Making, probing and understanding ultracold Fermi gases

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by  Wolfgang Ketterle

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Making, probing and understanding ultracold Fermi
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Wolfgang Ketterle and Martin W. Zwierlein
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1 Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of

Electronics,
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r Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA
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W.KetterleandM.W.Zwierlein,Making, probingandunderstandingultracoldFermi
gases, inUltracoldFermiGases,Proceedingsofthe InternationalSchoolofPhysics”En-
rico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006, edited by M. Inguscio, W.
Ketterle, and C. Salomon (IOS Press, Amsterdam) 2008
(cid:13)c Societa`ItalianadiFisica 1
Table of contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
.
11. State of thefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
.
12. Strongly correlated fermions - a gift of nature? . . . . . . . . . . . . . . . . 6
.
13. Some remarks on thehistory of fermionic superfluidity . . . . . . . . . . . 7
.
13.1. BCS superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
.
13.2. TheBEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . 9
.
13.3. Experimentson fermionic gases . . . . . . . . . . . . . . . . . . . . 11
.
13.4. High-temperaturesuperfluidity . . . . . . . . . . . . . . . . . . . . 13
.
14. Realizing model systems with ultracold atoms . . . . . . . . . . . . . . . . 13
.
15. Overview overthechapters . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2. Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
.
21. The atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
.
21.1. Hyperfinestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
.
21.2. Collisional Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 18
.
22. Cooling and trapping techniques . . . . . . . . . . . . . . . . . . . . . . . . 21
.
22.1. Sympatheticcooling . . . . . . . . . . . . . . . . . . . . . . . . . . 22
.
22.2. Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
.
23. RFspectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
.
23.1. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
.
23.2. Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . . . . . 29
.
23.3. Clock shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
.
23.4. Thespecial case of 6Li . . . . . . . . . . . . . . . . . . . . . . . . . 33
.
23.5. Preparation of a two-component system . . . . . . . . . . . . . . . 33
.
24. Using and characterizing Feshbach resonances . . . . . . . . . . . . . . . . 35
.
24.1. High magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 35
.
24.2. Methods for making molecules . . . . . . . . . . . . . . . . . . . . . 36
.
24.3. Observation of Feshbach resonances . . . . . . . . . . . . . . . . . . 39
.
24.4. Determination of thecoupling strength of Feshbach resonances . . 41
.
24.5. Therapid ramp technique . . . . . . . . . . . . . . . . . . . . . . . 44
.
25. Techniquesto observe cold atoms and molecules . . . . . . . . . . . . . . . 46
.
25.1. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
.
25.2. Tomographic techniques . . . . . . . . . . . . . . . . . . . . . . . . 47
.
25.3. Distinguishing atoms from molecules . . . . . . . . . . . . . . . . . 48
3. Quantitativeanalysis of densitydistributions . . . . . . . . . . . . . . . . . . . . 51
.
31. Trapped atomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
.
31.1. IdealBose and Fermigases in a harmonic trap . . . . . . . . . . . 51
.
31.2. Trapped,interacting Fermimixtures at zero temperature . . . . . . 54
.
32. Expansion of strongly interacting Fermi mixtures . . . . . . . . . . . . . . 59
.
32.1. Freeballistic expansion . . . . . . . . . . . . . . . . . . . . . . . . . 59
.
32.2. Collisionally hydrodynamicexpansion . . . . . . . . . . . . . . . . 61
.
32.3. Superfluidhydrodynamicexpansion . . . . . . . . . . . . . . . . . . 63
.
33. Fitting functions for trapped and expanded Fermi gases. . . . . . . . . . . 68
.
33.1. Non-interactingFermi gases . . . . . . . . . . . . . . . . . . . . . . 69
.
33.2. Resonantly interacting Fermi gases . . . . . . . . . . . . . . . . . . 72
.
33.3. Molecular clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4. Theory of the BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . . . . . . . 77
.
41. Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
.
42. Pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2
.
43. Cooper instability in a Fermi gas with attractive interactions . . . . . . . . 82
.
43.1. Two-body bound states in 1D, 2D and 3D . . . . . . . . . . . . . . 82
.
43.2. Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
.
43.3. Pairing of fermions – The Cooper problem . . . . . . . . . . . . . . 86
.
44. Crossover wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
.
45. Gap and numberequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
.
46. Discussion of the threeregimes – BCS, BEC and crossover . . . . . . . . . 94
.
46.1. BCS limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
.
46.2. BEC limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
.
46.3. Evolution from BCS to BEC . . . . . . . . . . . . . . . . . . . . . . 97
.
47. Single-particle and collective excitations . . . . . . . . . . . . . . . . . . . 99
.
47.1. Single-particle excitations . . . . . . . . . . . . . . . . . . . . . . . 100
.
47.2. RFexcitation into a third state . . . . . . . . . . . . . . . . . . . . 101
.
47.3. Collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 104
.
47.4. Landau criterion for superfluidity . . . . . . . . . . . . . . . . . . . 105
.
48. Finite temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
.
48.1. Gap equation at finitetemperature . . . . . . . . . . . . . . . . . . 107
.
48.2. Temperatureof pair formation. . . . . . . . . . . . . . . . . . . . . 107
.
48.3. Critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 108
.
48.4. “Preformed” pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
.
49. Long-range order and condensate fraction. . . . . . . . . . . . . . . . . . . 109
.
410. Superfluiddensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
.
411. Orderparameter and Ginzburg-Landau equation. . . . . . . . . . . . . . . 115
.
412. Crossing overfrom BEC to BCS . . . . . . . . . . . . . . . . . . . . . . . . 117
5. Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
.
51. History and experimental summary . . . . . . . . . . . . . . . . . . . . . . 120
.
52. Scattering resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
.
53. Feshbach resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
.
53.1. A model for Feshbach resonances . . . . . . . . . . . . . . . . . . . 124
.
54. Broad versusnarrow Feshbach resonances . . . . . . . . . . . . . . . . . . 127
.
54.1. Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
.
54.2. Criterion for a broad resonance . . . . . . . . . . . . . . . . . . . . 129
.
54.3. Coupling energy scale. . . . . . . . . . . . . . . . . . . . . . . . . . 131
.
54.4. Narrow Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . 132
.
55. Open channel resonance and the case of 6Li . . . . . . . . . . . . . . . . . 132
6. Condensation and superfluidityacross theBEC-BCS crossover . . . . . . . . . . 138
.
61. Bose-Einstein condensation and superfluidity . . . . . . . . . . . . . . . . . 138
.
62. Signatures for superfluidityin quantumgases . . . . . . . . . . . . . . . . 139
.
63. Pair condensation below the Feshbach resonance . . . . . . . . . . . . . . . 141
.
64. Pair condensation abovethe Feshbach resonance . . . . . . . . . . . . . . . 143
.
64.1. Comparison with theory . . . . . . . . . . . . . . . . . . . . . . . . 146
.
64.2. Formation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 148
.
65. Direct observation of condensation in the density profiles . . . . . . . . . . 149
.
65.1. Anomalous density profiles at unitarity . . . . . . . . . . . . . . . . 150
.
65.2. Direct observation of the onset of condensation in Fermi mixtures
with unequalspin populations . . . . . . . . . . . . . . . . . . . . . 152
.
66. Observation of vortex lattices . . . . . . . . . . . . . . . . . . . . . . . . . 155
.
66.1. Somebasic aspects of vortices . . . . . . . . . . . . . . . . . . . . . 155
.
66.2. Realization of vortices in superconductors and superfluids . . . . . 156
.
66.3. Experimental concept . . . . . . . . . . . . . . . . . . . . . . . . . 157
3
.
66.4. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
.
66.5. Observation of vortex lattices . . . . . . . . . . . . . . . . . . . . . 162
.
