Pairing and Superfluidity in Strongly Interacting
Fermi Gases
byChristian H. Schunck
Diplom, Physik (2003)
University of Heidelberg
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2008
c Massachusetts Institute of Technology 2008. All rights reserved.
°

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
May 22, 2008

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

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

Pairing and Superfluidity in Strongly Interacting
Fermi Gases
byChristian H. Schunck
Submitted to the Department of Physics
on May 22, 2008, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
This thesis describes experiments with superfluid spin mixtures of ultracold fermionic 6 Li
atoms. The properties of the strongly interacting gas are studied in the crossover regime
between Bose-Einstein condensation (BEC) of two-body bound molecules and a BardeenCooper-Schrieffer (BCS) superfluid of pairs bound by many-body interactions. We obtain
the homogeneous phase diagram of the two-component gas with resonant interactions. As
a function of temperature and spin polarization the phase diagram shows first and second
order phase transitions that merge at a tricritical point. At zero temperature a first orderphase transition from a superfluid with equal spin populations to a mixed normal phase
is observed at a critical spin polarization known as the Chandrasekhar-Clogston limit ofsuperfluidity.
Pairing correlations in the superfluid and normal phase are studied with radio-frequency(rf) spectroscopy. A signature of strong correlations is observed above the critical temperature but also at spin polarizations where superfluidity is quenched even at zero temperature.
Significant limitations for the interpretation of these experiments due to final state interactions are overcome by the creation of new superfluid spin mixtures. The asymmetric rf
dissociation spectra of the new mixture allow us to determine the spectroscopic pair size inthe crossover regime. The size of the resonantly interacting pairs is found to be on the orderof, but smaller than the interparticle spacing. Rf spectra of the majority component in an
imbalanced system show a signature of thermally excited quasiparticles and by comparison
to the minority spectra reveal changes in the nature of the binding as a function of spinpolarization.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

To my wife Alessandra,
and my parents Angelika and Heiner

Acknowledgments
My time as a PhD student at MIT was an incredibly intense, exciting and thrilling experience. First and foremost this was due to the extraordinary group of people from whom I
could learn and who made this work possible and enjoyable.
I would like to thank Wolfgang Ketterle for giving me the opportunity to join his groupand for his advise, guidance, encouragement and support. Wolfgang’s enthusiasm for physics
and his ability to condense it into easily graspable ideas were a constant source of inspiration.
I will cherish the memory of the many open ended brain storming sessions during groupmeetings.
Working in the BECI/Lithium team was simply an amazing experience. I would like tothank all members of the Lithium team, for these fantastic years, for teaching me physics
and for your friendship. When I joined the group Claudiu Stan, Zoran Hadzibabic, andSubhadeep Gupta introduced me to the lithium project as well as to life at MIT. With
Claudiu I share the experience of building a magnetic trap, directly followed by another(non magnetic) one. I learned a lot from Claudiu’s original ideas, his attention to detailand his patience. I am very glad that we stayed in contact over all the years even after hejoined a group at Harvard as a Postdoc: from the first day I set foot into MIT to the dateof my thesis defense I could count on Claudiu’s support.
Zoran and Deep formed a perfect duo and even graduated within the same week. They
showed me how to operate the laser systems and introduced me to the “dudes”. Bothof them taught me lots of physics through their vivid discussions. Deep and Zoran also
introduced me to the cuisine of their respective countries which I enjoyed immensely. I still
remember a wonderful summer night Deep, Zoran, Claudiu and me spend on Zoran’s porch
discussing physics and the world beyond until many hours past sunrise.
When Martin Zwierlein re-joined the group as a graduate student I got to know an
enthusiastic and brilliant physicist who would become a close friend. We spend crazy times
together both inside the lab and out. In countless run nights Martin taught me an incredible
amount of physics and how to “play” with the machine. It is hardly possible to describe
the many memorable and unique experiences we shared together from a trip to Pasadena(with Martin singing and playing on the various Steinways scattered around in the hotel at
2 am in the morning), eating a Thanksgiving turkey at 3 am after working without sleep
for a couple of days, to spontaneous parties in Boston. After finishing his PhD Martin left
MIT for a year - it is great to have him back again on the hallway in a double bay office!
Throughout the years we were the best customers Bertucci’s could ever hope to get and I
wish that everyone there will remember that the rolls should be served with butter (as wellas olive oil) and be accompanied by a diet coke.
With André Schirotzek another post-diploma student became part of the team andimmediately made a difference by flooding the lab not only with his boundless energy,
dedication and good humor but also with an eclectic choice of music. During numerous

runs we enjoyed many interesting discussions covering essentially any imaginable topic fromphysics, to politics, to religion, to ... etc. We survived a couple of interesting moments in thelab (for the record, André proved that a duster actually also works well as a fire extinguisher)
and we were lucky to be able to attend the Enrico Fermi Summer School on ultracold Fermi
gases in Varenna together.
The fact that we had not been working with a postdoc for many years, did not scare
Yong-il Shin to join our team after graduating from BEC III. We were very lucky to have
him with us during the last two years. The efficiency with which Yong works in the lab isincredible, the way he uncovers new physics is elegant and beautiful and he always extended
his help in a very kind and supportive way. Soon after joining the lithium project Yong got
married to his wife Miye and I wish him and his family all the best for the future.
Sebastian Raupach joined the lithium lab during a time of fast paced progress. We
soon became close friends and after a DAMOP meeting in Tuscon we hit the road driving
through Arizona, Utah and Nevada together with Sebastian’s wife Heike. Postdoc Jamie
Kerman joined the lab during the same year but unfortunately stayed with us only for ashort time.
Our nearest neighbors on the hallway are the members of BEC III, Aaron Leanhardt,
Tom Pasquini, Michele Saba, Gyu-Boong Jo, Caleb Christensen, Ye-yeoung Lee and Jae
Choi which deserve my apologies for ramping up the Feshbach fields far too often. I am
especially indebted to Michele Saba for advice in numerous matters and to Jae Choi forbeing a good friend and “just a boy”.
Whenever one needs a tool BEC II is a great place to look. Jamil Abo-Shaeer, Johnny
Vogels, Kaiwen Xu, Jit Kee Chin, Takashi Mukaiyama, Dan Miller, Yingmei Liu, Christian
Sanner, Aviv Keshet, Ed Su and Ralf Gommers deserve lots of credit for keeping their lab
well organized. Jamil always knew when it was time to cheer people up and how to do it
best. With Dan Miller I share the memory of many early morning coffee’s in Varenna anda fantastic crawfish dinner in New Orleans. Christian Sanner is yet another post-diploma
PhD student. When I shared a room with him during the summer school in Varenna he told
me many fascinating stories about his trip to Africa - hopefully I will be able to go there
myself sometime! Aviv Keshet fits well into BEC II keeping up with the lab’s tradition ofeating (or at least the attempt to eat) healthy. Aviv finished the new word generator (it’sawesome!) just in time to help me out during the last weeks of the PhD.
The members of BEC IV Micah Boyd, Eric Streed, Gretchen Campbell, Dominik
Schneble, Luis Marcassa, Patrick Medley, Jongchul Mun, David Hucul, David Weld, andHiro Miyake always had to struggle not to be left behind in building 36. Thanks to Gretchen
and Micah for complaining loud when we had forgotten them once again. With Micah I
spend a very relaxed OSA meeting at a pool in Tuscon and Micah, Tom and me spend two
nice days exploring Barcelona together.
Many thanks to Martin’s student Cheng-Hsun Wu for being a great office mate and thenew members of Martin’s group, Ariel Sommer and Christoph Clausen for relaxing dinners
at Cosi and Bertucci’s.
I would also like to thank the late Carol Costa for help in numerous occasions. It is
possible that I am one of the last students in the hallway who has been encouraged by her
to write the thesis acknowledgements first and I am joining the long list of students who
did not listen...and at this very moment I can tell that her advice was in place! Ellenor
Barish and Joanna Keseberg also helped whenever possible and deserve special thanks forprocessing receipts for hundreds of pounds of Starbucks coffee beans. Maxine Samuels wasalways there to help and has placed emergency orders late on Friday afternoons on morethan one occasion.
My parents and my sister Anne deserve my deep gratitude for encouraging me on myway, for their constant support and for providing me with good advice and help.
Finally, I would like to thank my wife Alessandra for her love and invaluable support
during challenging times, for cheering me up, for being understanding with my work andsometimes crazy working schedule and for being patient during the last couple of months.
I am looking forward to continuing our journey together in Europe.

Contents
1 Introduction
1.1

1.2

13

Ultracold Fermi gases and the BEC-BCS crossover . . . . . . . . . . . . . .

13

1.1.1

High temperature superfluidity . . . . . . . . . . . . . . . . . . . . .

14

1.1.2

The Fermion pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.1.3

Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Overview of the experiments in the BEC I lab at MIT . . . . . . . . . . . .

16

1.2.1

From strong interactions to fermion pair condensation . . . . . . . .

17

1.2.2

High temperature superfluidity . . . . . . . . . . . . . . . . . . . . .

17

1.2.3

Macroscopic properties of the superfluid: Population imbalanced mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.4
1.3

18

Microscopic properties of the superfluid: Exploring pairing correlations with rf spectroscopy . . . . . . . . . . . . . . . . . . . . . . . .

19Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2 BEC-BCS crossover in ultracold atomic Fermi gases

21

2.1

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

21

2.2

Quasiparticle excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3

Fermion pair wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.4

Rf excitation of fermion pairs . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3 The Experiment: Atoms, apparatus and techniques

30

3.1

6 Li

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2

Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.3

Preparation of ultracold 6 Li fermions . . . . . . . . . . . . . . . . . . . . . .

32

3.4

Creation of superfluid spin mixtures . . . . . . . . . . . . . . . . . . . . . .

32

3.4.1

Inducing strong interactions: Feshbach resonances . . . . . . . . . .

32

3.4.2

Feshbach resonances in 6 Li

. . . . . . . . . . . . . . . . . . . . . . .

34

3.4.3

Three superfluid mixtures . . . . . . . . . . . . . . . . . . . . . . . .

34

3.4.4

Population imbalance . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.4.5

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.4.6

Comparison of the mixtures . . . . . . . . . . . . . . . . . . . . . . .

38

7

3.5

The harmonic trap as a feature: Spatially resolved 3D density distributionsin imbalanced mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.5.1

Phase contrast imaging . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.5.2

Inverse Abel transformation . . . . . . . . . . . . . . . . . . . . . . .

41

3.5.3

Trapping potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.5.4

LDA and doubly integrated density difference profiles . . . . . . . .

43

3.5.5

Temperature determination . . . . . . . . . . . . . . . . . . . . . . .

44

4 Creating and exploring a high temperature superfluid of ultracold 6 Li
atoms

47

4.1

Fermion pair condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.1.1

Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.2

High temperature superfluidity . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.3

Superfluid expansion of a rotating Fermi gas . . . . . . . . . . . . . . . . . .

53

5 Superfluidity and phase separation in strongly interacting Fermi gaseswith density imbalance

57

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5.2

Superfluidity and direct observation of the superfluid phase transition in

5.3

5.4

imbalanced Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

5.2.1

Breakdown of superfluidity: The Clogston-Chandrasekhar (CC) limit

58

5.2.2

The experiment: Superfluidity and the CC limit at unitarity . . . .

59

5.2.3

Bimodal density profiles: Direct observation of the superfluid phasetransition at unitarity . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Probing the nature of the superfluid: The homogeneous phase diagram . . .

63

5.3.1

Density profiles in a harmonic trap . . . . . . . . . . . . . . . . . . .

63

5.3.2

Central density difference and superfluidity: finite vs zero temperature 65

5.3.3

Analysis of in situ density profiles . . . . . . . . . . . . . . . . . . .

66

5.3.4

The phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Changing the interactions: From a Bose-Fermi mixture to exotic superfluids? 72
5.4.1

The CC limit as a function of interaction strength . . . . . . . . . .

73

6 Pairing correlations of fermions in the superfluid and the normal phase

77

6.1

The Innsbruck experiment: A signature of pairing in the crossover . . . . .

78

6.2

Rf spectroscopy and the superfluid phase transition . . . . . . . . . . . . . .

82

6.3

Spatially resolved rf spectroscopy . . . . . . . . . . . . . . . . . . . . . . . .

85

6.4

Final state interactions and new superfluid spin mixtures . . . . . . . . . .

86

6.4.1

Effect of final state interactions . . . . . . . . . . . . . . . . . . . . .

87

6.4.2

(1,3) spectroscopy: |1i to |2i versus |3i to |2i transition. . . . . . . .

88

Determination of the fermion pair size in a resonantly interacting superfluid

90

6.5

8

6.5.1

Fermion pair size and rf spectroscopy . . . . . . . . . . . . . . . . .

90

6.6

Observation of quasiparticles in a polarized superfluid . . . . . . . . . . . .

94

6.7

Final state interactions revisited . . . . . . . . . . . . . . . . . . . . . . . .

96

7 Conclusion and Outlook

100

A Feshbach resonances in fermionic 6 Li

102

B Formation Dynamics of a Fermion Pair Condensate

107

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

112

D Superfluid Expansion of a Rotating Fermi Gas

123

E Pairing Without Superfluidity: The Ground State of an Imbalanced Fermi
Mixture128

F Tomographic rf Spectroscopy of a Trapped Fermi Gas at Unitarity

140

G Phase diagram of a two-component Fermi gas with resonant interactions145
H Determination of the fermion pair size in a resonantly interacting superfluid

155

Bibliography

164

9List of Figures
1-1 Critical temperature in the BEC-BCS crossover. . . . . . . . . . . . . . . .

15

1-2 Fermion pairs in the BEC-BCS crossover. . . . . . . . . . . . . . . . . . . .

16

2-1 Chemical potential µ and gap ∆ in the BEC-BCS crossover. . . . . . . . . .

22

2-2 vk and uk in the BEC-BCS crossover. . . . . . . . . . . . . . . . . . . . . .

24

2-3 Spatial pair wavefunctions in the BEC-BCS crossover. . . . . . . . . . . . .

25

2-4 Schematic illustration of rf spectroscopy. . . . . . . . . . . . . . . . . . . . .

27

2-5 Rf spectra in the BEC and BCS limit. . . . . . . . . . . . . . . . . . . . . .

28

2-6 Generic lineshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3-1 Hyperfine structure of the 6 Li ground state. . . . . . . . . . . . . . . . . . .

31

3-2 Feshbach resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3-3

Feshbach resonance in 6 Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3-4

Broad s-wave Feshbach resonances in 6 Li.

. . . . . . . . . . . . . . . . . . .

35

3-5 s-wave scattering lengths at fields below 600 G. . . . . . . . . . . . . . . . .

37

3-6 Phase contrast imaging of the density difference of two spin states. . . . . .

41

3-7 Double phase contrast imaging. . . . . . . . . . . . . . . . . . . . . . . . . .

42

3-8 Effect of anharmonicities on the in situ temperature determination. . . . . .

44

3-9 In situ temperature determination. . . . . . . . . . . . . . . . . . . . . . . .

45

4-1 Phase transition: Fermion pair condensation on the BEC side.

. . . . . . .

48

4-2 Phase transition: Fermion pair condensation at unitarity. . . . . . . . . . .

48

4-3 “Phase diagram” of fermion pair condensation. . . . . . . . . . . . . . . . .

49

4-4 Relaxation of a fermion pair condensate. . . . . . . . . . . . . . . . . . . . .

50

4-5 Schematics: process of vortex creation. . . . . . . . . . . . . . . . . . . . . .

51

4-6 Vortex lattices in the BEC-BCS crossover. . . . . . . . . . . . . . . . . . . .

52

4-7 Superfluid expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4-8 Determination of the critical interaction strength. . . . . . . . . . . . . . . .

54

4-9 Expansion at unitarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

5-1 Superfluidity in imbalanced Fermi gases. . . . . . . . . . . . . . . . . . . . .

58

5-2 Direct observation of the superfluid phase transition at unitarity. . . . . . .

61

10

5-3 Emergence of bimodal density distributions: a consequence of phase separation? 62
5-4 Schematics of density profiles in the harmonic trap. . . . . . . . . . . . . . .

64

5-5 In situ observation of the superfluid phase transition: Emergence of phaseseparation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5-6 First order vs second order phase transition. . . . . . . . . . . . . . . . . . .

67

5-7 n↑ and n↓ at the lowest temperature. . . . . . . . . . . . . . . . . . . . . . .

68

5-8 Temperature dependence of nc,↑ , nc,↓ , n↑ , n↓ and σ(r). . . . . . . . . . . . .

69

5-9 Homogeneous phase diagram at unitarity. . . . . . . . . . . . . . . . . . . .

70

5-10 ∆ vs. h at T=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5-11 Schematic illustration of a 3D phase diagram in the crossover. . . . . . . . .

73

5-12 CC limit in the crossover. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5-13 Density profiles with spin imbalance in the crossover. . . . . . . . . . . . . .

75

5-14 Phase diagram of the partially polarized normal phase in the crossover. . .

76

6-1 The Innsbruck experiment: rf spectroscopy on the BEC side. . . . . . . . .

79

6-2 The Innsbruck experiment: rf spectroscopy close to unitarity. . . . . . . . .

79

6-3 Rf spectra of a highly imbalanced superfluid. . . . . . . . . . . . . . . . . .

82

6-4 Rf spectra across the phase transition. . . . . . . . . . . . . . . . . . . . . .

84

6-5 Spatially resolved rf spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . .

85

6-6 Homogenous and inhomogenous rf spectrum. . . . . . . . . . . . . . . . . .

86

6-7 The problem with the (1,2) rf spectra. . . . . . . . . . . . . . . . . . . . . .

87

6-8 The solution: Rf spectroscopy of the (1,3) superfluid at unitarity. . . . . . .

88

6-9 Dramatic effect of final state interactions. . . . . . . . . . . . . . . . . . . .

89

6-10 Spectroscopy of the two magnetic field insensitive rf transitions in the (1,3)
mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

6-11 Evolution of the fermion pair size in the BEC-BCS crossover. . . . . . . . .

91

6-12 Rf dissociation spectra in the crossover. . . . . . . . . . . . . . . . . . . . .

92

6-13 Universal rf lineshape in the crossover. . . . . . . . . . . . . . . . . . . . . .

93

6-14 Density effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

6-15 Quasiparticle peak in the rf spectrum of the majority component. . . . . . .

95

6-16 Rf dissociation spectra in the (1,2) mixture. . . . . . . . . . . . . . . . . . .

97

6-17 Studying final state effects with the (1,3) mixture. . . . . . . . . . . . . . .

99

11

List of Tables
3.1

Feshbach resonances in the lowest three hyperfine states of 6 Li . . . . . . . .

6.1

Final state scattering lengths for rf transitions at unitarity for the (1,2), (2,3)
and (1,3) mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

35

87

Chapter 1

Introduction
1.1

Ultracold Fermi gases and the BEC-BCS crossover

The first superfluids of bosons and fermions were both realized in the laboratory of Heike
Kamerlingh Onnes at the University of Leiden in the Netherlands. A century ago, in 1908
Onnes created the coldest temperatures on earth thus far and cooled liquified 4 He below
the λ-point at 2.2 K into - what was later shown to be - the superfluid phase. Three years
later, by using 4 He as a coolant, Onnes observed superconductivity in mercury, when hefound that at temperatures below 4.2 K the resistivity of the metal essentially dropped tozero. At this time the concepts of quantum theory just started to emerge and the notion of“bosons” (4 He) and “fermions” (the electrons in the mercury) had not yet been conceived.
A century later, and again in systems cooled to the coldest temperatures on earth - now
about a million times colder then in Onnes’ laboratory in 1908 - the regimes of bosonic andfermionic superfluidity can be smoothly connected in experiments with ultracold atomic
gases.
In between lay the introduction of Bose-Einstein statistics in 1924/25 and the formulation of Fermi-Dirac statistics in 1926 which was first applied to electrons in a metal by
Arnold Sommerfeld in 1927. In 1939/40 Fierz and Pauli introduced the spin statistics
theorem that says that bosons have integer and fermions have half integer spin. Superfluidity of bosons was connected early to Bose-Einstein statistics (London, 1938) and thephenomenon of Bose-Einstein condensation (BEC). Superconductivity (i.e. superfluidity ofcharged fermions), however, remained a puzzle for decades until in 1957 Bardeen, Cooper
and Schrieffer (BCS) developed a full theory for the superconducting state. The discoveryof the proton (Rutherford, 1918) and the neutron (Chadwick, 1932) which, together withthe the electron (Thomson, 1897), are the fermionic building blocks of atoms, allowed theclassification of atoms as “bosons” and “fermions”: if the combined number of electron,
protons, and neutrons is even, atoms have integer spin and are bosons, if total number of
atomic constituents is odd, an atom has half-integer spin and is a fermion. Bose-Einsteincondensation of bosonic atoms in ultracold atomic gases was observed in 1995 and the first13

quantum degenerate Fermi gas of fermionic atoms was created in 1999.
It is interesting to note that usually the “bosons” made the first appearance: Superfluidity of 4 He was realized before superconductivity of electrons in a metal and superfluidityof 3 He, Bose-Einstein statistics was formulated before Fermi-Dirac statistics, superfluidityof bosons was understood before superconductivity of electrons, and ultracold Bose gases
were created before ultracold Fermi gases. The additional complexity involved in studying
fermions both theoretically and experimentally and particularly in the context of superfluidity lies in their quantum statistics: while undistinguishable bosons can occupy the same
quantum state, undistinguishable fermions must occupy orthogonal states.
Given the very different statistics bosons and fermions obey, how can the regimes ofbosonic and fermionic superfluidity be related? As a tightly bound pair of two fermions
has integer spin and can thus be regarded as a boson, the idea of a connection between
fermionic superfluidity and the formation of fermions pairs seems quite natural. Becauseof the Coulomb repulsion between electrons, however, it appeared impossible that electrons
could form a tightly bound pair in a metal. In 1950 it was realized that there actually is asmall, effectively attractive interaction between the electrons in superconductors mediatedby lattice vibrations. Six years later Cooper found that for any small attractive interaction
a bound state of fermions can form as a many body effect. While the idea of pair formation
proved to be correct, the fermion pairs in the BCS superfluid are far from tightly bound:they are correlated in momentum space and they do not obey Bose-Einstein statistics. The
BCS superfluid is not a Bose-Einstein condensate of fermion pairs. However, it turned outthat Bose-Einstein condensates of tightly bound fermion pairs can actually be described
as a limit of the BCS state. This observation is the basis of BCS-BEC crossover theories,
conceived and developed by Popov, Keldysh, Eagles [1], Legget [2], Noziéres and SchmittRink [3], which smoothly connect the BCS limit of superfluid, many-body bound fermion
pairs to the BEC limit of superfluid, two-body bound molecules.
This crossover between the well known limits of Bose-Einstein condensation and BCS
superfluidity can be realized and explored in ultracold atomic gases and the studies offermionic superfluidity in this regime are the subject of this thesis.

