, e-print condmat/0308306 [Phys. Rev. Lett. (to be published)].
[25] The condensate peak broadened for longer d and could
not be reliably discerned after 200 s.
[26] In the 40 K experiment, the ratio of the effective sweep
rate through resonance and Fermi energy was similar [12].

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Appendix D
Feshbach Resonances in Fermionic
6Li
This appendix contains a reprint of Ref. [217]: C. H. Schunck, M. W. Zwierlein, C.
A. Stan, S. M. F. Raupach, and W. Ketterle, A. Simoni, E. Tiesinga, C. J. Williams,
and P. S. Julienne, Feshbach resonances in fermionic 6 Li, Phys. Rev. A 71, 045601(2004).

208

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Feshbach resonances in fermionic 6Li
C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge,
Massachusetts 02139, USA

A. Simoni, E. Tiesinga, C. J. Williams, and P. S. Julienne
National Institute of Standards and Technology, 100 Bureau Drive stop 8423, Gaithersburg, Maryland 20899-8423, USA
sReceived 14 July 2004; published 5 April 2005d
Feshbach resonances in 6Li were experimentally studied and theoretically analyzed. In addition to two
previously known s-wave resonances, three p-wave resonances were found. Four of these resonances are
narrow and yield a precise value of the singlet scattering length. The position of the broad s-wave resonancenear 83 mT is mostly sensitive to the triplet potential. It was previously determined in a molecule-dissociation
experiment for which we, here, discuss systematic shifts.
DOI: 10.1103/PhysRevA.71.045601

PACS numberssd: 03.75.Ss, 32.80.Pj, 34.50.Pi

Interactions in ultracold atomic gases can be magneticallytuned using Feshbach resonances f1g. A Feshbach resonanceoccurs when the energy of two colliding atoms is nearly
degenerate to the energy of a bound molecular state. Tunableinteractions have been used to explore novel phenomena incollisional and many-body physics. Recently, Feshbach resonances have been used to control pairing processes in ultracold fermionic gases. This led to the achievement of BoseEinstein condensation sBECd of molecules in 40K f2g and 6Li
f3–5g and to the first studies of the BEC-BCS crossover, thecontinuous transition of fermion pairs from weakly boundmolecules to long-range Cooper pairs f5–11g.
Most experiments in 6Li have been carried out in the vicinity of the s-wave Feshbach resonance near 830 G
f5,7–11g s1 G = 10−4 Td. The quantitative interpretation ofthese experiments and the characterization of the BEC-BCS
crossover require a precise knowledge of the resonance location. However, its determination is not trivial since theresonance width is extremely large and the line shape isstrongly affected by many-body effects. In our previous work
we determined the position of this resonance by the onset ofmolecule dissociation to be 822± 3 G f8g.
In this paper we report on a detailed study of Feshbachresonances in the two lowest hyperfine states of 6Li with thegoal of accurately characterizing the interaction potential of
two 6Li atoms. In addition to two previously known s-wave
resonances, we find three p-wave resonances f12g. The positions of the p-wave resonances together with the location of
the narrow s-wave resonance near 543 G are used for a precise determination of the singlet s-wave scattering length.
These results, however, do not constrain the position of thebroad resonance, which also depends on the triplet scattering
length. An improved measurement of its location is presented
and the magnitude and the origin of possible systematic errors are discussed.
The experimental setup has been described in Ref. f13g.
Up to 4 3 107 quantum degenerate 6Li atoms in the uF , mFl= u3 / 2 , 3 / 2l state were obtained in a magnetic trap by sympathetic cooling with 23Na. The 6Li atoms were then transferred into an optical dipole trap sODTd formed by a focused1050-2947/2005/71s4d/045601s4d/$23.00

1064-nm laser beam with a maximum power of 9 W. In the
optical trap a single radio-frequency sweep transferred theatoms to state u1l suF , mFl = u1 / 2 , 1 / 2l at low fieldd. A subsequent Landau-Zener sweep at an externally applied magnetic
field of 565 G could then be used to either prepare the entire
sample in state u2l su1 / 2 , −1 / 2l at low fieldd or create anequal mixture of atoms in states u1l and u2l. Except for themeasurement of the broad s-wave Feshbach resonance, allresonances were observed by monitoring magnetic-fielddependent atom losses. Atom numbers were obtained fromabsorption images taken at zero field. The externally appliedfield was calibrated by driving microwave transitions fromstate u2l to state u5l su3 / 2 , 1 / 2l at low fieldd at several magnetic fields close to resonance positions and from state u2l tostate u3l su3 / 2 , −3 / 2l at low fieldd at high magnetic fields
around 800 G.
For spin-polarized samples either in state u1l or u2l s-wave
scattering is forbidden by symmetry; therefore, the observed
resonances occur in the p-wave channel. The same molecularstate that is responsible for these two resonances also causes
a p-wave resonance in the u1l + u2l mixture. The three p-wave
resonances were observed in clouds with typical temperatures T , 6 mK and T / TF . 0.5– 1.5, where TF is the Fermi
temperature. Radial and axial trap frequencies were typically
vr = 2p 3 1.0 kHz and va = 2p 3 6.9 Hz.
The position of the p-wave resonance in the collision of apair of state u1l atoms was determined by first ramping themagnetic field to approximately 5 G below the resonance.
Using an additional power supply to precisely change themagnetic field within a 10 G range, the field was thenswitched in 1 ms to a test value Btest. Here the atoms were
kept for 200 ms before the field and the optical trap were
switched off. Finally, atom number versus Btest was recorded.
Resonantly enhanced losses due to inelastic three-body decay led to a Lorentzian shaped feature as shown in Fig. 1sad.
Resonance positions and widths are summarized in Table I.
The same technique was used to determine the u1l + u2l
and u2l + u2l p-wave resonances. The resonance line shapes
are asymmetric ssee Fig. 1d, possibly due to threshold effects
f14,15g. The splitting of a p-wave resonance due to spin-spin
interactions f16g is for these resonances more than one order045601-1©2005 The American Physical Society

PHYSICAL REVIEW A 71, 045601 s2005d

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FIG. 1. p-wave resonances for u1l + u1l sad, u1l + u2l sbd, and u2l+ u2l scd collisions. Dashed lines are Lorentzian fits to the data. The
results are summarized in Table I.

TABLE I. Position of the Feshbach resonances. Given are theexperimentally and theoretically determined resonance locations
Bexpt and Btheory, respectively, and the measured resonance width.
The uncertainties for the experimental data in the first four rows are
dominated by magnetic field drifts between the measurement of theresonance and the field calibration for which we find an upperbound of 80 mG. For the u1l + u1l resonance an additional drift wasmonitored. The statistical error of determining the line center andthe estimated uncertainty due to asymmetric line shapes are negligible. The quoted linewidths are not corrected for source depletiondue to atom loss. We estimate that this effect reduces the linewidths
by 25%–40%. For the broad s-wave resonance sfifth rowd only arange is given. See the text for a discussion.
StatesWave

Bexpt fGg

Btheory fGg

Width fGg

u1l + u1l
u1l + u2l
u2l + u2l
u1l + u2l
u1l + u2l

p
p
p
s
s

159.14± 0.14
185.09± 0.08
214.94± 0.08
543.28± 0.08
822…834

159.15s4d185.15s4d214.90s4d
543.27s5d

0.4
0.2
0.4
0.4

of magnitude smaller than the width and could not be discerned with our sensitivity.
The position of the s-wave resonance near 543 G in the
u1l + u2l mixture was determined as presented above in clouds
with typical temperatures of 6 mK, but in a slightly deeper
optical trap and with an extended holdtime of 2900 ms at
Btest. The result of a fit to the Lorentzian lineshape is given in
Table I. This s-wave resonance was first reported in f17g andcalculated in f18g.
To determine the position of the broad s-wave Feshbachresonance near 830 G a different method was required. The
resonance was identified as the onset of molecular dissociation f6,8,19g. Molecules were first created on the repulsive
sBECd side of the Feshbach resonance and then dissociatedinto atoms when the magnetic field crossed the resonance.
However, this method is subject to systematic shifts in theresonance position that depend on the molecular density andthe speed of magnetic field ramps. To control the densitydependent shift, the molecular density was varied by using
different parameters for the optical dipole trap and by performing the dissociation at different times of flight.
The starting point of the experiment was an almost pure6
Li2 molecular BEC that was prepared at a magnetic field of
about 780 G in the optical trap as described in Ref. f8g. The
data shown in Fig. 2 were obtained by releasing the molecules from the optical trap at 780 G f20g. After 2 ms thefield was ramped to a test value Btest in 14 ms. In these first
16 ms time of flight the peak molecular density dropped bythree orders of magnitude to nmol = 5 3 109 cm−3. The magnetic field was held at Btest for another 5 ms before it was
ramped down. The critical field ramp, which can alter the
resonance position, is the initial phase of the magnetic fieldramp down in which the molecules are still in the resonanceregion. Here, fast ramps can dissociate weakly bound molecules. However, we could only use a limited time of flightwhile maintaining a good signal-to-noise ratio. Therefore thefield was ramped down in two steps: at an initial rate of
100 G / ms for 2 ms to leave the resonance region, followedby an exponential decay with time constant 30 G / ms whichbrought the field to zero in 3 ms. To better control the effectsof the field ramp, the experiment was repeated for different
initial switch off speeds. Finally, the sample was imaged
with light which was resonant only with unbound atoms; thepossible molecular transitions are far detuned from theatomic transition at zero field. By monitoring the atom number as a function of Btest the onset of molecule dissociation
was observed. The data in Fig. 2 show the onset at 821± 1 G.
The slow approach of the atomic signal to unity reflects thetime constant of dissociation and the possible reconversion
of atoms into molecules during the magnetic field switch off.
In our analysis only the onset of the atomic signal was evaluated.
We now consider the two sources of systematic errors
mentioned above in more detail. Few-body collisions might
dissociate molecules when their size, which near resonanceis on the order of the scattering length between the constituent atoms f21g, becomes comparable to the mean distance−1/3. The scattering length near
between the molecules, a , nmol
resonance is parametrized by a = abgf1 + DB / sB − B0g< abgDB / sB − B0d, where abg is the negative background scat-

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FIG. 2. Determination of the position of the broad s-wave
Feshbach resonance. sad Onset of dissociation of molecules into
atoms at 821± 1 G. sbd The resonance position was obtained byfitting two lines to the data points near the threshold, one horizontal
through the points showing no atomic signal and a second line
following the initial rise in atom number. The intersection of thetwo lines gives the resonance position; the estimated uncertainty ofthis point is ±1 G.

tering length, B0 is the resonance position, and DB is theresonance width. So molecule dissociation will become important at a magnetic field B for which abgDB / sB − B0d−1/3. For the broad resonance, this density-dependent,, nmolfew-body effect is expected to shift the observed resonanceposition to lower magnetic fields.
The second systematic error is a density-independent,
single-molecule effect. Switching off the magnetic field becomes nonadiabatic close to resonance and destroys very
weakly bound molecules f22g. If a molecule with binding
energy "v = "2 / sma2d is forced to change its size faster thanits oscillation frequency si.e., if ȧ / a @ vd, the molecule may
dissociate. With the magnetic field dependence of a given
above, the rate ȧ / a , Ḃ / sB − B0d becomes comparable to v
, sB − B0d2 at a magnetic field that is shifted from the resonance location B0 by DB = B − B0 , Ḃ1/3. This expression
gives the scaling of the ramp-induced systematic error withthe ramp speed Ḃ.
To find the order of magnitude of these shifts we havedetermined the resonance locations for three different ramp
rates at constant density and for three different densities atconstant ramp rate.
At a molecular density of nmol = 1.53 1010 the resonancepositions were measured at initial ramp speeds of 30 G / mssfastest possible switch offd, 100 G / ms sfastest externallycontrolled rampd, and 12.5 G / ms scontrolled rampd. For the
fastest switch off the onset of dissociation occurs at
793± 7 G, for the other two controlled ramps at 822± 3 G.
Assuming that no density shifts affect these data, one canextrapolate to zero ramp speed based on the sB − B0d ~ Ḃ1/3
dependence. In this way we find a resonance position of
825± 3 G.
For a fixed initial ramp speed of 100 G / ms the resonancelocations were determined at densities of 5 3 109 cm−3, 1.5
3 1010 cm−3, and 1.23 1012 cm−3 to be 821± 1 G, 822± 3 G,
and 800± 8 G, respectively. Here one can use the sB − B0d~ n1/3 dependence to extrapolate to a resonance position of
825± 3 G, neglecting effects due to nonadiabatic magnetic
field ramps.
Both systematic effects shift the maximum magnetic fieldvalue at which the molecules are stable to lower magnetic
fields. In a simple picture, one would expect the total shift tobe the larger of the two. However, if they are similar, as inour case, they may add or combine in a more complicatedway. We have measured the threshold position at low density
and slow ramp rates to be 822± 3 G and determined two
shifts of 3 ± 3 G. Therefore, we expect the position of the
Feshbach resonance to be between 822 and 834 G. A moreaccurate extrapolation requires measuring the dissociationthreshold for more ramp speeds and densities. However,technical limitations in varying magnetic field ramp speeds
and an unfavorable signal-to-noise ratio at lower densities
precluded this.
All Feshbach resonances discussed in this paper are dueto the v = 38 vibrational state of the singlet potential withtotal electronic spin S equal to zero. The p-wave resonances
have a total nuclear spin I = 1, while the 543 G and broads-wave resonances have I = 2 and I = 0, respectively.
The resonance locations are compared with results ofscattering coupled-channel calculations. We locate the resonance from the maximum of the elastic cross section as afunction of magnetic field. The collision energy is fixed at
E = kBT, where kB is the Boltzmann constant and T is theexperimental temperature. Our collision model, described indetail in Ref. f18g, treats the singlet and triplet scattering
lengths as adjustable parameters. The triplet state has a total
electron spin equal to one. It turns out that all narrow resonances, which could be accurately located, are insensitive tothe triplet scattering length. Only s- and p-waves are included in the calculation. Fitting the singlet scattering length
aS to the field locations given in the first four rows of Table
I yields a very accurate value of aS = 45.1591s16da0, wherea0 = 0.052 917 7 nm. With this value, the resonance positions
given in the third column of Table I were calculated at acollision energy equal to kBT. The agreement with the experimental values is excellent. The location of the s-wave
resonance is also in very good agreement with the determination of Ref. f23g, 543.26s10d G.
Our theoretical uncertainties do not include contributions
due to a thermal average. Moreover, there can be a discrepancy between the field values at which the observed threebody loss rate and the theoretical two-body elastic cross section are maximal. Experimental observations on 40K f14g are

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not conclusive about the magnitude of this shift, although
they suggest it is well within the linewidth of the observed
loss features. As an estimate of our possible systematic error,a shift in the resonance position dsGd will give rise to a shift
from our best aS of s−0.0365dda0.
The broad resonance is caused by a hyperfine-induced
mixing between a singlet vibrational level and an almostbound virtual state of the triplet potential, a situation analyzed in f24,25g. It is the virtual state that gives rise to thelarge and negative triplet scattering length aT of 6Li. Mixing
occurs for magnetic field values above 500 G. In fact, in
absence of the hyperfine mixing, the resonance would occur
around 550 G. The coupling shifts the resonance by a few
hundred gauss. For typical Feshbach resonances, these shifts
are no more than a few gauss. A consequence of the large

shift is that the resonance location depends critically on theless well known triplet potential.
In conclusion, we have found three p-wave Feshbachresonances in 6Li. Together with the narrow s-wave resonance they give a precise value of the singlet scattering
length. The position of the broad resonance could not be
constrained using the refined singlet potential. The determination of the position of the broad resonance via molecule
dissociation is subject to systematic errors, which shift theonset of dissociation to lower magnetic fields.

f1g E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 47,
4114 s1993d; W. C. Stwalley, Phys. Rev. Lett. 37, 1628
s1976d; S. Inouye et al., Nature sLondond 392, 151 s1998d; P.
Courteille et al., Phys. Rev. Lett. 81, 69 s1998d.
f2g M. Greiner, C. A. Regal, and D. S. Jin, Nature sLondond 426,
537 s2003d.
f3g S. Jochim et al., Science 302, 2101 s2003d.
f4g M. W. Zwierlein et al., Phys. Rev. Lett. 91, 250401 s2003d.
f5g T. Bourdel et al., Phys. Rev. Lett. 93, 050401 s2004d.
f6g C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92,
040403 s2004d.
f7g M. Bartenstein et al., Phys. Rev. Lett. 92, 120401 s2004d.
f8g M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 s2004d.
f9g J. Kinast et al., Phys. Rev. Lett. 92, 150402 s2004d.
f10g M. Bartenstein et al., Phys. Rev. Lett. 92, 203201 s2004d.
f11g C. Chin et al., Science 305, 1128 s2004d.
f12g We presented most of the results of the present paper at the
KITP workshop in Santa Barbara, 2004. Similar results wereindependently obtained in Paris f15g.

f13g Z. Hadzibabic et al., Phys. Rev. Lett. 91, 160401 s2003d.
f14g C. A. Regal et al., Phys. Rev. Lett. 90, 053201 s2003d.
f15g J. Zhang et al., Phys. Rev. A 70, 030702sRd s2004d.
f16g C. Ticknor et al., Phys. Rev. A 69, 042712 s2004d.
f17g K. Dieckmann et al., Phys. Rev. Lett. 89, 203201 s2002d.
f18g K. M. O’Hara et al., Phys. Rev. A 66, 041401sRd s2002d.
f19g T. Mukaiyama et al., Phys. Rev. Lett. 92, 180402 s2004d.
f20g The ODT power was 36 mW, yielding trap frequencies of vr= 2p 3 690 Hz radially and va = 2p 3 12.5 Hz axially. The
axial frequency is mostly due to magnetic field curvature.
f21g T. Kohler et al., Phys. Rev. Lett. 91, 230401 s2003d.
f22g J. Cubizolles et al., Phys. Rev. Lett. 91, 240401 s2003d.
f23g K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev.
Lett. 91, 080406 s2003d.
f24g B. Marcelis et al., Phys. Rev. A 70, 012701 s2004d.
f25g E. G. M.v. Kempen, B. Marcelis, and S. J. J. M. F. Kokkelmans, Phys. Rev. A 70, 050701sRd s2004d.

The MIT research is supported by NSF, ONR, ARO, andNASA. S.R. acknowledges financial support from the Dr.
Jürgen Ulderup Foundation.

045601-4

Appendix E
Formation Dynamics of a Fermion
Pair Condensate
This appendix contains a reprint of Ref. [270]: M.W. Zwierlein, C. H. Schunck, C. A.
Stan, S. M. F. Raupach, and W. Ketterle, Formation Dynamics of a Fermion Pair
Condensate, Phys. Rev. Lett. 94, 180401 (2005).