66.6. Vortexnumberand lifetime . . . . . . . . . . . . . . . . . . . . . . 165
.
66.7. A rotating bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
.
66.8. Superfluidexpansion of a rotating gas . . . . . . . . . . . . . . . . 167
7. BEC-BCS crossover: Energetics, excitations, and new systems. . . . . . . . . . . 170
.
71. Characterization of the equilibrium state . . . . . . . . . . . . . . . . . . . 170
.
71.1. Energy measurements . . . . . . . . . . . . . . . . . . . . . . . . . 170
.
71.2. Momentum distribution . . . . . . . . . . . . . . . . . . . . . . . . 171
.
71.3. Molecular character . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
.
72. Studiesof excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
.
72.1. Collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 173
.
72.2. Speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
.
72.3. Critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
.
72.4. RFspectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
.
73. New systems with BEC-BCS crossover . . . . . . . . . . . . . . . . . . . . 178
.
73.1. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
.
73.2. Population imbalanced Fermi mixtures . . . . . . . . . . . . . . . . 180
8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4
1. – Introduction
.
11.State of the field.–This papersummarizesthe experimentalfrontierofultracold
fermionicgases. ItisbasedonthreelectureswhichoneoftheauthorsgaveattheVarenna
summer schooldescribing the experimentaltechniques usedto study ultracoldfermionic
gases, and some of the results obtained so far. In many ways, the area of ultracold
fermionicgaseshasgrownoutofthestudyofBose-Einsteincondensates. Aftertheirfirst
experimental realizations in 1995 [1, 2], the field of BEC has grownexplosively. Most of
the exploredphysics was governedby mean-field interactions, conveniently described by
the Gross-Pitaevskii equation. One novel feature of trapped inhomogeneous gases was
the spatially varying density, that allowed for the direct observation of the condensate,
but also led to new concepts of surface effects and collective excitations which depended
on the shape of the cloud. The experimental and theoretical explorations of these and
other features have been a frontier area for a whole decade!
A major goal had been to go beyond mean field physics, which is in essence sin-
gle particle physics, and to find manifestations of strong interactions and correlations.
Three avenues have been identified: lower dimensions that enhance the role of fluctua-
tions and correlations, optical lattices that can suppress the kinetic energy in the form
of tunnelling [3, 4], and Feshbach resonances [5, 6, 7, 8] that enhance interactions by
resonantly increasing the interparticle scattering length. In bosonic systems, the tuning
of interactions near Feshbachresonances was of limited applicability due to rapid losses.
Feshbach resonances were used mainly to access molecular states of dimers and trimers.
In contrast, for fermions, losses are heavily suppressed (see below), and so most of this
review focuses on strongly interacting fermions near Feshbach resonances.
Byaddressingthephysicsofstronglycorrelatedmatter,thefieldofultracoldatomsis
entering a new stagewhere we expect major conceptionaladvances in, andchallenges to
many-body theory. We regardit as fortunate that BEC turned out to be a less complex
target(both experimentally and theoretically),and overa decade, important techniques
and methods have been developed and validated, including experimental techniques to
confine andcoolnanokelvinatoms,the use ofFeshbachresonancestomodify theirprop-
erties, and many theoretical concepts and methods to describe trapped ultracold gases
and their interactions. What we are currently experiencing is the application of these
powerful methods to strongly correlated systems, and due to the maturity of the field,
the developments have been breath-taking, in particular with bosons in optical lattices
andfermionsinteractingviaFeshbachresonances. Itispossiblethatthe mostimportant
conceptional advances triggered by the advent of Bose-Einstein condensation are yet to
be discovered.