1.1.1

High temperature superfluidity

Fermionic superfluidity has been observed over a huge range of temperatures from morethan 100 K in high-temperature superconductors, in the 1 K range for conventional superconductors, in the mK range for 3 He, down to 20 nK in ultracold quantum gases (asdescribed in chapter 4 of this thesis). However, the absolute temperature scales are deceiving as fermionic superfluidity is a many-body effect and therefore depends on the density
n of the surrounding fermions. An appropriate temperature scale for a comparison of thecritical temperatures TC across various systems is the Fermi temperature TF which is proportional to the Fermi energy kB TF = EF ∝ n2/3 (kB is Boltzmann’s constant). With
14

TC/TF
TBEC/TF

High-TC
Superfluid

TC/TF ≈ 0.2

∞

1

BEC

-1
0
Interaction Parameter 1/kFa-∞
BCS

Figure 1-1: Critical temperature in the BEC-BCS crossover as a function of the interaction
parameter 1/kF a.

this scaling Tc /TF is as low as 10−8 for metallic lithium at ambient pressure, in the rangeof 10−4 to 10−5 for conventional superconductors, 5 × 10−4 for 3 He, and 10−2 for high-TC
superconductors.
Figure 1-1 shows the critical temperature for an ultracold Fermi gas in the BEC-BCS
crossover. Here the strength of the coupling is measured by the dimensionless quantity
1/kF a, where kF is the Fermi wavenumber and a the experimentally tunable s-wave scattering length. In the BCS limit TC decays exponentially with the coupling or interaction− 2kπ astrength: Tc ' TF e

F

, the regime where most fermionic superfluids including high TC

superconductors have been observed. On the BEC side, the large critical temperature for
BEC of two-body bound molecules is approached. In between, for −1 < 1/kF a < 1, lies the
crossover where fermionic superfluidity in ultracold atomic Fermi gases has been realized [4].
Here the critical temperature is as high as Tc /TF ≈ 0.2. Scaled to the density of electrons
in a metal, this critical temperature corresponds to superconductivity far above room temperature (and the melting point of the metal, too). The crossover therefore permits thestudy of fermionic superfluidity in a new regime - and the strongly interacting Fermi gas at
20 nK forms in fact a high-temperature superfluid.

1.1.2

The Fermion pairs

The instability of a Fermi Sea with weak attractive interactions towards fermion pair formation gives rise to fermionic superfluidity in the BCS limit. It is the character of thesepairs that changes smoothly but -comparing the BEC and BCS limits - quite dramaticallyin the BCS to BEC crossover. One of the important quantities characterizing the pairs andwith significant influence on the properties of the superfluid is the pair size relative to theinterparticle spacing, see figure 1-2. In the BEC limit the fermion pairs are bound dimers,
stable in isolation and small compared to the interparticle spacing. In contrast, the pairs on
15

BEC of Molecules

Resonance Superfluid

BCS state

Figure 1-2: Fermion pairs in the BEC-BCS crossover.

the BCS side are significantly larger than the interparticle spacing and unstable in vacuum.
In the crossover regime the size of fermion pairs is on the order of the interparticle spacing.
The main chapters of this thesis focus on radio-frequency (rf) spectroscopy experiments
which probe the fermion pairs directly and on interesting macroscopic consequences of thepairing phenomenon: by imbalancing the populations in the two strongly interacting spinstates of gas superfluidity can be quenched which gives rise to an interesting and rich phase
diagram.

1.1.3

Universality

At unitarity, where the scattering length a far exceeds the interparticle spacing and theinteraction strength 1/kF a reaches zero (in the limit of a approaching infinity, see section 3.4) the only relevant length and energy scales in the fermion system are the inverse
Fermi wavenumber kF and the Fermi energy EF . In this regime the physics is expected to
be universal [5] and not to depend on the specific properties of the fermions used in theexperiment. For example the fermion pair size at unitarity should be a universal constanttimes 1/kF (which is proportional to the interparticle spacing n1/3 ), and the interactionenergy a universal fraction of EF . For this reason some of the experiments with ultracoldgases could be relevant to understand properties of neutron starts and even the quark gluon
plasma of the early universe although the density and temperature scales involved are vastlydifferent.

1.2

Overview of the experiments in the BEC I lab at MIT

This section gives a short summary of the experiments that were carried out by my colleagues and me with the BEC 1 machine in the group of Wolfgang Ketterle at MIT1 .
The BEC 1 experiment has a long history and some of its parts date back to the years
1

A general overview of the developments in the field are given in recent review papers [6, 7, 8].

16

well before 1995 when BEC in 23 Na was observed. The ultra high vacuum of the mainchamber has been maintained continuously since 1996. Of course, many upgrades havebeen introduced since then, including a double species oven to accommodate both 23 Na and
6 Li, the laser system for 6 Li, a new magnetic trap, etc.

1.2.1

From strong interactions to fermion pair condensation

When I joined the BEC I lab as an undergraduate student in 2001 the first quantum
degenerate cloud of 6 Li had been observed and the group was on the way to access a regimeof strong interactions via Feshbach resonances. We soon saw first signs of the s-wave
Feshbach resonance, which was used in most of the later experiments, via magnetic fields
dependent losses [9]. At this time we were puzzled that these losses occurred mainly at fields
closer to 700 G than to 800 G where the resonance was predicted. When I returned as a
graduate student in 2003 an rf spectroscopy experiment was in progress, which studied mean
field shifts in the gas as it entered the strongly interacting regime [10]. Again something“strange” seemed to be going on at fields around and above 700 G, where the mean field
shifts became small and the mixture unstable. In the summer of 2003 we learned that someof these features were due to molecule formation and that surprisingly stable molecules
started to form at fields above 750 G with lifetimes that could be sufficient for evaporative
cooling into the quantum degenerate regime [11, 12, 13, 14]. Bose-Einstein condensation(BEC) of these molecules2 was accomplished soon thereafter, in the fall of 2003 [15, 16, 17].
A bimodal density distribution - the “smoking gun” for condensation - could, however,not be observed in the regime of universal interactions close to the center of the Feshbachresonance and on the BCS side. Early in 2004 the JILA group introduced a rapid magnetic
field ramp technique that transformed the increasingly fragile fermion pairs at unitarityand on the BCS side into more tightly bound and stabile molecules. This enabled theobservation of fermion pair condensation in the whole BEC-BCS crossover at JILA andlater at MIT [18, 19].

1.2.2

High temperature superfluidity

The observation of fermion pair condensation reinforced the quest to observe superfluidity
in the system. Due to the strong interactions this turned out to be a challenging task since
the behavior of the gas in the normal phase is in most instances qualitatively similar to theone associated with superfluids. In spring 2005 we observed vortex lattices in a rotating
cloud in the entire crossover regime which demonstrated phase coherence and superfluidityof the strongly interacting system [4]. This was the starting point for most experiments
discussed in this thesis.
2
The term “molecule” might not be appropriate since in the regime where BEC was observed many-bodyinteractions are already important.

17

With the new system at hand we could now set out to explore its macroscopic andmicroscopic properties. On the macroscopic scale we studied the stability of the superfluidwith regard to changes in a number of experimental parameters. In one of the first experiments we observed for how long superfluidity survived the expansion of the gas using
vortices as “markers” for the superfluid region in the cloud [20]. Since the density of thesystem was reduced adiabatically during expansion this experiment addressed the stabilityof the superfluid as a function of the interaction strength (at finite temperature).

1.2.3

Macroscopic properties of the superfluid: Population imbalanced
mixtures

In a second experiment we looked at the stability of the superfluid with regard to an
imbalance in the populations of the two spin states from which it is formed. The questions
raised by this experiment stimulated a whole series of further studies. The initial, quite
playful experimental approach was to turn the knob on the Agilent frequency generator
used to control the populations in the spin mixture and then to check whether vortices
could be observed. This allowed us to establish superfluidity in imbalanced mixtures andto characterize the breakdown of superfluidity at the so-called Chandrasekhar-Clogston
limit [21].
We soon realized that in an imbalanced mixture the superfluid phase transition could
be observed directly in expansion without any magnetic field sweeps3 : a bimodal density
distribution emerged in the minority component as a function of temperature [22]. With
these observations new questions about the origin of the bimodal structure and the natureof the superfluid and normal phases in imbalanced mixtures emerged. The data indicatedthat superfluidity at zero temperature required equal densities, leading to phase separation
between the normal mixture and the superfluid. To confirm this hypothesis we had toobserve phase boundaries in the trapped cloud with high spatial resolution. This was
accomplished by a phase contrast imaging technique which allowed us to observe the phasetransition directly in the trap [23]. Subsequently we refined this method by taking twophase contrast images of the same sample in rapid succession. Since we found that the
local density approximation was valid, these two images gave immediate access to the local
physics at a full range of densities and spin imbalances. The non-interacting ideal Fermi
gas of the pure majority component in the spatial wings served as a reliable, ideal gas
thermometer. The result was the homogeneous phase diagram of a resonantly interacting
imbalanced Fermi gas, with first and second order phase transitions merging at a tricriticalpoint [24].
Already in the first paper about population imbalanced Fermi gases we studied how thecritical imbalance evolves with varying interaction strength. Due to the Chandrasekhar3

The observation of vortices at unitarity still required a magnetic field sweep to enhance the vortex
contrast.

18

Clogston limit there is no continuous crossover between different superfluid regimes as theinteractions are changed. Instead, there will be at least one, but possibly a series of phasetransitions involving exotic superfluids, between the limits of a Bose-Fermi mixture on the
BEC side and the (ultimately) normal gas in the BCS limit. With the newly developed
techniques for the analysis of in trap density distributions we recently explored the phase
diagram as a function of interaction strength and identified the Bose-Fermi mixture in the
BEC limit.

1.2.4

Microscopic properties of the superfluid: Exploring pairing correlations with rf spectroscopy

In another set of experiments we studied the microscopic physics of the fermion pairs inthe superfluid with radio-frequency (rf) spectroscopy. After the creation of molecular condensates the Innsbruck group had applied rf spectroscopy to a resonantly interacting Fermi
gas and observed pairing correlations at low temperatures [25]. Since no clear experimental
signature of the superfluid phase transitions was observed, we first started out to correlate
the rf spectra with the indicators for superfluidity. This confirmed that pairing set in at
a temperature larger than the superfluid transition temperature. Since we were alreadyworking with imbalanced systems we also took rf spectra above and below the critical imbalance and found that they did not show any differences. Our conclusion was that the
rf spectra do not reveal the onset of superfluidity within the experimental resolution but
provide evidence for pairing in a regime where the critical temperature is zero [26].
While performing this experiment we realized that we could resolve what parts of thetrapped cloud were resonant at a given radio-frequency. Focusing on an equal mixture andbuilding on our experience from resolving spatial features in imbalanced clouds, we were now
able to obtain “local” rf spectra, i.e. rf spectra at a given density [27]. All previous spectra
were averaged over the density distribution of atoms in the trap and therefore more difficult
to interpret. The spatially resolved spectrum, however, had a narrow, symmetric lineshapewhich appeared to be at odds with the expectations for both the BEC and BCS limits.
This observation indicated that so-called final state interactions might severely affect the rf
spectra: The rf transfers atoms in one state of the initial two state spin mixture (| ↑i, | ↓i)to a third state |3i. If this state interacts strongly with atoms in the initial state the rf
spectrum does not simply reflect the pairing correlations of the initial state but the possiblycomplex interplay of interactions in the initial and final state. To make progress we neededan efficient handle on the final state interactions while preserving the initial state of interest.
The solution emerged from a seemingly unrelated observation. The rf experiments wereassociated with rapid atom losses after the rf pulse was applied. The losses appeared to
involve the collision of three atoms, each of them in a different state |↑i, |↓i and |3i. Thisobservation implied that all three two-state spin mixtures were likely to be stable andenabled us to change final state interactions by exploring new strongly interacting spin
19

mixtures. A suitable mixture with negligible final state interactions could be created andthe observed rf spectra at unitarity were dramatically different from the ones reported in
the previous experiments. The essential features of the rf spectra could now be understoodwithin a simple model and we were able to determine the fermion pair size at unitarity. The
pairs turned out to be smaller than the interparticle spacing and are the smallest observedin fermionic superfluids.
With both the knowledge of the homogeneous phase diagram for imbalanced mixtures
and an ideal system for obtaining spatially resolved rf spectra, we were then able to systematically study the pairing correlations in the superfluid, polarized superfluid, and mixed
normal phase of an imbalanced mixture. This lead to the observation of a local double
peak structure in the rf spectra of the majority component - the first clear signature ofquasiparticles in the superfluid. A part of the experiments described above are the subjectof this thesis which is outlined in the following section.

1.3

Outline of the thesis

The main focus of this thesis are experiments with imbalanced spin mixtures and experiments probing the microscopic nature of the fermion pairs with rf spectroscopy. A few earlier publications with a somewhat different background (p-wave Feshbach resonances [28],timescales related to fermion pair condensation [29] as well as the expansion of a superfluid
Fermi gas [20]) will also be briefly discussed. Since a number of recent review papers are
now available [6, 7, 8] most of the general background will be summarized in short andprimarily to establish a common language. We then proceed to discuss the new results indetail.
The thesis is organized as follows: Chapter 2 gives a a brief introduction of BEC-BCS
crossover theory. In chapter 3 the new techniques and systems developed for the experiments
presented in this thesis will be discussed. Chapter 4 introduces the superfluid 6 Li system
as the starting point for the experiments. In chapter 5 the experiments leading to theobservation of the phase diagram for population imbalanced mixtures will be presented.
The rf spectroscopy experiments studying the fermion pairs are the subject of Chapter 6.
Publications directly related to this thesis are included in the appendices A to H.

20

Chapter 2

BEC-BCS crossover in ultracold
atomic Fermi gases
2.1

BEC-BCS crossoverThis section gives a brief summary of BEC-BCS crossover theory. The purpose is to establish
the notation and to illustrate a few concepts relevant to this thesis. For extensive anddetailed reviews of the topics discussed in this chapter please see references [6, 7, 8]. The
discussion presented here is based on ref. [7].
The many-body Hamiltonian for a two state mixture of interacting fermions in the
BEC-BCS crossover is:Ĥ =X²k c†kσ ckσ +k,σV0 X †c q c† q c 0 q c 0 qΩ 0 k+ 2 ↑ −k+ 2 ↓ k + 2 ↓ k − 2 ↑(2.1)

k,k ,qHere ckσ and c†kσ are creation and annihilation operators of a fermion with momentum k,mass m and spin σ = (↑, ↓) obeying the commutation relations[ckσ , c†k0 σ0 ]+ = ckσ c†k0 σ0 + c†k0 σ0 ckσ = δkk0 δσσ0

(2.2)[ckσ , ck0 σ0 ]+ = 0(2.3)[c†kσ , c†k0 σ0 ]+

(2.4)

= 0

Ω is the volume of the system and the coupling “constant” V0 is given in terms of thes-wave scattering length a (for a δ function potential V (r) = V0 δ(r) and ignoring higher
order corrections):
V0 =4π~2 am(2.5)

The first term of the Hamiltonian corresponds to the kinetic energy of the particles and
21

1
0

1.5

∆/E F

µ/E F

-1
-2

1.0

0.5
-3
-4

0.0
2

1

0
1/k F a

-1

-2

2

1

0
1/k F a-1

-2

Figure 2-1: Chemical potential µ and gap ∆ in the BEC-BCS crossover. In the BCS limit:− π
π~2 an~2
and |µ| À ∆. Theµ ≈ EF À ∆ and ∆ ≈ e82 e 2kF |a| . In the BEC limit: µ = − 2ma
2 +m
first term in µ is the binding energy per particle in a molecule, the second one reflects density
dependent molecule-molecule interactions (which the theory does not predict correctly [30]).
∆ ∝ √EkF a has no special significance in the BEC limit, however, ∆2 /µ is proportional toF2the molecular mean field 4π~m na .

²k = ~2 k 2 /2m. The second term describes the a dependent correlation energy of the interacting fermion system and includes density fluctuations, i.e. interactions between pairs
at finite momentum q 6= 0. By neglecting these density fluctuations - which is a strong
approximation for ultracold gases - one obtains the BCS Hamiltonian:Ĥ =X²k c†kσ ckσ +V0 X † †ck↑ c−k↓ ck0 ↓ c−k0 ↑
Ω

(2.6)

k,k

k,σ

The exact solution1 to this Hamiltonian is the BCS wave function:|ΨBCS i =Y
(uk + vk c†k↑ c†−k↓ ) |0i

(2.7)

k

with
vk2

=

u2k =

µ
¶
ξk
1
1−
2
Ek
µ
¶
1
ξk
1+
2
Ek
(2.8)

Ek and ξk depend on the gap ∆ and the chemical potential µ which vary in the BEC-BCS
1The BCS wavefunction is an exact solution in the sense that it provides the correct thermodynamic
quantities. There is an ongoing discussion how it is related to the the exact wavefunction for the groundstate of the BCS Hamiltonian [31]].

22

crossover as shown in figure 2-12 :Ekqξk2 + ∆2=ξk = ²k − µ(2.9)

The uk and vk give the probability amplitude for a pair state c†k↑ c†k↓ |0i to be empty or fully
occupied, respectively, and u2k +vk2 = 1). Several functions involving combinations of uk andvk (like u2k , vk2 , uk vk and vk /uk ) have special significance in the crossover and are important
for the description of the fermion pairs as discussed in the following sections. Figure 2-2
shows their k dependence for five different interactions strengths.

2.2

Quasiparticle excitations

The BCS state describes a wavefunction of pairs. In some experiments, however, excitedstates with unpaired fermions will be accessed. Such unpaired fermions also emerge as finite
temperature excitations. They are referred to as quasiparticles and their creation operators
are given by:†
γk↑= uk c†k↑ − vk c−k↓(2.10)†γ−k↓= uk c†−k↓ + vk ck↑(2.11)The BCS ground state can be regarded as the quasiparticle vacuumγk↑ |ΨBCS i = γ−k↓ |ΨBCS i = 0(2.12)

with excitations†
γk↑|ΨBCS i = c†k↑ (u2k + vk2 )Y(ul + vl c†l↑ c†−l↓ |0i
l6=k

=

c†k↑

Y
(ul + vl c†l↑ c†−l↓ |0i(2.13)

l6=k(2.14)
where a pair (k ↑,−k ↓) is removed and a single fermion in k ↑ added.
The energy cost for adding a quasiparticle relative to the chemical potential has twocontributions: the kinetic energy ξk = ²k − µ required to add a particle with momentum
k and the loss in pairing energy of the BCS state which can be shown to be Ek − ξk [7].
∆ and µ have to be determined simultaneously from the gap equation ∆ = − VΩ0
R d3 k 2
and from the number equation for the total particle density N/Ω = 2 (2π)
3 vk .
2

23

P

V0 P
∆
k uk v k = − Ω
k 2Ek

uk2

1/kFa =2

1

2

v 2
k

u v
k k

v /u
k k

3
0.2

0.2

0.15

0.15

0.04

0.99
0.98

0.03

0.97

0.02

0.96

0.01

1
1

2

3

4

2

3

4

0.1

0.1

0.05

0.05

5

5

1

2

3

4

5

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

2

3

4

5

0.5

1/kFa = 1

0.95
0.4
0.9
0.3
0.85
0.2
0.8

0.1
1

2

3

4

1

5

2

3

4

5

0.5

1/kFa = 0

1

2

3

4

5

2
0.2

0.4

0.15

0.3

0.1

0.2

0.8

1.5
0.6

1

0.4
0.05

0.5

0.1

0.2
1

1

1

2

3

4

5

1

2

3

4

5

0.5

1/kFa = -1

8
0.8

0.8

0.4

0.6

0.6

0.3

0.4

0.4

0.2

0.2

0.2

0.1

6

1

2

3

4

5

1

1

2

3

4

5

1

4
2

1

2

3

4

5

0.5

1/kFa = -2

40
0.8

0.8

0.4

0.6

0.6

0.3

0.4

0.4

0.2

0.2

0.2

0.1

30
20

1

2

k/kF

3

4

5

1

2

3

4

5

k/kF

10

1

2

k/kF

3

4

5

1

k/kF

Figure 2-2: vk and uk in the BEC-BCS crossover. Plotted are u2k , vk2 , uk vk and vk /uk forfive different interaction strengths. vk2 ∝ n and u2k = 1 − vk2 . In the BEC limit uk → 1
and vk → 0 for k → 0 whereas in the BCS limit uk → 0 and vk → 1 for k < kF .
uk vk = hΨBCS | c†k↑ c†−k↓ |ΨBCS i is the two-point correlation function, or the “Cooper pair”
wave-function in k space. In the BCS limit this function is sharply peaked around the
Fermi momentum kF whereas in the BEC limit momentum states from k = 0 to k >kF contribute to the wavefunction with monotonously decreasing weight. The BCS pair
wavefunction suggest another form for the pair wavefunction in k-space: vk /uk . This is(up to a normalization factor) the k-dependent pre-factor of the pair creation operatorsc†k↑ c†−k↓ in the BCS state. The identification of vk /uk as the pair wavefunction is basedon the interpretation of the BCS state as N/2 identical particles. The function vk /uk issimilar to uk vk in the BEC limit. In the BCS limit, however, vk /uk carries more weight at
low momentum components and sharply falls off to zero around kF . The function vk /uk isrelevant for rf spectroscopy. The plots are based on the following values for µ and ∆: 1)
1/kF a = 2, µ ≈ −4EF and ∆ ≈ 1.9EF ; 2) 1/kF a = 1 µ ≈ −0.8EF and ∆ ≈ 1.3EF ; 3)
1/kF a = 0 µ ≈ 0.6EF and ∆ ≈ 0.7EF ; 4) 1/kF a = −1 µ ≈ 0.95EF and ∆ ≈ 0.2EF ; 5)
1/kF a = −2 µ ≈ 0.997EF and ∆ ≈ 0.046EF .

24

Figure 2-3: Spatial pair wavefunctions φE (from the mathematically exact solution of the
BCS Hamiltonian), φP and φBCS for three different interaction strengths (here η = 1/kF a
and r is measured in units of 1/kF ). Reprinted from ref. [31].

The total energy cost is therefore Ek , with a minimal value either given by the gap ∆ forpξk = µ > 0 or if µ < 0 on the BEC side by µ2 + ∆2 for k = 0 .

2.3

Fermion pair wavefunction

The formation of fermion pairs is a prerequisite for fermionic superfluidity and many properties of the superfluid are related to the stability of these pairs. This becomes especiallyapparent in the BEC-BCS crossover where the pairs change from tightly bound molecules
to large and many-body bound Coper pairs.
The molecules in the BEC limit can be described straightforwardly in terms of a pair
wavefunction φm (r) ∝ e−r/b /r, where b is the size of the molecule. The situation is more
complex in the BCS limit, where there are several potential “candidates” for the pair wavefunction. In figure 2-2 the k-dependence of two of them (vk uk and vk /uk ) is plotted for
different interaction strengths. The first wavefunction is the two-point correlation functionhΨBCS | c†k↑ c†−k↓ |ΨBCS i = uk vk which is peaked at kF indicating that most of the correlations are maintained by the pairs on the Fermi surface. The BCS wavefunction which
describes N/2 equal pairs suggests vk /uk as an alternative description. A third possibility
are the wavefunctions obtained from the mathematically exact solution of the BCS Hamiltonian which describe N/2 different pairs [31].
25The spatial wavefunctions can be obtained as the Fourier transform of uk vk and uk /vk :Z
1
d3 k ∆ ikr
d3 k
ikr
u
v
e
=
e
φP (r) =
k
k
(2π)3
2
(2π)3 EkZZ
d3 k∆
d3 k vk ikr
e=eikr
φBCS (r) =33(2π) uk(2π) Ek + ξk
Z

(2.15)(2.16)

and are shown for three different values of kF a in figure 2-3 (reprinted from ref. [31]). Allwavefunctions decay exponentially as a function of distance everywhere in the crossover andbecome identical in the BEC limit. Since uk vk as well as vk /uk show sharp features in the
BCS limit at the Fermi wavenumber kF (a peak and an edge, respectively) their Fouriertransforms on the BCS side show a modulation around 1/kF .
It depends on the experiment which pair wavefunction appears most naturally. In the
rf spectroscopy experiments discussed in the next section, the rf excitation affects fermionsat all k and therefore the wavefunction vk /uk appears naturally in the description. The
two-point correlation function based on uk vk is best used to describe excitations at the
Fermi surface.