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Formation Dynamics of a Fermion Pair Condensate
M. W. Zwierlein, C. H. Schunck, C. A. Stan, S. M. F. Raupach, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT,
Cambridge, Massachusetts 02139, USA(Received 23 December 2004; published 9 May 2005)
The dynamics of pair condensate formation in a strongly interacting Fermi gas close to a Feshbachresonance was studied. We employed a phase-shift method in which the delayed response of the manybody system to a modulation of the interaction strength was recorded. The observable was the fraction ofcondensed molecules in the cloud after a rapid magnetic field ramp across the Feshbach resonance. The
measured response time was slow compared to the rapid ramp, which provides final proof that themolecular condensates reflect the presence of fermion pair condensates before the ramp.
DOI: 10.1103/PhysRevLett.94.180401

PACS numbers: 03.75.Ss, 05.30.Fk

Atomic Fermi gases close to a Feshbach resonance [1]offer the unique possibility of studying many-body phenomena in a strongly interacting system with tunable interactions. Recently a major focus has been on condensatesof pairs of fermionic atoms [2 –8]. By changing the magnetic field, the interaction strength between atoms in two
spin states can be varied. That way, condensates of either
tightly bound molecules or of extended pairs of fermionscan be created, whose size can become comparable or evenlarger than the interparticle spacing. The description of thisso-called BEC-BCS crossover [9] is an active frontier inmany-body physics with still controversial interpretations[10 –13].
The control of interactions via magnetic fields does not
only give access to very different physical regimes, it also
allows one to apply a time-varying interaction strength[14,15] and to study the dynamics of a many-body system
in novel ways. This was used in recent experiments inwhich molecular condensates were observed after a rapid
field ramp from the BCS to the Bose-Einstein condensate(BEC) side of the Feshbach resonance [6,7]. It was argued
that if the ramp time was faster than the formation time of amolecular condensate, its presence after the sweep necessarily reflected a preexisting condensate of fermion pairs.
However, without access to that formation time, secondary
evidence was gathered, namely, the invariance of the condensate fraction under variations of the sweep rate [6] or ofthe density immediately before the ramp [7]. This excluded
simple models of the molecular condensate formation
during the ramp, but left room for more sophisticatedmany-body effects. In particular, the time to cross the
Feshbach resonance in these experiments was not fasterthan the unitarity limited collision time / hE 1
F , and therefore dynamics during the sweep could not be ruled out.
Here we present an experimental study of the formation
dynamics of a fermion pair condensate on the BCS side of
the Feshbach resonance [16]. We employ a novel phaseshift method, which records the delayed response of the
many-body system to a modulation of the magnetic fieldthat changes periodically its interaction strength. The ob0031-9007=05=94(18)=180401(4)$23.00

servable is again the molecular condensate fraction after a
rapid sweep to the BEC side of the Feshbach resonance. Itssensitivity to changes in the scattering length on the BCS
side [6,7] arises through the dependence of the critical temperature for pair condensation on the interaction strength.
By showing that the delayed response time of the molecular condensate fraction is long compared to the sweep
times used in the present and previous experiments, we infer that the observed condensates could not have been created during the rapid transfer and thus must originate frompreexisting fermion pair condensates. However, we do findevidence that condensed pairs are more likely to be transferred into molecules than thermal pairs. Therefore, in contrast to assumptions made in previous work [6,7], the molecular condensate fraction after the ramp may not equal
the fraction of condensed atom pairs above resonance.
The experimental setup was the same as in our previous
work [7]. A degenerate cloud of 6 Li, sympatheticallycooled with 23 Na, was loaded into an optical dipole trap
to access a broad Feshbach resonance at 834 G [17,18]
between the two lowest hyperfine states of 6 Li, labeled j1i
and j2i. An equal mixture of these states was evaporatively
cooled at 770 G using an exponential ramp-down of the
optical trap to 15 mW. This resulted in an essentially pureBose-Einstein condensate of 3  106 molecules. An upper
limit for the temperature of the gas is TTF < 0:2, with the
Fermi temperature TF given by the zero-temperature, ideal
1=3
, !=2 is the geometric meangas relation TF  h!3N
of the trapping frequencies, and N is the total atom number.
Next, the trap was recompressed to 25 mW(trap frequenp
cies: x  y  580 Hz, z  12:1 Hz 0:2 B with themagnetic field B in kG) and the magnetic field was adiabatically increased in 500 ms to 1000 G, the starting point
for the following experiments. Here, in the wings of the
Feshbach resonance, the scattering length a was still sufficiently large and negative for the gas to be in the stronglyinteracting regime, with kF jaj  1:6 at a Fermi energy ofEF  2:0 K and a Fermi wave number kF  1=2700a0 .
The temperature at this point could therefore not be reli-

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 2005 The American Physical Society

PRL 94, 180401 (2005)

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PHYSICAL REVIEW LETTERS

ramp did not follow the magnetic field modulation instantaneously, but lagged behind. At a Fermi energy of EF 
2 K, the peak condensate fraction was delayed by R 
500 100 s with respect to the magnetic field’s closest
approach to resonance [22]. This time scale was independent of the modulation frequency [compare Figs. 2 and4(a), below]. This also rules out that our results are affectedby the excitation of collective modes. R equals 130 times
the unitarity limited collision time, hE
 1
F  3:8 s. The
rapid magnetic field ramp utilized here and in [7] traversed
the Feshbach resonance in less than 10 s, which is muchsmaller than R .
This delay time can be interpreted as the relaxation timeof the fermionic condensate. In a normal Fermi gas of N
particles at temperatures much smaller than the Fermi
temperature TF , relaxation occurs through collisions between the thermally excited particles close to the Fermi
surface, whose number scales as Nth ’ N kEBFT . The numberof available scattering states again being proportional tokB TF kET
2 . In general, if
EF , the relaxation time will be R ’ h
B
the Fermi surface is smeared out over an energy width E,EFthe relaxation time is ’ h E
2 . This formula with E  should apply also to the superfluid state [23] when the gap
parameter  is rapidly changed to a much smaller value.
Generally, one would expect E to be the larger of  andkB T. Using R  500 s, we obtain the estimate E 
0:1EF which may set an upper bound for both temperatureand pairing gaps.
A decay is superimposed to the periodic modulation of
the condensate fraction. It could be caused by heating due
to the nonadiabaticity of the process. Another source ofheating could be the excitation of the cloud via the small
accompanying variation of the magnetic field curvature.
0.6
0.5
0.4
0.3
840

0.2

880
920

FIG. 1. Imaging of molecular condensates. The rapid ramp tozero field after release from the trap created a cloud containing
both molecules and unpaired atoms. A Stern-Gerlach field
gradient separated atoms (magnetic moment 13 B for states
j1i and j2i, respectively) from molecules, which are purely
singlet at zero field. At the end of 5 ms of ballistic expansion,
the molecules were dissociated in a fast ramp (in 3 ms to1200 G) across the Feshbach resonance. After another 2 ms
expansion again at zero field, an absorption image of the separated clouds was taken. Condensate fractions were determinedfrom the molecular cloud, and the numbers in each component
were recorded. An absorption image is shown on the bottom, thefield of view is 3 mm  1 mm.

960
0

2

4

6

8

10

12

14

Magnetic Field [G]Molecular Condensate Fraction

ably determined, but is expected to be significantly lower
than the one on the BEC side due to adiabatic cooling [19].
Subsequently, the magnetic field and thus the interaction
strength in the gas were modulated at frequencies in therange of 100 –500 Hz, and an amplitude of about 50 G [20].
At a variable time t after the start of the modulation, thefraction of condensed fermion pairs was recorded by timeof-flight analysis.
To identify fermionic condensates across the resonanceregion, we proceeded as in [6,7]. Immediately after the
release of the cloud from the optical potential, the magnetic
field was switched to zero field (initial ramp rate
30 G=s), where further expansion of the cloud took
place. This rapid ramp out of the resonance region transformed large fermion pairs into deeply bound moleculeswith high efficiency [21]. Figure 1 details the imaging
procedure used to determine molecular condensate fractions and the number of unpaired atoms in each state after
the ramp. In our previous work, we showed that the condensate fractions had a peak around the Feshbach resonance and fell off on either side [7]. Here, this dependence
was exploited to observe the delayed response of thesystem to the magnetic field modulation on the BCS side.
Figure 2 shows the main result of this Letter: The
condensate fraction in the molecular clouds after the rapid

16

Time [ms]FIG. 2. Measurement of the relaxation time of fermionic paircondensates. Shown is the delayed response of the observed
condensate fraction (data points and thick line to guide the eye)to a 250 Hz magnetic field modulation (thin line) on the BCS
side of the Feshbach resonance at 834 G. The condensates weredetected as described in Fig. 1. Three measurements per point
were taken in random order, the size of the data points reflecting
the standard deviation. The vertical lines indicate the points ofmaximum condensate fraction, which are delayed with respect tothe times at which the magnetic field is closest to resonance.

180401-2

800
950
700
900

850
1

2

3

4
Time [ms]

5

6

7

8

FIG. 3. Total number of detected atoms (unbound atoms andmolecules) after the rapid ramp (same data set as in Fig. 4). It is
modulated in phase with the magnetic field. For initial fields
close to resonance, more atoms are ‘‘missing’’ after the rapid
ramp.

Fraction of Molecules Magnetic Field [G] Molecular Condensate Fraction

900

Magnetic Field [G]Total Number of Detected Atoms [arb. units]

1000

observed after the sweep and Ntotal the total number ofatoms before the sweep (this definition includes the disappeared molecules) [24]. If the molecule fraction would
follow the instantaneous magnetic field, it would again
reflect the two-body physics during the sweep. Instead,
we observe a delayed response in perfect correlation withthe condensate fraction (Fig. 4). Since the delay timereflects the many-body physics of condensate formation,
this is clear evidence that the molecule conversion efficiency depends on the initial many-body state.
One consistent explanation of these findings is thatfermion pairs in the condensate are more completely transferred into tightly bound molecules than thermal pairs.
With this assumption, we extrapolated the fitted line in
Fig. 4(b) to a zero condensate fraction to obtain the transfer
efficiency from thermal atom pairs into molecules (including the missing fraction) as pth  75% [21]. Extrapolating
towards the other limit, we do not expect any unpaired
atoms after the ramp already for a condensate fraction of
80% [25], suggesting a transfer efficiency for condensed
fermion pairs into molecules of p0  100%. This effectwould lead to an overestimate of the fermionic condensate
fraction before the sweep. Small condensate fractionsth
could be overestimated by as much as p0pp
 33%. The
thlargest absolute error occurs for an initial pair condensate

a)

0.40
0.35
0.30
0.25

0.88

0.20

0.86

0.15

0.84

0.10

0.82
0.80

850

Fraction of Molecules

Despite the decay of the condensate fraction, the relaxation
time was constant for subsequent cycles of modulation,
within the (limited) accuracy of our measurement.
In a compressed trap of p  150 mW, at a 1.8 times
higher Fermi energy of 3:6 K, the measured delay timewas R ’ 230 100 s. BCS theory predicts that therelaxation time should scale with density like R /=kF jaj, giving R ’ 200 s for this experiment perE1
F e
formed around 900 G. However, we regard this agreementwith observation as fortuitous since BCS theory cannot be
rigorously applied, and finite temperature effects may contribute to the relaxation.
We now discuss further observations regarding the efficiency of converting atoms into molecules. Since the relaxation time introduces some hysteresis, we observe thesame condensate fraction at two different magnetic fieldvalues. Therefore, in contrast to equilibrium experiments[6,7], we can distinguish the dependence of the conversion
efficiency on condensate fraction and magnetic field.
Figure 3 shows that the total number of detected atoms(in both the atom and the molecule channels) was modulated by the magnetic field. We assume that this instantaneous response reflects the two-body physics during themagnetic field sweep. In a simple two-state Landau-Zenermodel, the initial magnetic field and the sweep rate determine what fraction of the atoms appears as bound molecules. However, the total number of bound or unboundatoms should be constant in contrast to our observations.
This is evidence for the presence of other molecular states(e.g., lower lying vibrational states) which are populated
during the magnetic field sweep, and the population islarger for initial magnetic fields closer to the Feshbachresonance. Note that the determination of the condensate
fraction is immune against those ‘‘disappeared’’ molecules, since the two-body physics does not depend on thecenter-of-mass motion of the atom pair.
We now look at the molecular fraction which we define
as 1  Natom =Ntotal , where Natom is the number of atoms

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PHYSICAL REVIEW LETTERSPRL 94, 180401 (2005)

0.78

900
950
0

1

2

3

4

5

6

7

8

Time [ms]
1.0
0.9

b)

0.8
0.7
0.0

0.2

0.4

0.6

0.8

Molecular Condensate Fraction

FIG. 4. Correlation between the observed condensate fraction
and the molecular fraction. Shown are (a) the condensate fraction vs time during a 500 Hz field modulation (circles), thefraction of molecules (triangles) and the magnetic field. Unlike
the total detected signal (Fig. 3), the molecular fraction ismodulated not in phase with the magnetic field, but in complete
correlation with the condensate fraction. (b) The molecular
fraction vs the condensate fraction, together with a fitted line
through the data.

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PHYSICAL REVIEW LETTERS
13 MAY 2005
PRL 94, 180401 (2005)p
pth[8] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, andp  46% and would be about 7% in ourfraction of pp0
pth
J. E. Thomas, Phys. Rev. Lett. 92, 150402 (2004).
case.
[9] D. M. Eagles, Phys. Rev. 186, 456 (1969); A. J. Leggett, inThis effect can have several possible origins: One is that
Modern Trends in the Theory of Condensed Matter,
the atomic separation in a condensed atom pair is smaller
Proceedings of the XVIth Karpacz Winter School ofthan that of two uncondensed atoms. Also, the presence ofTheoretical Physics, Karpacz, Poland, 1980 (Springera large pair condensate increases the density of the cloud
Verlag, Berlin, Karpacz, Poland, 1980), pp. 13–27;
P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys.
[26]. Finally, if there are incoherent processes involved
59, 195 (1985).
during the rapid ramp, bosonic stimulation into the mo[10] G. M. Falco and H. T. C. Stoof, Phys. Rev. Lett. 92,
lecular condensate could play a role.
130401 (2004).
In conclusion, we have determined the intrinsic time
[11]
R.
A. Barankov and L. S. Levitov, Phys. Rev. Lett. 93,
scale for the growth of a fermion pair condensate by
130403 (2004).
observing the delayed response of the system to a change[12] Roberto B. Diener and Tin-Lun Ho, cond-mat/0404517.
in its interaction strength. For our trap parameters, the[13] S. Simonucci, P. Pieri, and G. C. Strinati, cond-mat/
response was delayed by  500 s. This time is far longer
0407600.
than the time spent within the resonance region during the[14] M. R. Matthews, D. S. Hall, D. S. Jin, J. R. Ensher, C. E.
conversion of fermion pairs into molecules. This provides
Wieman, E. A. Cornell, F. Dalfovo, C. Minniti, andfinal proof that the observed molecular condensates origiS. Stringari, Phys. Rev. Lett. 81, 243 (1998).
[15] M. Greiner, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 94,
nated from condensates of pairs of fermions above the070403 (2005).
resonance. Regarding the two-body physics of the rapid[16] We presented the main result of the present Letter at thetransfer, we found that there is a missing fraction of
KITP workshop in Santa Barbara, May 10 –14, 2004.
particles after the ramp, presumably transferred into un[17]C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F.
observed molecular states. We found evidence that conRaupach, W. Ketterle, A. Simoni, E. Tiesinga, C. J.
densed fermion pairs are more efficiently transformed into
Williams, and P. S. Julienne, Phys. Rev. A 71, 045601
molecules than thermal pairs during the rapid ramp. Thus,(2005).
the observed molecular condensate fractions tend to over[18] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen,
estimate the initial fermion pair condensate fraction.
S. Jochim, C. Chin, J. H. Denschlag, R. Grimm,
This work was supported by the NSF, ONR, ARO, and
A. Simoni, E. Tiesinga, C. J. Williams, and P. S.
NASA. We thank Michele Saba for the critical reading ofJulienne, Phys. Rev. Lett. 94, 103201 (2005).
[19] L. D. Carr, G. V. Shlyapnikov, and Y. Castin, Phys. Rev.
the manuscript. S. Raupach is grateful to the Dr. Jürgen
Lett. 92, 150404 (2004).
Ulderup foundation for financial support.

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47, 4114 (1993); W. C. Stwalley, Phys. Rev. Lett. 37, 1628
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D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys.
Rev. Lett. 81, 69 (1998).
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426, 537 (2003).
[3] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl,
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Science 302, 2101 (2003).
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[5] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang,
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[20] The instantaneous magnetic field was determined by probing the atoms using a Zeeman-sensitive optical transition.
For the 500 Hz modulation, the deduced field followed themodulation current with a time delay of 85 5 s and areduced amplitude of 95% compared to the dc situation.
This was due to induced eddy currents in the apparatus.
[21] The transfer probability depends on the ramp speed and on
the density of the cloud. In a tighter trap with 150 mW ofpower we cannot discern any unpaired atoms after theramp.
[22] This was far shorter than evaporation time scales, whichwere on the order of 100 ms.
[23] R. A. Barankov, L. S. Levitov, and B. Z. Spivak, Phys. Rev.
Lett. 93, 160401 (2004).
[24] Ntotal was determined in a different cycle of the experiment, for which the rapid ramp was omitted.
[25] Indeed, on resonance we observe almost pure condensates,and only a negligible amount of unpaired atoms after the
ramp. Note that fitting routines, saturated absorption, andimaging noise all tend to underestimate condensate fractions.
[26] This is not true in the BCS regime, where the atomic
density is independent of the presence of a condensate.
Still, an additional overestimation of the condensate fraction comes from the fact that the condensed pairs areconcentrated in the high density region of the cloud, where
the conversion efficiency is higher. However, this does notaffect the total molecular signal.

180401-4

Appendix F
Vortices and Superfluidity in a
Strongly Interacting Fermi Gas
This appendix contains a reprint of Ref. [265]: M. W. Zwierlein, J. R. Abo-Shaeer, A.
Schirotzek, C. H. Schunck, and W. Ketterle Vortices and Superfluidity in a Strongly
Interacting Fermi Gas, Nature 435, 1047 (2005), (in Nature Articles).

218

Vol 435|23 June 2005|doi:10.1038/nature03858

ARTICLES
Vortices and superfluidity in a stronglyinteracting Fermi gas
M. W. Zwierlein1, J. R. Abo-Shaeer1†, A. Schirotzek1, C. H. Schunck1 & W. Ketterle1
Quantum degenerate Fermi gases provide a remarkable opportunity to study strongly interacting fermions. In contrast toother Fermi systems, such as superconductors, neutron stars or the quark-gluon plasma of the early Universe, thesegases have low densities and their interactions can be precisely controlled over an enormous range. Previous
experiments with Fermi gases have revealed condensation of fermion pairs. Although these and other studies wereconsistent with predictions assuming superfluidity, proof of superfluid behaviour has been elusive. Here we reportobservations of vortex lattices in a strongly interacting, rotating Fermi gas that provide definitive evidence forsuperfluidity. The interaction and therefore the pairing strength between two 6Li fermions near a Feshbach resonancecan be controlled by an external magnetic field. This allows us to explore the crossover from a Bose–Einstein condensate
of molecules to a Bardeen–Cooper–Schrieffer superfluid of loosely bound pairs. The crossover is associated with a newform of superfluidity that may provide insights into high-transition-temperature superconductors.
The first observations of Bose–Einstein condensates (BECs) of
molecules consisting of loosely bound fermionic atoms1–3 initiateda series of explorations4–12 of the crossover between a BEC and a
Bardeen–Cooper–Schrieffer (BCS) superfluid13–15. When an external
magnetic field is varied across a Feshbach resonance, these molecules
transform adiabatically into the Cooper pairs of a BCS superfluid. All
physical properties are expected to vary smoothly throughout thiscrossover. For example, the size of fermion pair condensatessmoothly increases from the BEC- to the BCS-side of the resonance,
while the strength of the bond between two paired atoms smoothly
decreases. The similar size and shape of normal and condensed gas
clouds makes it difficult to detect condensation on the BCS-side.
However, using a rapid magnetic field sweep back to the BEC-side,
introduced by the Boulder group4 , pair condensation wasobserved4,5,12. Although Bose–Einstein condensation and superfluidity are intimately connected, they do not necessarily occur together.
In lower dimensions, superfluidity occurs in the absence ofBose–Einstein condensation16. An ideal Bose gas or disordered
three-dimensional system can have a condensate, but shows nosuperfluidity16. Phase fluctuations, which are probably present inshort-lived condensates of fermionic pairs1, can suppress superfluid
behaviour.
Several ground-breaking studies in Fermi gases of hydrodynamic
expansion7,17, collective excitations8,9, thermodynamic properties11
and the binding energy of pairs10 were suggestive of superfluid
behaviour or were consistent with theoretical calculations predicting
superfluidity, but did not provide unambiguous evidence. In themeantime, several theoretical papers18–22 emphasized that therotational properties of a gas of fermion pairs could directly reveal
superfluidity in such systems.
Quantized vortices in a rotating gas provide conclusive evidence
for superfluidity because they are a direct consequence of theexistence of a macroscopic wavefunction that describes the superfluid. The velocity field of the superfluid is proportional to thegradient of the wavefunction’s phase. In such a case, flow must be
irrotational and angular momentum can enter the system only in theform of discrete line defects (vortices). In contrast, for a normal gas
the lowest state of rotation corresponds to rigid body rotation.
Metastable vortex patterns have been observed in classical inviscid
fluids23. However, the final number and charge of the vortices
depends chaotically on the initial conditions, in contrast to theregular vortex lattices that we have reproducibly observed. Furthermore, vortex patterns in classical fluids are only stable at extremely
low viscosity (that is, for Reynolds numbers .105). For a Boltzmann
gas rotating close to the trap frequency, the Reynolds number isapproximately the cloud size divided by the mean free path, which
does not exceed 103 in our case. Pauli blocking can only decrease theReynolds number further.