Itisamusingtonotethatincertainlimits,stronglycorrelatedfermionpairsareagain
describedbyamean-fieldtheory. Theirwavefunctionisaproductofidenticalpairwave
functions (albeit correctly anti-symmetrized), that for strong binding of the pairs turns
intothestatedescribedbytheGross-Pitaevskiiequation. Thisisthesimplestdescription
oftheBEC-BCScrossover. Still,thefactthatpairinghasnowbecomeamany-bodyaffair
stands for the advent of a new era in ultracold atom physics.
5
.
12. Strongly correlated fermions - a gift of nature?. – It shows the dynamics of
the field of ultracold atoms that the area of strongly interacting fermions has not been
expected or predicted. This may remind us of the pre-BEC era, when many people
considered BEC to be an elusive goal, made inaccessible by inelastic interactions at the
densities required [9]. When Feshbach resonances were explored in bosonic systems,
strong interactions were always accompanied by strong losses, preventing the study of
stronglyinteractingcondensates[7,10,11]. ThereasonisthataFeshbachresonancecou-
ples the atomic Hilbert space to a resonantmolecular state which is vibrationally highly
excited. Collisions can couple this state to lower lying states (vibrational relaxation).
What occurred in Fermi gases, however, seemed too good to be true: all relaxation
mechanismsweredramaticallysuppressedbytheinterplayofthePauliexclusionprinciple
andthelargesizeoftheFeshbachmolecules. SowhatwehavegotisaHilbertspacewhich
consistsofatomiclevelsplusonesinglemolecularlevelresonantlycoupledtotwocolliding
atoms. Allothermolecularstatescoupleonlyweakly. Asaresult,paircondensationand
fermionic superfluidity could be realized by simply ramping down the laser power in an
optical trap containing 6Li in two hyperfine states at a specific magnetic field, thereby
evaporatively cooling the system to the superfluid state. Even in our boldest moments
we would not have dared to ask Nature for such an ideal system.
Before the discovery of Feshbach resonances, suggestions to realize fermionic super-
fluidity focused onlithium because ofthe unusually large andnegative triplet scattering
length[12,13,14]. However,amajorconcernwaswhetherthegaswouldbestableagainst
inelastic collisions. The stability of the strongly interacting Fermi gas was discovered in
Paris in the spring of 2003, when long-lived Li molecules were observed despite their
2
high vibrational excitation [15](1). This and subsequent observations [17, 18] were soon
explained as a consequence of Pauli suppression [19]. Within the same year, this un-
expected stability was exploited to achieve condensation of fermion pairs. This unique
surprise has changed the field completely. Currently, more than half of the research
program of our group is dedicated to fermions interacting near Feshbach resonances.
There is another aspect of Fermi gases, which turned out to be more favorable than
expected. Early work on the BCS state in ultracold gases suggested a competition
between superfluidity and collapse (for negative scattering length) or coexistence and
phase separation (for positive scattering length) when the density or the absolute value
of the scattering length a exceeded a certain value, given by k a = π/2, where k is
F F
| |
the Fermi wave vector [13, 20, 21]. This would have implied that the highest transition
temperatures to the superfluid state would be achieved close to the limit of mechanical
stability, and that the BCS-BEC crossover would be interrupted by a window around
theFeshbachresonance,wherephaseseparationoccurs. Fortunately,unitaritylimitsthe
maximum attractive energy to a fraction of the Fermi energy (βE with β 0.58),
F
≈ −
completely eliminating the predicted mechanical instability.
(1) The observation of long lifetimes of molecules outside a narrow Feshbach resonance [16] is
not yet understood and has not been used to realize a strongly interacting gas.
6
Finally,athirdaspectreceivedalotofattention,namelyhowtodetectthesuperfluid
state. Since no major change in the spatial profile of the cloud was expected [21], sug-
gesteddetectionschemesincludedachangeinthedecayrateofthegas[21],opticallight
scattering of Cooper pairs [22, 23], optical breakup of Cooper pairs [24], modification
of collective excitations [25, 26], or small changes in the spatial shape [27]. All these
signaturesare weakor complicatedto detect. Fortunately,muchclearerand moreeasily
detectable signatures were discovered. One is the onset of pair condensation, observed
through a bimodal density distribution in expanding clouds, observed either well below
the Feshbach resonance or after rapid sweeps of the magnetic field. Another striking
signature was the sudden change in the cloud shape when fermion mixtures with pop-
ulation imbalance became superfluid, and finally, the smoking gun for superfluidity was
obtained by observing superfluid flow in the form of quantized vortices.