2.4

Rf excitation of fermion pairs

Rf spectroscopy is a powerful tool to study the microscopic properties of the fermion pairs
in the BEC-BCS crossover. Important quantities that can - at least in principle - beobtained with this technique are the binding energy of the pairs, the pair size and possiblya characteristic temperature for pair formation.
In the rf spectroscopy experiments atoms in one state of the two state (↑, ↓) mixture(here chosen to be |↑i) are transferred into an unoccupied third state |3i (see figure 2-4). If
the mixture is non-interacting the rf transition occurs at an excitation energy ~ωrf equal
to the atomic hyperfine energy ~ωhf as it would be the case in absence of atoms in state|↓i (see also Fig, 2-4 a).
In the following we assume that the initial state is interacting and described by thecrossover wavefunction. The rf excitation can then “break” a pair resulting in a quasiparticle
in the initial state plus a free particle in state |3i. Since the momentum of the rf photon
is negligible the rf excitation only changes the internal state of the atom and preserves its
momentum. To transfer an atom from state |↑i to state |3i the rf photon must provide
the kinetic energy of the transferred particle ²k as well as the energy Ek − µ required for
creating the quasiparticle in addition to ~ωhf (figure 2-4 b):~ωrf = ~ωhf + ²k + Ek − µ(2.17)

It is convenient to express all energies relative to the bare hyperfine energy and therefore

26

b)

a)

c)

|3〉
Ñ ωhf

Ñ ωhf

Ñ ωhf + εk + Ek − µ

|↑〉

|↓〉

Figure 2-4: Schematic illustration of rf spectroscopy. Rf spectroscopy of two state fermion
mixtures requires a transition to a third state since interaction shifts cannot be detectedby driving transitions both in coherent [10] and incoherent [32] two state mixtures. a, in
the absence of interactions the rf is resonant with the bare “atomic” transition at ~ωrf .
b) If the initial state is interacting the frequency response changes. c) In the discussion in
the text we neglect “final state interactions” which will be discussed in detail in chapter 6.
Such interactions can have a significant impact on the rf spectrum.

we define ω ≡ ωrf − ωhf so that~ω = Ek + ξk

(2.18)The minimum or threshold energy required for an rf excitation Eth = ~ωth is given byp~ωth = ~ω(k = 0) = µ2 + ∆2 −µ. In the BEC limit Eth approaches the molecular binding
energy 2|µ| = ~2 /ma2 and in the BCS limit Eth → ∆2 /2EF which is the condensation(binding) energy of a fermion pair.
Here we have assumed that atoms in the final state |3i do not interact with atoms in
the initial states. In their most dramatic form such interactions can suppress pair dissociation but instead lead to so-called bound-bound transitions between pairs (as schematicallyillustrated in figure 2-4 c). The effect of such interactions will be discussed in detail inchapter 6).
The rf operator is given by:
HRF ∝

X †
ck3 ck↑ + h.c.

(2.19)kwhere only the first part contributes since state |3i is initially empty. It is convenient to†express ck↑ via quasiparticle creation and annihilation operators as ck↑ = uk γk↑ + vk γ−k↓.

With that we have
HRF |ΨBCS i ∝X ††ck3 (uk γk↑ + vk γ−k↓
) |ΨBCS i(2.20)k=X†
vk c†k3 γ−k↓
|ΨBCS ik27

(2.21)

a

0

1

k/kb

2

b

2.5

0

0.8

k/kF

0.9

0.93

1

1

Signal (arbitrary unit)

BEC-limit

BCS-limit

δEFWHM

δEFWHM

0

0
0

Eb 2

4

6

0

8

∆2/2E24

6

8

F

Pair dissociation energy in units of the binding energy

Figure 2-5: Rf spectra in the BEC and BCS limit.

The complete rf spectrum can be obtained with Fermi’s Golden Rule [7] and is proportional toq
Here ρ(²) ∝v2
I(ω) ∝ ρ(²k ) k2 |²k =²(ω)
uk
2m²k
~2

(2.22)

is the density of states and vk /uk is proportional to the fermion pair

wavefunction in the BCS state (see figure 2-2). Both ρ(²k ) and vk2 /u2k have to be evaluatedat the photon energy which can be obtained from 2.18 as²(ω) =
With that we find

ω − ωth(~ω + ~ωth + 2µ)
2ω
vk2
∆2
|²k =²(ω) = 2 2
2
~ ω
uk

and

p
√
ω − ωth ω + ωth + 2µ/~(2.23)
I(ω) ∝
ω 5/2
The rf spectra in the BEC and BCS limits are very similar (see figure 2-5): the spectral

response rises steeply around ωth and falls off as ω −3/2 for large ω. The second factor in√
the nominator reduces to ω in the BEC limit for molecular dissociation and approachespω + 2EF /~ in the BCS limit, where the Fermi energy enters as a new energy scale.
Compared to the BEC limit this leads to a smaller ratio between spectral linewidth andthreshold energy. The dissociation lineshape in the BEC limit is given by:√ω − ωthIm ∝ω2(2.24)

This lineshape is, in fact, quite generic. By introducing an additional frequency offset ω028

Signal [a.u.]

1

0
0

5

10

E/Eth

Figure 2-6: Generic rf lineshape. Simulated rf dissociation lineshape in the BCS limit for∆ ¿ EF (black solid line) with a fit based on Igen (dashed red). E = ~ω is given in unitsof Eth .

we can define:√ω − ωthIgen = const.
(ω − ω0 )2(2.25)

Figure 2-6 shows that a simulated rf spectrum in the BCS limit is well fit by Igen . Thisimplies that in the presence of uncalibrated shifts to the pair dissociation threshhold it isvirtually impossible to distinguish experimentally between BEC and BCS pair dissociationlineshapes. In the experiments the onset of rf spectrum may be indeed be subject to possibly
small, but unknown shifts due to Hartree terms [33, 10].

29

Chapter 3

The Experiment: Atoms,
apparatus and techniquesThis chapter gives a brief overview over the experiment and experimental techniques. Our
experiments are carried out in a gas of ultracold 6 Li atoms. The experimental setup andmost techniques have been discussed in a number of previous publications and PhD theses.
We will often refer to the Varenna notes [7] which provide a detailed review of this work.
Below we will focus on some newer developments, specifically the creation of two newsuperfluid spin mixtures in 6 Li and spatially resolved imaging techniques.

3.1

6

Li

6 Li consists of three protons, three neutrons and three electrons and is therefore a fermion.

The hyperfine structure of the 6 Li ground state is shown in figure 3-1. All experiments
discussed here are carried out in two state mixtures of atoms in the three lowest 6 Li hyperfine
states which are labeled in the order of increasing energy as |1i, |2i and |3i.

3.2

Equipment

The major equipment involved in the preparation of a strongly interacting 6 Li gas in our
experiments involves:• an ultrahigh vacuum system• a double species oven creating a combined 23 Na and 6 Li atomic beam [34, 35]• the laser systems for 23 Na and 6 Li providing the photons for laser cooling and imaging
the atoms [36, 37]• a high current coil assembly (magnetic trap) for trapping the atoms and for generatingexternal magnetic fields up to 1000 G [36, 38]
30

6Li

|6>
|5>
|4>

400

F = 3/2

mS = +1/2

Energy [MHz ]2000mS = -1/2-200

F = 1/2

|3>

-400

|2>
|1>
0

50

100

150

200

250

300

Magnetic Field [G ]Figure 3-1: Hyperfine structure of the 6 Li ground state. The hyperfine splitting at zero
field between the F = 3/2 and F = 1/2 levels is 228 MHz (F is the total - nuclear plus
electron - spin and a good quantum number at low field, mS (mI ) are the projections of theelectron (nuclear) spin on the z-axis and good quantum numbers at high fields). During
sympathetic cooling with 23 Na in the magnetic trap 6 Li is in the stretched state |6i. The
experiments described in this thesis are carried out in two state spin mixtures of the threelowest hyperfine states (shown in red). At fields larger than 300 G the splitting between
these states is on the order of 80 MHz.

31

Energy

EnergyBound State∆µ B

Interatomic distance

B0

Magnetic Field

Figure 3-2: Singlet and triplet potentials for two scattering atoms depending on the spin
configuration of the valance electrons. If the atoms scatter, for example, in the triplet potential (the “open channel”) they cannot access the continuum states of the singlet potential(the “closed channel”). A Feshbach resonance arises when a bound state in the singlet
potential is tuned into resonance with the scattering atoms via an external magnetic field.

• a 1064 nm infrared laser for creating the optical trap in which the various spin mixture
are realized and the experiments are carried out [39].
Detailed descriptions of these components and associated equipment are given in theabove references. Further references and information can also be found in [40].

3.3Preparation of ultracold 6 Li fermions

After pre-cooling in a magneto optical trap (MOT) 6 Li is sympathetically cooled withbosonic 23 Na in a magnetic trap resulting in a pure sample of typically 30 · 106 quantum
degenerate 6 Li atoms at a temperature of T /TF = 0.3. The sympathetic cooling with
23 Na is necessary since the collisions required for rethermalization would “freeze out” ina pure, spin polarized gas of 6 Li atoms at temperatures below 6 mK. In this regime pwave collisions are energetically suppressed and s-wave collisions are forbidden for identicalfermions. A detailed summary of the cooling procedure and optical trapping together withfurther references are given in [41, 39, 7].

3.4

Creation of superfluid spin mixtures

3.4.1

Inducing strong interactions: Feshbach resonances

The observation of fermionic superfluidity in ultracold Fermi gases and the exploration of
the BEC-BCS crossover are based on the ability to tune the interaction strength 1/kF a.
The inverse Fermi wavenumber 1/kF under typical experimental conditions is on the orderof 10000a0 , and typical scattering lengths do not exceed 100a0 . These conditions do not
32

5

20

Energy [MHz]

0

0

-10

-5

Scattering length [a 0 ]10-20x10
543

3

650
834
Ma gnetic Field [G]Figure 3-3: Feshbach resonance in 6 Li for the lowest two hyperfine states |1i and |2i in thes-wave channel. Shown in red (solid) is the binding energy of the bound molecular statein the singlet potential as it approaches the zero energy threshold at B12 = 834 G. The
continuum of scattering states is shaded in grey. The blue dashed line shows the s-wave
scattering length as a function of the magnetic field. The scattering length is positive on
the “BEC side”, where the bound molecular state exists in the potential, negative on the“BCS side” and diverges at “resonance” (834 G). Apart from the broad resonance at 834
G the (1,2) mixture exhibits a narrow s-wave resonance at 543 G (thick dashed line).

permit the observation of superfluidity at experimentally attainable temperatures nor dothey provide a sufficient tuning range for the interactions to observe crossover physics. The
crossover regime can be accessed experimentally via Feshbach scattering resonances thatallow the adjustment of the s-wave scattering length as a function of an externally appliedmagnetic field B. Feshbach resonances emerge when a molecular bound state in the “closed
channel” of the interatomic potential is brought into resonance with atoms colliding in the“open channel” (see figure 3-2). The energy of the bound state relative to the energy of thescattering atoms can be controlled by an externally applied magnetic field B.
Figure 3-3 shows the B12 = 834 G Feshbach resonance in 6 Li. At fields below B12 thebinding energy of weakest bound state in the singlet potential 1 , approaches the resonancesas ~2 /ma2 where a is the s-wave scattering length. This bound molecular state is strongly
coupled to the continuum of scattering states shifting the resonance from the uncoupled
threshhold at 543 G by about 300 G to 834 G. Such a huge shift is unusual as is the large
background scattering length of −2100a0 at high fields. Both is due to an virtual state in
1

The quantum numbers are X 1 Σ+
g , v = 38. This state has two total nuclear spin projections I = 0 and
I = 2. Here we refer to the state with I = 2. The I = 0 state gives rise to a second resonance close to 543
G.

33

the 6 Li triplet potential just in the continuum of scattering states. At fields above the B12 ,where a is negative, no bound molecular state exists.
The Feshbach resonance gives rise to the dispersive shape of the s-wave scattering length(shown in figure 3-3), and therefore allows the tuning of the interaction parameter 1/kF aover a large range from negative to positive values. For |1/kF a| . 1 the superfluid phasetransition can be reached at experimentally accessible temperatures. All experiments discussed in this thesis were performed in the vicinity of “broad” Feshbach resonances, where
the effective range of the interaction potential is small compared to the inverse Fermi
wavevector. In this regime the specifics of the molecular state in the closed channel do
not affect the physics and at unitarity the inverse Fermi wavenumber is the only relevant
length scale. See ref.[7] for an extensive discussion of Feshbach resonance and the relevantenergy scales.
Feshbach resonances appear to be the ideal “tool” to manipulate atoms and access strong
interactions. However, when Feshbach resonances were first observed in bosonic 23 Na [42]they were found to be associated with rapid and strong losses [43]. In contrast to bosons,fermion pairs were found to be stable in the vicinity of the Feshbach resonance which isa direct consequence of the Pauli principle [30]. Only this stability made the experiments
described in the following chapters of the thesis possible.

3.4.2

Feshbach resonances in 6 Li

In the three two state mixtures of the three lowest hyperfine states in 6 Li four s-wave [9, 10,
13, 44] and three p-wave [45, 28] resonances have been observed at magnetic fields between
0 and 1000 G. One additional p-wave resonance is predicted at 225 G [46]. Table 3.1 gives
the resonance locations and widths. Most experiments in 6 Li have been carried out in thevicinity of the broad (1,2) resonance at 834 G. Recently we have also created superfluids
in the (1,3) and (2,3) spin combinations around the (1,3) and (2,3) Feshbach resonances
respectively (see figure 3-4 and chapter 6). The preparation of the superfluid spin mixtures
for all three cases will be discussed in the next section.

3.4.3

Three superfluid mixtures

6 Li is sympathetically cooled in the stretched state |6i. The cloud is then transferred into

an optical trap. A low magnetic bias field is applied and the atoms are transferred into
state |1i via an adiabatic Landau-Zener sweep across the single photon |6i → |1i hyperfine
transition at 228 MHz (further details are given in [7]). The optically trapped sample instate |1i is the starting point for creating the various superfluid spin mixtures.

34

States(1, 1)(1, 2)(2, 2)(1, 3)(1, 2)(1, 3)(2, 3)(1, 2)

Wave
p
p
p
p
s
s
s
s

B [G]
159.1
185.1
214.9
225
543.3
690
811
834

Width [G]
< 0.4
< 0.2
< 0.4
< 0.4
≈ 120
≈ 220
≈ 300

Table 3.1: Feshbach resonances in the lowest three hyperfine states of 6 Li . The resonances
are sorted in the order of increasing magnetic field. The first three p-wave resonances werereported in [45, 28]. The fourth p-wave resonance has been predicted [46] but has not yet
been observed. The three lowest s-wave resonances have a huge width and all three of them
have now been used to create fermionic superfluids (see chapter 6). The width and positionof these resonances has been reported in [44].

Scattering length [1000 a0]

10

5

0

-5

-10
600

700

800
900
Magnetic field [G]

1000

1100

Figure 3-4: Broad s-wave Feshbach resonances in 6 Li (calculation by A. Simoni [46] basedon the data in ref. [44]). Except for our most recent rf spectroscopy experiments all studies
in the strongly interacting regime with 6 Li were carried out in vicinity of the (1,2) Feshbachresonance at 834 G (solid black line). However, also around the (1,3) resonance at 690 G(dashed blue) and the (2,3) resonance at 811 G stable fermionic superfluids in the respective
spin mixtures can be created.

35

(1,2) Mixture
To produce the superfluid in the (1,2) system the magnetic field is ramped to values in thevicinity of the (1,2) Feshbach resonance at 834 G. At this field a non-adiabatic LandauZener sweep around the |1i to |2i transition frequency of about 80 MHz is applied, creating
an equal mixture of atoms in state |1i and |2i. The mixture is then evaporatively cooledby lowering the trap depth of the optical trap in several seconds to about 1 µK.
(2,3) Mixture
The preparation of a (2,3) superfluid follows essentially the same procedure as described
for the (1,2) mixture. The only difference is that instead of applying a Landau Zenertransfer that creates an equal (1,2) mixture, the atoms are adiabatically transferred into
state |2i followed by a second non-adiabatic sweep that creates an equal (2,3) mixture.
(1,3) Mixture
The (1,3) mixture is prepared around 568 G, where both a12 and a13 are small. First an
equal (1,2) mixture is created with a non-adiabatic Landau-Zener rf sweep. The atoms instate |2i are then transferred into state |3i with an adiabatic Landau-Zener sweep, yielding
an equal (1,3) mixture. To access the strongly interacting regime the magnetic field isadjusted in about 100 ms to fields close to the (1,3) Feshbach resonance at 690 G where thesystem is evaporatively cooled into the superfluid phase.

3.4.4

Population imbalance

The probability for transferring atoms via a Landau-Zener sweep from state |ai to |bi is
given by:
P|ai→|bi = 1 − exp(−2π

Ω2R
)
ω̇

If ω̇ À Ω2R the rf sweep is adiabatic and the transferred fraction from |ai to |bi approaches
100%. A smaller transfer can be accomplished by performing a non-adiabatic sweep andchoosing appropriate values for either ω̇ by adjusting the sweep time or ΩR by adjusting
the rf power. The Landau-Zener sweep is a coherent process. The initial state after thetransfer is therefore a coherent superposition and not yet the desired incoherent mixtureof distinguishable atoms in state |ai and |bi. Decoherence, however, occurs within about
ten milliseconds after the sweep due to the motion of the atoms in the externally applied
inhomogeneous magnetic field.

36

Scattering length [a0]

400

0

-400

-800
0

200
400
Magnetic field [G]600

Figure 3-5: s-wave scattering length at fields below 600 G in 6 Li (calculation by A. Simoni [46] based on the data in ref. [44]): a13 (dashed blue), a23 (dotted red) as well as a12(black solid). The zero crossings of the scattering length are at magnetic fields of about: 527
G (a12 = 0), 568 G (a13 = 0), and 588 G (a23 = 0). The minima of the scattering length areat about 320 G (where a12 = −290a0 and a13 = −895a0 ) and 363 G (where a23 = −476a0 ).
The narrow s-wave resonance in the (1,2) mixture at 543 G is also indicated.

3.4.5

Stability

All three mixtures show lifetimes at unitarity that exceed 10 seconds. This is surprisingsince one would expect that close to a Feshbach resonance where interactions of the atoms at
short distances are enhanced, inelastic collisions are facilitated as well. The potentially bad,inelastic two-body collisions are dipolar- and spin relaxation. Dipolar relaxation involves
the transfer of angular momentum from the relative motion of the atoms to the electrons
and nuclei. Spin relaxation occurs when angular momentum is exchanged between electrons
and nuclei without affecting the relative motion of the atoms.
Both processes are forbidden for fermions if the internal state of two the outgoing atoms
is identical. In this case anti-symmetrization requires an odd (ex. p-wave) outgoing spatialwave, whereas the incoming spatial wavefunction is s-wave. Inelastic spin relaxation processes are therefore excluded. Also dipolar relaxation is not allowed since this process canonly connect spatial waves of the same parity.
The (1,2) mixture with only one possible exothermic relaxation process (1, 2) → (1, 1)
must therefore be stable. For the (1,3) and (2,3) mixtures the allowed exothermic collisionsare (1, 3) → (1, 2), (2, 3) → (1, 3), and (2, 3) → (1, 2). For all processes spin relaxation is notpossible since the z-projection of the total angular momentum is not conserved. However,
dipolar relaxation is allowed with an outgoing d-wave in the final state. Since the initial37

state is s-wave, the released energy in the relaxation process occurring at small distances
must be sufficient to overcome the d-wave barrier. We can estimate the height of this barrier
as

6~2
C6
V (rmax ) = − 6 +2rmax 2µrmax³

with rmax =

C6 µ
2~2

´1
4

(3.1)

. Here C6 ≈ 1393 a.u. (1 a.u.= 9.57344×10−26 J nm6 ) is the coefficientof the 6 Li van-der-Waals potential and µ = mLi /2. We find V (rmax ) ≈ 600 MHz which islarge compared to the 80 MHz Zeeman splitting between the hyperfine levels. This might
compensate for any collisional enhancement of dipolar relaxation close to the Feshbachresonance. The observed lifetimes are consistent with typical dipolar relaxation rates on
the order of 10−15 cm−3 s−1 .

3.4.6

Comparison of the mixtures

What is the “mixture of choice” for experiments? Since all three s-wave resonances arebroad, universality requires that at unitarity the mixtures show exactly the same behavior.
For most experiments it is therefore not important which mixture is chosen. Until our
realization of the two new superfluid mixtures, all experiments in 6 Li have been carried out
in the (1,2) system.
Even at unitarity, however, the choice of the mixtures matters in experiments involving
a third state. This is the case in rf spectroscopy experiments where one drives rf transition
from the initial mixture to a third state. Here it is desirable that the final state interacts
only weakly with the initial one. The (1,3) mixture (with |2i as the final state) fulfills thisrequirement but not for the (1,2) and (2,3) systems (with |3i and |1i as the final states,
respectively). The (1,3) mixture is indeed the right choice for rf spectroscopy experiments(see chapter 6).
One noticeable difference between the three large s-wave resonances is their width which
differs by almost a factor of three between the (1,3) and (1,2) resonances. The smaller widthof the (1,3) resonance together with its location at lower magnetic fields might be of someadvantage if one would like to access the more weakly interacting regime on the BCS side
at lower externally applied magnetic fields (at 1000 G a13 = −2560a0 , which is comparable
to a12 = −2580a0 at 1500 G).
At fields below the broad s-wave resonances the (1,3) mixture is also very interesting(see figure 3-5). At 320 G the s-wave scattering length a13 = −895a0 which should permit
efficient evaporation. In optical traps providing tight radial confinement with trap frequencies in the kHz range values of kF a13 ≈ −1 might be within experimental reach. This could
be sufficient to enter the superfluid regime at experimentally attainable temperatures. By
increasing the magnetic field one can change the interactions from attractive to repulsive
which is interesting for the observation of antiferromagnetism in an optical lattice.
Apart from the two-state mixtures, a one-state “mixture” of atoms in state |1i and38

a three-state mixture of atoms in states |1i, |2i, and |3i show some promise. In a puresample of |1i atoms the p-wave resonance close to the 160 G could be used to form p-wave
molecules. Since only fermions in the ground state are involved, these molecules might bereasonably stable and there could be an opportunity to create a superfluidity with p-wave
pairing. Very recently the creation p-wave molecules at the three lowest p-wave resonancesin 6 Li has been reported together with measurements of their elastic and inelastic collision
rates [47].
The ternary mixture shows lifetimes exceeding 30 ms within a broad range of magnetic
fields and for strong interactions. If this mixture proves to be sufficiently stable [48] future
work could focus pairing competition in multi-component Fermi gases and spinor Fermi
superfluids (with analogies to quark matter) [49, 50, 51, 52, 53, 54].