Figure 1 | Observation of a vortex lattice in a molecular condensate.
a, Fixed field. Stirring for 800 ms, followed by 400 ms of equilibration, andimaging after 12 ms time-of-flight all took place at 766 G. The vortex core
depletion of the integrated density profile is barely 10%, as indicated by the
5-mm-wide cut on top. b, Fourier filter applied to a to accentuate the vortex
contrast. Spatial frequencies with an absolute value of about the inverse
vortex core size were enhanced by a factor of four. c, Varying field. The
vortex lattice was created at 766 G and imaged at 735 G following theprocedure outlined in the text. The vortex core depletion is now about 35%(see 5-mm-wide cut on top). The field of view is 780 mm £ 780 mm.

1

Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA. †Present address:
Lawrence Berkeley National Laboratory, One Cyclotron Road, MS 88R0192, Berkeley, California 94720, USA.

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Figure 2 | Vortices in a strongly interacting gas of fermionic atoms on the
BEC- and the BCS-side of the Feshbach resonance. At the given field, thecloud of lithium atoms was stirred for 300 ms (a) or 500 ms (b–h) followedby an equilibration time of 500 ms. After 2 ms of ballistic expansion, themagnetic field was ramped to 735 G for imaging (see text for details). The
magnetic fields were 740 G (a), 766 G (b), 792 G (c), 812 G (d), 833 G (e),843 G (f), 853 G (g) and 863 G (h). The field of view of each image is
880 mm £ 880 mm.

Experimental procedure
To create a strongly interacting Fermi gas, spin-polarized fermionic6
Li atoms were sympathetically cooled to degeneracy by 23Na atoms
in a magnetic trap24. The Fermi cloud was then loaded into an optical
dipole trap, and an 875 G external magnetic field was applied. Herea 50%/50% spin mixture of the two lowest hyperfine states of 6Li
was prepared. Between these two states, labelled j1l and j2l, there is a300-G-wide Feshbach resonance located at 834 G (refs 25, 26).
Evaporative cooling (achieved by reducing the laser power)
accompanied by a magnetic field ramp to 766 G on the BEC-sideof the resonance typically produced a BEC of 3 £ 106 molecules3.
Previous experiments studying the rotation of atomic BECs
employed magnetic traps operating at low bias fields27–31. Because
the Feshbach resonance in our system occurs between two high-fieldseeking states that cannot be trapped magnetically, an optical dipole
trap operating at high magnetic bias fields was necessary. Our set-up
employed a trapping beam with a 1/e2 radius of 123 mm (wavelength
1,064 nm), radially confining the gas with a trap frequency of 59 Hz ata power of 145 mW. Axial confinement with trap frequencyn z ¼ 23 Hz was provided by an applied magnetic field curvaturethat decreased the radial trap frequency to n r ¼ 57 Hz. The aspectratio of the trap was 2.5. In this trap, at a field of 766 G, condensates of1 £ 106 molecules (the typical number in our experiment after
rotating the cloud) have Thomas–Fermi radii of about 45 mm radiallyand 110 mm axially, a peak molecular density of 2.6 £ 1012 cm23, a
chemical potential of about 200 nK, and a characteristic microscopic
length scale of 1/k F < 0.3 mm. Here, the Fermi wavevector k F isdefined by the Fermi energy (E F) of a non-interacting two-state
mixture of 6Li atoms of mass m with total atom number N in a
1=3;
harmonic trap of (geometric) mean frequency q; EF ¼ "qð3NÞ
"2 k2F =2m: Throughout this Article we will estimate the interaction
parameter 1/k Fa using the average number of fermion pairs
N/2 ¼ 1 £ 106. Here, a is the scattering length between atoms instates j1l and j2l: At a field of 766 G, 1/k Fa ¼ 1.3. Because this gas isstrongly interacting, it is difficult to extract a temperature from thespatial profile. For weaker interactions (at 735 G) the condensate
fraction was in excess of 80%, which would isentropically connect toan ideal Fermi gas32 at T/T F ¼ 0.07. The BEC–BCS crossover(1=kF jaj , 1) occurs in the region between 780 G and 925 G.
The trapped cloud was rotated about its long axis using a bluedetuned laser beam (wavelength 532 nm)28,29,33. A two-axis acoustooptic deflector generated a two-beam pattern (beam separationd ¼ 60 mm, gaussian beam waist w ¼ 16 mm) that was rotatedsymmetrically around the cloud at a variable angular frequency Q.

The two beams with 0.4 mW power each produced a repulsive
potential of 125 nK for the 6Li cloud, creating a strongly anisotropic
potential. This method was first tested using a weakly interacting,atomic BEC of 23Na in the stretched upper hyperfine state in an
optical trap with n r ¼ 60 Hz, n z ¼ 23 Hz. Fully equilibrated lattices
of up to 80 vortices were observed. The vortex number decayed with a
1/e lifetime of 4.2 ^ 0.2 s, while the atom number decayed, owing tothree-body losses and evaporation, with a lifetime of 8.8 ^ 0.4 s. The
roundness of the optical trap and its alignment with both the opticalstirrer and the axes of the magnetic potential were critical. Any
deviation from cylindrical symmetry owing to misalignment, opticalaberrations, or gravity rapidly damped the rotation. The generationof vortices in sodium was comparatively forgiving, and had to be
optimized before vortices in 6Li2 could be observed.
Observation of vortex lattices
In experiments with 6Li close to the Feshbach resonance, theinteraction strength between atoms in states j1l and j2l can be freely
tuned via the magnetic field. Thus, it is possible to choose different
magnetic fields to optimize the three steps involved in the creation ofa vortex lattice: stirring of the cloud (for 800 ms at a typical stirring
frequency of 45 Hz), the subsequent equilibration (typically 500 ms)
and time-of-flight expansion for imaging. To stay close to the

Figure 3 | Optimized vortex lattices in the BEC–BCS crossover. After avortex lattice was created at 812 G, the field was ramped in 100 ms to 792 G(BEC-side), 833 G (resonance) and 853 G (BCS-side), where the cloud washeld for 50 ms. After 2 ms of ballistic expansion, the magnetic field wasramped to 735 G for imaging (see text for details). The field of view of each
image is 880 mm £ 880 mm.

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NATURE|Vol 435|23 June 2005

analogous case of an atomic condensate, our search for vortices
started on the BEC-side of the resonance, at a fixed magnetic field of
766 G. To image the cloud, the trapping beam was switched off andthe cloud expanded in the residual magnetic potential. After 12 mstime-of-flight, a molecular absorption image along the axial direction was taken with light resonantly exciting atoms in state j2l3
(Fig. 1a). Although the contrast of the vortex cores was low, a regularlattice pattern containing about 25 vortices was visible (see Fourier
filtered image in Fig. 1b). This establishes superfluidity for molecularcondensates.
Subsequently, it was found that the contrast of the vortex cores
could be enhanced by the following procedure: the magnetic fieldcurvature was reduced by a factor of five during the first millisecondof time-of-flight. After 2 ms expansion at the initial field, themagnetic field was ramped down over 2 ms to 735 G. The cloud
was imaged after an additional 9 ms of expansion at this field.
Attempts to image at even lower fields did not enhance the contrast.
Further improvements were achieved by forcing the cloud to expand
faster by increasing the power of the optical trap by a factor of 4.5
during the last 2 ms of trapping (Fig. 1c). We suspect that owing to
the residual magnetic field curvature, a faster expansion was superior
to a longer time-of-flight.
Using this procedure, we observed vortices that were created not
only on the BEC- but also the BCS-side of the Feshbach resonance, atmagnetic fields between 740 G and 863 G (Fig. 2). On the BCS-side,
isolated fermion pairs are unstable. As the cloud expands from thetrap and the density decreases, the pairs will become more fragile anddissociate at a certain point in the time-of-flight. Information on thecentre-of-mass wavefunction of the pairs, and hence the vortex
contrast, will be gradually lost. The field ramp to the BEC-side
during expansion protects the pairs by transforming them into stablemolecules. At 853 G, this ramp could be delayed by 6 ms into ballistic
expansion before the vortex contrast was lost. It is not clear why thisramp was found to be necessary already at 812 G, on the BEC-side of
the resonance, where isolated fermion pairs are stable.
It is impossible for vortex lattices to form during ballistic expansion at the imaging field (735 G). We show below that the formationof vortex lattices even at the high density of the trapped cloud takes
several hundred milliseconds. Furthermore, even if there was some

unpredicted fast formation mechanism for vortices, they could notform a regular array with long-range order in a cloud that expands atthe speed of sound of the trapped gas. The observation of vortex
lattices on the BCS-side of the Feshbach resonance above 834 G
demonstrates superfluidity of fermions at magnetic fields where theycannot form stable, isolated molecules.
The highest number of vortices (,40) was obtained by stirring
at 766 G and then equilibrating close to resonance at 812 G(1/k Fa ¼ 0.35). We suspect that the violent nature of the stirringproduced more heating near the Feshbach resonance where the pairs
are loosely bound. On the other hand, fields closer to resonance werefavourable for equilibration owing to suppression of vibrational
relaxation. After preparing such a vortex lattice at 812 G, themagnetic field was ramped over 100 ms to a test field. After a holdtime of 50 ms, the vortex lattice was imaged as discussed above(Fig. 3). Vortices were observed for test fields between 700 G(1/k Fa ¼ 3.8) and 954 G (1/k Fa ¼ 21.2) (Fig. 4).
The regularity of the lattice proves that all vortices have the same
vorticity. From their number, the size of the cloud and the quantumof circulation h/2m for each vortex, we can estimate the rotationalfrequency of the lattice. For an optimized stirring procedure, we findthat it is close to the stirring frequency. This excludes a quantum of
circulation of h/m or doubly charged vortices.

Figure 4 | Vortex number versus magnetic field and interaction strength in
the BEC–BCS crossover. The open triangles show the number of vortices
obtained after stirring and equilibration at the given, fixed magnetic field, as
in Fig. 2. For the filled circles, a procedure similar to the one in Fig. 3 wasused, where the vortex lattice was prepared at 812 G, and then the field wasramped to the test field. The position of the Feshbach resonance26 is marked
with the vertical dash-dotted line. The data points and error bars give theaverage and standard deviation of several measurements.

Figure 5 | Vortex number versus stirring frequency in the BEC region fordifferent interaction strengths. Vortices were efficiently created over abroad range of stirring frequencies. The open triangles and filled circles
correspond to stirring (for 800 ms) and equilibration (for 500 ms) at 766 G
and 812 G, respectively. The data points and error bars (if larger than thesymbol) show the average and standard deviation of three measurements.
The radial trap frequency (57 Hz) is indicated by the small arrow on thehorizontal axis.

Formation and lifetime of vortex lattices
Before we found the detection scheme described above, we studied
the formation and decay of the vortex lattices using a different
procedure. The magnetic field was lowered to 735 G (1/k Fa ¼ 2.3)
in the last 30 ms before expansion. Reduction of the magnetic fieldcurvature by a factor of five took place during the last 5 ms before
expansion, avoiding undesired compression of the cloud in the axial
direction. As before, the trap was compressed by increasing thetrapping power by a factor of 4.5 during the last 2 ms before theswitch-off. Imaging was done after 12 ms expansion at 735 G.
When a superfluid is rotated, the creation of quantized vortices isenergetically favoured only above a certain critical rotation frequency. In some cases, a higher rotational frequency is necessary toactually nucleate the vortices34. This occurs through a dynamic
instability of surface excitations33,35,36. Figure 5 shows that vortices
in the BEC region were created over a large range of stirring
frequencies, as opposed to only near the quadrupole surface mode
resonance28,29. Similar non-resonant behaviour was observed in
atomic sodium condensates with small stirring beams36. The strong
dependence of the observed stirring efficiency curve on the magnetic

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Figure 6 | Formation and decay of a vortex lattice in a fermion paircondensate on the BEC-side close to the Feshbach resonance. A molecularcondensate, prepared at 766 G as shown in a, was stirred for 800 ms. The
field was then ramped to 812 G in 20 ms for equilibration. At this field,
1/k Fa ¼ 0.35, and the condensate was deep in the strongly interacting
regime. To observe the vortex lattice, the field was ramped in 25 ms to 735 G(1/k Fa ¼ 2.3), where the condensate was released from the trap and imaged
after 12 ms time-of-flight. The equilibration times after the end of thestirring were 40 ms (b), 240 ms (c), 390 ms (d), 790 ms (e), 1,140 ms (f),1,240 ms (g) and 2,940 ms (h). Owing to stirring, evaporation andvibrational relaxation, the number of fermion pairs decayed from 3 £ 106(a) to 1 £ 106 (b–h). The field of view of each image is 830 mm £ 830 mm.

field strength could be related to the increase of the speed of sound in
the condensate for stronger interactions, leading to a higher criticalvelocity.
Figure 6 shows the formation of a vortex lattice close to the
Feshbach resonance, on the BEC-side at 812 G (1/k Fa ¼ 0.35).
Immediately after stirring, the cloud was in a turbulent state(Fig. 6b). It took several hundred milliseconds for the vortices tofully crystallize into a lattice (Fig. 6c–e). The vortices arranged
themselves in a hexagonal Abrikosov lattice to minimize theirinteraction energy29,37. Because the trap potential was not perfectlyround (trap asymmetry ðn2x 2 nyy Þ=ðn2x þ n2y Þ < 0:03), the vortex latticeslowly decayed on a timescale of several seconds (Fig. 6f–h). Theseobservations are fully analogous to those already made in atomic
BECs29,35,38. The formation time of several hundred milliseconds is inagreement with these experimental studies, as well as with a recent
theoretical study on the vortex lattice formation in a stronglyinteracting Fermi gas39. Note that this timescale was found to be
independent of temperature38, and seems to represent an intrinsic
timescale of superfluid hydrodynamics. The long formation time
excludes the possibility that ordered vortex lattices can be createdduring the 30 ms field ramp to 735 G, used for imaging. This holds
even more strongly for the imaging method described first, where themagnetic field was switched only in time-of-flight. We are not awareof any possible formation mechanism that could create regularvortex lattices during ballistic expansion.
In principle, the presence of vortices can be used to map out thesuperfluid regime as a function of temperature and interaction
strength, even in regions where fermion pairs are much larger than
the interparticle spacing and can no longer be detected by transforming them into stable molecules4,5,12. As a first step, the lifetime of thevortex lattice was studied at different magnetic fields and interaction
strengths. After preparing a fully crystallized vortex lattice containing
about 30 vortices in the BEC-region at 812 G, the magnetic field wasramped to a chosen point in the crossover. After a variable hold time,
the cloud was imaged at 735 G as described above, and the remaining
number of vortices was counted. The results of this measurement aresummarized in Fig. 7. The longest lifetime, 3.4 s, was obtained formagnetic fields slightly below resonance, near 810 G (1/k Fa ¼ 0.4).
Here, we observed four vortices even after 7 s. During this time thefluid at the vortex core (with characteristic size 1/k F) rotated morethan 50,000 times, displaying truly superfluid behaviour. As
expected, the lifetime was reduced at low magnetic fields where themolecules heat up owing to vibrational relaxation. In addition, andunexpectedly, a narrow dip in lifetime at approximately 831 G with awidth of 8 G was observed. We speculate that this decrease in lifetime
so close to resonance is caused by a coupling of the external motion of
the loosely bound pairs to their internal motion, causing pairbreaking or rotational excitation. Indeed, at 831 G, the binding
energy for isolated molecules divided by h is about 35 Hz, comparable to the initial rotational frequency of the lattice. The reduced
lifetime on the BCS-side could be caused by an increasing fraction ofthermal fermion pairs5,38 due to the decreasing critical temperaturefor superfluidity. In a two-fluid model, angular momentum is stored
as a vortex lattice in the superfluid component. Friction is provided
by the normal component, which increases as we move further awayfrom resonance on the BCS-side. At 925 G (1/k Fa ¼ 21.0) the vortex
lifetime was reduced to only 10 ms, with a large error bar indicating

Figure 7 | Decay rate and lifetime of the vortex lattice versus magnetic field
and interaction strength. The data points and error bars give the decay rate
or lifetime and one standard deviation extracted from exponential fits to thevortex number decay. The dashed and dotted lines are gaussian andexponential fits to guide the eye, the dashed line including a lorentzian fit for
the narrow feature near resonance. The position of the Feshbach resonance26
is marked with the vertical dash-dotted line.

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strong fluctuations in the vortex number. Here we might beapproaching the region where the superfluid-to-normal transition
takes place.
Conclusions
We have detected long-lived vortex lattices in a strongly interacting
Fermi gas over the entire BEC–BCS crossover region by imaging
them after switching to lower magnetic fields during ballistic expansion. This provides the first direct signature of superfluidity in thesesystems. We expect that vortices in rotating Fermi gases will serve as
an important starting point for future studies on superfluid
dynamics.
Received 9 May; accepted 31 May 2005.
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Acknowledgements We thank P. Zarth for experimental assistance and C. Stanfor contributions in the early stages of the experiment. We also acknowledge
discussions with the participants of the OCTS conference in Ohio, and thankJ. Anglin, Z. Hadzibabic, D. Kleppner and A. Leanhardt for a critical reading of
the manuscript. This work was supported by the NSF, ONR, ARO and NASA.
Author Information Reprints and permissions information is available at
npg.nature.com/reprintsandpermissions. The authors declare no competing
financial interests. Correspondence and requests for materials should be
addressed to M.W.Z. (zwierlei@mit.edu).

1051© 2005 Nature Publishing GroupAppendix G
Superfluid Expansion of a Rotating
Fermi Gas
This appendix contains the preprint Ref. [216]: C. H. Schunck, M. W. Zwierlein, A.
Schirotzek, and W. Ketterle, Superfluid Expansion of a Rotating Fermi Gas, preprintcond-mat/0607298.

224

Superfluid Expansion of a Rotating Fermi Gas
C.H. Schunck, M.W. Zwierlein, A. Schirotzek, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
MIT, Cambridge, MA 02139(Dated: October 5, 2006)
We study the expansion of a rotating, superfluid Fermi gas. The presence and absence of vortices
in the rotating gas is used to distinguish superfluid and normal parts of the expanding cloud. We findthat the superfluid pairs survive during the expansion until the density decreases below a criticalvalue. Our observation of superfluid flow in the expanding gas at 1/kF a = 0 extends the range
where fermionic superfluidity has been studied to densities of 1.2 × 1011 cm−3 , about an order ofmagnitude lower than any previous study.
PACS numbers: 03.75.Ss, 03.75.Hh, 05.70.Fh

Ultracold atomic gases have been used to create
novel quantum many-body systems ranging from BoseEinstein condensates and Mott insulators in optical lattices to high-temperature superfluids of strongly interacting fermions. These systems offer a high degree of control
over physical parameters including interaction strengthand density. Many important features in these gases havea spatial scale too small to be resolved while the gas istrapped. A standard technique to reveal this physics is toswitch off the confining potential and release the gas fromthe trap. A non-interacting gas expands ballistically andthe expansion reveals its momentum distribution. The
expansion dynamics of an interacting gas is modified by
the effect of collisions. This can result in classical hydrodynamic flow and in this case the expansion serves as a(not necessarily linear) magnifying glass for the trappedstate. In contrast to classical hydrodynamics, superfluidhydrodynamic flow does not rely on collisions. Whena weakly interacting Bose-Einstein-Condensate (BEC) is
released from an anisotropic trapping potential, superfluid hydrodynamics leads to an inversion of the aspectratio, often regarded as a hallmark of Bose-Einstein condensation [1].
The expansion dynamics of strongly interacting Fermi
gases has been the subject of a longstanding debate. Fora weakly interacting ultracold Fermi gas anisotropic expansion has been proposed as a probe for superfluidity,
analogous to the case of weakly interacting BECs [2].
Anisotropic expansion has been experimentally observedin strongly interacting Fermi gases [3–5]. In this case,however, the inversion of the aspect ratio can occur due
to collisions between the expanding atoms even if they
were initially at zero temperature [6, 7]. So far experiments have not been able to discriminate between superfluid and collisional hydrodynamics in expansion andindeed one would expect both effects to contribute: In
the BCS-regime, the superfluid transition temperature
TC depends exponentially on the density. Starting at
T < TC , the superfluid gas first expands according tosuperfluid hydrodynamics. As the density drops, T approaches TC and superfluidity cannot be maintained.