Our ultimate goal is to control Nature and create and explore new forms of matter.
But inthe end, it is Nature whosets the rules,andin the case ofultracoldfermions, she
has been very kind to us.
.
13. Some remarks on the history of fermionic superfluidity. –
.
13.1. BCSsuperfluidity.Manycoldfermioncloudsarecooledbysympatheticcooling
with a bosonic atom. Popular combinations are 6Li and 23Na, and 40K and 87Rb. It
is remarkable that the first fermionic superfluids were also cooled by a Bose-Einstein
condensate. KamerlinghOnnesliquefied4Hein1908,andlowereditstemperaturebelow
thesuperfluidtransitionpoint(theλ-point)atT =2.2K.InhisNobellecturein1913,he
λ
notes “thatthe density ofthe helium, whichatfirstquicklydropswiththe temperature,
reaches a maximum at 2.2 K approximately, and if one goes down further even drops
again. Suchanextremecouldpossiblybeconnectedwiththequantumtheory”[28]. But
insteadofstudying,whatweknownowwasthefirstindicationofsuperfluidityofbosons,
he firstfocusedonthe behaviorof metalsatlow temperatures. In1911,Onnes used4He
to cooldownmercury,finding thatthe resistivityofthe metal suddenly droppedto non-
measurablevaluesatT =4.2K,itbecame“superconducting”. Tin(atT =3.8K)and
C C
lead(atT =6 K)showedthe sameremarkablephenomenon. This wasthe discoveryof
C
superfluidity in an electron gas.
The fact that bosonic superfluidity and fermionic superfluidity were first observed
at very similar temperatures, is due to purely technical reasons (because of the avail-
able cryogenicmethods) and rather obscures the very different physics behind these two
phenomena.
Bosonic superfluidity occurs at the degeneracy temperature, i.e. the temperature T
at which the spacing between particles n 1/3 at density n becomes comparable to the
−
thermaldeBrogliewavelengthλ= 2π~2 ,wheremistheparticlemass. Thepredicted
mkBT
transition temperature of T 2qπ~2n2/3 3 K for liquid helium at a typical density
BEC ∼ m ≈
of n=1022 cm 3 coincides with the observed lambda point.
−
In contrast, the degeneracy temperature (equal to the Fermi temperature T
F
≡
E /k ) for conduction electrons is higher by the mass ratio m(4He)/m , bringing it
F B e
7
up to several ten-thousand degrees. It was only in 1957 when it became clear why
in fermionic systems, superfluidity occurs only at temperatures much smaller than the
degeneracy temperature.
Of course,the main difference to Bose gases is that electrons,being fermions, cannot
be in one and the same quantum state but instead must arrange themselves in different
states. An obvious scenario for superfluidity might be the formation of tightly bound
pairsofelectronsthatcanactasbosonsandcouldformacondensate. Butapartfromthe
problemthatthecondensationtemperaturewouldstillbeontheorderofE /k ,thereis
F B
noknowninteractionwhichcouldbesufficienttoovercomethestrongCoulombrepulsion
and form tightly bound electron pairs (Schafroth pairs [29]). The idea itself of electrons
forming pairs was indeed correct, but the conceptual difficulties were so profound that
it took several decades from the discovery of superconductivity to the correct physical
theory.