3.5

The harmonic trap as a feature: Spatially resolved 3D
density distributions in imbalanced mixtures

The ability to reconstruct the three dimensional (3D) density distribution of the trapped
atomic gas was crucial for many of the experiments discussed in this thesis. Usually, theharmonic confinement of the atoms leads to experimentally determined quantities that are
averaged over the density distribution of atoms in the trap. For the purpose of obtaining
quantities at a given density a box potential appears to be more suitable.
If it is possible, however, to obtain spatially resolved 3D densities the trap actuallyactually becomes a great feature (under the assumption that the local density approximation
is valid). This is the case in our system, where a single harmonically confinement sample
allows us to observe the behavior of the gas for whole range of local densities and spinpolarizations. In imbalanced mixtures the harmonic confinement also provides an ideal gas
thermometer by spatially separating the non-interacting wings of the majority component
from the strongly interacting mixed system.
Complete information about the 3D density n(x, y, z) can be obtained from images of
the atoms. We assume that the imaging light E0 propagates into the y direction enteringand existing the cloud at the same x and z coordinates. The transmitted light Et will
be attenuated and phase shifted according to: Et = tE0 eiφ , where t is the transmission
coefficient and φ is the phase shift.
In most experiments we were interested in situations where the two components ofthe gas |↑i and |↓i had different density distributions n↑ (x, y, z) and n↓ (x, y, z) (caused
for example by a global population imbalance, or a radio-frequency pulse that selectively
removed atoms in one component). To obtain both n↑ or n↓ several problems needed to beaddressed.
R

Both t and φ do not directly depend on n but on the column density nc (x, z) =n(x, y, z)dy due to line-of-sight integration along the y direction. Therefore the 3D density
39

distributions need to be reconstructed form nc using the so-called inverse Abel transformation as detailed below.
In order to obtain information about both n↑ and n↓ , two images of the cloud need tobe taken separately. In principle this could be done by using light resonant with atoms instate |↑i and |↓i for the first and second image, respectively. Resonant light, however, will
be absorbed which leads to heating of the cloud and therefore a potentially blurred second
image. Signal to noise considerations make absorption imaging even more undesirable. The
imaging signal has to be particularly sensitive to the difference n↑ − n↓ since abrupt featuresin this quantity clearly indicate changes in the behavior of the system and possibly phasetransitions. As absorption images only contain information about n↑ or n↓ separately, they
do not provide this sensitivity. Especially if n↑ and n↓ are both large and fluctuating thiswill lead to large fluctuations in n↑ − n↓ . The solution was to use phase contrast imaging,a dispersive imaging method sensitive to the phase shift φ of the transmitted light.

3.5.1

Phase contrast imaging

For phase contrast imaging off-resonant light at large detunings δ À 1 is used, with δ
measured in units of the natural linewidth of 6 Li (ΓLi = 6 MHz). In this limit absorption
3 2
λ is the resonant cross section2 .
can be neglected, t ≈ 1 and φ = −nc σ0 /δ where σ0 = 2π

Since φ is linear in nc we would like the recorded intensity on the CCD camera to be linear
in φ as well.
Any dispersive imaging method requires that the scattered and phase shifted light isseparated from the transmitted light. The simplest solution is to block the unscattered lightin the Fourier plane of the imaging system where it comes to a focus. The recorded intensity
is then proportional to |E0 eiφ − E0 |2 ∝ 1 + φ2 /2 for φ ¿ 1, which depends quadraticallyon φ. The situation can be modified by not blocking, but instead just phase shifting theunscattered light. For a phase shift of ±π/2, one finds for the intensity in the limit φ ¿ 1,
|E0 eiφ + E0 (e±iπ/2 − 1)|2 ∝ 1 + 2φ, which shows the desired linear dependency on φ andtherefore nc .
Let ν be the probe frequency of the imaging beam and ν↑0 and ν↓0 the optical resonancefrequency for the atoms in state |↑i and state |↓i respectively. The optical signal is therefore
proportional to:
nc,↓
nc,↑
+
0
ν − ν↑
ν − ν↓0
Choosing ν = ν1/2 = (ν↑0 + ν↓0 )/2 as shown in figure 3-6, the optical signal directlyreflects the density difference nc1 = nc,↑ − nc,↓ . With this choice for the imaging frequency
atoms in state |↑i cause exactly the opposite phase shift from atoms in state |↓i and any
common mode signal cancels out. In principle this can be verified in an equal mixture,2

Valid for a two level atom in the rotating wave approximation.

40

Figure 3-6: Phase contrast imaging of the density difference of two spin states. a) Schematicsof the imaging setup. b) Phase contrast images of a spin polarized sample in state |1i (left),|2i (right), or an equal mixture (center). The phase shift of the light for the two spin states
has the opposite sign and cancels for an equal mixture.

where no phase contrast signal should be observed. In practice, the imaging frequency ν0 isdetermined experimentally by canceling the phase contrast signal for an equal mixture. We
find that ν0 usually deviates from v1/2 by about 1 to 2 MHz probably due to imperfectionsin the imaging setup.
For the second phase contrast image the imaging frequency is detuned from ν0 by one
or two natural linewidths. The image then reflects the weighted density difference nc2 =α↑ nc,↑ − α↓ nc,↓ . To determine the coefficients α↑ and α↓ we use the signal ratio between thefirst and second image of highly imbalanced clouds. An example of the two phase contrast
images, together with cuts through the column densities nc1 (0, z) and nc2 (0, z) are shownin figure 3-7 (a-c). The column densities nc,↑ and nc,↓ can now be determined from nc1 andnc2 as nc,↑ = (α↓ nc1 − nc2 )/(α↓ − α↑ ) and nc,↓ = (α↑ nc1 − nc2 )/(α↓ − α↑ ). In the next stepthe 3D densities n↑ and n↓ are reconstructed from nc,↑ and nc,↓ .

3.5.2

Inverse Abel transformation

The inverse Abel transform can be applied to obtain a 3D density distribution from line-ofsight integrated images for systems with cylindrical symmetry3 . The details of the trapping
potential used in our experiments are given below. In short, the trap is cigar shaped and thetwo transverse trapping frequencies are equal to better than 2%. The criterium of cylindricalsymmetry is therefore met to a sufficiently high degree. The inverse Abel transformation is
given by:
3
The inverse Abel transformation is a special case of the so-called inverse Radon transformation, which
does not require cylindrical symmetry, but the knowledge of line-of-sight integrated distributions for an
infinite number of angles. The cylindrical symmetry reduces this requirement to one distribution obtained
from line-of-sight integration along an axis perpendicular to the symmetry axis.

41Column density (a.u.)

a

b

c

1

0

-1

Linear density (a.u.)

1

d

0
-400

-200

0

200

400

Axial (z) position (µm)
6

e

3

z
0

x

-60

-30

0

30

60

Radial (x) position (µm)

Figure 3-7: Double phase contrast imaging. The images show a) the density difference andb) the weighted density difference of an imbalanced cloud; c) the column density difference(black) and the weighted column density difference (red) along the dashed lines of the phase
contrast images; d) the radially and e) the axially intergrated column density difference.
The profiles in d) and e) show a flat top distribution indicating that the local density
approximation is valid.

n(r, z) = −

1
π

Z ∞
r

dnc (x, z)
1
dx √
2
2
dx
x −r

As one would expect, the transformation depends on the derivative of column density
nc , which causes the inverse Abel transformation to be very sensitive to noise. An excellent
signal to noise ratio for nc is therefore required. Any “kinks” in the density profiles willcause jumps in the density distribution after the transformation. Averaging the column
density profiles along elliptical equipotential lines λ2 x2 + z 2 = r2 , where λ is the aspectratio of the trapping potential, helps to reduce noise. The averaging has to be performed
with carefully chosen restrictions to avoid washing out sharp features caused, for example,
by phase boundaries [24].

3.5.3Trapping potential

Here we describe the trapping configuration for typical experimental conditions after preparation of the strongly interacting sample, i.e. at magnetically applied bias fields in the rangeof 700 to 900 G and after evaporation in the optical trap. We specifically state the parameters relevant for the experiments based on in situ imaging. The trapping potential iscigar shaped, with an aspect ratio λ ∼ 6. The confinement of the atoms is only to firstorder harmonic and due to a combined optical and magnetic potential. The optical trap
is generated by a 1064 nm gaussian laser beam with a waist w of about 125 µm, creating
42

2

2

a trapping potential Vo (ρ, z) = U0 e−(2ρ /w ) , with ρ =px2 + y 2 . The magnetic potential

provides essentially all of the axial confinement along with some weak radial anti-trapping:
2

2

ρ
2
z
Vm (ρ, z) = mω2 (− 2 + z ). The combined potential is then given by

¶
µ 2
¶
µ
mωz2
ρ
2ρ2
2
− +z
V (ρ, z) = U0 exp − 2 +
w
2
2

(3.2)

yielding a trap depth of
·
µ
¶¸
ω2
1
2ω 2
U = ωr2 w2 1 − z2 ln 1 + 2r
4
2ωr
ωz

(3.3)

Here ωr is the radial trapping frequency in the central harmonic region of the potential.
Typical values of ωr , ωz and U are: ωr ∼ 2π×130 Hz, ωz ∼ 2π×23 Hz and 2 µK respectively.
Typical Fermi energies are on the order of 1 µK. Therefore U ∼ EF and the anharmonicitiesof the trapping potential in the radial direction (for ρ ∼ w) affect the spatial wings of thetrapped cloud. Although the overall confinement is predominantly harmonic, the effects ofanharmonicities must therefore still be considered when information is obtained from thespatial wings. An important example concerns the temperature determination from thenon-interacting wings of the majority cloud in an imbalanced gas.

3.5.4

LDA and doubly integrated density difference profiles

As we have pointed out above, spatially resolved imaging techniques turn the harmonic
confinement of the atoms into a feature since the images provide a “cut” through thephysics in the cloud at various densities. An important requirement, however, is that the
local density approximation (LDA) is valid, i.e. that all relevant quantities at a given point(ρ, z) in the trapping potential (here assumed to be axially symmetric as in the experiment)
depend only on the local value of V (ρ, z). It turns out that imbalanced clouds actuallyprovide a way to check the validity of the LDA for their description. As we will discusslater in this thesis the superfluid core in an imbalanced mixture has equal densities inthe zero temperature limit (at the expense of a larger density imbalance in the normal
outer regions of the cloud). Here we will show that this leads to a characteristic “flat top”
distribution of doubly integrated density difference profiles in an imbalanced mixture if theconfinement is harmonic and the LDA applies.
In an axial symmetric harmonic trap, equipotential shells are given by ellipses λ2 ρ2 +z 2 =c; here λ = ωr /ωz is the aspect ratio, ρ2 = x2 + y 2 as above, and c is a constant. If the local
density approximation is valid, the densities on the elliptical shells are constant and depend
only on the value of c. The doubly intergrated density difference profile with ∆n = n↑ − n↓is then given by:Z ∞∆ndbl (z) =0πρ∆n(λ ρ + z )dφdρ = 2
λ
2 2

2

43

Z ∞
∆n(a)daz2(3.4)

Fitted temperature parameter (T/TF0)

0.08

2θ
0.04

z
x

30

40

50

60

70

80

Averaging angle (θ)

Figure 3-8: Effect of anharmonicities on the in situ temperature determination. The temperature of the cloud was determined for various angles θ of the averaging sector. For larger
values of θ, spatial wings at larger values of ρ contribute and the radial anharmonicities
become increasingly important in the averaged profile. This results in a broadening of thespatial wings and consequently higher value of the fitted temperature. The red line shows
the results of a simulation using the same parameters as the experiment (λ = 6.15, TF 0 = 1µK and the trap depth U/kB = 2 µK).

The density difference ∆n is zero in the balanced superfluid core up to a certain criticalλ2 ρ2 + z 2 ≡ a2c > z 2 . So for z < ac we have:π∆ndbl (z < ac ) = 2λZ ∞∆n(a)da = const

(3.5)

a2c

Figure 3-7 d) and e) indeed show the flat top distribution of the doubly integrated density difference profiles, demonstrating that the LDA and harmonic confinement are good
approximations for our system.

3.5.5

Temperature determination

The absolute temperature of strongly interacting Fermi gases is difficult to determine since
the exact relation between the density distribution of the atoms and the temperature of thesystem is not known. A comparison with theory is limited to the assumptions underlying
the theoretical description and their accuracy.
For a balanced mixture in the strongly interacting regime the temperature has been
determined as the derivative of entropy with energy [55] which resulted in temperatures
averaged over a certain range. Furthermore some assumptions and approximations had to
be made with regard to the energy and entropy measurements [55].
To obtain the local temperatures in the trapped, imbalanced cloud we therefore have torely on a different temperature calibration. Usually, when there is the need to characterize
44Averaged column density
difference (a.u.)

a

b

1

1

0

0
0

200

R

400

0

200

R

400

Radial position (µm)

Figure 3-9: In situ temperature determination. The relative temperature T 0 = T /TF 0 is
determined from the outer region (r > R↓ ) of the averaged column density difference profile(black line) fitted to a finite temperature Fermi-Dirac distribution in a harmonic trap (redline). The radius of the minority cloud R↓ was determined from a fit of the wing profile of theminority component (black dashed line) to a zero temperature Thomas-Fermi distribution(red dashed line). a, T’ = 0.03(1) and δ = 44(4)% b, T’ = 0.08(1) and δ = 46(4)%.

a system with unknown properties, it is helpful to bring it in contact with a simpler, wellunderstand system for calibration. In imbalanced mixtures such a situation arises naturally
since the spatial wings of the majority cloud (where the minority density has dropped tozero) constitute a non-interacting ideal Fermi gas in equilibrium with the interacting cloud.
In other words the majority wings can serve as an ideal gas thermometer, an idea that was
first applied in ref. [22]. In this experiment, however, the temperature was determined bya fit to the wings of the expanded majority cloud after release from the trap. For small
imbalances this lead to an overestimate of the temperature (of not more than 20%) due tothe collisional dynamics during expansion.
For an in situ temperature calibration expansion dynamics is of course not a problem,
but instead the trap anharmonicities have to be taken into account. To obtain the low noise
profiles required for an accurate temperature calibration, the column density profiles areaveraged along lines of constant (λ2 ρ2 + z 2 ). For a full average, the trap anharmonicities
which affect the wings of the majority cloud, lead to a systematic overestimate of thetemperature. This systematic shift can be controlled by restricting the averaging range tosmall angles where only small values of ρ contribute and the trap asymmetry only has anegligible influence (see figure 3-8).
As shown in figure 3-9 the temperature of the system can then be obtained from a fit
to the outer wing of the majority cloud, i.e. to the region r > R↓ where the density ofthe minority has dropped to zero. Here R↓ , the radius of the minority cloud, is determinedfrom a zero-temperature Thomas-Fermi fit to the wing profile of the minority component4 .
4See ref. [7] for an extensive discussion on the quantitative analysis of density distributions

45

The fit to the majority cloud yields the relative temperature
T 0 ≡ T /TF 0 = ~2

kB T

(3.6)

2/3
2
2m (6π n0 )where kB TF 0 the Fermi energy of a non-interacting, zero temperature, harmonically trapped
Fermi gas with central density n0 and the same density distribution in the outer wings as0
the majority cloud. The local temperature Tlocal(r) can then be defined with the density ofthe majority cloud n↑ (r):0
Tlocal
(r) =

T
kB T
T
= 2
2 =
~
2
TF ↑ (r)TF0(6π n↑ (r)) 3
2m

46

µ

n0
n↑ (r)

¶2
3

(3.7)

Chapter 4

Creating and exploring a hightemperature superfluid of ultracold
6Li atoms
This chapter gives a brief summary of the experiments that lead to the observation of hightemperature superfluidity. See ref. [7] for an extensive review.

4.1

Fermion pair condensation

An important step on the way to superfluid Fermi gases was the creation of the conceptuallysimplest fermion pairs: the weakly bound “molecules” on the BEC side of the Feshbach resonance [56, 11, 12, 13]. The stability of these molecules came as a surprise since Feshbach
associated molecules from bosonic atoms were too short lived to reach thermal equilibrium in the quantum degenerate regime [57, 58, 59]. The molecules created from fermionic
atoms, however, were protected from decay into lower lying states by Fermi statistics [30]and showed lifetimes which exceeded 10 s in the (1,2) mixture of 6 Li. With such favorable lifetimes direct evaporation from a two component (1,2) mixture into a Bose-Einsteincondensate of weakly bound fermion pairs proved possible (figure 4-1) [16, 17]. In 40 K
molecular lifetimes were considerably shorter (100 ms) and the observation of BEC relied
on a sweep across the Feshbach resonance [15].
The observation of fermion pair condensation in the entire BEC-BCS crossover wasachieved a few month later (see figure 4-2) [18, 19]. Close to resonance and on the BCS
side bimodal density profiles could not be observed in situ. Here the chemical potentialµ becomes comparable to kB Tc and the condensate does not spatially separate from thethermal cloud. The solution was a rapid non-adiabatic magnetic field sweep across the
Feshbach resonance towards lower magnetic fields shortly after the optical trapping potential
is turned off. This sweep transfers the many-body bound fermion pairs into stable molecules47

Figure 4-1: Phase transition: Fermion pair condensation on the BEC side at 770 G. As thetemperature is lowered a bimodal density distribution emerges in a gas of weakly bound
6 Li molecules, the hallmark of Bose-Einstein condensation.
2

Figure 4-2: Phase transition: Fermion pair condensation at unitarity. The observation ofbimodal density distribution in the strongly interacting regime required a rapid magnetic
field ramp to the BEC side of the Feshbach resonance. Fermion pair condensates could
then be observed as a peak of condensed, zero momentum pairs surrounded by a cloud ofthermal molecules.

while preserving their center of mass momentum. The interaction energy of the pairs islowered by up to two orders of magnitude as the molecular scattering length between thepairs is drastically reduced. After further expansion the momentum distribution of the pairs
before the sweep can be observed: the condensed, zero momentum pairs are now clearly
separated from the thermal finite momentum pairs.
With the “rapid ramp” technique the “phase diagram” of fermion pair condensationin the BEC-BCS crossover shown in figure 4-3 was obtained. Note that the rapid ramp
method is an excellent qualitative indicator for the onset of fermion pair condensation
but very difficult to interpret quantitatively due to the complicated dynamics during themagnetic field sweep. For example the relation between the observed condensate fraction
and the original one is not well understood [60, 61, 62, 63].
The rapid ramp methods gives rise to an important concern: could the observed paircondensates be created during the rapid ramp and not result from a pair condensate in the48

Condensate Fraction at 820 G

0

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

Figure 4-3: “Phase diagram” of fermion pair condensation. Condensates are observed in theentire crossover regime. The highest condensate fractions are obtained close to resonance.
The decreasing condensate fraction on the BCS side is due to the decreasing interaction
strength in this regime. The sharp decay of the condensate on the BEC side is caused byheating owing to vibrational relaxation of the molecules.

initial state? While the experiments provided some evidence that such concerns were notjustified we decided to explicitly study the relevant timescales [29].

4.1.1

Timescales

The condensates detected after the rapid ramp can be expected to reflect condensation of
the original fermion pairs if the ramp is fast compared to the many-body physics leading
to the formation of a fermion pair condensate. The formation dynamics of the condensate
can be studied by observing the relaxation of the system in response to a fast change in
the interaction strength. By periodically changing the interactions via a magnetic field
modulation on the BCS side of the Feshbach resonance and monitoring how the measured
condensate fraction adjusted to this perturbation the relaxation time of the fermion paircondensate could be determined [29]. Figure 4-4 shows the delayed response of the fermion
pair condensate to the magnetic field modulation. The relaxation time is more than anorder of magnitude larger than the time required to cross the Feshbach resonance with therapid ramp.
In later experiments we have actually seen direct signs of the phase transition at unitarity
in high resolution images of expanding clouds of an equal mixture [7]. The column density
profiles showed small deviations from an ideal Thomas-Fermi profile close to the criticaltemperature. However, these very faint features could have hardly been interpreted as
signatures of pair condensation without the dramatic changes revealed by the rapid ramp
method or the observation of superfluidity via vortex lattices which is the subject of the
49

0.5
0.4
0.3
840

0.2

880
920
960
0

2

4

6

8

10

12

14

Magnetic Field [G]Molecular Condensate Fraction

0.6

16

Time [ms]Figure 4-4: Relaxation of a fermion pair condensate. Shown is the response of the condensate fraction to a modulation of the external magnetic field on the BCS side of the Feshbachresonance. The higher the magnetic field the smaller is the interaction strength and thesmaller is the expected steady state condensate fraction (see figure 4-3). The response of thefermion pair condensate to the magnetic field modulation (and therefore of the interaction
strength) is delayed by about 500 µs. This is about 130 times the unitarity limited collisiontime ~EF−1 and large compared to the 10 µs long magnetic field ramp through the Feshbachresonance.

following sections.

4.2

High temperature superfluidity

The observation of fermion pair condensation did not demonstrate superfluidity in ultracold
Fermi gases. Although in three dimensions superfluidity and condensation usually occur
together, the observation of long range phase coherence and superfluid flow are necessary
to establish superfluidity. For example the formation of a quasi condensates can occur insystems that have not yet reached full three dimensional equilibrium and therefore do notexhibit superfluidity.
A number experiments showed features that were consistent with superfluidity but did
not provide sufficient prove as the strongly interacting normal phase often showed at least
a qualitatively similar behavior. Examples are superfluid hydrodynamics (probed in expansion [64] and via collective oscillations [65, 66]), pair formation [25], or a measurement of
the heat capacity of the gas [67]. An experiment that is directly sensitive to the emergenceof a macroscopic wavefunction (and therefore long range phase coherence and superfluidflow) is to rotate the gas. Since the flow field of a superfluid is rotation free the superfluid
can contain angular momentum only in the form of quantized vortices. The observation ofordered and stable vortex lattices in a rotating gas is therefore a “smoking gun” for superfluidity. Note that classical vortices can spontaneously form in the normal phase. However,in this case the vortices are not an equilibrium property of the system, they are not quan50

Figure 4-5: Schematics: process of vortex creation. Angular momentum is imparted onto
the gas with the help of two rotating stirring beams (in green), that create an repulsive
potential for the atoms. The atoms are confined in an optical dipole trap (in pink) withadditional axial confinement provided by the magnetic field generated by the circular coils
shown in blue. The stirring beams excite a surface mode of the superfluid that decays via theformation of vortices. The vortices penetrate into the superfluid and arrange themselvesin a triangular Abrikosov lattice, a process that takes several 100 ms after the stirrer isturned off. The vortex cores are too small to be imaged in situ. Therefore the optical trap
is switched off and the gas is allowed to expand before imaging. The expansion serves asa (non-linear) magnifying glass for the vortices. To enhance the vortex contrast and tostabilize the fermion pairs during expansion the magnetic field is usually switched to lower
fields after the optical trap is turned off.