From this point on, the gas should expand according
to collisional hydrodynamics or enter a regime intermediate between collisional hydrodynamic and collisionless
expansion.
In this paper we study the expansion of a superfluid
Fermi gas, in the regime where pairing is purely a manybody effect. We have observed superfluid flow even after
5 ms of expansion, when the cloud size had increased bymore than a factor of 4 and the peak density had dropped
by a factor of 17 compared to the in-trap values.
Superfluidity in Fermi gases has previously been established through the observation of vortex lattices [8, 9]. To
detect vortices in a rotating fermion pair condensate thepairs are transferred into stable molecules by sweeping
an external magnetic field across a Feshbach resonanceshortly after the gas is released from the trap. Vortices
can be observed only when the gas is still a superfluidat the moment of the magnetic field sweep [10]. At the
final magnetic field (on the BEC side of the Feshbachresonance) the interactions are much weaker. Therefore
the vortex core has higher contrast and is larger thannear resonance. If the gas is no longer superfluid at thetime of the field ramp, we expect the vortex core to fill inquickly and disappear. The observed vortex cores therefore serve as markers for the regions which are superfluidat the time of the magnetic field ramp.
Our experimental setup has been described earlier [11,12]. 6 Li fermion pair condensates containing 5 × 106
fermions were created in an optical dipole trap at a magnetic field of 812 G. This is on the BEC-side of a Feshbach resonance at B0 = 834 G. At magnetic fields below(above) B0 , on the BEC (BCS) side, the scattering lengtha is positive (negative) and a nearby molecular boundstate exists (does not exist). The radial and axial trapping frequencies were ωr = 2π × 120 Hz and ωa = 2π × 23
Hz, respectively. To observe vortices as a probe of superfluid flow, the gas was set in rotation: two blue-detuned
laser beams were rotated symmetrically around the cloudfor 1 s at an angular frequency of 2π × 80 Hz [8]. We allowed 500 ms of equilibration before the magnetic field
was ramped (in 500 ms) to several probe fields on the2

FIG. 1: Superfluid expansion of a strongly interacting rotating Fermi gas. Shown are absorption images for different expansion
times on the BCS-side of the Feshbach resonance at 910 G (0.0, 1.0, 2.0, 3.0, 3.5, 4.0, and 4.5 ms) and 960 G (0.0, 0.5, 1.0,
1.5, 2.0, 2.5, and 3 ms), before the magnetic field was ramped to the BEC-side for further expansion. The vortices served as
markers for the superfluid parts of the cloud. Superfluidity survived the expansion for several milliseconds and was gradually
lost from the low density edges of the cloud towards its center. Compared to 910 G (a = −7200 a0 ), superfluidity decayed
faster at 960 G (a = −5000 a0 ) due to the reduced interaction strength. The total expansion time remained constant [14]. The
field of view of each image is 1.2 mm × 1.2 mm.

BCS side of the resonance. Finally, we studied the expansion of the rotating superfluid: The gas was released fromthe optical trap and expanded at the probe field for avariable “BCS-expansion” time tBCS , that was increased
in 500 µs steps. To transfer the remaining fermion pairs
into stable molecules the magnetic field was then lowered in 400 µs to 680 G [13]. Here, the cloud was given
several milliseconds of “BEC-expansion”. For absorption
imaging the magnetic field was raised to 730 G in 500 µs
before the last 2 ms of time-of-flight. For most of the data
the total time-of-flight was chosen to be 11 ms [14]. An
absorption image of the gas was obtained separately attBCS to determine the peak density and the peak Fermi
momenta kF before the magnetic field sweep.
Fig. 1 shows absorption images taken as outlined above
for seven different BCS-expansion times at both 910 G
and 960 G. The presence of vortices proves that superfluid fermion pairs survived in the expanding gas for several milliseconds. As the density of the gas dropped during the BCS-expansion the vortices were gradually lostfrom the low density edges of the cloud towards its center.
After 4.5 ms time-of-flight at 910 G and 3 ms at 960 G allof the vortices had decayed. If we regard the number ofvortices as an indicator of the superfluid fraction of thegas, we can draw the “phase diagram” of Fig. 2. Here thenumber of vortices is shown as a function of the inverse
scattering length 1/a and the inverse peak Fermi momentum 1/kF . As 1/kF increases at a given magnetic field,
corresponding to the decrease in density during time-offlight, vortices are lost. The reduction in the numberof vortices for decreasing |a| reflects the decrease of thesuperfluid fraction for smaller attractive interactions at
a given temperature. In addition, the increase in thenormal fraction leads to higher damping of the remain-

ing vortex number [8]. Most importantly, however, we
see that vortices are lost earlier in time-of-flight as theinteractions are reduced.
At all magnetic fields, we find that the peak interaction strength at the point where all vortices were lostis about constant, kF a ∼ −0.8 (see Fig. 3). As shownin Fig. 1 the loss of vortices occurred gradually and thesurviving vortices were located within a circle of decreasing radius. We assume that the critical value of kF a forwhich superfluidity was lost, was first reached at the edgeof the cloud and subsequently further inward. However,we were not able to confirm this picture quantitatively
without a model that describes how the shape of thecloud and the bimodality develop during and after themagnetic field sweep.
It is remarkable that the observation of superfluidityand fermion pair condensation for trapped gases has also
been limited to values of kF |a| larger than 1 [8, 12, 17].
This suggests that the underlying reason for this limitation is the same for a trapped and an expandinggas. One obvious scenario for the decay of the vortex
lattice during expansion is the breakdown of superfluidity due to finite temperature when a critical interaction strength is reached. As the density decreases,
TC /TF drops while T /TF remains constant (since thephase space density n × T −3/2 is invariant during expansion). Therefore TC eventually becomes smaller than T
everywhere in the cloud and superfluidity is lost. The
critical interaction strength can be estimated by equating 1 ≡ T /TC = (T /TF )(TF /TC ) = 1.76(T /TF )(EF /∆).
Here ∆ = (2/e)7/3 EF exp(−π/2kF |a|) is the pairing gap
in the BCS limit (valid for kF |a| <∼ 1) [18], where thepeak Fermi energy EF = ~2 kF 2 /2m and kF are density dependent. For an estimate of our lowest tempera-

3

FIG. 2: (color online)“Phase diagram” of an expanding, rotating Fermi gas: At a given magnetic field the number ofvortices served as a measure for the size of the superfluid region in the gas. The number of vortices is plotted versus
1/kF and 1/|a|. The contour plot was created from a total of
53 data points. In this diagram lines of constant kF a correspond to hyperbolas. The vortices decayed when the density(increasing 1/kF ) or the scattering length (increasing 1/|a|)was reduced. For weaker interactions, at smaller scattering
lengths |a|, vortices were lost already at higher densities. The
four data points shown mark the breakdown of superfluidity
and are the same as the squares in Fig. 3.

FIG. 3: (color online) The peak interaction strength during
superfluid expansion. Starting at a peak kF a in the opticaltrap (triangles) vortices survived up to a critical peak kF a of
-0.8 +/- 0.1 (squares), almost independent of the magnetic
field (scattering length). Filled circles correspond to partiallysuperfluid, open circles to normal clouds. The observed number of vortices is color coded. The critical kF a was obtainedfor each magnetic field separately by taking the average ofthe peak kF of the last partially superfluid and the first completely normal cloud. The error in kF a is about 10% anddominated by the systematic error in the atom number.

FIG. 4: Disappearance of bimodality. Zero temperature
Thomas-Fermi profiles (dotted) were fit to the density profiles(solid) obtained after BCS-expansion at 885 G and subsequent
BEC-expansion at 680 G. The χ2 of the fit was monitored asa function of the BCS-expansion time tBCS . A high χ2 indicates a bimodal density distribution. Vortices were still observed after 5 ms of expansion (indicated by the dashed line in
the figure) while the bimodality had already disappeared (for
χ2 values smaller than 0.01 bimodality cannot be discerned).
Hence, the absence of bimodality does not imply the absenceof superfluidity.

tures of T /TF = 0.05 [19] this gives kF a = −0.9 close to
the observed value. This finite-temperature scenario implies that the superfluid state evolves adiabatically during expansion, which is plausible: Even when the criticalkF a is reached, the pair binding energy still changes at˙
a slower rate, ∆/∆,
than the rate at which the pairs can
respond to this change, ∆/~ [20]. For weakly interacting
BECs, the decay of vortex lattices at finite temperature
was studied theoretically in [21], and remarkably similar
structures are found.
In analogy to the critical magnetic field Hc2 in typeII superconductors, superfluidity can also break down
in response to rapid rotation [22, 23]. Superfluidity is
quenched when the vortex core size becomes comparable
to the separation between the vortices. Both the critical temperature as well as the vortex core size in the
BCS limit depend exponentially on kF a and therefore
the effects of rotation might not be negligible. In [23],however, it was found that superfluidity should be stable in the strongly interacting regime (kF |a| >∼ 1.0) at all
rotation frequencies.
Another explanation for the loss of vortices is a possible failure of the transfer of correlated fermion pairs intomolecules since the size of the fermion pairs increases
with decreasing density. When the fermion pair size becomes larger than the interparticle spacing, molecules
might be formed out of uncorrelated nearest neighbors
rather than out of correlated pairs. The magnetic fieldsweep then destroys the coherent many-body wavefunction.
Vortices [8, 9] and bimodal density distributions [12,17] have been used as indicators for superfluidity and4FIG. 5: Loss of vortex contrast on resonance at 834 G. Shown are absorption images after a fixed total time-of-flight, but fordifferent expansion times on resonance (2, 2.5, 3, 3.5, 4, 5, and 6 ms) before the magnetic field was swept to the BEC-side forfurther expansion. A gradual loss of the vortex contrast from about 15% (after 2 ms of expansion on resonance) to 3% (after
5 ms) was observed across the whole cloud. The field of view of each image is 1.2 mm ×1.2 mm.

pair condensation, respectively. If a fermion pair condensate is transferred to the BEC side before its interaction energy has been converted into kinetic energy, it
continues to expand with the drastically reduced meanfield energy of a molecular BEC at 680 G. This results ina clear separation of condensate and thermal cloud after
further BEC-expansion. If the transfer of fermion pairs
into molecules is delayed after releasing the gas from thetrap, the fermion pair condensate initially expands justlike the normal part of the cloud. This eventually leads toa loss of bimodality in the density profiles after the transfer. We can now study how the two indicators, vortices
and bimodality, are related in this experiment. For short
BCS-expansion tBCS our data showed bimodality as well
as vortices. However, the bimodality was gradually lostand could not be discerned after a longer BCS-expansion
although vortices were still visible (see Fig. 4 for details).
The absence of bimodality therefore does not indicate abreakdown of superfluidity.
So far we have studied the expansion of the gas on the
BCS side of the Feshbach resonance. On the BEC side
and on resonance, TC is proportional to TF so that T /TC
is constant during expansion. Therefore, one would not
expect to observe a breakdown of superfluidity in expansion. Fig. 5 shows absorption images that were obtained
after an initial expansion of the cloud on resonance at
834 G. In contrast to the situation on the BCS-side of
the resonance no vortices were lost. Instead, the vortex
contrast decreased uniformly across the cloud for longer
expansion times. Vortices have been detected at total
densities as low as 1.2 × 1011 cm−3 in the wings of the
expanded cloud. Here the critical temperature TC of approximately 0.2 TF [15, 16] was below 20 nK (kB TF is the
local Fermi energy). We believe that the decrease in thevortex contrast is due to the low density of the gas after
long BCS-expansion: after the magnetic field sweep thevortex cores cannot adjust quickly enough to the highcontrast and large size they would have in equilibriumon the BEC-side. This reduction of contrast limited ourstudy of the breakdown of superfluidity to magnetic fields
above 880 G.
In conclusion we have shown that superfluid pairs cansurvive during the expansion of a strongly interacting
Fermi gas. This is the first observation of non-equilibrium
superfluid flow in such systems. It has allowed us to observe fermionic superfluidity at total densities as low as
1.2 × 1011 cm−3 . Our results show that future experiments with expanding, superfluid Fermi gases can be
carried out in situ, i.e. without magnetic field sweeps to
the BEC side. An intriguing question is whether fermion
pairs expanding from two clouds can coherently interfere.
We thank Gretchen Campbell for a critical reading of
the manuscript. This work was supported by the NSF,
ONR, and NASA.

[1] W. Ketterle et al., in “Bose-Einstein Condensation in
Atomic Gases”, IOS Press, Amsterdam, 1999 pp. 67–176.
[2] C. Menotti et al., Phys. Rev. Lett. 89, 250402 (2002).
[3] K. M. O’Hara et al., Science 298, 2179 (2002).
[4] C. A. Regal et al. Phys. Rev. Lett. 90, 230404 (2003).
[5] T. Bourdel et al., Phys. Rev. Lett. 91, 020402 (2003).
[6] S. Gupta et al., Phys. Rev. Lett. 92, 100401 (2004).
[7] B. Jackson et al., Europhys. Lett. 67, 524 (2004).
[8] M. W. Zwierlein et al., Nature 435, 1047 (2005).
[9] M. W. Zwierlein et al., Science 311, 492 (2006).
[10] The sweep time is much faster than the formation time ofa vortex lattice in the trap, which is several hundred milliseconds. Detection of vortices after the ramp therefore
proves their presence before the ramp [8].
[11] Z. Hadzibabic et al., Phys. Rev. Lett. 91, 160401 (2003).
[12] M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403(2004).
[13] The ramp time was 200 µs for the data taken at 834, 865
and 885 G. We have checked that the rate of the magnetic
field sweep to 680 G had no influence on the number ofobserved vortices within our measurement accuracy.
[14] To increase the visibility of the vortices for tBCS = 0 and
500 µs the total time-of-flight was increased to up to 15
ms and/or the power of the optical trap was increased bya factor of 4.5 during the last 2 ms of trapping [8].
[15] A. Bulgac et al., Phys. Rev. Lett. 96, 090404 (2006).
[16] E. Burovski et al., Phys. Rev. Lett. 96, 160402 (2006).
[17] C. A. Regal et al., Phys. Rev. Lett. 92, 040403 (2004).
[18] L. P. Gor’kov et al., Sov. Phys. JETP 13, 1018 (1961).
[19] M. W. Zwierlein et al., Nature 442, 54 (2005).

Appendix H
Fermionic Superfluidity with
Imbalanced Spin Populations
This appendix contains a reprint of Ref. [268]: Martin W. Zwierlein, André Schirotzek, Christian H. Schunck, and Wolfgang Ketterle, Fermionic Superfluidity with
Imbalanced Spin Populations, Science 311, 492 (2006), published online on Science
Express on 21 December, 2005 (10.1126/science.1122318) (in Science Research Articles).

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RESEARCH ARTICLES
Fermionic Superfluidity with
Imbalanced Spin Populations
Martin W. Zwierlein,* André Schirotzek, Christian H. Schunck, Wolfgang Ketterle
We established superfluidity in a two-state mixture of ultracold fermionic atoms with imbalanced state
populations. This study relates to the long-standing debate about the nature of the superfluid statein Fermi systems. Indicators for superfluidity were condensates of fermion pairs and vortices inrotating clouds. For strong interactions, near a Feshbach resonance, superfluidity was observed fora broad range of population imbalances. We mapped out the superfluid regime as a function ofinteraction strength and population imbalance and characterized the quantum phase transition to
the normal state, known as the Pauli limit of superfluidity.
ermionic superfluidity, whether it occurs
in superconductors, helium-3, or inside
a neutron star, requires pairing of fermions, particles with half-integer spin. In an
equal mixture of two states of fermions (Bspinup[ and Bspin down[), pairing can be complete and the entire system will become superfluid. When the two populations of fermions
are unequal, however, not every particle canfind a partner, raising the question of whether
superfluidity can persist in response to such aFDepartment of Physics, Massachusetts Institute of Technology (MIT)–Harvard Center for Ultracold Atoms, andResearch Laboratory of Electronics, MIT, Cambridge, MA
02139, USA.
*To whom correspondence should be addressed. E-mail:zwierlei@mit.edu

population imbalance. This problem arises inmany different fields of physics—for example, in the study of superfluidity of quarks inthe dense matter of the early universe, wherecharge neutrality and differing masses impose
unequal quark densities. In superconductors,
an applied magnetic field could in principle
imbalance the densities of spin up and spin
down electrons. As first discussed by Clogston
in 1962 (1), there exists an upper limit for thismagnetic field, beyond which superconductivity with equal spin densities will break down.
Fulde and Ferrell (2), and independently Larkin
and Ovchinnikov (3), proposed a more stableconfiguration of the superconductor that allows
for unequal densities, the FFLO or LOFF state
containing nonzero-momentum Cooper pairs.

Fig. 1. Superfluidity in a strongly interacting Fermi gas with imbalanced
populations. The upper (lower) pair of rows shows clouds prepared at
812 G, on the BEC side (853 G, BCS side), where 1/kF a 0 0.2 (1/kF a 0
j0.15). In each pair of rows, the upper image shows state k1À, thelower one state k2À. For the 812-G data, the population imbalance d 0(N2 j N1)/(N1 þ N2) between N1 atoms in state k1À and N2 in state k2À was

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The true ground state of imbalanced fermionicsuperfluids has been the subject of debate for
decades (4, 5), and experimental studies arehighly desirable. However, superconductors arecharged fermionic superfluids, and imbalancing the electron densities by applying magnetic fields is hindered by the Meissner effect.
The fields are either fully shielded from thesuperconductor, or they enter in the form ofquantized flux lines or vortices. Only in special materials can these effects be suppressed,
such as in heavy-fermion superconductors (6–8)
or in quasi–two-dimensional (2D) organic superconductors (6). In the neutral superfluid
helium-3, one can mismatch the Fermi surfaces by a magnetic field and thus destroy
interspin pairing. However, superfluidity persists due to (p-wave) pairing between equal
spins (9).
Fermionic superfluids of atom pairs. The
recently discovered fermionic superfluids inultracold atomic gases (10–19) provide an exciting new possibility of exploring unequal mixtures of fermions, because populations in two
hyperfine states of the fermionic atom can be
freely chosen. In addition, the (s-wave) interactions between two atoms in different states
and the binding energy of atom pairs can betuned via Feshbach resonances. In equal mixtures of fermions, this tunability gives accessto the crossover from a Bose-Einstein Condensate (BEC) of molecules to a BardeenCooper-Schrieffer (BCS) superfluid of looselybound pairs (13–19). At zero temperature, thiscrossover is smooth (20–22), the system stays(from left to right) 100, 90, 80, 62, 28, 18, 10, and 0%. For the 853-G
data, the imbalance was 100, 74, 58, 48, 32, 16, 7, and 0%. For different
values of d, the total number of atoms varied only within 20% around N 0
7  106, with the exception of the end points d 0 100% (N 0 1  107)and d 0 0% (N 0 1.2 107). The field of view of each image is 1.4 mm
by 1.4 mm.

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RESEARCH ARTICLES
superfluid even for arbitrarily weak interactions, and no phase transition occurs. In thecase of unequal mixtures, the phase diagram ispredicted to be much richer (23–28). In themolecular limit of tight binding, all fermions in
the less populated spin state will pair up withatoms in the other state. The resulting molecular condensate will spatially coexist with theremaining Fermi sea of unpaired atoms. As therepulsive interaction between atoms and molecules is increased, the condensate will start toexpel unpaired atoms, leading to a phase separation of the superfluid from the normal phase(24–26, 29, 30). This picture is expected to extend into the BCS limit of weakly bound pairs,
where the pairing gap D prevents unpaired atoms
from entering the BCS superfluid (24–26, 31).
As the binding energy and hence the pairing
gap is further reduced, D will eventually become small compared to the chemical potential difference dm 0 m2 j m1 between the two
spin states, allowing unpaired excess atoms toenter the superfluid region. Close to this point,superfluidity will cease to exist. In the weaklyinteracting BCS limit, the pairing gap is exponentially small compared to the Fermi energy;
Fig. 2. Vortex number versus population imbalance for different interaction strengths. Results are shownfor 812 G or 1/kF a 0 0.2 ( ), 853 G(1/kFa 0 j0.15, )), 874 G (1/kFa 0
j0.3, r), 896 G (1/kFa 0 j0.4, g),and 917 G (1/kFa 0 j0.5, 0).