In 1950, it became clear that there was indeed an effective attractive interaction be-
tween electrons, mediated by the crystal lattice vibrations (phonons), that was respon-
sible for superconductivity. The lattice vibrations left their mark in the characteristic
variation T 1/√M of the critical temperature T with the isotope mass M of the
C C
∝
crystalions,the isotopeeffect[30,31]predictedbyH.Fr¨ohlich[32]. Vibrationalenergies
in the lattice are a factor m /M smaller than the typical electronic energy(2) E , on
e F
the order of k several 100 K (the Debye temperature T of the metal). While the
B× p D
isotope effect strongly argues for T being proportional to T , the Debye temperature
C D
is still one or two orders of magnitude higher than the observed critical temperature.
A breakthrough came in 1956, when L. Cooper realized that fermions interacting
via an arbitrarily weak attractive interaction on top of a filled Fermi sea can form a
bound pair [33]. In other words, the Fermi sea is unstable towards pair formation.
However, unlike the tightly bound pairs considered before, the “Cooper” pair is very
large, much larger than the interparticle spacing. That is, a collection of these pairs
necessarily needs to overlap very strongly in space. In this situation, it was far from
obvious whether interactions between different pairs could simply be neglected. But
it was this simplifying idea that led to the final goal: Bardeen, Cooper and Schrieffer
(BCS) developed a full theory of superconductivity starting from a new, stable ground
state in which pair formation was included in a self-consistent way [34]. Using the
effectivephonon-mediatedelectron-electroninteractionV,attractiveforenergiessmaller
than k T and assumed constant in this regime, the pair binding energy was found to
B D
be ∆ = 2kBTDe−1/ρF|V|, with ρF = mekF/2π2~2 the density of states at the Fermi
energy and ρ V assumed small compared to 1. The bound state energy or the pairing
F
| |
gap depended in the non-analytic fashion e−1/ρF|V| on the effective electron-electron
(2) Theaveragedistancebetweenelectronsr isontheorderofatomicdistances(severalBohr
0
radiia ),theFermienergyE ~2/m r2 isthusonthescaleoftypicalCoulombenergiesinan
0 F ∼ e 0
atom. Vibrational energies of thelattice ions arethenon theorder~ωD ≈~q∂2UC∂oru2lomb/M ∼
~ E /Mr2 m /ME .
F 0 ∼ e F
p p
8
Fig.1.–TheBEC-BCScrossover. Bytuningtheinteractionstrengthbetweenthetwofermionic
spinstates,onecansmoothlycrossoverfromaregimeoftightlyboundmoleculestoaregimeof
long-rangeCooperpairs,whosecharacteristicsizeismuchlargerthantheinterparticlespacing.
In between these two extremes, one encounters an intermediate regime where the pair size is
comparable to theinterparticle spacing.
interaction V, explaining why earlier attempts using perturbation theory had to fail.
Also, this exponential factor can now account for the small critical temperatures T
C
≃
5K: Indeed,itisaresultofBCStheorythatk T issimplyproportionalto∆ ,thepair
B C 0
binding energy at zero temperature: k T 0.57∆ . Hence, the critical temperature
B C 0
≈
TC TDe−1/ρF|V| is proportional to the Debye temperature TD, in accord with the
∼
isotope effect, but the exponential factor suppresses T by a factor that can easily be
C
100.
.
13.2. The BEC-BCScrossover.Earlywork on BCStheory emphasized the different
natureofBECandBCStype superfluidity. Alreadyin1950Fritz Londonhadsuspected
thatfermionicsuperfluiditycanbeunderstoodasapaircondensateinmomentumspace,
in contrast to a BEC of tightly bound pairs in real space [35]. The former will occur
for the slightest attraction between fermions, while the latter appears to require a true
two-body bound state to be available to a fermion pair. Schrieffer points out that BCS
superfluidity is not Bose-Einstein condensation of fermion pairs, as these pairs do not
obey Bose-Einstein statistics [36]. However, it has become clear that BEC and BCS
superfluidity are intimately connected. A BEC is a special limit of the BCS state.