51

Magnetic field [G]
833

730

1.6
BEC

0
Interaction parameter 1/k F a

935

-0.7
BCS

Figure 4-6: Observation of vortex lattices in a rotating superfluid in the BEC-BCS crossover.

tized and they decay due to finite viscosity. In rotating atomic Bose-Einstein condensate
as well as rotating superfluid 4 He and 3 He the observation of vortex lattices was regarded
as unambiguous proof of superfluidity.
Rotating the ultracold fermion gas in the optical trap and observing the vortices required a considerable effort the details of which are documented in Martin Zwierlein’s PhD
thesis [39] and in [7]. The experimental procedure is illustrated and summarized in figure 4-5. Figure 4-6 shows the vortex lattices observed in the BEC-BCS crossover. Thisestablished phase coherence and superfluidity for strongly interacting gases of fermions andfor molecular gases of weakly bound molecules.
The observation of vortices at resonance and on the BCS side was not possible in situ. Ina narrow window on the BEC side stirring, equilibration and expansion could be performedat the same magnetic field. In general, however, the final step of the process - the expansionof the vortex lattice - had at least in part to take place on the BEC side: On resonanceand further in the strongly interacting regime on BCS side (where kF a < −1) the size ofthe vortices is essentially given by 1/kF which is on the order of 0.2 µm for typical valuesof kF . This size is smaller than the resolution one could reasonably hope to achieve withimaging light resonant with the 6 Li atoms at a wavelength of 0.67 µm. During expansion at
a stationary magnetic field the vortex size will at best increase as the condensate radius [7].
In addition the vortex contrast on resonance and on the BCS side is reduced as EF /∆ due
to quantum depletion which leads to a finite density of non-coherent atoms inside the vortex
cores. During expansion on the BCS side the fraction of atoms inside the core will evenincrease further. Finally superfluidity will cease if the density of the system drops below a
critical value during expansion (for finite temperature and interaction strength).
These problems can all be solved with a sweep to the BEC side where the superfluid is
stabilized, the size of the vortices relative to the condensate radius increases and quantum
depletion is drastically reduced. The only requirement is that the condensate density at thetime of the magnetic field sweep is still sufficiently high for the condensate to adjust to the52change of interaction strength.
Since the observation of vortex lattices in the crossover relies in part on a magnetic
field sweep the concern that the system became superfluid during the sweep and the vortex
lattices formed afterwards has to be considered. However, the time required for vortices tocrystalize in a regular vortex lattice is several 100 ms 1 and more than an order of magnitudelarger than the duration of the sweep and the remaining expansion on the BEC side of about
10 ms. The equilibrated, triangular vortex lattices observed in the experiments cannot haveformed in this short time frame and in particular not in a cloud expanding at the speed ofsound.
As mentioned above, superfluidity can break down during expansion on the BCS sideof the resonance: as the density decreases in expansion T /Tc increases and superfluidity
will eventually be lost. We have studied the stability of the expanding fermionic superfluidon resonance and the BCS side as a function of density and interaction strength. This
experiment is the subject of the next section.

4.3

Superfluid expansion of a rotating Fermi gas

From preceding discussion the ultimate fate of an expanding superfluid on the BCS sideat finite temperature is known: the gas will turn normal. But how long can superfluidity
survive the expansion and at what densities or interaction strengths will superfluidity break
down?
These questions can be addressed quite elegantly by releasing the gas from the trap
and delaying the sweep to the BEC side for a variable BCS expansion time tBCS . If thesuperfluid turns normal during the BCS expansion the vortex core will fill in quickly anddisappear. Therefore the vortices can be used as “markers” for the superfluid regions in the
expanding cloud [20].
Figure 4-7 shows the expansion of the superfluid gas for a fixed total, but variable BCS
expansion time at 910 and 960 G. The presence of vortices indicates that the superfluid
initially survives the expansion. Vortices are first lost in the lower density wings of thecloud and finally in its central region. The total BCS expansion time until superfluidity is
quenched depends on the initial interaction strength. The “healthier” the system starts out,
(i.e. the larger the initial kF |a|) the longer the superfluid survives the expansion. Figure 4-8
demonstrates that superfluidity is lost at a critical kF a ∼ −0.8 independently of the initialinteractions strength or the magnetic field.
The most plausible explanation for the breakdown of superfluidity is that the cloud turns
normal during expansion. As the cloud expands T /TF stays³constant´ ³ ´while T /T³c increases.
´³ ´
We can estimate the critical kF a by equating 1 ≡ TTc =with ∆ = 2e
1

7/3

T
TF

TF
Tc

= 1.76T
TF

EF∆

EF e−π/2kF |a| which is valid in the BCS limit for kF |a| ≤ 1. For our lowest

This is independent of temperature and interaction strength and holds for atomic BECs as well.

53

0 ms

4.5ms

0 ms

3.0 ms

910 G

960 G

Figure 4-7: Superfluid expansion of the rotating gas. Shown are absorption images fordifferent 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 themagnetic field was ramped to the BEC-side for further expansion. The total expansion timeremained constant. The vortices served as markers for the superfluid parts of the cloud.
Superfluidity survived the expansion for several milliseconds and was gradually lost fromthe low density edges of the cloud towards its center. The field of view of each image is 1.2
mm × 1.2 mm.

Peak k Fa

-1.0

-2.0
Number of Vortices
40

30

20

10

0

-3.0
880

900

920

940

960

Magnetic field [G]Figure 4-8: Central interaction strength kF a during superfluid expansion. The triangles
indicate the initial kF a of the trapped superfluids which decreases as |a| decreases for higher
magnetic fields. During expansion vortices survived up to a critical interaction strength of
kF a = −0.8 + / − 0.1 (squares) almost independent of of the magnetic field (or scattering
length a). Filled circles correspond to partially superfluid, open circles to normal clouds.
The observed number of vortices is color coded.

54

Figure 4-9: Expansion at unitarity. Shown are absorption images after a fixed total time-offlight, but for different 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 for further expansion. While the gas remains
superfluid the vortex contrast is lost gradually across the whole cloud from about 15% to
3% after 2 and 5 ms of expansion on resonance respectively. Vortices could still be detected
at densities as low as 1.2 × 1011 cm−3 in the wings of the expanded cloud. At this point thecloud size had increased by more than a factor of four and the peak density had dropped
by a factor of 17 compared to in traps values. The field of view of each image is 1.2 mm
×1.2 mm.

estimated temperatures of 0.05TF this gives kF a = −0.9 close to the observed value.
Also the increasing size of the fermion pairs for decreasing density could in principle
lead to the observed loss of superfluidity. If the size of the pairs exceeds the interparticle
spacing the sweep methods which relies on the efficient transfer of correlated fermion pairs
into molecules might fail. We will consider these issues when we discuss our determinationof the fermion pair size in the strongly interacting regime. Note, however, that the loss ofvortices in the cloud due to an increasing pair size would likely be more gradual and notlead to the sharp boundaries between superfluid and normal regions seen in the experiment(see figure 4-7).
Could superfluidity be quenched due to rapid rotation? When the size of the vortex cores
becomes comparable to the separation between the vortices, superfluid flow is expected tobreak down as superconductivity in type II superconductors at the critical field Hc2 . Since
the size of the vortex cores depends exponentially on kF a the effects of rotation might not
be negligible. However, in ref. [68] it is found that the superfluid should be stable in thestrongly interacting regime at all rotation frequencies.
So far we have focused on the breakdown of superfluidity on the BCS side. At resonance
Tc /TF = const. and therefore the gas should remain superfluid during expansion. Indeed
figure 4-9 shows that vortices are not lost after the gas is released from the trap. However,the vortex contrast decreases uniformly across the cloud with increasing resonant expansion.
The low densities the gas reaches during expansion could prevent the vortex cores to adjust
quickly enough to the high contrast and larger size they would have in equilibrium on the
BEC side after the sweep.
In this experiment we have observed vortices at densities as low as 1.2 × 1011 cm−3 ,demonstrating fermionic superfluidity in a system 100 million times more dilute than air
and at a temperature of less than 20 nK. At the other extreme fermionic superfluidity inneutron stars is expected to occur at temperatures of 10 billion K and densities nearly 20
55

orders of magnitude higher than in air. Even higher temperatures and densities exist in thequark gluon plasma of the early universe. Although this state of matter is not a superfluid,recent experiments demonstrated that it shows very small viscosity similar to normal but
strongly interacting ultracold Fermi gases [64, 69]. It is amazing to see that systems undersuch extremely conditions share universal properties.

56

Chapter 5

Superfluidity and phase separationin strongly interacting Fermi gaseswith density imbalance
5.1

Introduction

The creation of high-temperature superfluid in fermionic gases allows the study fermionicsuperfluidity in an entirely new regime. An interesting way to get insight into the new
system is to study its stability against a change of parameters that can ultimately quench
superfluidity. In the previous chapter we have looked at the stability of the superfluid
many-body state as a function of the interaction strength. An even more powerful approach is to break a symmetry of the superfluid state and to observe the response of thesystem. In ultracold Fermi gas this can be easily achieved by creating an imbalance in thenumber of atoms in the two hyperfine states that form the superfluid. The first experiments with strongly interacting imbalanced Fermi gases established superfluidity in thesesystem [21] and lead to the direct observation of the superfluid phase transition at unitarity [22]. Furthermore the so-called Chandrasekhar-Clogston limit of superfluidity which
gives the critical population imbalance for a first order quantum phase transition from thesuperfluid to the normal phase was established [21]. These experiments were based on dataobtained from the gas after expansion. A more detailed characterization of the superfluidand normal phases was carried out by observing the trapped gas in situ with the methods
described in chapter 3 [23]. This lead to the determination of the homogenous phase diagram of a strongly interacting Fermi gas at unitarity as a function of temperature and spinpolarization [24]. The phase diagram proved to be very rich with first and second orderphase transitions that merge at a tricritical point. In the last part of the chapter we will
discuss imbalanced Fermi gases in the BEC-BCS crossover.

57

BEC side|1〉

B = 812 G
1 / kF a = 0.19
|2〉

BCS side|1〉

B = 853 G
1 / kF a = -0.15|2〉

Figure 5-1: Superfluidity in imbalanced Fermi gases. The presence of vortex lattices in
the rotating gas demonstrates the stability of the superfluid against substantial imbalancesin the strongly interacting regime. Superfluidity breaks down at a population imbalance
N −N
δN,c = N↑↑ +N↓↓ of about 75%. This is known as the Chandrasekhar-Clogston (CC) limit ofsuperfluidity. For the 812 G (853) data δN was from left to right 100%, 90%, 80%, 62%,28%, 18%, 10% and 0% (100%, 74%, 58%, 48%, 32%, 16%, 7% and 0%).

5.2

Superfluidity and direct observation of the superfluid
phase transition in imbalanced Fermi gases

5.2.1

Breakdown of superfluidity: The Clogston-Chandrasekhar (CC)limit

The standard BCS theory describes fermionic superfluidity in a two component fermion
mixture with the same mass, chemical potentials and particle numbers and the Fermi energies of the two components are equal. Although pairing costs kinetic energy there is a gain√
)3/2 EF ) which makesin potential energy of (1/2)ρF ∆2 (where ρF ≡ ρ(EF )/Ω = 4π1 2 ( 2m~2
the formation of superfluid pairs favorable even for arbitrarily small attractive interactions.
The creation of an imbalance in the spin populations (N↑ > N↓ ) leads to different Fermi
energies of the two components. Since in the imbalanced system the Fermi energies are thennot at “eye level” there is in an additional energy cost for the formation of superfluid pairs.
In a simple T = 0 model one can assume that superfluidity requires equal densities. The
imbalanced system may then form an equal density superfluid in one part of the volume
and a normal mixture at a local imbalance that is larger than the global one in the otherpart of the volume. This requires the superfluid to expel majority atoms which costs kinetic energy. If this energy cost exceeds the superfluid stabilization energy superfluidity is
quenched. This is known as the Chandrasekhar-Clogston (CC) limit of superfluidity [70, 71].
58

N −N

Assuming that δN = N↑↑ +N↓↓ ¿ 1 one can show [7] that the CC limit is reached at a criticaldensity difference ∆n =N↑ −N↓of2ΩρF ∆∆nc = √
2

(5.1)

It is convenient to define the parameter h proportional to the chemical potential difµ↑ −µ↓
2 .
µ↑ +µ↓
EF = µ = 2 and h ¿ µ. With that we have1 :ference between the two components h =

∆n =

1
12π 2

µ

2mµ
~

For definiteness we assume ∆ ¿ EF ,¶ 3 "µ
¶3 µ
¶3 #
2
h 2
h 2
1+
− 1−
µ
µ

≈ hρF

(5.2)
(5.3)

In terms of h the CC limit is therefore reached when1hc ≈ √ ∆2(5.4)

Similarly one finds for the critical population imbalanceδN,c ≈

3∆
2µ

(5.5)

The above discussion assumed a homogenous system. However, the gases we are studying are trapped in a harmonic potential and therefore the densities n↑ (r), n↓ (r), the chemical
potentials µ↑ (r), µ↓ (r) and the gap ∆(r) vary spatially. Note that in chapter 3 we havedemonstrated that the local density approximation is valid for our system.
In the following we will use global as well as local quantities. Most importantly the
population imbalance δN is a global quantity since it is based on the total number ofminority and majority atoms in the trap. We will refer to the local population imbalanceas the “spin polarization”:
σ(r) =

5.2.2

n↑ (r) − n↓ (r)n↑ (r) + n↓ (r)(5.6)

The experiment: Superfluidity and the CC limit at unitarity

Superfluidity in population imbalanced Fermi gases was established with the same methods
as in equal mixtures by rotating the gas and observing vortex lattices [21] (see figure 51). Vortices are always found within the parts of the cloud that also show the presence ofa fermion pair condensate. In very small fermion pair condensates no vortices have been
observed as it is difficult to nucleate them in small condensates surrounded by large thermal
clouds. We therefore conclude that the presence of a fermion pair condensate shows that thesystem is superfluid and assume that fermion pair condensation is a more sensitive indicator
1

With n = 6π1 2` 2mµ ´ 23
~2and µ↑ = h + µ and µ↓ = h − µ59for the onset of superfluidity than vortices. Careful measurements of the emergence offermion pair condensation as a function of imbalance and temperature [22] demonstrated a
critical population imbalance at unitarity of δN,c = 74(5)% for an harmonically trapped gas.
This value agrees with results from more recent quantum Monte-Carlo calculations [72].

5.2.3

Bimodal density profiles: Direct observation of the superfluid phasetransition at unitarity

As we have mentioned in the previous chapter the density profiles of a balanced mixture at
unitarity show only very subtle signs of the superfluid phase transition (see figure 5-2 a-c).
Bimodal density distributions, the hallmark of BEC for weakly interacting Bose gases, were
only revealed after a rapid ramp to the BEC side of the Feshbach resonance that suddenly
reduced the chemical potential of the condensate relative to the thermal cloud [18, 19].
In imbalanced mixtures this situation changes dramatically: At low temperatures theminority component develops a dense central core inside a thermal cloud which is directly
visible at unitarity and the BCS side (figure 5-2 e-f). Rapid ramp experiments confirm thatthe emergence of bimodal density profiles on resonance coincides with the superfluid phasetransition. The density profiles of the minority component therefore reveal the superfluid
phase transition without the quantitatively poorly understood magnetic field sweeps [22].
What phenomenon leads to the emergence of a bimodal density distribution in the minority cloud? To gain insight into this question we studied the normal to superfluid phasetransition as a function of population imbalance at the lowest temperatures. Figure 5-3
correlates the emergence of bimodal density profiles, i.e. the onset of superfluidity, withan estimate of the central densities of minority and majority component2 . The data suggest that superfluidity indeed sets in when the densities of minority and majority become
equal. The densities profiles also show that the superfluid core is surrounded by a mixed,
imbalanced normal gas. Only in the wings of the cloud is a pure gas of majority atoms
observed.
One explanation for the bimodal density distribution of the minority component isphase separation: the superfluid of equal densities (T = 0) spatially separates from themixed normal component giving rise to a first order phase transition and a discontinuity
in the density profile of the minority cloud. We have actually used this phase separation
scenario, first suggested by Bedaque, Caldas and Rupak in 2003 [73], when motivating theCC limit.
The characterization of the superfluid and normal phases including the determinationof phase boundaries requires the knowledge of the in situ density profiles of both majority
and minority components. A detailed description of the experimental techniques required
to obtain these profiles including phase contrast imaging and the important aspects of2

Since the column density profiles were obtained after expansion a precise reconstruction of the central
densities in the trapped cloud was not possible.

60Balanced mixture:bcOptical 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]1.0Minority component of an imbalanced mixture:
d

e

f

>

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 5-2: Direct observation of the superfluid phase transition at unitarity in the minority
component of an imbalanced mixture. When a balanced gas is cooled through Tc thedensity profiles do not change significantly (a-c). In contrast, the minority component of
an imbalanced mixture clearly shows a bimodal density distribution below Tc (d-f). The
density profiles of the majority component (shown in blue) are considerably less affected by
the onset of superfluidity. Only a careful analysis of the density profiles for the balanced
gas and the majority component in the imbalanced case reveals small changes as a functionof temperature which can be attributed to the superfluid phase transition. See refs. [22]and [7] for details.
61

Population imbalanceaCentral density (1012 cm-3)

Fraction outside Thomas-Fermi profile

1.0

0.9

0.8

0.7

0.6

0.5

0.4

5
4
3
2
1

0.06
0.03
0.00

b

Figure 5-3: Emergence of bimodal density distributions: a consequence of phase separation?
As the population imbalance decreases the CC limit is approached and at δc the minority
cloud becomes bimodal (b). The onset of bimodality/superfluidity is measured by thefraction outside of a Thomas-Fermi profile fitted to the wings of the cloud. At the criticalimbalance δc this quantity suddenly increases (a). The inset in (a) gives an estimate of
the central densities of the minority and majority clouds. As the imbalances decreases theminority density increases until it reaches the almost constant majority density at δc . Thisobservation provides evidence for a phase separation scenario where a superfluid of equal
densities (at T = 0) is spatially separated from a normal mixed cloud of unequal densities.

62

temperature calibration has been given in chapter 3. In the following we will focus on thethe density profiles to answer the new questions raised by the experiments:• Does superfluidity at zero temperature require equal densities?
• Is there phase separation between a balanced superfluid and the mixed normal phase
at T=0?
• What is the phase diagram of the system at finite temperature?

5.3

Probing the nature of the superfluid: The homogeneous
phase diagram

5.3.1

Density profiles in a harmonic trap

Due to the confinement of the gas in an optical trap which provides an approximately
harmonic trapping potential V (r) the chemical potentials of minority (↓) and majority (↑)
component vary spatially asµ↑ (r) = µ↑0 − V (r)(5.7)µ↓ (r) = µ↓0 − V (r)(5.8)

Here µ↑0 and µ↓0 are the global chemical potentials. The total chemical potential µ(r) andthe chemical potential difference h are given byµ(r) =h =µ↑ (r) + µ↓ (r)2µ↑0 − µ↓0µ↑ (r) − µ↓ (r)== const2
2(5.9)(5.10)The ratios η(r) = µ↓ (r)/µ↑ (r) and h/µ(r) vary spatially over the trapped sample andprovide a cut through the homogenous phase diagram (since the local density approximation
is valid).
In a harmonic trapping potential V (r) ∝ r2 (figure 5-4 a) the chemical potentials ofmajority and minority components can be expressed as:Ãµ↑ (r) = µ↑0Ãµ↓ (r) = µ↑0

r2
1− 2
R↑

!

r2
η0 − 2R↑(5.11)!
(5.12)

Here R↓ and R↑ mark the radii where the densities of minority and majority component
become zero respectively and η0 = µ↓0 /µ↑0 = R↓2 /R↑2 .
63

V(r)
Potential
energy (a.u.)aµ0µ 0 +δµ0I

Density (a.u.)

b

II

III

1

c

Spin polarization

0
1
0.5

σC
σS

0
0

RC

R

R

r

Radial position

Figure 5-4: Schematics of density profiles in the harmonic trap assuming phase separation
between a balanced superfluid and a normal mixed region. As a function of radius fromthe trap center three distinct regions can be identified: I) balanced superfluid, II) partiallypolarized normal gas, III) fully polarized normal gas. See the text for a discussion. µ↑0 andµ↓0 are the global chemical potentials, δµ↓ measures the shift for the minority component
due to interactions.

64

Figure 5-4 b) schematically shows the density distributions of majority and minority
components in a harmonic trapping potential. We assume that the phase separation scenario between a balanced superfluid and a partially polarized normal gas yields a correct
description of the system at T=0. The experiments described in the following sections willshow that the observed density profiles indeed have a similar structure.
In the center of the cloud (region I) one expects the balanced (n↓ = n↑ ) superfluid (atT=0). At finite temperature this region can have a small polarization due to thermally
excited quasiparticles. According to equation 5.4 with a spatially varying ∆(r) the CC
limit is reached when hc ≈ ∆(Rc ). At r = Rc the minority density suddenly jumps to lower
densities, indicating a sharp boundary between the superfluid and the partially polarized
normal phase. This is a consequence of phase separation which is associated with a firstorder phase transition [73]. At radii R↓ < r < R↑ the gas is fully polarized: only majority
atoms are present. This part of the cloud serves as an ideal gas thermometer and also yields
an absolute density calibration.
The jump in the density of the minority cloud at r = Rc corresponds to a sudden change
in the local polarization σ. The maximum polarization of the stable superfluid is given byσs and the minimum polarization of the stable normal gas by σc . For σs < σ < σc the gas
is unstable. Of particular interest for the comparison between theory and experiments are
the value of the critical polarization σc0 at T=0 (the CC limit) and the polarization andtemperature where σs = σc , which corresponds to a tricritical point in the phase diagram.

5.3.2

Central density difference and superfluidity: finite vs zero temperature

In a first experiment we obtained phase contrast images that (after 3D reconstruction)
showed the density difference n↑ (r) − n↓ (r) [23]. The in situ phase contrast images directly
reveal how the superfluid core emerges inside the trapped Fermi gas as the temperature
decreases (figure 5-5). While the rapid ramp and vortex experiments described in theprevious chapter were required to independently establish that this indeed is a signature of
the superfluid phase transition it is very satisfying to finally observe the onset of superfluidity
so dramatically without further “ado”.
From images like the ones in figure 5-5 the three dimensional central density differencen↑ (0) − n↓ (0) was reconstructed and then correlated with the independently measured onsetof fermion pair condensation (see figure 5-6). This demonstrated that in the T = 0 limit thesuperfluid is balanced (see figure 5-6) while at finite temperature the superfluid is polarized(the yellow shaded region in figure 5-6 b)). If T > 0 the superfluid can tolerate a certain
access of majority atoms in the form of quasiparticle excitations.
Importantly, the central density difference shows a very different behavior at T ∼ Tc , δ ¿δc compared to T ¿ Tc , δ ∼ δc . In the first limit the onset of condensation occurs at a
finite central density difference which smoothly decreases to zero (figure 5-6 b)). This is a
65

3.7

3.3

2.7

2.1

1.7

1.3

1.0

0.8

Figure 5-5: In situ observation of the superfluid phase transition: Emergence of phaseseparation at unitarity. Shown are phase contrast images of the density difference n↑ − n↓(δN = 56%). As the temperature (final trap depths for evaporation) decreases a superfluid
core forms inside the cloud. The field of view for each image is 160 µm × 940 µm.

signature of a second order phase transition. The onset of superfluidity around the criticalimbalance, however, is associated with a rapid drop of the central density difference to zero,a discontinuous feature typical for first order phase transitions. A phase diagram of theimbalanced superfluid at unitarity is therefore expected to show both first and second orderphase transitions. Since the experiment described here was only sensitive to n↑ (r)−n↓ (r) we
could not yet extract quantities like µ(r) and σ(r) which are required to fully characterize
the phase diagram.