&Fig. 3. Radial density profiles of the two components of a strongly interacting Fermi gas mixture
with unequal populations. The profiles are azimuthal averages of the axially integrated density. (Aand B) Profiles of the component in state k1À and k2À, respectively, originating from 883 G (1/kF a 0
j0.27). The imaging procedure, as detailed in the text and in (32), involves a magnetic-field sweep
and ballistic expansion. The population imbalance was d 0 0% (red), d 0 46% (blue), and d 0 86%(green). (C) Difference between the distributions in state k1À and k2À. The total number of atoms was
N 0 2.3  107. The clear dip in the blue curve caused by the pair condensate indicates phaseseparation of the superfluid from the normal gas. (D) Color-coded profiles of clouds prepared atthree different interaction strengths. The condensate is clearly visible as the dense central partsurrounded by unpaired fermions or uncondensed molecules. Spin-polarized clouds (d 0 T100%)are not in thermal equilibrium, owing to Pauli suppression of collisions. OD, optical density.

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SCIENCEVOL 311hence, an exponentially small population imbalance can destroy superfluidity.
This superfluid-to-normal transition is an
example of a quantum phase transition, which
occurs even at zero temperature, when all thermal fluctuations are frozen out and only quantum fluctuations prevail. It can also be driven
by increasing the mismatch in chemical potentials between the two spin states to the criticalvalue of dm , D, inducing collapse into thenormal state. In this context the phase transition is known as the Pauli or Clogston limit ofsuperfluidity (1). However, its exact nature—whether there is one or several first- and/or
second-order transitions—remains the subjectof debate (6, 27, 28).
Imbalanced spin populations. As the starting point of our experiments, we prepared adegenerate Fermi gas of spin-polarized 6Li
atoms, using methods of laser cooling, sympathetic cooling by sodium atoms, and opticaltrapping (32). A radiofrequency sweep with an
adjustable sweep rate created a variable spin
mixture of the two lowest hyperfine states,labeled k1À and k2À. Interactions between these
two states are strongly enhanced around a 300G-wide Feshbach resonance located at B0 0
834 G. At lower values of the magnetic-bias
field B, two isolated fermions can bind into astable molecule (BEC side), whereas at higher
values fermion pairs can exist only in the stabilizing presence of the surrounding gas (BCSside). The interaction is described by the parameter 1/kFa, where a is the scattering lengthand kF is defined as the Fermi momentum of anoninteracting, equal spin mixture.
For the study of vortices and superfluid
flow as a function of population imbalance,
the spin mixture was set in rotation on the
BEC side, using two laser beams that rotatedsymmetrically around the cloud (19, 32). Starting with either a rotating or a nonrotating cloud,we then varied the interaction strength by ramping the magnetic field B to several values
around the Feshbach resonance. To image thefermion pair condensates, the cloud was released from the optical trap and the binding
energy of the pairs was increased by switching the magnetic field to the BEC side, faraway from resonance (13, 14, 19, 32). Thisrevealed the center-of-mass wave function of
the pairs and thus, for rotating clouds, the eventual presence of vortices.
Figure 1 shows images of the two spinstates for varying population imbalance, originating from the BEC side (B G 834 G) andfrom the BCS side (B 9 834 G) of the resonance. Starting with a pure Fermi sea instate k1À, we see how gradually, for increasing
numbers in the second spin state k2À, first a
normal (uncondensed) cloud of fermion pairs
emerges, then a condensate peak appears
within the normal cloud (see also Fig. 3, A
and B). The condensate can be clearly distinguished in the minority cloud as the dense

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494

allowing unequal densities, a careful analysis of
the 3D density, reconstructed from the integrated
optical densities, will be necessary.
We did not observe (by simultaneously imaging along the long and short axis) a modulation in the condensate density as would be
predicted for the FFLO state (23, 38, 39). However, this state is predicted to be favored only ina narrow region of parameter space and might
have escaped our attention.
The condensate fraction was determinedfrom the minority component, which in allcases is very well fit by a Gaussian for thermal molecules and unpaired atoms, plus a
Thomas-Fermi profile for the condensate (fig.
S2) (32). Figure 4 shows the condensate fraction obtained for varying population difference and temperature, and for several magnetic
fields (i.e., interaction strengths) around resonance. The data for 754 G, on the BEC sideof the resonance, show condensation over almost the entire range of population imbalance.
As the interaction strength is increased toward
resonance, the condensate fraction for equal mixtures grows (14). However, for large population

0.4

754 G
1/kF a = 0.75

asymmetries, the condensate disappears. The
window of condensation shrinks further as wecross the resonance and move to the BCS side
(Fig. 4, 833 to 924 G).
The temperature varied with number imbalance, as indicated in the insets of Fig. 4. The
temperature maximum for equal mixtures at
754 G is likely due to the greater energy release
when more deeply bound molecules were formed
and explains the smaller condensate fraction forequal mixtures found at this field. For higher
fields, the temperature changes much less with thespin composition (32). The observed critical population imbalance was only weakly dependent on
temperature. This may reflect the fact that well
below the critical temperature for superfluidity,
the pairing gap is only weakly dependent on temperature (40). The critical imbalance at our coldest
temperatures will thus essentially coincide with its
value at zero temperature.
On resonance, where the scattering length adiverges, the system is in the unitary regime (41),where the only remaining energy scales of thesystem are the Fermi energies EF,1 and EF,2 of thetwo spin components (42). The breakdown of
786 G
1/kF a = 0.39

EKin [nK]
600

818 G
1/kF a = 0.11

400
200

0.3
0.2

Condensate fraction

central region (appearing as white in the image) surrounded by the lower density normalcomponent (appearing as gray). As the condensate size increases and the friction due tothe normal component decreases, vortices appear in the rotating cloud, a direct and unambiguous signature of superfluid flow. As
expected, the largest condensates with thelargest number of vortices are obtained for an
equal mixture. However, superfluidity in thestrongly interacting Fermi gas is clearly not
constrained to a narrow region around the perfectly balanced spin mixture, but is observed
for large population asymmetries.
Figures 1 and 2 summarize our findings
for rotating spin mixtures and displays thenumber of detected vortices versus the population imbalance between the two spin states.
The vortex number measures qualitatively how
deep the system is in the superfluid phase: The
higher the nonsuperfluid fraction, the faster thecondensate’s rotation will damp given the nonvanishing anisotropy [(wx j wy)/(wx þ wy) ,1.5%] of our trap (19, 33, 34). Figure 2 therefore shows the shrinking of the superfluid region with decreasing interaction strength on the
BCS side, closing in on the optimal situationof equal populations.
The fraction of condensed fermion pairs.
Close to the breakdown of superfluidity, vortices
are strongly damped and difficult to observe.
Therefore, the presence of vortices provides only alower bound on the size of the superfluid window.
A more detailed map of the superfluid phase as afunction of interaction strength and temperature
was obtained from a study of condensate fractions,determined from cloud profiles such as thoseshown in Fig. 3. Throughout the whole crossover
region, pair condensation occurred for a broad
range of population imbalances. As expected,
this range was even wider than that obtained
from the observation of vortices.
An intriguing property of the superfluid statewith imbalanced populations is the clear depletionin the excess fermions of the majority component(Fig. 3C). The profiles in Fig. 3 present theaxially integrated density; hence, the true
depletion in the 3D density is even stronger.
The condensate seems to repel the excess
fermions. This feature was observed after
expansion at 690 G, where interactions are still
strong (initially 1/kFa , 2.0). The expansion, atleast in the region around the condensate, ishydrodynamic and should proceed as a scalingtransformation (35, 36). Therefore, the depletion
observed in expansion hints at spatial phaseseparation of the superfluid from the normal
state. This effect was observed throughout theresonance region, and on resonance even when
no magnetic field ramps were performed during
expansion. After submission of our work,
depletion of excess fermions in the center of
the trap was reported (37). However, to distinguish a phase-separated state with equal densities
in the superfluid region from more exotic states

0.1
0.0
-1.0

-0.5

0.0

0.5

1.0 -1.0

-0.5

0.0

0.5

1.0 -1.0

-0.5

0.0

0.5

1.0

EKin [nK]
400
300

0.4

883 G
1/kF a = - 0.27

833 G
1/kF a = 0

0.3

924 G
1/kF a = – 0.44

0.2
0.1
0.0
-1.0

-0.5

0.0

0.5

1.0 -1.0

-0.5

0.0

0.5

1.0 -1.0

-0.5

0.0

0.5

1.0

Population imbalance

Fig. 4. Condensate fraction versus population imbalance for several temperatures and interaction
strengths. The total number of atoms N 0 2.3  107 is constant to within 20% for all data points(TF 0 1.9 mK for an equal mixture) (32). For a given population imbalance, the uppermost curves
for different magnetic fields are approximately isentropically connected. The different symbols
correspond to different evaporation ramps. The average radial kinetic energy per molecule ofthermal clouds in the minority component serves as an indicator for temperature and is shown in
the insets for 754 G (upper) and 833 G (lower) for the coldest data. On resonance, for a population asymmetry of 50%, we measure an energy of kB  300 nK (circles) (kB is the Boltzmann
constant), 345 nK (inverted triangles), 390 nK (squares), 420 nK (triangles), and 505 nK(diamonds). The critical population imbalance dc for the breakdown of condensation at 754 G isabout dc754 , 96%, and at 786 G it is dc786 , 95%. For the data at higher magnetic fields, we
determine dc through a threshold fit to the first three data points with nonzero condensate
fraction for each sign of asymmetry. Although we could have used any reasonable threshold
function, empirically, it was found that the function nc(1 j kd/dck3.3) (nc j maximum condensate
fraction) provided a good fit to all data points. Therefore, it was used for the threshold fits and isshown as a guide to the eye.

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superfluidity occurs for a certain universal ratioof these two or equivalently, in a harmonic trap,
for a certain critical population imbalance. We
determine this universal number to be dc ,T70(3)% for our approximately harmonic trapping
potential. In (37), depletion of excess fermions
was reported up to an imbalance of 85% and wasinterpreted as indirect evidence for superfluidity.
However, superfluidity was not directly observed,
and our data show that the system is normal atthis imbalance.
The critical imbalance dc corresponds to a
Fermi energy difference dEF 0 EF,2 j EF,1 0[(1 þ dc)1/3 j (1 j dc)1/3]EF 0 0.53(3)EF,
where EF is the Fermi energy of an equal mixture of noninteracting fermions. The standard
BCS state is predicted (1) to break downpfor
ffiffi a
critical chemical-potential difference dm 0 2 D.
On resonance, however, Monte-Carlo studies
predict (24) the superfluid breakdown to occur
when dm 0 2.0(1)D 0 1.0(1)EF. Only in the
weakly interacting regime do the chemical potentials equal the Fermi energies. Quantitative
agreement with the Monte-Carlo study would
require that dm , 2dEF. This is not unreasonablegiven that interactions will reduce the chemical potential of the minority component. In apreliminary analysis, we indeed find close
agreement with theory.
Figure 5 summarizes our findings, showing
the critical mismatch in Fermi energies for which

we observed the breakdown of superfluidity as
well as the pairing gap D versus the interaction
parameter 1/kFa. Far on the BEC side of theresonance, the superfluid is very robust withrespect to population imbalance. Here, pairing is
dominantly a two-body process: The smallest
cloud of atoms in state k1À will fully pair withatoms in state k2À and condense at sufficiently
low temperatures. On the BCS side of theresonance, however, pairing is purely a manybody effect and depends on the density of the two
Fermi clouds. As the density of the minority
component becomes smaller, the net energy gainfrom forming a pair condensate will decrease.
Even at zero temperature, this eventually leads tothe breakdown of superfluidity and the quantum
phase transition to the normal state. We haveexperimentally confirmed the qualitative picturethat fermionic superfluidity breaks down when
the difference in chemical potentials between thetwo species becomes larger than the pairing gap.
Concluding remarks and outlook. We haveobserved superfluidity with imbalanced spin
populations. Contrary to expectations for the
weakly interacting case, superfluidity in theresonant region is extremely stable against
population imbalance. As the asymmetry is increased, we observe the quantum phase transition to the normal state, known as the Paulilimit of superfluidity. Our observation opens
up intriguing possibilities for further studies on

Fig. 5. Critical difference in Fermi energies dEF between the two spin states for which thesuperfluid-to-normal transition is observed. dEF for each interaction strength and temperature is
obtained from the critical population imbalance determined in Fig. 3 using dEF /EF 0 (1 þ dc)1/3 j(1 j dc)1/3. The symbols are defined in Fig. 3. The line shows the expected variation of the pairing
gap D, where the value on resonance has been taken from (24) and the exponential behavior in the
BCS regime, D È ejp/2kFkak, was assumed. Although the trend of dEF is expected to follow that of D,
the close agreement is coincidental. Representative density profiles illustrate the quantum phasetransition for fixed interaction and for fixed population imbalance along the dashed lines.

www.sciencemag.org

SCIENCEVOL 311

Fermi systems with mismatched Fermi surfaces. One important aspect concerns the density
distribution in the superfluid regime. Standard
BCS theory allows only equal spin densities,
which would entail complete phase separation of
the superfluid from the normal density. Moreexotic solutions (6) allow superfluidity also withimbalanced densities, most notably the FFLO
state. A more detailed scan of the parameter
space and precise measurements of spatial
profiles might resolve the long-standing questionof the true ground state. Equally fascinating is thenature of the strongly correlated normal stateslightly below resonance. For sufficient population imbalance, we have the remarkable situation in which bosonic molecules, stable evenin isolation, do not condense at zero temperature, owing to the presence of the Fermi sea.
References and Notes
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8. A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso,
J. L. Sarrao, Phys. Rev. Lett. 91, 187004 (2003).
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47 (2003).
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040403 (2004).
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J. E. Thomas, Phys. Rev. Lett. 92, 150402 (2004).
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C. H. Schunck, W. Ketterle, Nature 435, 1047 (2005).
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and X.-G. Wen, E. Demler, and S. Sachdev for stimulating

Community Genomics Among
Stratified Microbial Assemblages
in the Ocean’s Interior
Edward F. DeLong,1* Christina M. Preston,2 Tracy Mincer,1 Virginia Rich,1 Steven J. Hallam,1
Niels-Ulrik Frigaard,1 Asuncion Martinez,1 Matthew B. Sullivan,1 Robert Edwards,3Beltran Rodriguez Brito,3 Sallie W. Chisholm,1 David M. Karl4
Microbial life predominates in the ocean, yet little is known about its genomic variability,
especially along the depth continuum. We report here genomic analyses of planktonic microbialcommunities in the North Pacific Subtropical Gyre, from the ocean’s surface to near–sea floor
depths. Sequence variation in microbial community genes reflected vertical zonation of taxonomic
groups, functional gene repertoires, and metabolic potential. The distributional patterns of
microbial genes suggested depth-variable community trends in carbon and energy metabolism,attachment and motility, gene mobility, and host-viral interactions. Comparative genomic analysesof stratified microbial communities have the potential to provide significant insight intohigher-order community organization and dynamics.
icrobial plankton are centrally involvedin fluxes of energy and matter in the sea,
yet their vertical distribution and functional variability in the ocean_s interior is still only
poorly known. In contrast, the vertical zonation of
eukaryotic phytoplankton and zooplankton in theocean_s water column has been well documented
for over a century (1). In the photic zone, steep
gradients of light quality and intensity, temperature,and macronutrient and trace-metal concentrations
all influence species distributions in the watercolumn (2). At greater depths, low temperature,
increasing hydrostatic pressure, the disappearanceof light, and dwindling energy supplies largely
determine vertical stratification of oceanic biota.
For a few prokaryotic groups, vertical distributions and depth-variable physiological properties
are becoming known. Genotypic and phenotypic
properties of stratified Prochlorococcus Becotypes[
for example, are suggestive of depth-variable
adaptation to light intensity and nutrient availability (3–5). In the abyss, the vertical zonation of
deep-sea piezophilic bacteria can be explained inM

1
Massachusetts Institute of Technology, Cambridge, MA
02139, USA. 2Monterey Bay Aquarium Research Institute,
Moss Landing, CA 95064, USA. 3San Diego State University, San Diego, CA 92182, USA. 4University of Hawaii
Honolulu, HI 96822, USA.

*To whom correspondence should be addressed. E-mail:delong@mit.edu

496

part by their obligate growth requirement forelevated hydrostatic pressures (6). In addition,
recent cultivation-independent (7–15) surveys haveshown vertical zonation patterns among specific groups of planktonic Bacteria, Archaea,
and Eukarya. Despite recent progress however,a comprehensive description of the biologicalproperties and vertical distributions of planktonic microbial species is far from complete.
Cultivation-independent genomic surveys
represent a potentially useful approach for characterizing natural microbial assemblages (16, 17).
BShotgun[ sequencing and whole genome assembly from mixed microbial assemblages has been
attempted in several environments, with varying
success (18, 19). In addition, Tringe et al. (20)compared shotgun sequences of several disparate
microbial assemblages to identify communityspecific patterns in gene distributions. Metabolic
reconstruction has also been attempted with environmental genomic approaches (21). Nevertheless, integrated genomic surveys of microbialcommunities along well-defined environmental
gradients (such as the ocean_s water column)
have not been reported.
To provide genomic perspective on microbial
biology in the ocean_s vertical dimension, wecloned large EÈ36 kilobase pairs (kbp)^ DNA
fragments from microbial communities at different depths in the North Pacific Subtropical Gyre

27 JANUARY 2006

VOL 311SCIENCEdiscussions. This work was supported by the NSF, Office ofNaval Research, Army Research Office, and NASA.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/1122318/DC1
Materials and Methods
Figs. S1 and S2References and Notes
7 November 2005; accepted 14 December 2005
Published online 22 December 2005;
10.1126/science.1122318
Include this information when citing this paper.