It was Popov [37], Keldysh and collaborators [38] and Eagles [39] who realized in
different contexts that the BCS formalism and its ansatz for the ground state wave
function provides not only a gooddescription for a condensate of Cooper pairs,but also
foraBose-Einsteincondensateofadilutegasoftightlyboundpairs. Forsuperconductors,
Eagles [39] showed in 1969 that, in the limit of very high density, the BCS state evolves
into a condensate of pairs that can become even smaller than the interparticle distance
and should be described by Bose-Einsteinstatistics. In the language of Fermi gases, the
scattering length was held fixed, at positive and negative values, and the interparticle
spacing was varied. He also noted that pairing without superconductivity can occur
abovethesuperfluidtransitiontemperature. Usingagenerictwo-bodypotential,Leggett
9
showed in 1980 that the limits of tightly bound molecules and long-range Cooper pairs
are connected in a smooth crossover [40]. Here it was the interparticle distance that
was fixed, while the scattering length was varied. The size of the fermion pairs changes
smoothly from being much larger than the interparticle spacing in the BCS-limit to the
small size of a molecular bound state in the BEC limit (see Fig. 1). Accordingly, the
pair binding energy varies smoothly from its small BCS value (weak, fragile pairing) to
the large binding energy of a molecule in the BEC limit (stable molecular pairing).
The presence of a paired state is in sharp contrast to the case of two particles in-
teracting in free (3D) space. Only at a critical interaction strength does a molecular
state become available and a bound pair can form. Leggett’s result shows that in the
many-body system the physics changes smoothly with interaction strength also at the
point where the two-body bound state disappears. Nozi`eres and Schmitt-Rink extended
Leggett’s model to finite temperatures and verified that the critical temperature for su-
perfluidity varies smoothly from the BCS limit, where it is exponentially small, to the
BEC-limit where one recoversthe value for Bose-Einsteincondensationof tightly bound
molecules [41].
The interest in strongly interacting fermions and the BCS-BEC crossover increased
with the discovery of novel superconducting materials. Up to 1986, BCS theory and its
extensionsandvariationswerelargelysuccessfulinexplainingthepropertiesofsupercon-
ductors. The record critical temperature increased only slightly from 6 K in 1911 to 24
K in 1973 [42]. In 1986, however,Bednorz and Mu¨ller [43] discovered superconductivity
at 35 K in the compound La Ba CuO , triggering a focused search for even higher
2 x x 4
−
critical temperatures. Soon after, materials with transition temperatures above 100 K
werefound. Duetothestronginteractionsandquasi-2Dstructure,theexactmechanisms
leading to High-T superconductivity are still not fully understood.
C
The physics of the BEC-BCS crossover in a gas of interacting fermions does not di-
rectlyrelatetothecomplicatedphenomenaobservedinHigh-T materials. However,the
C
two problems shareseveralfeatures: In the crossoverregime,the pair size is comparable
to the interparticle distance. This relates to High-T materials where the correlation
C
length(“pairsize”)isalsonotlargecomparedtotheaveragedistancebetweenelectrons.
Therefore,wearedealingherewithastronglycorrelated“soup”ofparticles,whereinter-
actions between different pairs of fermions can no longer be neglected. In both systems
the normal state above the phase transition temperature is far from being an ordinary
Fermi gas. Correlations are still strong enough to form uncondensed pairs at finite mo-
mentum. In High-T materials, this region in the phase diagram is referred to as the
C
“Nernst regime”, part of a larger region called the “Pseudo-gap”[44].
One point in the BEC-BCS crossover is of special interest: When the interparticle
potentialisjustaboutstrongenoughtobindtwoparticlesinfreespace,the bondlength
of this molecule tends to infinity (unitarity regime). In the medium, this bond length
will not play any role anymore in the description of the many-body state. The only
length scale of importance is then the interparticle distance n 1/3, the corresponding
−
energy scale is the Fermi energy E . In this case, physics is said to be universal [45].
F
The average energy content of the gas, the binding energy of a pair, and (k times)
B
10