5.3.3

Analysis of in situ density profiles

To map out the phase diagram of an imbalanced superfluid at unitarity n↑ and n↓ haveto be known separately. According to the discussion in section 5.3.1 the first order phasetransition may then be characterized by determining σs (T /TF ↑ ) and σc (T /TF ↑ ) from thejump in n↓ at the phase boundary between the mixed normal and the superfluid phase (we0define T /TF ↑ ≡ Tlocal(Rc ), see section 3.5.5).

The phase diagram was determined in an improved experiment where a second phase
contrast image, yielding a weighted density difference αn↑ (r) − βn↓ (r), was taken shortly
after the first image with α = β = 1 (see chapter 3). From the two images the three
dimensional densities n↑ (r) and n↓ (r) as well as the spin polarization σ(r) were obtained.
Figure 5-7 shows n↑ (r) and n↓ (r) at T 0 = 0.03. In the density profiles the three regions fromthe schematic profile of figure 5-4 can be clearly identified: the balanced superfluid in thecenter (I) a mixed normal region (II) and the wings of the cloud where only majority atoms
are present (III). At the interface of region I and II the minority density changes abruptly,
the signature of the first order phase transition. The corresponding spin polarization σ(r)

66

1

0.5
B = 834 G(1/kFa = 0)

0.25

0.5

0.0

0
0.0

0.2

0.6

0.8

1.0

Population Imbalance

b
2

S

PS

N

6
4
2

0.1
6
4

1

2

0.01
6
4

0

Condensate Fraction

Central density differnce(1012/cm3)

0.4

Normalized central
density difference

Condensate Fraction

a

2

1

2

3

Figure 5-6: First order vs second order phase transition: Correlation of the central density
difference with the formation of a fermion pair condensate. (a) At low temperatures theonset of fermion pair condensation (red triangles) as a function of the population imbalanceis associated with a sharp drop in the central density difference (filled circles), indicating afirst order phase transition. (b) As a function of temperature three regimes can be identified:i) An imbalanced normal (N) state at high temperatures. ii) For intermediate temperatures
fermion pair condensation emerges at a finite central density difference (n↑ (0) − n↓ (0) > 0)
which gradually drops to zero as the temperature decreases with decreasing trap depth. In
this regime the superfluid is polarized (PS). iii) A balanced superfluid (S) at the lowest
temperatures. The smooth decrease in the central density difference as superfluidity sets in(see the yellow shaded region in the figure) is typical for a second order phase transition.

67

Normalized density

I

II

III

1.5
1

0.5

0
0

RC

1

Radial position (r/R )Figure 5-7: n↑ (red; the upper trace in region II and III) and n↓ (blue; the lower trace
in region II and III) at the lowest temperature (T 0 = 0.03). The dashed line is a zerotemperature Thomas-Fermi fit to the non-interacting wings of the majority component.
The three regions indicated in the schematic profile of figure 5-4 can be clearly identified.

is displayed in figure 5-8 i). From σ(r), σc and σs were determined as σc = σ(Rc ), σs =σ(Rc −0.05R↑ ). Here Rc is given by the kink in the column density difference profile (figure 58 a-c)). The definition of σs prevents σc and σs to become equal at high temperature andis therefore a lower bound for the maximum polarization of the superfluid. By determining
σs and σc as a function of temperature (see figure 5-8 i-l) the phase boundary between the
superfluid and normal phase can be traced at low temperature until σc ≈ σs , where the jumpin σ disappears. From the previous experiment we already know that at higher temperatures
the superfluid phase transition is second order without sharp features in the density profiles.
In this regime we have used the rapid ramp method to determine the onset of superfluidity
and then evaluated the central density difference for the same parameters. The combined
data give the critical lines in the homogeneous phase diagram of an imbalanced Fermi gas
at unitarity shown in figure 5-9.

5.3.4

The phase diagram

The phase diagram is quite rich: the thermodynamically unstable region for σs < σ < σcgives rise to the first order phase transition observed in the density profiles. At higher
temperatures the superfluid to normal transition is second order. The first and second
order critical lines meet at a tricritical point. The phase diagram shows two further points ofinterest the critical temperature Tc0 at zero imbalance and the critical spin polarization σc0
at zero temperature. In the absence of a theory for the critical line between the superfluid

68Increasing temperature

Normalized
column density

1.5

a

b

c

d

e

f

g

h

1

0.5

Normalized density

0

1.5
1

0.5

0

RC 0.50Spin polarization

0.8

1

0

i

0.5

1

0

0.5

1

0

k

j

0.5

1

l

0.6
0.4

σC

0.2

σS

0
0

0.2

0.4

0.6

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0

0.2

0.4

0.6

Radial position (r/R )Figure 5-8: Temperature dependence of nc,↑ , nc,↓ , n↑ , n↓ and σ(r). Shown are the density
profiles for four different temperatures T 0 given by: 0.03(1) (a,e,i); 0.05(2) (b,f,j); 0.07(1)(c,g,k); 0.10(1) (d,h,l). T 0 was determined as discussed in section 3.5.5. As the temperature increases the jump in σ(r) decreases until it vanishes in i). Note that beyond theanalysis presented in this section, the density profiles allow the determination of the zerotemperature equation of state in a polarized Fermi gas [74].

69

a
0.2

b

Temperature (T/TF )Normalcd0.1

Tricritical point

SuperfluidfUnstable

e

0.0
0.0

0.1

0.2

0.3

0.4

Spin polarization
Figure 5-9: Homogeneous phase diagram at unitarity. The filled black (grey) circles givethe experimentally determined values for σc (σs ) and trace the phase boundary of thethermodynamically unstable region σs < σ < σc . The critical line at temperatures above
Ttc (black square) was determined by the onset of fermion pair condensation. Note thatfor small population imbalances a reliable fit to the non-interacting wings of the majority
cloud for temperature calibration was not possible and therefore no further data yielding
a stronger constraint for Tc0 could be obtained. The phase diagram also shows several
theoretical (and one experimental)√predictions for Tc , 0 and σc0 : a) ref. [75]; b) ref. [55], theoriginal data have been scaled by ξ with ξ = 0.42 from ref.[76]; c) ref. [77]; d) ref. [78]; e)ref. [72]; f) The location of the tricritical point in the diagram has recently been obtained
theoretically in ref. [79] and is found to be (σtc , Ttc /TF ↑ ) = (0.24, 0.06) in good agreementwith our experimental determination.

70

and normal phases, we have applied a linear fit to the data points. This extrapolationsuggest that (σtc , Ttc /TF ↑ ) ≈ (0.2, 0.07), σc0 ≈ 0.36 and Tc0 ≈ 0.15 (the constraint on Tc0
is not very strong as the in situ temperature calibration could not be performed for small
populations imbalances).
The values for both Tc0 and σc0 have been the subject of long standing debates. With
regard to Tc a number of approaches [75, 78, 77] (including two Quantum Monte Carlocalculations) and one measurement [55] obtained different results within a range of 0.15 to
0.23 of local T /TF . Our extrapolation seems to favor rather the lower critical temperatures.
The critical σc0 has been discussed controversially for more than two years. While all
our previous observations [21, 22, 23] and Monte Carlo calculations [72] were consistent witha low σc0 < 0.4, a number of theoretical studies obtained values for σc exceeding 0.9 [76,80, 81, 82, 83, 84]. Furthermore experiments of the Rice group [85, 86] were considered tobe consistent with a large σc0 > 0.9.
The results presented here clearly confirm a σc,0 < 0.4 and rule out considerable higher
values. This is a dramatic consequence of the strong interactions in the normal phasewhich stabilize an imbalanced normal two-state mixture against the formation of a balanced
superfluid. The important role of these interactions is a qualitative feature of stronglyinteracting Fermi gases and theories that obtain high values for the CC limit do not capture
this physics appropriately. For example Bogliubov-de Gennes theory includes interactions
only in the form of pairing correlations in the superfluid phase. The stability of the mixed
normal phase at T=0 at unitarity has first been demonstrated theoretically in ref. [87]based on very general arguments and was further supported in ref. [88] and by Monte Carlocalculations [72, 89, 90].
The interpretation of the experiments at Rice is still an open issue. While surface
tension and finite size effects [91, 92] seem to play an important role in these experiments3 ,they cannot explain an apparently higher Clogston limit but should rather lead to a further
reduction compared to the value established in our experiments. Further experiments arenecessary to gain a better understanding of these effects and their relation to superfluidity.
So far we have considered the critical line separating the superfluid from the normal
phase. However, there is another interesting point in the phase diagram which is the
crossing of the critical line σs with the σ axis at T = 0. If this crossing occurs at a finite
value of σ the superfluid can tolerate a finite polarization even at T = 0. While the data do
not permit an accurate extrapolation to zero temperature there is another way to address
this question. As illustrated in figure 5-10, the answer is simply determined by the relative
magnitude of ∆ and hc . If hc > ∆ excess fermions have a lower energy in the superfluid(as polarized quasiparticles) than in the mixed normal phase and the superfluid at T = 0
should show a finite polarization. From the in situ profiles we find that hc ≈ 0.95µ [24].
Since µ < ∆ at unitarity as shown in [77, 76, 93] this implies hc < ∆ and therefore we can
3

The experiments were carried out in very elongated traps and with comparatively low atom numbers.

71

µ1

Δ
2h

µs
µ2
S

N

Figure 5-10: ∆ vs h at T=0. Only if ∆ < hc can the superfluid be polarized at zerotemperature. The experiment provides evidence that this is not the case and the superfluidis balanced.

conclude that the superfluid at T = 0 is unpolarized and balanced.

5.4

Changing the interactions: From a Bose-Fermi mixture
to exotic superfluids?

In the previous sections we considered the imbalanced gas with universal interactions at
unitarity. Imbalanced mixtures on the BEC and BCS side of the resonance are, however, ofgreat interest in themselves and here we will give an overview of our experiments studying
these systems [21, 22, 23, 94]. With the phase diagram at unitarity in mind, an important
goal of experiments in imbalanced mixtures is to obtain the three dimensional phase diagramof strongly interacting fermions as a function of temperature, imbalance, and interaction
strength (see figure 5-11).
In the BEC limit the imbalanced system is a weakly interacting Bose-Fermi mixture ofbosonic dimers and unpaired fermions. This mixture can be described within mean field
theory using a Bose-Fermion scattering length abf to capture the atom-dimer interactions
and a Bose-Bose scattering abb for the dimer-dimer interactions. The value of abf has been
predicted more than fifty years ago in the context of neutron-deuteron scattering to be
abf = 1.18a [95], but has not been experimentally confirmed since then. The Bose-Bose
scattering length abb = 0.6a has been obtained about five years ago [30] and studies inbalanced Fermi gases were found to be consistent with this value [96, 97].
As the interactions increase towards unitarity an interesting situation emerges: close toresonance, fermion pairs - that are weakly bound and stable bosonic molecules in isolation cannot undergo Bose-Einstein condensation if they are imbedded in a Fermi sea of majority
atoms at a sufficiently high spin polarization. This is just a consequence of the CC limit that72

Temperature

N

SF
BEC

U
BCS
Interaction strength

Polarization

Figure 5-11: Schematic illustration of a three dimensional, homogeneous phase diagram ofa strongly interacting Fermi gas with spin imbalance as a function of polarization, temperature and interaction strength. The normal (N, green), superfluid (SF, blue) and unstable(U,white) regions in the phase diagram are color coded. In the previous section we havedetermined the tricritical point at unitarity (red circle) and the critical spin polarization(blue circle) [24]. The yellow circle marks the critical point on the BEC side, where thenormal mixed phase disappears and fermion pairs will start to condensate at T=0 irrespective of the spin polarization of the sample (see section 5.4.1). The phase diagram does notshow possible states of exotic superfluids discussed in the text.

increases from zero in the BCS limit to σc0 ∼ 0.4 at unitarity and eventually reaches 1 ata critical point on the BEC side. In the following we will determine this critical interaction
strength.
In the BCS limit the imbalanced gas will ultimately become normal. There are, however,
several suggestions for exotic superfluid phases like the breached pair or Sarma state [98, 99],
the Fulde-Ferrel-Larkin-Ovchninikov (FFLO) state [100, 101, 102, 103] or a state with adeformed Fermi surface [104] that can stabilize the superfluid close to the CC limit on the
BCS side. So far none of this phases have been observed in ultracold Fermi gases.

5.4.1

The CC limit as a function of interaction strength

Figure 5-12 shows that the CC limit occurs at increasingly high imbalances as the interactions become stronger from the BCS to the BEC side [21, 22, 23]. To precisely determine
how the critical imbalance approaches unity on the BEC side we will again take full advantage of spatially reconstructed density profiles which are displayed in figure 5-13.
At 834 G we observe the density profiles and spin polarizations as discussed in the previous section. In the center of the cloud (r < Rc ) is the balanced superfluid, at intermediateradii (Rc < r < R↓ ) we find the mixed normal region and for R↓ < r < R↑ the minority
density is zero. As in the previous experiments the non-interacting wings of the majority

73

1

0.5
BEC
(1/kFa = 0.59)

0.5

Condensate Fraction

0.0

0
1

0.5
0.25

Resonance(1/kFa = 0)

0.5

0.0

0
1

0.5
0.25

Normalized Central Density Difference

0.25

0.5

BCS
(1/kFa = -0.36)

0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

Population Imbalance

Figure 5-12: CC limit in the crossover determined via the onset of fermion pair condensationand correlated with the central density difference in the cloud. Note that the x-axis refers
to the global population imbalance δN and not the spin polarization σ. At unitarity the
CC limit occurs at δN c ≈ 0.75 and σc ≈ 0.4.

cloud for r > R↓ provide an absolute temperature and density calibration. The extend of
the mixed normal region (shaded in blue) is largest on the BCS side and decreases with increasing interaction strength and critical spin polarization. It disappears when R↓ becomesequal to Rc at a critical interaction strength 1/(kF ↑ a)c . For larger interaction strength
no mixed normal region exists, implying that all minority atoms are fully paired and thatthese pairs can undergo Bose-Einstein condensation independently of the spin polarizationof the sample. Therefore one may argue that at this point the pairs fulfill the minimum
requirement to be called “bosonic” and that the two spin state fermion mixture can now
be justifiably called a “Bose-Fermi” mixture, though still strongly affected by interactions.
In the regime where R↓ approaches Rc it becomes increasingly difficult to determine
σc directly from the density profiles with high precision (see figure 5-14 a). Another goodmeasure for the extend of the normal mixed phase is the ratio κ of the chemical potentialof the majority component at R↓ and Rc :R↑2 − R↓2µ↑ (R↓ )= 2
κ=µ↑ (Rc )R↑ − Rc2(5.13)The smaller the value of κ, the larger is the normal mixed region and κ approaches unity
when R↓ = Rc . The linear extrapolation of κ to unity yields the critical interactions strength
1/(kF,↑ a)c = 0.71(5).
74

BEC

B=740 G
Normalized
column density

2.0

777 G

800 G

Resonance

BCS

834 G

851 G

a

b

c

d

e

f

g

h

i

j

1.5
1.0
0.5

Normalized density

0.0

3.0
2.0
1.0
0.0

Spin polarization

1.0

k

l

m

n

o

0.8
0.6
0.4

σC

0.2
0.0
0

0.5

1

0

0.5

1

0

0.5

1

0

Radial position (r/R )

0.5

1

0

0.5

RC

R

1

Figure 5-13: Density profiles and spin polarization (nc,↑ , nc,↓ , n↑ , n↓ σ) of a trapped Fermi
mixture as a function of interaction strength (green: majority, blue: minority, black: difference). The black dotted line is a zero-temperature Thomas-Fermi distribution fit to thewings of the majority component (r > R↓ ). R↑ , R↓ (dashed dot lines), and Rc (dashed lines)
are the radii of the majority, the minority cloud, and the superfluid core, respectively. The
critical polarizations σc at the phase boundary r = Rc are indicated by the right arrows.
The values for R↑ (in µm), Rc /R↑ , and R↓ /R↑ were respectively: for (a,f,k), 381, 0.33, 0.33;
for (b,g,l), 380, 0.33, 0.33; for (c,h,m), 362, 0.35, 0.59; for (d,i,n), 371, 0.44, 0.72; for (e,j,o),367, 0.41, 0.76. T /TF 0 ≤ 0.05 and TF 0 ≈ 1.0 µK.

75(b) 1.0

0.8

0.9

0.6

0.8

κ

Critical polarization(a) 1.0

0.7

0.4

0.6

0.2

0.5
0.0
0.8

0.6

0.4

0.2

0.0

-0.2

0.8

0.6

0.4

0.2

0.0

-0.2

Interaction strength (1/kF a)

Figure 5-14: Phase diagram of the partially polarized normal phase in the crossover regime.
(a) The critical polarization σc as a function of the interaction strength 1/kF ↑ a at the phaseboundary. (b) κ as a function of the interaction strength. The critical 1/(kF ↑ a)c for σc = 1(red square) is found by linear extrapolation of κ to unity. The red solid line is a guide to
the eye for the critical line that exponentially connects (1/(kF ↑ a)c = 0.71, σc = 1) and thecritical value on resonance (1/kF ↑ a = 0, σc0 = 0.36) (as determined experimentally [24]),indicated by the red open circle.

The density profiles of the spin mixture reveal even more details about the system. On
the BEC side above the critical interaction strength 1/(kF,↑ a)c both the Bose-Bose andBose-Fermi scattering lengths can be extracted [94]. The observed values of abf = 1.23(3)aand abb = 0.55(1)a are close to the theoretical predictions [95, 30]. The Bose-Bose scattering
length actually starts to increase (from its exact value of 0.6) in the vicinity of the criticalinteraction strength. Interestingly, if beyond mean field effects via the Lee, Huang andYang (LHY) correction [105] are included in the model 4 , the value of abb remains constant
up to 1/(kF,↑ a)c , implying that mean field theory plus the LHY correction yields a good
description even of strongly interacting Bose-Fermi mixtures. With the determination of
abf a prediction made in the context of neutron-deuteron scattering fifty years ago, has now
be tested with ultracold atoms. This stresses again the universality of interacting fermion
systems and shows the power of the tools provided by atomic physics to study them.

4

The effects of the LHY correction which reduces the compressibility of the Bose gas has previously been
observed in strongly interacting Fermi gases via an upshift of the radial compression mode frequency [106].

76

Chapter 6

Pairing correlations of fermions inthe superfluid and the normalphase
In the previous chapters we have discussed the superfluid properties and the phase diagramof strongly interacting Fermi gases with (and without) population imbalance. This has
provided us with insights into the properties of the strongly interacting gas on a macroscopic
scale. We now proceed to address the underlying physics at a microscopic level by studying
the fermion pairs directly.
Pair formation is the prerequisite for condensation and superfluidity in fermionic superfluids. The fermion pairs play a central role in BEC-BCS crossover [1, 2, 3] and show a very
different behavior in both limits with important ramifications for the superfluid systems.
In the BEC limit the fermion pairs are tightly bound molecules, small compared to theinterparticle spacing and stable in isolation. The molecules start forming around temperatures T ∗ well above the critical temperature Tc for Bose-Einstein condensation. In the BCS
limit the fermion pairs are weakly bound, large compared to the interparticle spacing andunstable in isolation. Here the pairs form only once the system undergoes the superfluid
phase transition.
With the realization of a crossover superfluid fermionic pairing can be studied in a new
regime. In particular at resonance the high temperature superfluid is expected to show
universal behavior with the only relevant energy and length scales given by EF and 1/kF ,respectively. The comparison between BEC and BCS limits points to some of the interestingquestions about the resonantly interacting pairs: Do pre-formed pairs exist at temperatures
T ∗ > Tc ? What is the pair size compared to the interparticle spacing? Further questions
concern imbalanced systems: do fermion pairs also exist at imbalances δ > δc i.e. in a
regime where Tc = 0, or is the phase transition as a function of spin polarization associatedwith “Pauli pair breaking” [107, 108, 109]? To summarize these questions we would like77

to understand where in the phase diagram of figure 5-9 fermion pairs are present andto investigate their nature. The “tool” employed for these studies is radio-frequency (rf)spectroscopy.
Rf techniques provide access to the hyperfine structure of atoms and are widely usedin atomic physics from precision spectroscopy in atomic clocks to evaporative cooling inultracold atomic gases or as an output coupler for atom lasers [110, 111]. As discussed inchapter 2 signatures of pairing correlations in rf spectroscopy experiments are a characteristic pair dissociation lineshape and shifts of the rf transition line with regard to the bareatomic line. Such interaction shifts are often referred to as “clock shifts” since they are
the dominant source of systematic errors for atomic clocks. While being a vice for precise
clocks, these shifts are a great feature to study and understand correlations in interacting
systems.
In this chapter we will follow the progress made in rf spectroscopy experiments ofstrongly interacting fermions over the past years and will address a number of the fundamental questions with regard to pairing as we go along. In the course of the experiments
we realized, however, that so-called final state interactions severely affected the rf spectra
obtained in the standard (1,2) mixture. The results of these experiments have therefore to
be interpreted with caution. We then realized a new superfluid spin mixture where final
state interactions are negligible. This was decisive for finally being able to easily interpretthe observed rf spectra in terms of pair dissociation and to determine the size of the fermion
pairs at unitarity. In ongoing experiments imbalanced mixtures were studied which lead tothe spectroscopic observation of quasiparticles.

6.1

The Innsbruck experiment: A signature of pairing in the
crossover

The first rf spectroscopy experiments with ultracold fermions have been carried out at
JILA [33] and MIT [10] even before molecules and fermion pair condensation were observed. These experiments demonstrated the effects of attractive and repulsive interactionsin the gas by studying mean field shifts. Shortly afterwards the JILA group published rf
dissociation spectra of the newly created weakly bound molecules [56].
A first signature of pairing in the many-body regime was observed by the Innsbruckgroup [25] in 2004. We will discuss the Innsbruck results in some detail to motivate thequestions we have addressed in our subsequent experiments. In the Innsbruck experiment
the initial state is a (1,2) mixture of 6 Li atoms and the rf is tuned around the |2i to|3i transition. The spectra are obtained by monitoring losses in state |2i after the rf isapplied.
Let us first consider the rf spectra for weakly interacting molecules on the BEC side of the
Feshbach resonance shown in figure 6-1. At high temperatures (figure 6-1a) only unbound78

Fractional loss in |2〉

Rf offset [kHz]Figure 6-1: Rf spectra on the BEC side of the 834 G Feshbach resonance reported by
the Innsbruck group [25]. The three spectra were obtained for different temperatures. At
high temperatures no molecules have formed only the “atomic” peak is visible. As thetemperature is lowered molecules start forming and a molecular peak emerges. The peakis located at higher radio frequencies due to the additional energy required to break themolecule. At the lowest temperatures all atoms are paired into molecules and the molecularpeak is the only remaining feature in the spectrum. The gap between the atomic peak andthe onset of the asymmetric molecular peak corresponds to the molecular binding energy.
Reprinted from [25].