(NPSG) at the open-ocean time-series station
ALOHA (22). The vertical distribution of microbial genes from the ocean_s surface to abyssal
depths was determined by shotgun sequencing of
fosmid clone termini. Applying identical collection,cloning, and sequencing strategies at seven depths(ranging from 10 m to 4000 m), we archived
large-insert genomic libraries from each depthstratified microbial community. Bidirectional DNA
sequencing of fosmid clones (È10,000 sequences
per depth) and comparative sequence analyses
were used to identify taxa, genes, and metabolic
pathways that characterized vertically stratified
microbial assemblages in the water column.
Study Site and Sampling Strategy
Our sampling site, Hawaii Ocean Time-series(HOT) station ALOHA (22-45’ N, 158-W),
represents one of the most comprehensively
characterized sites in the global ocean and has
been a focal point for time series–oriented oceanographic studies since 1988 (22). HOT investigators have produced high-quality spatial andtime-series measurements of the defining physical, chemical, and biological oceanographic parameters from surface waters to the seafloor. These
detailed spatial and temporal datasets present
unique opportunities for placing microbial genomic depth profiles into appropriate oceanographic context (22–24) and leverage these data
to formulate meaningful ecological hypotheses.
Sample depths were selected, on the basis ofwell-defined physical, chemical, and biotic characteristics, to represent discrete zones in the watercolumn (Tables 1 and 2, Fig. 1; figs. S1 and S2).
Specifically, seawater samples from the upper
euphotic zone (10 m and 70 m), the base of thechlorophyll maximum (130 m), below the base of
the euphotic zone (200 m), well below the upper
mesopelagic (500 m), in the core of the dissolved
oxygen minimum layer (770 m), and in the deep
abyss, 750 m above the seafloor (4000 m), werecollected for preparing microbial community DNA
libraries (Tables 1 and 2, Fig. 1; figs. S1 and S2).
The depth variability of gene distributions was
examined by random, bidirectional end-sequencingof È5000 fosmids from each depth, yielding È64
Mbp of DNA sequence total from the 4.5 Gbp
archive (Table 1). This represents raw sequence
coverage of about 5 (1.8 Mbp sized) genome
equivalents per depth. Because we surveyed
È180 Mbp of cloned DNA (5000 clones bywww.sciencemag.org

Fermionic Superfluidity with Imbalanced Spin
Populations
Martin W. Zwierlein, André Schirotzek, Christian H. Schunck,
Wolfgang Ketterle

Materials and Methods
Creation of ultracold imbalanced spin mixtures. To map out the superfluid regime as afunction of population imbalance, we use two complementary techniques: the detection of vortices in a rotating cloud (S1) and the determination of the fraction of condensed fermion pairs
in non-rotating mixtures (S2–S4). The two experimental methods require slightly different procedures for imaging the pair condensate wavefunction after release from the trap. To determine
the fraction of condensed vs uncondensed pairs, the condensate must separate well from thethermal cloud and should therefore remain small. For the detection of rotating clouds, the condensate should expand to a large size in order to magnify the vortices. In the following, we givethe parameters used to determine the condensate fraction (experiment B) in parentheses after
those used for vortex detection (experiment A).
In the first stage of the experiment, fermionic 6 Li atoms were sympathetically cooled todegeneracy by 23 Na atoms in a magnetic trap (S5). The ultracold cloud was subsequently loaded
into an optical dipole trap (waist w ≈ 120 µm) at a maximum trap depth of about 8 µK. At amagnetic bias field of 875 G, a variable spin-mixture of the two lowest hyperfine states (labelled|1i and |2i) was created via a Landau-Zener radiofrequency sweep with an adjustable sweep
rate. The spin mixture was evaporatively cooled further by lowering the trap depth to 1.6 µK
resulting in radial and axial trap frequencies of νr = 110 Hz and νa = 23 Hz, respectively.
At the same time, the magnetic field was ramped to about 815 G, which is on the BEC-side
close to the Feshbach resonance at 834 G (S6), and deep in the strongly interacting regime. The
rather moderate evaporation still leaves room for thermal molecules in an equal mixture, but waschosen to efficiently cool highly asymmetric mixtures, avoiding spilling of large Fermi clouds.
This ensured that the total number of atoms was approximately constant and independent of theasymmetry between the two spin states (see Fig. S1).
For the vortex experiment, we set the spin mixture in rotation using two blue-detuned laser
beams (wavelength 532 nm) rotated symmetrically around the cloud at angular frequency Ω =2π 70 Hz) (S1). After 800 ms of stirring, the rotating cloud was left to equilibrate for several
hundred ms (S7).
1Tuning the interactions and imaging the cloud. Starting with either the rotating or the nonrotating cloud, we varied the interaction strength between the two spin states in the gas byramping the magnetic field in 100 ms (experiment B: 500 ms) to several values around the Feshbach resonance (for the condensate fraction experiment B, the trap depth was simultaneously
increased to 4 µK (νr = 192 Hz)). After 50 ms (B: 100 ms) of hold time, an image of thecloud was taken following the procedure outlined in (S1): After releasing the cloud from theoptical trap the binding energy of fermion pairs was rapidly increased by ramping the magnetic
field within 2 ms (B: 200 µs) to 690 G, in the far wings of the resonance on the BEC side.
Here, fermion pairs were stable throughout further expansion. After a total of 11 ms (B: 14ms) of expansion (in the remaining magnetic saddle-point potential) an image of either state |1i
or state |2i was taken. In experiment A on vortices, the clouds were imaged at 690 G. For thecondensate fraction data (B), the magnetic field was suddenly switched to 800 G right before
imaging. At this field the molecules absorb the probe light with the same strength as free atoms(reduced to 75% at 690 G). The images revealed the center-of-mass wavefunction of the pairs
and, for rotating clouds, eventually the presence of vortices. For the condensate fraction experiment, the 200 µs fast ramp to the BEC-side immediately after release from the trap ensured thateven large condensates separated well from the normal, uncondensed component. A rapid ramp
to zero field would result in an even better separation (S3, S4), but we found that field sweeps
into regions far outside the resonance cause loss of observed atoms (S4). Since the ramp was
fast compared to the radial trapping period, the size of the expanded condensate was mostly
governed by the residual mean-field interaction at 690 G, where a = 1400 a0 . This expanded
size was actually smaller than that of a condensate released from equilibrium at 690 G, becauseof the lower mean-field energy before expansion.
The described approach to detect fermion pair condensates originating from the BCS-side
was theoretically discussed by several authors (S8–S12). In some cases (S8, S10, S11) evenquantitative agreement with experiments (S2, S3) was reached. It is reasonable to assume thatcondensed fermion pairs on the BCS-side can transform into zero-momentum molecules, aslong as their original pair size is smaller than the interparticle spacing. Experimentally, ourgroup showed that the formation dynamics of these condensates is slow compared to the timeneeded to cross resonance in the rapid ramp (S4), excluding growth of the condensate during
the ramp. The ramp employed in our present work does not result in loss of atoms, previously
encountered for ramps to zero field (S4). The condensate fraction is thus a direct measure of thesuperfluid component in the mixture.

2

Determining the condensate fraction. Fig. S2 shows typical optical density profiles originating from resonance. Azimuthal averaging resulted in very good signal-to-noise in the opticaldensity, with relative fluctuations of 10−3 . The total signal fluctuated from shot-to-shot by about
5%, due to frequency fluctuations of the imaging laser, but the profile’s shape was not affected.
Imperfections in the polarization of the light, stray light etc. tend to systematically lower the
optical density. Indeed, the total signal obtained by imaging with circularly polarized lightalong the magnetic field axis was 20% smaller than the (more robust) number obtained fromimaging perpendicular to the magnetic field axis. The latter coincided with the total number ofatoms, which was calibrated by measuring the trap frequencies and the Fermi radius of almost
perfectly spin-polarized degenerate Fermi clouds (a pure spin-polarized Fermi gas would notreach thermal equilibrium due to Pauli suppression of collisions).
The profile of the minority component was very well fit by a bimodal fit, consisting of a22
gaussian e−r /2Rth plus a Thomas-Fermi-profile of the form (1 − x2 )3/2 for the condensate. The
standard deviation in the condensate fraction for three measurements taken over the course of anhour was 1%, while the standard deviation of the population imbalance was 3%. A more refined
fitting function, accounting for depletion of the thermal cloud in the presence of the condensate,increased the condensate fraction by at most 10% (for a condensate fraction of 50%), withoutchanging the quality of the fit (see Fig. S2).
The highest condensate fractions of 40% were limited by the chosen evaporation ramp,
which avoided spilling of large Fermi clouds but did not result in the coldest attainable temperatures for equal mixtures (see Fig. S1). Essentially pure condensates, with no appreciable
thermal component, were achieved in equal mixtures by evaporating to a lower final trap depth.
For asymmetric mixtures, such a ramp strongly reduces the population imbalance by preferentially removing unpaired atoms.
Thermometry in the strongly interacting regime. The width of the expanded thermal molecular cloud Rth after the rapid ramp to the BEC-side provides an upper limit for the system’s
temperature (see Fig. S1). The average radial kinetic energy per molecule in the thermal cloud2
is given by Ekin = 2mRth/t2 , where m is the mass of a lithium atom and t is the expansion
time. A standard correction was applied to this formula accounting for the cloud’s expansion in
the harmonic saddle-point potential. For a non-interacting cloud, Ekin /kB equals the temperature of the gas, where kB is Boltzmann’s constant. Interactions increase Ekin . Note, however,that the mixture immediately after the rapid ramp to the BEC-side is more dilute than an equilibrium sample at 690 G, due to the large initial size of the Fermi gas on the BCS-side. Ekin /kBshould therefore be a good estimate of the system’s temperature. Indeed, for equal mixtureson resonance, a condensate was observed when Ekin /EF (EF is the Fermi energy of an equal
mixture) was lowered from 0.34 (diamonds in Fig. 4, 833 G) to 0.28 (triangles in Fig. 4). Thisis close to the predictions TC /TF = 0.29 (S13), TC /TF = 0.31 (S14) and TC /TF = 0.3 (S15)for the ratio of the critical temperature of superfluidity on resonance and the Fermi temperature
TF = EF /kB .
3

Population imbalance

6

Atom number [x 10 ]

20

15

10

5

0.5

0.0

-0.5

0.2

0.4
0.6
Landau-Zener probability

0.8

1.0

350
300
-1.0

-0.5

0.0
Population imbalance

0.5

1.0

-0.5

0.0
Population imbalance

0.5

1.0

6

Atom number [x 10 ]EKin [nK]

0.0

20
16
-1.0

Figure 1: Upper two panels: Dependence of the spin populations after evaporation and theirimbalance as a function of the rate of the radiofrequency sweep applied to control the spin
populations at full trap depth. These sweeps are characterized by their Landau-Zener transfer
probability. Lower panels: Dependence of the average radial kinetic energy and the total numberof atoms in the mixture as a function of population imbalance. The clouds originate from 883
G (BCS-side, see Fig. 4). Red closed circles: Atoms in state |1i. Open blue circles: Atoms instate |2i. The data-points are averages of three measurements taken over the course of one hour.
4

-2

Residual [x 10 ]

2
1
0
-1
-2
-3

Optical density

1.5

1.0

0.5

0.0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Distance from center [mm]Figure 2: Radial density profiles of an unequal spin mixture originating from resonance. The
density profile of the minority component (red curve) was fit using two different models for thethermal cloud: the non-interacting case (gaussian, dotted line) and the strongly interacting caseof a thermal cloud that is fully separated from the condensate (leading to a flat distribution in
the axially integrated profile at points where there is a condensate, dash-dotted line). The twofits are indistinguishable on the scale of this graph. The residual of the gaussian plus ThomasFermi-fit after subtraction of the original profile is shown on top.

5

References and Notes
S1. M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, W. Ketterle, Nature
435, 1047 (2005).
S2. C. A. Regal, M. Greiner, D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004).
S3. M. W. Zwierlein, et al., Phys. Rev. Lett. 92, 120403 (2004).
S4. M. W. Zwierlein, C. H. Schunck, C. A. Stan, S. M. F. Raupach, W. Ketterle, Phys. Rev.
Lett. 94, 180401 (2005).
S5. Z. Hadzibabic, et al., Phys. Rev. Lett. 91, 160401 (2003).
S6. M. Bartenstein, et al., Phys. Rev. Lett. 94, 103201 (2004).
S7. Note that due to damping in our not perfectly round trap, the mixture will rotate at a speed
lower than Ω. The vortex number in an equal mixture around 875 G (BCS-side) decayedwith a time constant of 3 s. Asymmetric spin mixtures should spin down faster, due to theincreased friction exerted by the non-superfluid component.
S8. T.-L. Ho. Preprint cond-mat/0404517.
S9. A. V. Avdeenkov, J. L. Bohn, Phys. Rev. A 71, 023609 (2005).
S10. A. Perali, P. Pieri, G. C. Strinati, Phys. Rev. Lett. 95, 010407 (2005).
S11. L. Salasnich, N. Manini, A. Parola, Phys. Rev. A 72, 023621 (2005).
S12. E. Altman, A. Vishwanath, Phys. Rev. Lett. 95, 110404 (2005).
S13. J. Kinast, et al., Science 307, 1296 (2005).
S14. A. Perali, P. Pieri, L. Pisani, G. C. Strinati, Phys. Rev. Lett. 92, 220404 (2004).
S15. G. M. Bruun, H. Smith, Phys. Rev. A 72, 043605 (2005).

6

Appendix I
Direct Observation of theSuperfluid Phase Transition in
Ultracold Fermi Gases
This appendix contains a reprint of Ref. [269]: Martin W. Zwierlein, Christian H.
Schunck, André Schirotzek, and Wolfgang Ketterle, Direct Observation of the Superfluid Phase Transition in Ultracold Fermi Gases, Nature 442, 54-58 (2006)242

Vol 442|6 July 2006|doi:10.1038/nature04936

LETTERS
Direct observation of the superfluid phasetransition in ultracold Fermi gases
Martin W. Zwierlein1, Christian H. Schunck1, André Schirotzek1 & Wolfgang Ketterle1

Phase transitions are dramatic phenomena: water freezes into ice,
atomic spins spontaneously align in a magnet, and liquid helium
becomes superfluid. Sometimes, such a drastic change in behaviour is accompanied by a visible change in appearance. The
hallmark of Bose–Einstein condensation and superfluidity intrapped, weakly interacting Bose gases is the sudden formationof a dense central core inside a thermal cloud1–7. However, instrongly interacting gases—such as the recently observed fermionic superfluids8 —there is no longer a clear separation between the
superfluid and the normal parts of the cloud. The detection offermion pair condensates has required magnetic field sweeps9–11
into the weakly interacting regime, and the quantitative descriptionof these sweeps presents a major theoretical challenge. Here we
report the direct observation of the superfluid phase transition in astrongly interacting gas of 6Li fermions, through sudden changes inthe shape of the clouds—in complete analogy to the case of weaklyinteracting Bose gases. By preparing unequal mixtures of the two
spin components involved in the pairing12,13, we greatly enhance thecontrast between the superfluid core and the normal component.
Furthermore, the distribution of non-interacting excess atoms
serves as a direct and reliable thermometer. Even in the normal
state, strong interactions significantly deform the density profileof the majority spin component. We show that it is these interactions that drive the normal-to-superfluid transition at thecritical population imbalance of 70 6 5 per cent (ref. 12).
The dramatic signature of Bose–Einstein condensation in weaklyinteracting gases in atom traps derives from a natural hierarchy ofenergy scales: the critical temperature for condensation, T C / n 2/3 atparticle density n, is much larger than the chemical potential(divided by the Boltzmann constant k B) of a pure condensate,m / na, which measures the interaction strength between particles(a is the scattering length). Hence, for weak (repulsive) interactions(a . 0, na 3 ,, 1), the condensate is clearly distinguished from thecloud of uncondensed particles through its smaller size and higher
density. However, as the interactions are increased, for example bytuning a using a Feshbach resonance, this hierarchy of energy scales
breaks down, as m can now become comparable to k BT C. In Fermi
gases with weak attractive interaction (a , 0, njaj3 ,, 1), thechemical potential is given by the Fermi energy E F and will evenfar exceed the superfluid transition temperature kB T C / EF e2p=2kF jaj(where k F / n 1/3 is the Fermi wave vector). Both the normal and thecondensed cloud will here be of the same size and shape, dependent
only on E F and the trapping potential.
The phase transition from the normal to the superfluid state,although dramatic in its consequences, is thus not revealed by amajor change in the appearance of the gas. Indeed, in stronglyinteracting Fermi gases no deviation from a normal cloud’s shape
has so far been detected, either in the unitary regime, where adiverges, or on the attractive Bardeen–Cooper–Schrieffer (BCS)side of a Feshbach resonance. Theoretical works predicted small
1‘kinks’14–16 or other slight deviations17 in the density profiles of thegas in the superfluid regime, but after line-of-sight integration theseeffects have so far been too small to be observable. Condensates could
only be observed via rapid magnetic field ramps to the Bose–Einsteincondensate (BEC) side (a . 0) of the Feshbach resonance, performed during expansion9,10. This suddenly reduced the condensate’s
chemical potential, and let the thermal fraction grow beyond thecondensate size. A similar ramp was used to detect vortices onresonance and on the BCS side in the demonstration of fermionicsuperfluidity8. However, these magnetic field ramps are difficult tomodel theoretically, and a satisfactory quantitative comparison of,for example, the condensate fraction with experiments has not beenaccomplished18–21.
In this work we demonstrate that the normal-to-superfluid phasetransition in a strongly interacting Fermi gas can be directly observedin absorption profiles, without the need for any magnetic field ramps.
As in the case of weakly interacting BECs, preparation, expansion anddetection of the sample all take place at the same, fixed magnetic fieldand scattering length. As for BECs, the phase transition is observed asa sudden change in the shape of the cloud during time-of-flight
expansion, when the trap depth is decreased below a critical value. To
clearly distinguish the superfluid from the normal component, we
break the number symmetry between spin-up (majority atom
number, N ") and spin-down (minority atom number, N #) andproduce an unequal mixture of fermions (imbalance parameterd ¼ ðN " 2 N # Þ=ðN " þ N # ÞÞ. Standard BCS superfluidity requires
equal densities of the two spin components. Hence, when cooled
below the phase transition the cloud should show a sudden onset of asuperfluid region of equal densities. Indeed, below a critical temperature, we observe how the density distribution of the minority
component becomes bimodal.
Breaking the symmetry in atom numbers thus produces a direct
and striking signature of the superfluid phase transition22–24. A
similar situation has been encountered in Bose–Einstein condensation, where breaking the symmetry of a spherical trap resulted indramatic anisotropic expansion of the condensate, now a hallmark of
the BEC phase transition.
Figure 1 shows column density profiles of the two imbalanced spinstates for different points along the evaporation path corresponding
to different temperatures, and for three magnetic fields that correspond to the BEC side, exact resonance and the BCS side of theresonance. For large final trap depths (upper panels in Fig. 1), thesmaller cloud has the expected shape of a normal, non-superfluid gas:
it is very well fitted using a single, finite temperature Thomas–Fermiprofile (with central optical density, radius and the fugacity as
independent fit-parameters). However, below a critical trap depth,
a second, denser feature appears in the centre of the minority
component (lower panels in Fig. 1). This onset of bimodality occurs
very suddenly as the trap depth is lowered, as can be seen from Fig. 2:
Around the critical point, the atom number (Fig. 2a) and population

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

54

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NATURE|Vol 442|6 July 2006

imbalance (Fig. 2b) are practically constant, and the temperature(Fig. 2c) varies in a smooth linear way with the trap depth. In contrast,
below the critical trap depth, the shape of the smaller cloud starts todeviate drastically from the Thomas–Fermi distribution of a normalgas, as quantified in Fig. 2d. This sudden increase in the standard
deviation of a fit to a single-component fitting function is a standard
way of identifying the BEC phase transition in a model-independent
way2.
Figure 2e displays the fact that below the critical trap depth a new,third radius is required to describe the two clouds. As we will see below,
the appearance of this central feature coincides with the appearance ofthe fermion pair condensate in experiments involving the magnetic fieldramp technique10–12. It is this condensate that contains the superfluid
vortices in refs 8 and 12. We are thus naturally led to interpret the central
core as the condensate of fermion pairs, and the outer wings as thenormal, uncondensed part of the cloud. This constitutes, to ourknowledge, the first direct observation of the normal-to-superfluid
phase transition in resonantly interacting Fermi gases on resonance andon the BCS side (that is, without a magnetic field sweep that so far
cannot be quantitatively accounted for).
Already at high temperatures, above the phase transition, thelarger cloud’s profile is strongly deformed in the presence of thesmaller cloud, a direct signature of interaction. Indeed, on resonancethe cloud size of the minority component is significantly smaller thanthat of a non-interacting sample with the same number of atoms (seeFig. 2e). At the phase transition, the outer radii of the clouds do notchange abruptly. This demonstrates that interactions, not superfluidity, are the main mechanism behind the reduced cloud size of aninteracting Fermi gas.
On the BEC side, the condensate is clearly visible in the larger cloud.

On resonance, however, the condensate is not easily discernible in thelarger component’s profiles at the scale of Fig. 1. Nevertheless, wehave found a very faint but reproducible trace of the condensate
when analysing the curvature of these column density profiles (seeSupplementary Fig. S1). On resonance and on the BCS side, the onsetof bimodality in the smaller cloud can be clearly observed for
imbalances larger than ,20% (but below a certain critical imbalance,
see below), for which the condensate is small compared to theminority cloud size. With increasing magnetic field on the BCS
side (that is, with decreasing interaction strength), the bimodality
becomes less pronounced and is not clearly discerned beyond 853 G(interaction parameter 1/k F a , 20.15).
Thermometry of strongly interacting Fermi gases has always been amajor difficulty in experiments on strongly interacting fermions25. A
thermometer can only be reliable if the working substance is notaffected by the sample to be measured. In equal mixtures of fermions,the two overlapping atomic clouds are strongly interacting throughout. Temperatures determined from a non-interacting Thomas–
Fermi fit to these clouds need calibration based on approximate
theoretical calculations25. In addition, as will be reported elsewhere,
we find that those fits do not describe the profiles of a partiallysuperfluid Fermi gas as well as they do in the normal state, inagreement with theory14–17. In the case of imbalanced mixtures, thewings of the larger component, where the spin-down species areabsent, are non-interacting and thus serve as a direct thermometer(see Fig. 2c). For an imbalance of d ¼ 75 ^ 3% we determine thecritical temperature for the phase transition on the BEC side at 1/k F
a ¼ 0.46 to be T/T F ¼ 0.18(3) ðkB T F ¼ "qð3ðN " þ N # ÞÞ1=3 ;"2 k2F =2m is the Fermi energy of a non-interacting, equal mixture
with the same total number of fermions N " þ N #, q/2p is the

Figure 1 | Direct observation of the phase transition in a stronglyinteracting two-state mixture of fermions with imbalanced spin
populations. Top (a–c) and bottom (d–f) rows show the normal and thesuperfluid state, respectively. Panels a and d were obtained in the BEC
regime (at B ¼ 781 G), b and e on resonance (834 G), and c and f on the BCS
side of the Feshbach resonance (853 G). The profiles represent the azimuthal
average of the column density after 10 ms (BEC side) or 11 ms (on resonanceand BCS side) of expansion. The appearance of a dense central feature in thesmaller component marks the onset of condensation. The condensate causes
a clear depletion in the difference profiles (bottom of each panel). Both in
the normal and in the superfluid state, interactions between the two spinstates are manifest in the strong deformation of the larger component. The
dotted lines show Thomas–Fermi fits to the wings of the column density.
The radii R " and R # mark the Fermi radius of a ballistically expanding,
non-interacting cloud with atom number N ", N #. The trap depth U (in mK),the atom numbers, the population imbalance d (in %), the interaction
parameter 1/k F a, the temperature T (in nK) and the reduced temperature
T/T F were respectively: a, 4.8, 1.8 £ 107 and 2.6 £ 106, 75, 0.42, 350, 0.20;
b, 3.2, 1.8 £ 107 and 4.2 £ 106, 63, 0 (resonance), 260, 0.15; c, 2.5, 1.5 £ 107
and 4.5 £ 106, 52, 20.13, 190, 0.12; d, 0.8, 6.5 £ 106 and 1.5 £ 106, 62, 0.67,
50, #0.05; e, 1.1, 1.5 £ 107 and 3.8 £ 106, 60%, 0 (resonance), 70, 0.06; f, 1.2,
1.3 £ 107 and 4.4 £ 106, 50, 20.15, 100, 0.08. a.u., arbitrary units.

© 2006 Nature Publishing Group55

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NATURE|Vol 442|6 July 2006

geometric mean of the trapping frequencies, and m is the mass of 6Li).
This corresponds to T/T C,# ¼ 0.55(9) when comparing the temperature to the critical temperature T C,# for Bose condensation in a noninteracting gas with N # bosons. The reduction in the critical temperature is a direct consequence of strong repulsive interactions between themolecules. On resonance, at d ¼ 59 ^ 3%, we find T/T F ¼ 0.12(2),and on the BCS side (1/k F a ¼ 20.14) for d ¼ 53 ^ 3% we obtain T/
T F ¼ 0.11(2). These are, to our knowledge, the first directly measured
and reliable temperatures for the superfluid transition in stronglyinteracting Fermi gases. They may serve as a checkpoint for theoretical models.

We note that the critical temperature will in general depend on the
population imbalance. For example, for large enough imbalance on
resonance or on the BCS side, no condensate will form even at zerotemperature12, as we discuss below. Here, the critical temperature forsuperfluidity will be zero.
An important qualitative difference distinguishes the BEC sidefrom resonance at the lowest temperatures. On the BEC side, the gas
consists of only two parts—the superfluid core surrounded by a fullypolarized degenerate Fermi gas of the excess species. On resonanceand on the BCS side, however, there exists a third region, a normal
state in which both species are mixed. Several recent theories describe

Figure 2 | Characterization of the phase transition. a–e, The datacharacterize the evolution of the fermion mixture as the cloud isevaporatively cooled by lowering the trap depth. The chosen magnetic fields
are identical to those in Fig. 1. Data obtained from the majority (minority)
cloud are shown as diamonds (circles). a, The atom number; b, thepopulation imbalance between the two spin states; and c, the temperature of
the spin mixture as determined from the non-interacting wings of thelarger cloud’s profile. d, A finite temperature Fermi–Dirac (for resonanceand the BCS side) or gaussian (for the BEC side) distribution is fitted tothe minority cloud; the phase transition is marked by a sudden increase inx 2 as the condensate starts to appear. e, Outer radii of the majority andminority cloud (for the minority cloud on the BEC side, thermal cloud
radius; all other cases, Thomas–Fermi radius) as well as the condensate
radius (open circles), defined as the position of the ‘kink’ in the minority
profile (see Fig. 1). The majority (minority and condensate) cloud size isnormalized by the Fermi radius R " (R #) of a non-interacting cloud with N "(N #) atoms, and adjusted for ballistic (hydrodynamic) expansion. Notethat the imbalance decreases during evaporation because the larger
majority cloud incurs stronger evaporative losses. For the data, three(BEC and resonance) to five (BCS) independent measurements wereaveraged.

56© 2006 Nature Publishing GroupLETTERS

NATURE|Vol 442|6 July 2006

density profiles of imbalanced Fermi mixtures26–32. Mean-field theories
that neglect interactions in the normal cloud and between the normal
and condensed cloud are only in qualitative agreement with our
results. Descriptions that exclude the mixed region or find superfluidity on resonance at all population imbalances are ruled out by
our observations.
To elucidate the origin of the clear separation between condensate
and normal components, we varied the population imbalance at ourcoldest temperatures and on resonance. Figure 3b shows several
resulting profiles after 11 ms expansion from the trap. For large
imbalances, d . 70%, the minority cloud is not bimodal and well fitted
by a (unconstrained) Thomas–Fermi profile. At a critical imbalanceof d < 70%, the condensate appears and then grows further as
the imbalance is reduced (for the cloud radii, see Supplementary
Fig. S2).
To characterize the appearance of the condensate for imbalances
around d ¼ 70%, a Thomas–Fermi profile is fitted to the wings of theminority cloud. The fraction of atoms not contained in this fit is ameasure of the condensate fraction (see Fig. 3). We find a criticalimbalance of d c ¼ 70(5)% above which the condensate disappears.
This agrees with our previous work12, where we employed a rapid
ramp method to the BEC side to extract the condensate fraction. We

Figure 3 | Quantum phase transition to superfluidity for decreasing
population imbalance. a, Main panel, the ‘condensate fraction’ of excess
minority atoms, not contained in the Thomas–Fermi fit, versus population
imbalance on resonance. b, Column density profiles of majority (blue) andminority (red) clouds, azimuthally averaged, for varying population
imbalance. The condensate is clearly visible in the minority component as
the dense central feature on top of the normal background (finitetemperature Thomas–Fermi fit, dotted lines). Below the critical imbalanced c ¼ 70%, the condensate starts to form. The inset in a shows the central
densities of the larger (black diamonds) and smaller (grey circles) cloud in
the normal state above d c. This demonstrates that here the central densities
are unequal, suppressing superfluidity. The densities were calculated fromthe central optical density and the fitted size of the clouds, assuming local
density approximation and adjusting for ballistic (hydrodynamic)
expansion of the outer radii of majority (minority) clouds. The data points
for the condensate fraction show the average of several independentmeasurements.

observed the quantum phase transition from the superfluid to thenormal state as a critical population imbalance of d c ¼ 70% wasexceeded. This strongly suggests that the bimodality observed here
directly in the minority component, and the bimodality observed inmolecular clouds after a magnetic field sweep, are signatures of thesame phase transition.
The transition at d c is known as the Clogston limit of superfluidity12,33, and occurs when the chemical potential difference dm
becomes larger than a constant times the (local) superfluid gap D(r)(see Supplementary Information). Here we present a simple picture
for the character of this phase transition in a harmonic trap.
Thomas–Fermi fits for the normal clouds beyond d c allow a simple
estimate of the central three-dimensional density of the gas (with anestimated accuracy of 20% for the relative density difference), shownin the inset of Fig. 3. For large imbalances, we find that the threedimensional densities differ significantly, as is expected for two
weakly interacting Fermi clouds. As the imbalance is reduced towards
the critical d c , the central densities approach each other and become
approximately equal around d c. This is a direct consequence of strong
interactions in the normal state. In a non-interacting Fermi mixture
with an imbalance of d c, the central densities would differ by a factorof 2.4.
This observation now offers an intriguing insight into the nature ofa fermionic superfluid on resonance or on the BCS side. Already inthe normal state above T C or beyond d ¼ d c , interactions between
the two spin states are strong. Indeed, this is directly seen in the
deformation of the majority cloud due to the presence of theminority species (see Figs 1, 3). However, here these interactions
are not strong enough to let the central densities of the two clouds
become comparable. At the critical imbalance the Clogston criteriondm ¼ cD(r ¼ 0) is fulfilledin the centre of the trap (here, c is apconstant that equals 2 in the BCS limit33). For smaller imbalance, acentral superfluid region can form: the condensate. Its borders aredefined by dm , cD(r). The simple density estimate in Fig. 3 suggeststhat in this region, the two clouds will have equal densities, althoughmore refined techniques to measure small density differences have tobe developed to finally settle this question. Outside the superfluid
region there is still a normal state with unequal densities of minority
and majority components. The discontinuity in the clouds’ densities
at the normal-to-superfluid phase boundary gives rise to the visible
kink in the column density profiles. Such a density discontinuity is
characteristic of a first-order phase transition.
Interestingly, most of the ‘work’ needed to build the superfluid
state has already been done in the normal component by decreasing
the density difference. Consequently, the critical population difference needed to form the superfluid is largely determined by theinteractions in the normal gas.
In conclusion, we have observed the normal-to-superfluid phasetransition through the direct observation of condensation in an
imbalanced Fermi mixture—on the BEC side, on the BCS side, andright on the Feshbach resonance. Unequal mixtures offer a direct
method of thermometry by analysing the non-interacting wings ofthe majority species. Strong interactions are already visible in thenormal cloud as marked deformations of the majority profile. It isthese interactions in the normal gas that squeeze the two components
and eventually, at the critical imbalance, let them reach almost equal
densities in the centre, aiding the formation of the superfluid. Our
method of direct detection of the condensate is a powerful new tool
to characterize the superfluid phase transition. At the current level ofprecision, the appearance of a condensate after magnetic field sweeps
and the direct observation of the central dense core occur together, andindicate the normal-to-superfluid phase transition. An intriguingquestion is whether further phases are possible, including a more
exotic superfluid state with unequal densities. Several theories predictthat the Fulde–Ferrell–Larkin–Ovchinnikov state, a superfluid statewith oscillating order parameter, should be present for imbalanced spin
populations24,26,28.

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NATURE|Vol 442|6 July 2006

8.

METHODS
Experimental procedure. Our experimental setup is described in previous
publications8,12. A spin-polarized cloud of 6Li fermions is cooled to degeneracy
using a combination of laser cooling and sympathetic cooling with sodium
atoms in a magnetic trap. After transfer into an optical trap, a variable spin
mixture of the lowest two hyperfine states, labelled j " l and j # l, is prepared at amagnetic bias field of 875 G. Interactions between the two spin states can be
freely tuned via a 300-G-wide Feshbach resonance located at B 0 ¼ 834 G. At
fields below B 0 , two-body physics supports a stable molecular bound state (BECside), while at higher fields (BCS side), no such bound state exists for two isolated
atoms. Our trap combines a magnetic saddle potential with a weakly focused (waist
w < 120 mm) infrared laser beam (wavelength l ¼ 1,064 nm), leading to aharmonic axial confinement with oscillation frequency of n z ¼ 22.8(0.2) Hz anda gaussian radial potential with variable trapping frequency n r in the central
harmonic region. The trap depth U is related to n r and n z by:


 2
2 nr þ n2z =2
1
n2
U ¼ mð2pnr Þ2 w2 1 2 z2 ln
:
n2z
4
2nr
The initial degeneracy of the spin mixture is about T/T F < 0.3. The stronglyinteracting gas is further cooled by decreasing the laser power of the optical trap
over several seconds and evaporating the most energetic particles. During thefirst few seconds, the magnetic field is adiabatically ramped to a chosen final
field in the resonance region where the last stage of the evaporation (shown in
Fig. 2) takes place. For detection, the optical trap is switched off and the gas
expands in the remaining magnetic saddle-point potential. After a variable
time-of-flight, an absorption image of atoms either in state j " l or j # l is taken
along the axial direction of the trap (the direction of the optical trappingbeam). The cloud’s radial symmetry allows for azimuthal averaging of theresulting column densities, leading to low-noise profiles12.
For preparing clouds at the coldest temperatures (as shown in Fig. 3) withvarying population imbalance, the spin mixture is evaporated down to a trap
depth of 1 mK over several seconds on resonance, after which the trap depth isincreased again to 1.4 mK for more harmonic confinement (trap frequencies:n r ¼ 115(10) Hz and n z ¼ 22.8(0.2) Hz). The temperature of the gas is determined
to be T/T F # 0.06 for all d . 15%, and appears to smoothly rise to T/T F ¼ 0.11
for an equal mixture, although thermometry in the interacting wings is
problematic. The total atom number was 1.5 £ 107 and constant to within
15% for all values of d.
Errors. The error in the critical temperature T C /T F for the phase transition isdominated by the uncertainty in the atom number entering the determination ofT F, which we estimate to be 30% (ref. 12). For T F we use the harmonic
approximation for the radially gaussian trapping potential, with the measured
trapping frequencies reflecting the average curvature of the gaussian potential.
The phase transition is observed above U ¼ 2 mK, where anharmonicities
contribute only 3% to the error in T F . Note that anharmonicities do not affectthe temperature measurement performed on the majority wings: ballistic
expansion of non-interacting atoms reveals their momentum distribution,
regardless of the shape of the trap.
Received 6 March; accepted 23 May 2006.
1.

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3.
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Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements We thank the participants of the Aspen winter conference
on strongly interacting fermions for discussions. This work was supported by
the NSF, ONR and NASA.
Author Information Reprints and permissions information is available at
npg.nature.com/reprintsandpermissions. The authors declare no competing
financial interests. Correspondence and requests for materials should be
addressed to M.W.Z. (zwierlei@mit.edu).

© 2006 Nature Publishing Groupzwierlein 2006-03-02346A Supplemental Information

Direct Observation of the Superfluid Phase Transition in
Ultracold Fermi Gases
Martin W. Zwierlein, Christian H. Schunck, André Schirotzek, and Wolfgang Ketterle

Supplementary Information
Supplementary Figures

Supplementary Figure 1: (Color online) Signatures of the condensate on resonance in the spatial
profiles. The curvature of the observed column density is encoded in shades of gray with white(black) corresponding to positive (negative) curvature. The outer radii of the two components andthe condensate radius are shown as an overlay in the lower panel. As a direct consequence ofstrong interactions, the minority component causes a pronounced bulge in the majority densitythat is reflected in the rapid variation of the profile’s curvature. The condensate is clearly visible inthe minority component (δ > 0), but also leaves a faint trace in the majority component (δ < 0).
The image was composed out of 216 individual azimuthally averaged column density profiles,
smoothed to reduce technical noise. Data close to the cloud’s center suffer from larger noise dueto the lower number of averaged points. The central feature of about 50µm width is an artefact ofsmoothing in this region of increased noise.

1

1.0

R / RF,δ = 0

0.8

0.6

0.4

0.2

0.0

0.2

0.4
0.6
Population imbalance

0.8

1.0

Supplementary Figure 2: Outer radii of the two cloud profiles and condensate radius versus
population imbalance. Data obtained from the majority (minority) cloud are shown as diamonds(circles). The outer radii of the clouds (black) are determined from Thomas-Fermi fits to the profiles’ wings, where the results of a zero-temperature and a finite temperature fit were averaged. For
the minority cloud, the representative error bars indicate the difference between these two results.
The position of the ”bulge” in the majority profile (white diamonds) naturally follows the outerminority radius. The condensate radius is defined as the position of the ”kink” in the minority
profiles. It was obtained by a) fitting an increasing portion of the minority wings until a significant
increase in χ2 was observed (grey circles), and b) the position of the minimum in the profile’s
derivative (white circles). All sizes are scaled by the Fermi-radius of a non-interacting equal mixture. The minority radii were adjusted for the observed hydrodynamic expansion (expansion factor
11.0). The non-interacting wings of the majority cloud expand ballistically (expansion factor 9.7),
as long as they are found a factor 11/9.7 = 1.13 further out than the minority radius. For small
imbalances (δ < 20%), also the majority wing’s expansion will be affected by collisions. The grey
diamonds give the majority cloud’s outer radius if hydrodynamic expansion is assumed.

2Supplementary Methods
Hydrodynamic vs. ballistic expansion
A non-interacting cloud of atoms simply expands ballistically from a trap. However, stronglyinteracting equal Fermi mixtures, above and below the phase transition, are collisionally dense andtherefore expand according to hydrodynamic scaling laws 1–3 . These scaling laws only depend on
the equation of state of the gas, ² ∝ nγ , with γ = 1 for the BEC-side, γ = 2/3 for resonance(a direct consequence of unitarity) and γ = 2/3 for the BCS-side, away from resonance. In
an unequal spin mixture of fermions, the expansion does not follow a simple scaling law. The
minority cloud is always in contact with majority atoms and thus strongly interacting throughout
the expansion, which is therefore hydrodynamic. The excess atoms in the wings of the larger
cloud are non-interacting and will expand ballistically, as we have checked experimentally. The
absorption images after expansion are taken along the axial direction of the trap (the direction of
the optical trapping beam). In order to compare the expanded cloud sizes to the in-trap Fermi radiiof non-interacting clouds (see Fig. 2 and Fig. S2 below) we scale the majority cloud with theballistic factor for the radial direction
r
³
³
√ ´
√ ´
√
2
2
2
cosh 2πνz t/ 2 + ( 2νr /νz ) sinh 2πνz t/ 2 ,√where t is the expansion time and νz / 2 gives the radial anti-trapping curvature of the magnetic
saddle-point potential. The scaling factor for the hydrodynamic expansion of an equal mixture isgiven by the solution to a differential equation 2, 3 . A priori, the minority cloud in unequal mixtures
could expand with a different scaling, since the equation of state now depends on two densities.
However, by imaging the cloud in trap and at different times during expansion, we found thatthe minority cloud’s expansion is very well described by the scaling law for an equal mixture.
In particular, the aspect ratio of the minority cloud did not change as a function of population
imbalance (within our experimental error of 5%), and was equal to that of a balanced mixture.
For the data on resonance in Figs. 3, S1 and S2, which were obtained after 11 ms expansion
out of a trap with radial (axial) frequency of νr = 113(10) Hz (νz = 22.8(0.2) Hz), the ballistic(hydrodynamic) expansion factor for the radial direction is 9.7 (11.0).
Supplementary Discussion
Signature of the condensate
Fig. S1 demonstrates that on resonance, the condensate is visible not only in the minority component, but also in the larger cloud as a small change in the profile’s curvature. In the condensate
region, the majority profile is slightly depleted when compared to the shape of a normal Fermi
cloud. This effect is still significant on the BCS-side (see Fig. 1): Although here, the condensate
is less visible in the smaller component than on resonance, the larger cloud’s central depletion still
produces a clear dip in the difference profile.