Figure 6-2: Rf spectra close to unitarity at 837 G reported by the Innsbruck group [25].
Again the spectra were obtained at different temperatures and show the emergence of adouble peak structure similar the molecular spectra on the BEC side. Note that the pairing
peak is fairly symmetric in contrast to the molecular dissociation peak in the spectra of
fig 6-1. Reprinted from [25].

79atoms are present and the spectrum shows no shift from the bare atomic transition line (i.e.
the |2i to |3i transition in the absence of atoms in state |1i ). As the temperature is lowered(1,2) molecules start forming and a second, asymmetric peak emerges in the spectrum at
higher frequencies (figure 6-1b). At the lowest temperatures all atoms form molecules andthe molecular or “pairing” peak is the only remaining feature in the spectrum (figure 6-1c).
The molecular dissociation spectrum spectrum is characterized by the sharp increase of thesignal at a threshold value and a slow decay at high energy, the typical features of moleculardissociation spectra that were discussed in chapter 2.
The spectra obtained by the Innsbruck group close to unitarity are shown in figure 6-2.
As a function of temperature they display a similar behavior as the molecular spectra in the
BEC limit: at high temperatures the system responds at the atomic transitions frequency,
as the temperature is lowered a pairing peak emerges and at the lowest temperatures only
this peak remains in the spectrum. There are, however, certain differences: the onset ofthe pairing peak occurs at the frequencies of the atomic transition and the pairing peak
appears to be fairly symmetric. Furthermore the shift of the rf peak at unitarity was foundto be proportional to the Fermi energy.
The emergence of a double-peak structure in the rf spectra and the shifted spectral
response at the lowest temperature made a strong case for the formation of fermion pairs at
unitarity. The Innsbruck rf spectra were also regarded as a key indication for superfluidity
in the strongly interacting regime (before superfluidity was established via the observationof vortex lattices [4] about a year later).
We were motivated to revisit rf spectroscopy of strongly interacting fermions in part byquestions the Innsbruck experiment had left open and also by new questions that emerged
from the creation of imbalanced mixtures:• Rf spectroscopy, pair formation and the superfluid phase transition. Since no signature
for the superfluid phase transition was observed in the experiment [25] the connection
between the observed rf spectra, pair formation and superfluidity remained unclear.
– Do the rf spectra indicate the normal to superfluid phase transition?
– Do pre-formed pairs exist in the normal phase above Tc ?
• Effects of the trapping geometry. At unitarity fermion pairs form due to many bodyinteractions and consequently the response of the system to the rf pulse is density
dependent. Since the experiments are carried out in a harmonic trapping geometry
the observed spectra show the response averaged over the density distribution oftrapped cloud.
– In the BEC limit molecules and unbound atoms coexist locally. Is the double
peak structure at unitarity an artifact of the trapping geometry or does it imply
the local coexistence of fermion pairs and unbound atoms?
80

– Does inhomogeneous broadening prevent a clear observation of a gap between
the atomic line and the onset of the pairing peak?
• Effect of final state interactions and interpretation of the spectral lineshape. The early
MIT experiment [10] demonstrated the absence of mean field shifts in the (1,2) mixture
at magnetic fields beyond 700 G and temperatures on the order of TF . This was
attributed to the fact that all interactions between atoms in states |1i , |2i , and |3i are
simultaneously unitarity limited1 . However, this also implies strong interactions in the
final state which any detailed interpretation of the spectra needs to take into account.
– The pairing peak at unitarity is fairly symmetric in contrast to the pair dissociation spectra expected in the BEC and BCS limits. Is this a consequence of final
state interactions?
• Pairing correlations at imbalances above the CC-limit. The CC-limit is also referred
to as the “Pauli pair breaking” limit [107, 108, 109].
– Is the CC limit associated with pair dissociation?
– Does rf spectroscopy reveal the normal to superfluid phase transition as a function of imbalance at very low temperatures?
• Majority versus minority spectra. The correlation between the spectra of minority andmajority components can give important information about the nature of the pairs.
– Are the pair dissociation spectra of minority and majority component the sameor are they different? How does this depend on the phase of the system?
– Will the majority spectrum show “local” double peaks i.e. two peaks at a given
density? In other words can rf spectroscopy distinguish between paired andunpaired majority atoms at a finite spin polarization?
This collection of questions indicates that the research in this area both experimentallyand theoretically is a major frontier. Issues as fundamental as what should be called a“pair” in the many-body regime are still under debate. In the following we will describe
our experiments that were designed to address some of these questions.
The two experiments we will discuss first [26, 23] have been carried out in the standard(1,2) mixture of 6 Li as was the Innsbruck experiment. However, as already mentioned above
we have demonstrated in later studies [112, 113] that final state interactions severely affected
all rf spectra obtained from this mixture in the strongly interacting regime [25, 26, 27].
Since the experiments in the (1,2) mixture have motivated the later experiments we will
1

The exact range of magnetic fields where the interactions are unitarity limited depends on the densities
involved. For the typical densities in the center of the trapped gas at 834 G 1/kF aij ≤ 1 where ij refers tostate |ii and |ji.

81

0T/TF = 1.9

Atom number in |3〉[a. u.]

A

250

500

B
T/TF = 0.9
T/TF = 1.0
Decreasing temperature

0.03

D

0.025

0.020

0.04
T/TF = 0.5

CAtom number fraction in |2〉

0.02

0.03

0.02
-40

0

40

80

120

Radio-frequency offset [kHz]Figure 6-3: Rf spectra of the minority component in a highly imbalanced superfluid (δN ∼0.9), well above CC limit of superfluidity. As the temperature is lowered a “pairing peak”
develops in the absence of superfluidity. Note that in most cases losses in state |2i weremonitored as a function of the rf. The rf spectra therefore show dips where the response of
the system is strongest. In the Innsbruck experiment a different normalization was chosen
and the strongest response appears as a peak. The spectra were taken for the following
parameters: A) δ = 0.87, EF = h × 260 kHz, T /TF = 1.9; B) δ = 0.94, EF = h × 360 kHz,T /TF = 1.0; C) δ = 0.94, EF = h × 360 kHz, T /TF = 0.9; D) δ = 0.93, EF = h × 340 kHz,T /TF = 0.5.

nevertheless discuss them in some detail. But the reader should be cautioned up front
that any quantitative interpretation of the rf spectra obtained in the (1,2) mixture close tounitarity is difficult.

6.2

Rf spectroscopy and the superfluid phase transition

To study the relation between the signature of pairing in the rf spectra (in the (1,2) mixture)and the superfluid phase transition we set out to• Obtain rf spectra in the normal phase of an imbalanced gas above the CC limit.
• Correlate the rf spectra at lower imbalances with the onset of superfluidity via theobservation of fermion pair condensation as a function of temperature and imbalance.
These experiments probe for the presence of pre-formed pairs at temperatures T > Tc >0, as well as pairing correlations in a regime where Tc = 0 (at imbalances above δc ). The
82

experiments were carried out by preparing imbalanced Fermi gases at unitarity (833 G) andapplying the rf to the minority component [26].
Minority rf spectra for four different temperatures at an imbalance of δN ∼ 0.9, i.e.
well above the CC-limit, are displayed in figure 6-3. The spectra exhibit the same behavior
as the ones obtained for an equal mixture in the Innsbruck experiment (figure 6-2): As
the temperature is lowered a shifted pairing peak emerges in the rf spectrum. Based on
the assumptions made in the interpretation of the Innsbruck data this demonstrates theformation of fermion pairs in a regime where pairing is not a precursor to superfluidity
at lower temperatures. Furthermore the experiment shows that the emergence of a pairing
peak in the rf spectrum at unitarity is not necessarily related to superfluidity but yet anotherconsequence of the strong interactions in the normal phase.
To correlate the rf spectra with the onset of superfluidity we repeated the experiments
for smaller imbalances where the superfluid phase transition can be observed both as afunction of temperature and population imbalance. The corresponding rf spectra are given
in figure 6-4. Already above Tc as well as above δN,c the spectra show full pairing in
the normal phase and there is now indication for the onset of superfluidity within theexperimental resolution. The spectra demonstrate the presence of pre-formed pairs above
Tc i.e. a T ∗ > Tc as well as pairing above the critical population imbalance δN,c where
Tc = 0.
These conclusions have to be scrutinized again in the light of the more recent experiments which are not subject to strong final state interactions and will be discussed below.
Important questions concern a more detailed understanding of the signature of “pairs” inthe rf spectrum versus other forms of “pairing”. In this context it will be important to study
the majority rf spectra both in the polarized superfluid and in the normal phase. This willallow us to spectroscopically distinguish unbound majority atoms (which are quasiparticles
in the superfluid phase) from paired majority atoms.
It has been suggested that a single minority atom embedded in a Fermi sea of majority
atoms with resonant interactions, can be described as a “polaron” [87, 72, 88] with aninteraction energy on the order of −0.6EF . This implies that somewhere in the phase
diagram “pairs” turn into “polarons” for increasing imbalance. A spectroscopic signatureof this change in the nature of the pairs might help a to improve our understanding of therelation between pairing and superfluidity.
These ideas show that it may be possible to observe a signature of the superfluid phasetransition in the rf spectra. An experiment to observe such a signature must likely be
performed at temperatures well below the tricritical point. This is the regime where thephase transition is first order and the minority component shows a sizable jump in density
at the phase boundary. Reaching these temperatures consistently in the several hundred
subsequent runs of the experiment required to obtain a high resolution rf spectrum will,
however, pose a considerable challenge.

83

A

D

0.05

F
0.50

T/TF = 0.14

δ = 0.73

0.10

E

0.30

0.20

0.40

T/TF = 0.09

δ=0

0.35

Decreasing temperatureδ = 0.87

0.25
Atom number fraction in |2〉

C

0.30

Decreasing imbalance

B

Atom number fraction in |2〉

0.04

T/TF = 0.18

0.06

0.30

0.25
-20

0

20

40

-20

Radio-frequency offset [kHz]

0

20

40

Radio-frequency offset [kHz]Figure 6-4: RF spectra of the minority component obtained while crossing the phase transition by reducing either the population imbalance (A−C) or temperature (D−F). The rf
spectra do not reveal the phase transition within the experimental resolution. The onset ofsuperfluidity is indirectly observed by fermion pair condensation. The spectra were taken
for the following parameters A−C: A) δN = 0.87, EF = h × 27 kHz, T /TF = 0.08; B)
δN = 0.73, EF = h × 27 kHz, T /TF = 0.10; C) δN = 0.00, EF = h × 23 kHz, T 0 /TF = 0.10.
D−F: D) δN = 0.37, EF = h × 38 kHz, T /TF = 0.18; E) δN = 0.32, EF = h × 38 kHz,T /TF = 0.14; F) δN = 0.29, EF = h × 35 kHz, T /TF = 0.09.

84

(b)

(a)

z
x

Spectral response (a.u.)

(c)
1

0
0

5

10

15

20

25

Radio frequency offset (kHz)

Figure 6-5: Spatially resolved rf spectroscopy in an equal (1,2) mixture at unitarity. The
rf is resonant with the |2i to |3i transition thus creating a local “imbalance” in the (1,2)
mixture which can be precisely detected in situ with the phase contrast imaging techniquedescribed in chapter 3. a) Response of the cloud to different radio frequencies: the lower
density wings become excited at smaller frequencies than the high density parts in the centerof the cloud. b) Absorption image of an equal mixture without rf excitation indicating thesize of the cloud (see also the dashed line in a). c) Inhomogeneous rf spectrum obtained byintegrating the response in the phase contrast images.

6.3

Spatially resolved rf spectroscopy

The rf spectra obtained in the previous section suffered from the problem that they were not“spatially resolved” i.e. they sampled the response of the minority atoms across the varying
density of the majority component. On may argue that at high population imbalances thisshould not be such a serious problem as the density of the majority atoms changes only
very little across the small minority cloud. In general, however, a quantitative comparison
between the rf spectra and theory is straightforward only if one can spatially resolve wherein the trap atoms are transferred as a function of the applied rf frequency. This is especially
important if the trapped gas is in different (superfluid/normal) phases depending on thelocal densities in the trap.
We have been able to spatially resolve the response of an equal, superfluid mixture as afunction of the applied rf (see figure 6-5) [27]. After 3D image reconstruction one obtains
the rf signal at a given density of the system. In figure 6-6 the inhomogeneous signal andthe response at the center of the trapped cloud are compared. The homogenous spectrum
shows a clear gap from the zero offset and contributes only to the the high frequency partof the inhomogenous spectrum.
The two rf spectroscopy experiments we have discussed in this and the preceding sec85

Signal [a.u.]

1

0
0

5

10
15
Rf offset [kHz]20

Figure 6-6: Homogenous rf spectrum from the center of the trapped cloud (black circles) andthe inhomogenous spectrum from the integrated response (red triangles). The spectra arenormalized to the same maximum amplitude. Compared to the inhomogenous spectrum,
the central spectrum shows a reduction in width by about a factor of two and a clear gap
with regard to the zero frequency offset.

tion [26, 27], have renewed theoretical interest in the interpretation of the rf spectra andtheir relation to fermionic pairing in the normal and superfluid phases. Final state interactions have been explicitly considered in references [114, 115, 116, 117] and rf spectroscopy
in imbalanced mixtures in references [114, 118, 119, 120, 121, 122, 123].

6.4

Final state interactions and new superfluid spin mixtures

The potentially dramatic effect of final state interactions on the rf spectra obtained inthe (1,2) mixture becomes apparent when one compares the observed spatially resolved
spectrum with the theoretical expectations for both the BEC and BCS limits (see figure 6-7).
While the calculated rf spectra in both limits show a lineshape typical for pair dissociation
spectra, the experimentally obtained spectrum at unitarity does not: it is symmetric andnarrow. Since the physics in the BEC-BCS crossover is expected to evolve smoothly fromone limit to the other this is a surprising observation. Furthermore the properties of theinitial state at unitarity should be universal and therefore independent of the specific systemused in the experiment. Final state interactions, however, are non-universal and could very
well cause this unexpected behavior.
Final state interactions arise when atoms transferred from state |bi (in an initial (a,b)
mixture) to a third state |ci interact with atoms in the initial state. In rf spectroscopy
experiments with fermionic atoms only interactions of atoms in state |ci with atoms instate |ai can contribute [10, 32]. A conceptually very simple and dramatic effect of final
state interactions arises when (using a molecular picture for simplicity) the rf does notdissociate an (a,b) molecule but transfers it to another (a,c) molecule. If the wave functionof the (a,b) and (a,c) molecules are very similar, the Franck-Condon overlap for such atransition is very high and pair dissociation is suppressed. The lineshape of such a “bound-

86

1

1

1

Signal (arbitrary unit)

BCS-limit

Resonance?

BEC-limit

δEFWHM

δEFWHM

0
0

0
0

2

4

6

8

0

4

8

E/Eth

12

16

0

E/Ew

2

4

6

8

E/Eth

Figure 6-7: The problem: the rf spectrum of the (1,2) mixture at resonance is symmetricand narrow, while the spectra expected in both the BEC and BCS limits show a highly
asymmetric pair dissociation lineshape. The frequency axis of the experimental spectrum
has been scaled by the full width at half maximum since the spectral onset can be subject
to shifts due to Hartree terms (see also section 6.5).
Mixture
(1,2)
(2,3)
(1,3)
(1,3)

B0 [G]
834
811
691
691

Rf transition|2i →|3i|2i →|1i|1i →|2i|3i →|2iaf [a0 ]a13 ≈ −3300
a13 ≈ −3560
a23 ≈ +1140
a12 ≈ +1450

Table 6.1: Final state scattering lengths for magnetic field insensitive rf transitions at
unitarity for the (1,2), (2,3) and (1,3) mixtures. For a typical value of kF ≈ 3000a0 in our
experiments the final state for the (1,2) and (2,3) mixtures is strongly interacting. This is
not the case for the (1,3) mixture where the two possible final scattering lengths are morethan a factor of two smaller and positive. This mixture is therefore an ideal candidate forrf spectroscopy experiments.
bound” transition is symmetric and narrow in contrast to the asymmetric lineshape for pairdissociation or “bound-free” transitions.

6.4.1

Effect of final state interactions

In principle there are several ways to change the final state interactions without affecting
the initial state. At unitarity the final state interaction strength can be reduced by lowering the density and therefore kF af , while the initial state remains resonantly interacting(1/kF ai ≈ 0). Decreasing the density by a large factor while maintaining the same lowtemperature T /TF is, however, experimentally difficult. Alternatively one might attempt
to spectroscopically access a different final state. However, the (1,2) mixture in 6 Li does notprovide other rf transitions insensitive to the magnetic field. Magnetic field insensitivity
is necessary to obtain the required spectral resolution in the kilohertz regime. Instead, we
created resonantly interacting superfluids in new combinations of initial hyperfine states:(1,3) and (2,3) by utilizing the broad Feshbach resonances exhibited by these mixtures at
87

1

1

1

Signal (arbitrary unit)

BEC-limit

Resonance

BCS-limit

δEFWHM

δEFWHM

0

0

0
0

2

4

E/Eth

6

8

0

4

8
E/Ew

12

16

0

2

4

6

8

E/Eth

Figure 6-8: The solution: Rf spectroscopy of the (1,3) superfluid at unitarity. With negligible final state effects the rf spectrum displays the expected asymmetric lineshape. The
frequency axis of the experimental spectrum has been scaled by the full width at half maximum Ew (see also section 6.5).

691 G and 811 G respectively. Table 6.1 summarizes the location of the Feshbach resonances
together with the allowed magnetic field insensitive rf transitions and final state scattering
lengths. Details about the experimental realization of these mixtures and related information are given in chapter 3. The comparison of the final state scattering lengths between thethree mixtures clearly shows that the (1,3) mixture is the ideal candidate for rf spectroscopy
experiments. Indeed, the rf spectra obtained with a (1,3) mixture at unitarity show thetypical pair dissociation lineshape one would expect from a simple crossover picture (seefigure 6-8).
A direct comparison between the rf spectra obtained at unitarity in the (1,2) and the(1,3) mixtures reveals the dramatic effect of finite state interactions which suppress theasymmetric “tails” of the (1,2) rf spectrum and shift the spectral peak. We will discusspossible explanations for the strong effects of these interactions on the rf spectra in thecontext of further experiments later in this chapter.

6.4.2(1,3) spectroscopy: |1i to |2i versus |3i to |2i transition.

As has been indicated in table 6.1 the (1,3) mixture has two magnetic field insensitive
rf transitions, associated with positive final state scattering lengths (that differ by about30%). Figure 6-10 shows the rf spectra at unitarity for these two transitions. Apart from theasymmetric “bound-free” transition to higher energies all spectra show a second “boundbound” transition from fermion pairs in the initial state to more deeply bound (1,2) or(2,3) molecules for the |3i to |2i and |1i to |2i transitions respectively. Note that thebound-free spectra are very similar for both |3i to |2i and |1i to |2i transitions. The boundbound spectra, however, show different shifts indicating that the final (2,3) molecule is morestrongly bound than the (1,2) molecule. This is a consequence of the smaller width of the(2,3) Feshbach resonance at 811 G [44] compared to the width of the (1,2) resonance at 834
88Signal (arbitrary units)

1

0
0

2

4

6

E/EF

Figure 6-9: Dramatic effect of final state interactions: direct comparison of spatially resolved
rf spectra obtained at unitarity from the (1,2) mixture (open triangles) [27] and from the(1,3) mixture (solid circles). The frequency axis is normalized by the local Fermi energies.
While the (1,3) spectrum shows the expected asymmetric lineshape, the (1,2) spectrum issymmetric, narrow, and shifted to lower energies. All spectra in the strongly interacting
regime reported in references [25, 26, 27] were subject to these significant distortions.

1 |3〉
|2〉

a
Signal (arbitrary units)

|1〉

0
+600
1 |3〉

+400

+200

0

-200

-200

0

200

|2〉

b

|1〉

0
-600

-400

Rf offset (kHz)

Figure 6-10: Spectroscopy on the two magnetic field insensitive rf transitions in the (1,3)
mixture at unitarity. The insets in (a) and (b) schematically indicate the two rf transitions
employed in the experiment. Since state |3i has a higher energy than state |2i, the dissociation energy for the |3i to |2i transition is always less than the transition frequency for theatomic resonance E0 /h and therefore the |3i to |2i dissociation spectra appear at negative
energies compared to E0 .

89

G (see Tab. 3.1 and figure 3-4).
The spectra also show directly that final state interactions can only weakly affect the
dissociation spectrum: the separation between the bound-bound and bound-free transitions
is large compared to the linewidth of the dissociation spectrum. As discussed in detail inref. [112] this implies that the Franck-Condon overlap between the (1,3) pair in the initial
and the (1,2) or (3,2) pairs in final state is small and the bound-bound transition does nothave a significant spectral weight [124, 117].
Rf spectra obtained from the (1,3) mixture are therefore the ideal starting point forexperiments to probe the microscopic structure of the superfluid fermion pairs. Using the(1,3) mixture we can go ahead and record spatially resolved rf dissociation spectra (like theones shown in figure 6-10) that can be directly compared to theory.

6.5

Determination of the fermion pair size in a resonantlyinteracting superfluid

The fermion pair size at unitarity (measured in units of 1/kF ) is one of the fundamental,
non-trivial and universal parameters characterizing the superfluid system. In contrast toother important universal parameters at unitarity like the critical temperature Tc /TF , theClogston limit of superfluidity δc , or the ratio of µ/EF - which are more closely related
to the macroscopic properties of the superfluid gas - the pair size directly describes its
microscopic “building blocks”.
Interestingly, small fermion pair sizes have been linked to high critical temperatures by
the Uemura plot for a wide class of fermionic superfluids [125]. Since the critical temperature
Tc /TF in a balanced, resonantly interacting Fermi gas is an order of magnitude larger thanfor any other known fermionic superfluid (including high-temperature superconductors), it
is of fundamental interest to determine the fermion pair size in this new system and tocompare it to the pair size in other fermionic superfluids.
In the following we will show that both the width and the onset of the rf dissociation
spectrum are directly and simply related to pair size in the entire BEC-BCS crossover. We
have discussed rf spectroscopy and the rf lineshape in chapter 2 and we will repeatedly referto these results in the discussion below.