3

Radii in the unequal Fermi mixture
Fig. S2 shows the outer radii of the majority and minority cloud, together with the condensate
radius (on resonance, for the deepest evaporation compatible with constant total atom number versus imbalance). As was the case for the phase transition at finite temperature, the outer cloud sizeschange smoothly with imbalance. No drastic change is seen at the critical population imbalance.
The radii are obtained by fitting the profiles’ wings to the Thomas-Fermi expression for the radialcolumn density n(r):´³22
Li2 −λ1−r /R,n(r) = n0
Li2 (−λ)with the central column density n0 , the fugacity λ and the Thomas-Fermi radius R as the freeparameters. Li2 (x) is the Dilogarithm. The zero-temperature expression reduces to n(r) = n0 (1 −r2 /R2 )2 .
Lower and upper bounds for the critical chemical potential difference at δc
For the clouds at the critical imbalance δc , we now want to extract a lower and upper bound forthe difference in chemical potentials δµc of the majority and minority component. This difference
allows us to conclude that BCS-type superfluidity with imbalanced densities is not possible.
The chemical potential difference δµ ≡ 2h = (µ↑ − µ↓ ) measures the energy cost, relative
to µ = (µ↑ + µ↓ )/2, to add a particle to the cloud of excess fermions. ∆, the pairing gap, is theenergy cost for this additional majority particle to enter the superfluid. Both the critical temperature
TC and the critical chemical potential difference δµc provide a measure of the superfluid gap:
The superfluid can be either destroyed by raising the temperature or by increasing the population
imbalance. If hc ≡ δµc /2 < ∆, excess atoms will always stay outside the superfluid, in the phaseseparated normal state. For hc > ∆, excess atoms can enter the superfluid for hc > h > ∆.
Hence, superfluidity with unequal densities, if allowed via hc > ∆, would be favored at large
population imbalance, contrary to the interpretation in 4 , where such a state was proposed for small
population imbalance. A recent Monte-Carlo calculation 5 for the Clogston limit on resonancegives hc = 1.00(5)∆ = 0.50(5)EF and can thus not decide on the question of superfluidity withimbalanced densities.
We can attempt to extract the chemical potential from the cloud sizes R↑,↓ - taking into
account hydrodynamic expansion for the minority cloud and ballistic expansion for the excess
fermions. For the majority cloud, we find µc,↑ = 1/2 mωr2 R↑2 = 1.21(6)EF . For the minority
cloud, we find 1/2 mωr2 R↓2 = 0.39(10)EF . Throughout the smaller cloud, minority atoms arealways strongly attracted by majority atoms. This strong attractive interaction likely reduces theirchemical potential from the above upper limit. The difference of the chemical potentials δµc ≡ 2hc
is thus given by hc = (µc,↑ − µc,↓ )/2 ≥ 0.41(6)EF = 0.51µ, our lower bound. Another condition
on hc concerns whether the normal state can be mixed, hc < µ, (minority and majority atoms inthe same spatial region) or whether the normal state is always completely polarized hc > µ. Our
observation of the mixed region in Fig. 1 immediately results in hc < µ, the upper bound.

4

On resonance, ∆ = 1.16µ in BCS-theory, while a recent Monte-Carlo study 5 obtains∆ = 1.2µ. If ∆ > µ holds true, our finding of the upper bound on hc would imply hc < ∆and hence would exclude a superfluid with unequal spin densities (at least on the basis of BCStheory, see 6 for a recent suggestion which goes beyond BCS).

Supplementary Notes
1. O’Hara, K. M., Hemmer, S. L., Gehm, M. E., Granade, S. R. & Thomas, J. E. Observation of astrongly interacting degenerate fermi gas of atoms. Science 298, 2179 (2002).
2. Menotti, C., Pedri, P. & Stringari, S. Expansion of an Interacting Fermi gas. Phys. Rev. Lett.
89, 250402 (2002).
3. Castin, Y. Exact scaling transform for a unitary quantum gas in a time dependent harmonic
potential. Comptes Rendus Physique 5, 407–410 (2004).
4. Partridge, G. B., Li, W., Kamar, R. I., a. Liao, Y. & Hulet, R. G. Pairing and Phase Separation
in a Polarized Fermi Gas. Science 311, 503 (2006). Published online 21 December 2005
(10.1126/science.1122876).
5. Carlson, J. & Reddy, S. Asymmetric Two-Component Fermion Systems in Strong Coupling.
Phys. Rev. Lett. 95, 060401 (2005).
6. Ho, T.-L. & Zhai, H. Homogeneous Fermion Superfluid with Unequal Spin Populations.
Preprint at <http://arxiv.org/cond-mat/0602568> (2006).

5

Appendix J
Observation of Phase Separation ina Strongly Interacting Imbalanced
Fermi Gas
This appendix contains a reprint of Ref. [223]: Y. Shin, M. W. Zwierlein, C. H.
Schunck, A. Schirotzek, and W. Ketterle, Observation of Phase Separation in a
Strongly Interacting Imbalanced Fermi Gas, Phys. Rev. Lett. 97, 030401 (2006).

253

PRL 97, 030401 (2006)

PHYSICAL REVIEW LETTERS

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Observation of Phase Separation in a Strongly Interacting Imbalanced Fermi Gas
Y. Shin,* M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA(Received 15 June 2006; published 18 July 2006; corrected 21 July 2006)
We have observed phase separation between the superfluid and the normal component in a stronglyinteracting Fermi gas with imbalanced spin populations. The in situ distribution of the density difference
between two trapped spin components is obtained using phase-contrast imaging and 3D image reconstruction. A shell structure is clearly identified where the superfluid region of equal densities is surrounded
by a normal gas of unequal densities. The phase transition induces a dramatic change in the density
profiles as excess fermions are expelled from the superfluid.
DOI: 10.1103/PhysRevLett.97.030401

PACS numbers: 03.75.Ss, 03.75.Hh, 05.70.Fh

Cooper pairing is the underlying mechanism for the
Bardeen-Cooper-Schrieffer (BCS) superfluid state of an
equal mixture of two fermionic gases. An interesting situation arises when the two components have unequal populations. Does the imbalance quench superfluidity, does it
lead to phase separation between a balanced and an imbalanced region, or does it give rise to new forms ofsuperfluidity? A search for exotic superfluid states is promising in imbalanced mixtures, since the imbalance destabilizes BCS-type s-wave pairing, which is usually thestrongest pairing mechanism [1– 4]. Recently, this problem
has been experimentally addressed in ultracold atomic
Fermi clouds with controlled population imbalances [5–7]. Superfluidity was observed in a strongly interacting
regime with a broad range of imbalances, and theClogston limit of superfluidity [8] was characterized [5,7].
The phase separation scenario suggests that unpaired
fermions are spatially separated from a BCS superfluid of
equal densities due to the pairing gap in the superfluid
region [9–11]. In our previous experiments [5,7], we observed a strong central depletion in the difference profiles
of expanding clouds indicating that excess atoms are expelled from the superfluid region. Reference [6] reports
depletion of excess atoms at the trap center. None of theseexperiments answered the questions of whether the densities of the two spin components are equal in the superfluid region and whether phase separation or rather
distortions of the cloud due to interactions have occurred[12,13].
Here we report the direct observation of phase separation between the superfluid and the normal region in astrongly interacting Fermi gas with imbalanced spin populations. The density difference between the two spincomponents is directly measured in situ using a special
phase-contrast imaging technique and 3D image reconstruction. We clearly identify a shell structure in an imbalanced Fermi gas where the superfluid region of equal
densities is surrounded by a normal gas of unequal densities. This phase separation is observed throughout thestrongly interacting regime near a Feshbach resonance.
Furthermore, we characterize the normal-to-superfluid
0031-9007=06=97(3)=030401(4)phase transition of an imbalanced Fermi mixture using
in situ phase-contrast imaging. The onset of superfluidity
induces a dramatic change in the density profiles as excess
fermions are expelled from the superfluid.
A degenerate Fermi gas of spin-polarized 6 Li atoms was
prepared in an optical trap after laser cooling and sympathetic cooling with sodium atoms [14,15]. The population
imbalance  of the two lowest hyperfine states j1i and j2i
was adjusted with a radio-frequency sweep [5]. Here  
N1  N2 =N1  N2 , where N1 and N2 are the atom
numbers in j1i and j2i, respectively. Interactions between
these two states were strongly enhanced near a broad
Feshbach resonance at B0  834 G. The final evaporative
cooling was performed at B  780 G by lowering the trap
depth. Subsequently, the interaction strength was adiabatically changed to a target value by adjusting the value of themagnetic-bias field B with a ramp speed of  0:4 G=ms.
For typical conditions, the total atom number was Nt 
N1  N2  1 107 and the radial (axial) trap frequency
was fr  130 Hz (fz  23 Hz).
The condensate fraction in the imbalanced Fermi mixture was determined via the rapid transfer technique[16,17]. Immediately after turning off the trap, the magnetic field was quickly ramped to B  690 G (1=kF a 
2:6, where kF is defined as the Fermi momentum of anoninteracting equal mixture with the same total atom
number and a is the scattering length) in approximately
130 s. The density profile of the expanding minority
cloud was fit by a Gaussian for normal components (thermal molecules and unpaired atoms) and a Thomas-Fermi(TF) profile for the condensate [18].
The density difference between the two components was
directly measured using a phase-contrast imaging technique (Fig. 1). In this imaging scheme, the signs of thephase shifts due to the presence of atoms in each state areopposite so that the resulting phase signal is proportional tothe density difference nd of the two states [19]. This
technique allows us to directly image the in situ distribution of the density difference nd ~r and avoid the shortcoming of previous studies [5–7] where two images weresubtracted from each other.

030401-1© 2006 The American Physical Society

PRL 97, 030401 (2006)

PHYSICAL REVIEW LETTERS

FIG. 1. Phase-contrast imaging of the density difference of
two spin states. (a) The probe beam is tuned to the red for the
j1i ! jei transition and to the blue for the j2i ! jei transition.
The resulting optical signal in the phase-contrast image is
proportional to the density difference nd  n1  n2 , where n1
and n2 are the densities of the states j1i and j2i, respectively.
(b) Phase-contrast images of trapped atomic clouds in state j1i(left) and state j2i (right) and of an equal mixture of the twostates (middle).

For a partially superfluid imbalanced mixture, a shellstructure was observed in the in situ phase-contrast image(Fig. 2). Since the image shows the column density difference (the 3D density difference integrated along the y
direction of the imaging beam), the observed depletion in
the center indicates a 3D shell structure with even stronger
depletion in the central region. The size of this inner coredecreases for increasing imbalance, and the core shows a
distinctive boundary until it disappears for large imbalance. We observe this shell structure even for very small
imbalances, down to 53%, which excludes a homogeneous superfluid state at this low imbalance, contrary to theconclusions in Ref. [6].
The reconstructed 3D profile of the density differenceshows that the two components in the core region haveequal densities. We reconstruct 3D profiles from the 2D
distributions n~d x; z of the column density difference using the inverse Abel transformation (Fig. 3) [20]. The
reconstruction does not depend on the validity of the local
density approximation (LDA) or a harmonic approxima-

FIG. 2. In situ direct imaging of trapped Fermi gases forvarious population imbalance . The integrated
2D distributions
Rof the density difference n~d x; z  nd ~rdy were measured
using phase-contrast imaging at B  834 G for total atom number Nt  1 107 . For   75%, a distinctive core was observed
showing the shell structure of the cloud. The field of view foreach image is 160 m 800 m. The three leftmost images are
displayed with different contrast levels for clarity. The image
with   53% was taken for Nt  1:7 107 .

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tion for the trapping potential [12,21,22] and assumes only
cylindrical symmetry of the trap. The two transverse trap
frequencies are equal to better than 2% [15]. A 1D profile
obtained by integrating n~d along the axial direction[Fig. 3(d)] shows a flattop distribution, which is the expected outcome for a shell structure with an empty inner
region in a harmonic trap and assuming LDA.
The presence of a core region with equal densities forthe two components was correlated with the presence of apair condensate. The density difference at the center nd0
along with the condensate fraction is shown as a functionof the imbalance  in Fig. 4. As shown, there is a criticalimbalance c where superfluidity breaks down due to large
imbalance [5,7]. In the superfluid region, i.e.,  < c , nd0
vanishes, and for  > c , nd0 rapidly increases with asudden jump around   c . We observe a similar behavior throughout the strongly interacting regime near the
Feshbach resonance, 0:4 < 1=kF a < 0:6. This observation clearly demonstrates that for this range of interactions
a paired superfluid is spatially separated from a normalcomponent of unequal densities.
The shell structure is characterized by the radius of the
majority component, the radial position Rp of the peak innd r~, the size Rc of the region where nd is depleted, andthe ‘‘visibility’’  of the core region (Fig. 5). The sudden
drop of  around   c results from the sudden jump ofnd0 . The comparison between Rp and Rc shows that theboundary layer between the superfluid and the normal
region is rather thin. It has been suggested that the detailed
shape of profiles in the intermediate region could be used
to identify exotic states such as the Fulde-Ferrell-LarkinOvchinnikov state [23–25]. This will be a subject of future
research.

FIG. 3. Reconstruction of 3D distributions from their integrated 2D distributions. (a) An integrated 2D distribution n~dwith   58% at B  834 G. (b) A less noisy distribution was
obtained by averaging four quadrants with respect to the central
dashed lines in (a). (d) The 1D profile obtained by integrating the
averaged distribution along the z direction shows the ‘‘flattop’’
feature. (c) The 2D cut nd x; y  0; z of the 3D distributionnd ~r was reconstructed by applying the inverse Abel transformation to (b). (e) The radial profile of the central section of
the reconstructed 3D distribution in the xy plane. The dashedlines in (d) and (e) are fits to the profiles’ wings using a TF
distribution.

030401-2

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1

0.5
0.25

B = 834 G(1/kFa = 0)

0.5

0.0

0
1

0.5
0.25

(b)

0.0

0.2

0.4

0.6

0.8

2 RP
2 RC

-60

0

60

-60

0

60

-60

0

60

-60

0

60

0.4

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1.0

(c)

80

1.0

60
40
20

0.5

0.0
0
0.0

0.2

0.4

0.6

0.8

1.0

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Population Imbalance

0

0.0

98%

Radial Position (µm)

0.5

B = 884 G(1/kFa = -0.36)

82%

Visibility

Condensate Fraction

0.0

Radial Position (µm)

0.5Normalized Central Density Difference

B = 780 G(1/kFa = 0.59)

70%

δ=41%

Density Difference

(a)
1

0.5
0.25

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PHYSICAL REVIEW LETTERSPRL 97, 030401 (2006)

1.0

Population Imbalance

FIG. 4 (color online). Phase separation in a strongly interacting Fermi gas. Normalized central density difference  (black
circles) and condensate fraction (red triangles) as functions ofimbalance  for various interaction strengths at our lowest
temperatures (T & 0:06TF ).   nd0 =n0 , where nd0 was measured as the average over the central region of 7 m 40 m
in a 2D cut of the 3D distribution [Fig. 3(c)] and n0  3:3
1012 cm3 is the calculated central density of a fully polarizedFermi gas with Nt  1 107 . The condensate fraction wasdetermined from the minority component (see the text). The
solid line is a fit for the condensate fraction to a threshold
function / 1  j=c jn . The critical imbalances c indicatedby the vertical dashed lines were 96% (exponent n  21), 77%(n  3:1), and 51% (n  3:4) for B  780 G, B  834 G, andB  884 G, respectively.

The superfluid requires equal central densities in thestrongly interacting regime at our coldest temperatures.
A normal imbalanced Fermi mixture will have unequal
densities. Thus, one should expect that a visible change
in the density difference occurs as the temperature islowered across the normal-to-superfluid phase transition[7,9]. In situ phase-contrast images of a cloud at various
temperatures are shown in Fig. 6. The temperature T of thecloud is controlled with the final value of the trap depth in
the evaporation process. The shell structure appears andbecomes prominent when T decreases below a certain
critical value. This shell structure gives rise to the bimodal
density profile of the minority component that we observed
after expansion from the trap in our recent work [7]. Here
we show via in situ measurements that the onset of superfluidity is accompanied by a pronounced change in thespatial density difference.
The phase transition is characterized in Fig. 7. As T islowered, nd0 gradually decreases from its plateau value andthe condensate fraction starts to increase. From the point of

FIG. 5 (color online). Characterization of the shell structure.
(a) Reconstructed 3D radial profiles at B  834 G. (b) Radius of
the majority component R (black circles), peak position of thedensity difference Rp (red circles), and radius of the empty coreRc (blue triangles) as a function of population imbalance . R
was determined from the profiles’ wings using a fit to a zerotemperature TF distribution. The solid line indicates the TF
radius of the majority component in an ideal noninteracting
case. R at   0 (open circle) was measured from an image
taken with a probe frequency preferentially tuned to one component. Rc is defined as the position of the half-peak value in theempty core region for  < c . (c) Visibility of the core region isdefined as   nd Rp   nd0 =nd Rp   nd0 . For low ,nd Rp  is small, and small fluctuations of nd0 around zero leadto large fluctuations in .

condensation (condensate fraction >1%) and deformationof the minority clouds [2 in Fig. 7(b)], we determine thecritical temperature Tc  0:132TF for the imbalance of  563%. TF  1:7 K is the Fermi temperature of anoninteracting equal mixture with the same total atom
number. The rise in 2 , the drop in nd0 , and the onset of

FIG. 6. Emergence of phase separation in an imbalanced Fermi
gas. The temperature of the cloud was controlled by varying thefinal value of the trap depth Uf in the evaporation process.
Phase-contrast images were taken after adiabatically ramping
the trap depth up to kB 3:7 K (fr  192 Hz). The wholeevaporation and imaging process was performed at B  834 G(Nt  1:7 107 ,   56%). The field of view for each image is
160 m 940 m. The vertical and the horizontal scale of theimages differ by a factor of 1.5.

030401-3

PHYSICAL REVIEW LETTERS(a)

Population
Imbalance

PRL 97, 030401 (2006)

(b)

0.4
0.2

250

10

200
150

6
4

2

2

0.1
6
4

1

2

0.01
6
4

0

Condensate Fraction

Density Difference(1012/cm3)(c)

1

χ2 (arb.units)

Temperature(nK)

In conclusion, we have observed phase separation of thesuperfluid and the normal component in a strongly interacting imbalanced Fermi gas. The shell structure consisting of a superfluid core of equal densities surrounded by anormal component of unequal densities was clearly identified using in situ phase-contrast imaging and 3D image
reconstruction. The technique is a new method to measure
the in situ density distribution, allowing direct comparison
with theoretical predictions.
We thank G. Campbell for critical reading of the manuscript. This work was supported by the NSF, ONR, and
NASA.

0.6

300

2

1

2

3

Trap Depth (kBµK)

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FIG. 7 (color online). Phase transition in an imbalanced Fermi
gas. The phase transition shown in Fig. 6 was characterized bymeasuring (a) population imbalance , (b) temperature T (black
circles), 2 for fitting the minority cloud with a finite temperature Fermi-Dirac distribution (red triangles), (c) central density
difference nd0 (black circles), and condensate fraction (redtriangles). The solid line is a guide to the eye for 2 .  decreased
mainly due to loss in the majority component, which was
reduced by 14% between Uf =kB  3 and 1 K and more
rapidly below 1 K. The number of minority atoms was almost
constant (N2  3:7 106 ). T was determined from the noninteracting outer region of the majority cloud after 10 ms ofballistic expansion [7]. T and nd0 are averaged values of three
independent measurements. The open (solid) arrow in (c) indicates the position for Tc (T ). See the text for the definitions of
Tc and T .

condensation are all observed at about the same temperature. Better statistics are needed to address the question ofwhether some weak expulsion of majority atoms from thecenter occurs already slightly above Tc .
Below a certain temperature T , nd0 reaches zero while
the condensate fraction is still increasing, implying that thesuperfluid region of equal densities continues to expand
spatially with decreasing T (see Fig. 6). Full phase separation does not occur until this temperature T < Tc isreached. We interpret the state between T and Tc as asuperfluid of pairs coexisting with polarized quasiparticle
excitations. This is expected, since at finite temperatures
the BCS state of an equal mixture can accommodate excess
atoms as fermionic quasiparticle excitations [23,26 –28].
There is a finite energy cost given by the pairing gap T
for those quasiparticles to exist in the superfluid. The
assumption that excess atoms should have thermal energy
kB T > T to penetrate the superfluid region suggests therelation between T and T to be kB T  T . From
our experimental results, T  0:09TF and T   h
3:3 kHz.

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the bimodal distribution used for fitting does not accountfor the repulsion between the condensate and the normalcomponents.
[19] The probe frequency was detuned by about 3 MHz fromthe exact center between the two resonance frequencies(frequency difference 80 MHz, natural linewidth  
6 MHz) in order to compensate the effect of absorption,
zeroing the optical signal in an equal mixture (Fig. 1).
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