6.5.1

Fermion pair size and rf spectroscopy

A characteristic size of the fermion pairs in the crossover can be defined via the two-particle
correlation length ξpair given by:sξpair =

hφ|r2 |φi
hφ|φi

90

(6.1)

ξth/ξpair and ξw/ξpair

Pair size kFξ

100

2

1

10

2

1

0

-1

-2

1/kFa

1

2

1
0
-1
Interaction parameter 1/kFa-2

Figure 6-11: Evolution of the fermion pair size in the BEC-BCS crossover [126, 127, 31, 7].
ξw (solid blue), ξth (dashed red) and ξpair (dotted black) are displayed as a function of theinteraction parameter 1/kF a. The inset shows the ratios ξw /ξpair (solid blue) and ξth /ξpair(dashed red).

where φ is the pair wavefunction.
The weakly interacting molecules in the BEC limit are described by a wavefunction
φm (r) proportional to e−r/b /r, and have a binding binding energy Eb = ~2 /mb2 [7]. For√
φm equation 6.1 yields a pair size of ξpair = b/ 2. Both the onset energy Eth and the full
width at half maximum Ew of the rf spectrum (see chapter 2) are proportional to Eb andwe find Eb = Eth ≈ 1.89 × Ew ∝ 1/b2 . The pair size can therefore be directly obtained bothfrom the width and the onset of the rf spectrum as:
~2
2mEth
~2
= γ×
2mEw

2
ξth
=

(6.2)

2
ξw

(6.3)

where we choose γ = 1.89 so that ξpair = ξth = ξw in the BEC limit.
2∆
which is the binding or condensation energy of the N/2
In the BCS limit Eth = 2EFpairs in the superfluid. The fermion pair size is related to Eth and Ew via Ew ≈ 1.27×Eth =
2

2

~ kF
∆
∝ 1/ξc2 where ξc = πm∆
1.27× 2Eis the Pippard coherence length which is often associatedFwith the fermion pair size2 . We find ξth = 2 × ξpair and ξw ≈ 2.44 × ξpair . So up to factors

on the order of unity Eth and Ew are directly related to the fermion pair size both in the
BEC and BCS limits. Figure 6-11 displays ξth , ξw and ξpair as a function of interaction
2

The value√ of ξpair depends on the choice of the pair wavefunction φ; Here we chose φP and thenξpair = π/(2 2)ξc .

91

1

BEC

a

|3〉
|2〉
|1〉

0
1

0
0

-50

-100

-150

BCS

Signal (arbitrary units)cUnitarity01b

Rf offset (kHz)

Figure 6-12: Rf dissociation spectra in the crossover. Below, at and above resonance thespectrum shows the typical asymmetric lineshape of a pair dissociation spectrum. The
signal is proportional to the three dimensional local response at the center of the cloud. (a)670 G (BEC side), ²F = h × 24 kHz, T /TF ≈ 0.2, 1/kF ai = 0.4; (b) 691 G (resonance),²F = h×21 kHz, T /TF =0.1, 1/kF ai ∼ 0; (c) 710 G (BCS side), ²F = h×20 kHz, T /TF =0.1,
1/kF ai = −0.3.

strength. Although ξpair changes by orders of magnitude from the BEC to the BCS limit ξthand ξw show the same behavior and deviate from each other by not more than 22%. Thisillustrates that the pair size can be reliably determined from the rf dissociation spectrum
throughout the whole BEC-BCS crossover.
Figure 6-12 shows the rf dissociation spectra obtained in the BEC-BCS crossover withthe (1,3) superfluid. Three dimensional image reconstruction was applied to obtain thelocal spectrum at the center of the cloud. As expected, both Eth and Ew decrease withdecreasing interaction strength 1/kF ai . In chapter 2 we have seen that the rf lineshape is
expected to show only small changes in the crossover regime. Indeed, when the frequency
axes for the spectra in figure 6-12 are scaled by Ew and the spectra are shifted to show thesame onset all spectra overlap as shown in figure 6-13. This confirms experimentally the“universal” lineshape of the rf dissociation spectrum. The fit to the spectra in figure 6-12
used the simple dissociation lineshape Igeneric convolved with the lineshape of the square
excitation pulse [112].
As discussed above the pair size can in principle be obtained from both Eth and Ew .
However, since the whole spectrum may be subject to shifts due to Hartree terms [33, 10],92

Signal (arbitrary units)

1

0
0

-4
E/Ew

-8

Signal (arbitrary units)

Figure 6-13: Universal rf lineshape in the crossover. Same spectra as in figure 6-12 butwith the frequency axis scaled by Ew and shifted so that the spectral onsets overlaps with
the BEC side spectrum. BEC side: black circles; Resonance: red triangles; BCS side: blue
squares.
10050
Rf offset (kHz)

Figure 6-14: Density effects at unitarity for the (1,3) mixture at 691 G. The figure shows
the spectral response in the center (open circles, same spectrum as in figure 6-12b) as wellas the lower density wings (filled triangles) of the cloud. In this regime the cloud might
have turned normal.

we focus in the following only on the width of the spectrum. At unitarity we determine thefull width at half maximum of the fit, (which takes the lineshape of the square excitation
pulse into account) to be Ew = 0.28(5)²F corresponding to a spectroscopic pair size of√ξw = 2.6(2)/kF . Here ²F is the local Fermi energy and kF = 2m²F /~. The fermion pairs
are smaller than the interparticle spacing given by l = n1/3 = (3π 2 )1/3 /kF ∼ 3.1/kF (n isthe total density) and are in units of 1/kF the smallest observed in fermionic superfluids.
In high-temperature superconductors the pair size ξ at optimal doping is in the range of 5
to 10/kF [125]. The high temperature superfluid therefore confirms the relation between
small pair size and high transitions temperatures found in many fermionic superfluids.
The strong narrowing of the spectrum in figure 6-12 (a) to (c) demonstrates that thefermion pair size increases from strong to weak coupling. The decreasing width corresponds
to a twofold increase in the spectroscopic pair size from ξw = 1.4(1)/kF at 670 G to ξw =93

3.6(3)/kF at 710 G. A change of the absolute pair size with density at unitarity can inprinciple be observed by comparing the spectral width in the center and the outer region of
the cloud. As the density decreases the spectrum shifts to lower energies (see figure 6-14).
However, the spectral onset also becomes increasingly softer and the asymmetry of the pairdissociation peak less pronounced, possibly due to atomic diffusion during the excitation
pulse. This prevents a reliable determination of the pair size in the spatial wings where thedensity is changing rapidly and the system may already be in the normal phase.
We have shown here that the (1,3) rf spectra contain detailed information about the
microscopic properties of the fermion pairs at unitarity and in the crossover. In future
experiments the microscopic structure of the pairs can now be studied both in the superfluidand normal phase as a function of interaction strength, temperature and spin imbalance.
In the next section we will discuss rf spectra of the majority and minority component in an
imbalanced Fermi gas at unitarity.

6.6

Observation of quasiparticles in a polarized superfluid

In section 6.2 we discussed the first rf spectroscopy experiment with imbalanced mixtures
reported in [26]. The experiment was carried out in the (1,2) mixture and without spatial
resolution. The spectra were therefore subject to strong final state effects and only rf spectraof the minority component could be obtained at a satisfactory signal to noise ratio.
At this point, however, we are in the position to take full advantage of rf spectroscopy
in the (1,3) mixture [112], spatially resolved techniques [27] and our knowledge of the phase
diagram for imbalanced mixtures [24]. By spectroscopically probing both the majority andthe minority components it is also possible to directly correlate the spectral response ofmajority and minority atoms. Note, that the (1,3) mixture permits two rf transitions (from|1i to |2i and from |3i to |2i, see figure 6-10) so that majority and minority spectra can beobtained from identically prepared samples.
The data obtained in this experiment are very rich and we will only provide a small“taste” of what the full analysis may yield [113]. Figure 6-15 shows three majority andminority spectra obtained at a temperature of about 0.05 T /TF for three different spinpolarizations as indicated in the phase diagram. The most impressive feature is the clear
double peak structure in the majority cloud observed for intermediate imbalances in thevicinity of the first order normal to superfluid phase transition (figure 6-15 b). The spectral
peak at high frequencies overlaps with the pair dissociation peak of the minority spectrum.
At the second, lower frequency peak in the majority spectrum the minority component
shows no spectral response. This feature in the spectrum is therefore due to unboundmajority atoms, i.e. thermally excited quasiparticles.
This “double peak structure” is the first direct signature of quasiparticles in superfluid
Fermi gases. The observation of quasiparticles is facilitated in imbalanced systems since

94

1

a

0.2

b

Temperature (T/TF )Signal [arbitrary units]01Normal0.1

Superfluid

0
1

a

c

c

b
Unstable

0.0
0.0

0.1

0.2

0.3

0.4

Spin polarization

0
0

1

2

3

Rf offset/εFFigure 6-15: Quasiparticle peak in the rf spectrum of the majority component of an imbalanced mixture at unitarity. The majority (filled red circles) and minority (open black
circles) rf spectra were obtained for three sets of parameters as indicated in the phase diagram. (a) In the balanced case both components show the rf dissociation spectra discussed
in previous sections. (b) At higher spin polarizations, in the vicinity of the first order normal to superfluid phase transition, the majority spectrum has two peaks. The asymmetricpair dissociation peak at higher frequencies overlaps with the corresponding peak in the minority spectrum. The second peak at lower frequencies has no counterpart in the minority
cloud and stems from “unbound” majority atoms, i.e. thermally exited quasiparticles. (c)At high imbalances above the CC limit of superfluidity the majority spectrum is dominatedby the peak close to zero offset, but it shares the asymmetric high-frequency “tail” with theminority component. Importantly, the peak of the minority atoms now has contributions
that exceed the signal in the majority component at the same rf frequencies, possibly indicating a state where the pairing correlations become more polaron like. The frequency axisof the rf spectra is normalized by the local Fermi energy ²F .

95

the imbalance helps to “inject” a significant population of quasiparticles into the superfluid
at low temperatures. While double peak structures have also been found in previous rf
spectroscopy experiments (see figures 6-2, 6-3 and references [25, 26]), these rf spectra were
not spatially resolved and the two peaks may have originated from different locations inside
the trap. In this experiment, however, the double peaks imply the local coexistence of pairs
and quasiparticles.
Surprisingly, the quasiparticle peak appears on the same, positive side of the frequency
axis as the pair dissociation peak. By removing a quasiparticle from the superfluid, energy
is gained and therefore one would expect the quasiparticle peak to be located at negative
frequencies. However, this description does not take contributions from Hartree terms bothin the initial and final states into account, which can lead to shifts of the spectrum relative
to the zero frequency offset.
At the imbalances considered so far, the pair dissociation peak of minority and majority
components overlap. This changes at higher imbalances: in figure 6-15 c) we observe that at
frequencies close to the peak in the minority cloud, the majority atoms shows a significantly
weaker response. The majority spectrum is dominated by a peak close to zero frequency andonly at high frequencies, in the small, asymmetric wings of the spectrum, the signal fromboth majority and minority components are well overlapped. The fact that the spectral
weight in the minority peak is not anymore “balanced” by the spectral weight of the majority
spectrum at the same frequencies could indicate the development of a more “polaron-like”
type of pairing.
A more detailed analysis of the data will show whether the rf spectra reveal a signatureof the normal to superfluid phase transition and may also yield a better characterization of
the “pair” to “polaron” transition.

6.7

Final state interactions revisited

The advances reported in the previous two sections could not have been obtained with rf
spectra from the (1,2) mixture due to the dramatic effects caused by final state interactions.
In this section we will try to better understand why these interactions had such a significant
impact.
Figure 6-16 displays three rf spectra from the Innsbruck group taken on the BEC sideof the (1,2) Feshbach resonance at 661, 695 and 764 G [124]. At 661 G the spectrum isdominated by a “bound-bound” peak from (1,2) to (1,3) molecules due to a high FranckCondon overlap between these molecular states. The second spectrum was obtained at 695
G just on the BCS side of the (1,3) resonance. Here the final state is strongly interacting but
no final bound molecular state is accessible. Accordingly the rf transition is predominantly
bound-free, i.e. the (1,2) pair is dissociated by the rf pulse. Compared to regular dissociation
spectra the lineshape shows modifications owing to the strongly interacting final state. As

96

a

b

3) 764 G
ai ≈ + 4500 a0
af ≈ − 4700 a02) 695 G
ai ≈ + 1600 a0
af ≈ − 48000 a0

5000
Scattering length [a0]Franck-Condon factor

0.04

10000

1) 661 G
a ≈ + 1000 a0i
af ≈ + 5400 a0

0.02

1
0

3
2
-5000

0.00

-10000
700

0

200

400

600

800

800
Magnetic field [G]900

Rf offset [kHz]Figure 6-16: a) Rf dissociation spectra in the (1,2) mixture on the BEC side of the (1,2)
Feshbach resonance. The figure has been adopted (with modifications) from ref. [124], thedata were obtained by the Innsbruck group. b) Plot of a12 (red) and a13 (blue) as a functionof the magnetic field between 630 and 900 G. The dashed lines mark the resonance positions.
In the shaded region both the (1,2) and (1,3) mixtures are strongly interacting for typicalvalues of kF . The arrows indicate where the rf spectra shown in a) have been obtained.

the (1,2) resonance is approached for higher magnetic fields |af | decreases and the final stateinteractions therefore are expected to become weaker. The third rf spectrum in figure 6-16
was observed at 764 G, where the initial (1,2) mixture enters the strongly interacting regime(ai ≈ 4500a0 ). Interestingly, this rf spectrum is comparable, to the rf dissociation spectra werecorded in the BEC-BCS crossover of the (1,3) mixture (figure 6-12.) With the knowledge
from the (1,3) system one would therefore conclude that the (1,2) rf spectra should actually
not change significantly from 764 G towards unitarity, especially since final state interactionsin terms of |af | continue to decrease further. However, quite to the contrary, the observed(1,2) spectra are subject to the significant distortions discussed above (see figure 6-9). This
indicates that a better understanding of the interplay between the interactions in the final
and in the initial state is required.
The explanation for the strong impact of final effects must be related to the fact thatat 750 G and higher fields the interactions in the initial state enter the strongly interacting
regime and therefore become comparable to the ones in final state (see the shaded regionin figure 6-16 b). For identical interactions in the initial and final state rf spectroscopy
measures a delta function at zero frequency offset. If the interactions in the initial andfinal state are similar but different, one would expect a bound-bound peak with a certain
offset from zero frequency to dominate the spectrum and strongly suppressed bound-freecontribution to emerge. This scenario is consistent with the narrow and symmetric lineshape(characteristic for bound-bound transitions) observed in the (1,2) mixture at unitarity. Of
course, the sign of a in initial and final state is different, but there is evidence that in thestrongly interacting regime certain interactions are largely independent of the sign of the97scattering length [128, 129, 130, 10].
The (1,3) mixture allows us to check these ideas experimentally. Figure 6-17 shows (1,3)
rf spectra as the final state interactions change from weak to strong with increasing magnetic
field. The initial state stays in the strongly interacting regime, but may turn normal aroundkF ai ∼ 1. Compared to the (1,2) mixture the role of initial and final state are exchanged sothat the initial state is always strongly interacting while the final state enters this regime
at fields around 750 G. Here and at higher fields the bound-free transitions strongly looses
in spectral weight until it cannot be resolved any more at 833 G, where only the narrow andsymmetric bound-bound peak remains. This demonstrates the dominance of bound-boundtransitions in a regime where |1/(kF ai ) − 1/(kF af )| ≤ 1.5. The spectra from the (1,3)
mixture therefore provide further evidence for the dominance of bound-bound transitionsin the (1,2) mixture close to unitarity.
In a recent paper bound-bound and bound-free transition have been studied theoreticallyin the strongly interacting regime [117]. It was found that the spectrum of the superfluid atresonance shows bound-bound peaks for positive but also for negative values of 1/kF af . The
theoretical treatment in ref. [117] is in qualitative agreement with our observations in the(1,3) mixture but underestimates the region where bound-bound transitions are dominant
by about a factor of two. We infer from ref. [117] that for a system in the unitarity limit it ismuch more difficult to spectrally resolve bound-bound and bound-free transitions if af < 0.
When one approaches resonance for the (1,2) system from the BEC side the bound-free
spectrum narrows and smoothly turns into a bound-bound dominated one.
We conclude that the (1,2) to (1,3) rf spectra in the vicinity of 833 G reported in
ref [25, 26, 27] can therefore not be simply interpreted in terms of a pair dissociation
process and a pairing gap [25, 131, 132, 133, 27, 26] and are likely to be dominated bybound-bound transitions.

98670 G00

1 e
1/k’Fai = -0.9
1/k’Faf = 0.6

0

710 G

+400

+200

0

+600

+400

Rf offset (kHz)Scattering length (a0)

h

+200

0

BCS

0

0
+600

833 G

1 f
1/k’Fai = -1.2
1/k’Faf ≈ 0

1 c
1/kFai = -0.3
1/kFaf = 2.0

780 G

691 G

1 b
1/kFai ≈ 0
1/kFaf = 2.6

Signal (arbitrary units)0
Signal (arbitrary units)

750 G

1 d
1/k’Fai = -0.7
1/k’Faf = 1.1

BEC

1 a
1/kFai = 0.4
1/kFaf = 3.3

Rf offset (kHz)

10000

f

5000

a

e

0

d
c

-5000

b

-10000
700

800

900

Magnetic field [G]Figure 6-17: Effect of final state interactions on rf spectroscopy: bound-bound and boundfree spectra in the BEC-BCS crossover of the (1,3) mixture. While the initial (1,3) stateis strongly interacting at all fields the final state interactions change from weakly (a-c) tostrongly interacting (d-f ). See also the diagram in h. At the higher magnetic fields for
1/kF ai ≈ −1 the initial state may have turned normal. (a-c) Same bound-free spectra andparameters as in figure 6-12. The bound-bound peaks have been fit by a Gaussian. The relative weight of the bound-bound and bound-free peaks were not determined experimentally.
(d-f ) All peaks have been fit by a Gaussian. (d) 750 G, EF = h × 22 kHz, T /TF =0.09; (e)780 G, EF = h × 23 kHz, T /TF =0.09; (f ) 833 G, EF = h × 20 kHz, T /TF =0.06.

99Chapter 7

Conclusion and Outlook
The past five years were a fantastic time to do experiments with ultracold fermionic atoms.
Looking back to the early beginnings of learning about Feshbach resonances and molecule
formation, which are problems that mostly concerned “atomic physics”, it becomes apparent
at what amazing rate the field has matured. The current experiments locally probe stronglyinteracting fermions in regimes that are very difficult to realize in other systems and start
to study many-body problems of fundamental importance to various areas in physics.
The best part, however, is that in many ways one may still regard the field to be at a
beginning. This becomes apparent when one compiles a list of just a few ideas for future
experiments that are currently pursued:• p-wave Superfluidity. All ultracold atom experiments so far have studied superfluidity with s-wave pairing. However p-wave Feshbach resonances have been observed
and p-wave molecules have been created. In the future it might be possible to stabilize these molecules sufficiently to cool them into the superfluid state with anisotropicinteractions.
• Simulating condensed matter physics Hamiltonians with fermionic atoms.
All experiments described in this thesis studied interacting fermionic atoms in a harmonic trap. Superimposing an optical lattice allows one to explore a whole arrayof new phenomena closely related to condensed matter physics. A challenging goal
for future experiments is to study fermions with repulsive interactions and to realize
antiferromagnetic order in a lattice at half filling. This could be the first step towards
observing d-wave superfluidity at even lower filling. Such experiments will be able todirectly address some of the important open questions in condensed matter physics.
• Two species fermion-fermion mixtures and molecular BECs with dipolarinteractions. Two species mixtures of fermions have recently been created and may
allow studies of mass imbalanced fermionic superfluids. Furthermore, it could be possible to form heteronuclear bosonic molecules from two fermionic atoms by Feshbach100association and then transfer them into the molecular ground state. This could leadto the creation of molecular Bose-Einstein condensates with long range dipolar interactions which are of interest not only in the context of many-body physics but alsowith regard to quantum computation schemes.
• Bose-Fermi mixtures and Boson mediated pairing. As in Fermi-Fermi mixtures
also Bose-Fermi mixtures can be the starting point for the creation of heteronuclear
ground state molecules with the difference that these molecules will be fermionic.
Furthermore, if strongly interacting Bose-Fermi mixtures are sufficiently stable, it
might be possible to observe boson-mediated fermionic pairing and superfluidity. In
three-dimensional optical lattices at half filling and with weak and repulsive BoseFermi interactions the formation of a supersolid phase might be observable.
• Ternary mixtures of fermions The experiments with different spin mixtures in 6 Li
indicate that even a ternary mixtures might prove to be sufficiently stable at least fora certain range of interactions. Such a system would allow a wide range of studies
from pairing competition in multicomponent Fermi gases, to the observation of color
superfluidity (analogous to a color superconducting phase in QCD) in combination
with optical lattices.
Some of these ideas will likely become reality in the very near future. More than ten
years after the discovery of Bose-Einstein condensation the field of ultracold atoms is still
progressing at an amazing rate and in various new directions. The possibilities for future
experiments based on the combination of new “tools” and techniques with different fermionicmixtures are numerous. There are exciting years ahead of us.

101

Appendix A

Feshbach resonances in fermionic
6Li
This appendix contains a reprint of Ref. [28]: C. H. Schunck, M. W. Zwierlein, C. A.
Stan, S. M. F. Raupach, W. Ketterle, A. Simoni, E. Tiesinga, C. J. Williams, and P. S.
Julienne Feshbach resonances in fermionic 6 Li, Phys. Rev. A 71, 045601 (2005).

102

PHYSICAL REVIEW A 71, 045601 s2005d

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

BRIEF REPORTS

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, nmol. For the broad resonance, this density-dependent,
few-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

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.
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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.
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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.

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

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.
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045601-4

Appendix B

Formation Dynamics of a Fermion
Pair Condensate
This appendix contains a reprint of Ref. [29]: 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).

107Appendix C

Direct Observation of theSuperfluid Phase Transition in
Ultracold Fermi Gases
This appendix contains a reprint of Ref. [22]: Martin W. Zwierlein, Christian H. Schunck,
André Schirotzek & Wolfgang Ketterle, Direct Observation of the Superfluid Phase Transition in Ultracold Fermi Gases, Nature 442, 56 (2006).

112Appendix D

Superfluid Expansion of a Rotating
Fermi Gas
This appendix contains a reprint of Ref. [20]: C. H. Schunck, M.W. Zwierlein, A. Schirotzek,
and W. Ketterle, Superfluid Expansion of a Rotating Fermi Gas, Phys. Rev. Lett. 98,
050404 (2007).

123Appendix E

Pairing Without Superfluidity:
The Ground State of an
Imbalanced Fermi MixtureThis appendix contains a reprint of Ref. [26]: C. H. Schunck, Y. Shin, A. Schirotzek, M. W.
Zwierlein, W. Ketterle, Pairing Without Superfluidity: The Ground State of an Imbalanced
Fermi Mixture, Science 316, 867 (2007).

128Appendix F

Tomographic rf Spectroscopy of aTrapped Fermi Gas at Unitarity
This appendix contains a reprint of Ref. [27]: Y. Shin, C. H. Schunck, A. Schirotzek, and
W. Ketterle, Tomographic rf Spectroscopy of a Trapped Fermi Gas at Unitarity, Phys. Rev.
Lett. 99, 090403 (2007).

140Appendix G

Phase diagram of a two-component
Fermi gas with resonantinteractions
This appendix contains a reprint of Ref. [24]: Y. Shin, C. H. Schunck, A. Schirotzek & W.
Ketterle, Phase diagram of a two-component Fermi gas with resonant interactions, Nature
451, 689 (2008).

145Appendix H

Determination of the fermion pairsize in a resonantly interacting
superfluid
This appendix contains a preprint of Ref. [112]: C. H. Schunck, Y. Shin, A. Schirotzek &W. Ketterle, Determination of the fermion pair size in a resonantly interacting superfluid.
Preprint, arXiv:0802.0341v2 [cond-mat.supr-con].

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