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Superﬂuidity in a Gas of Strongly Interacting Fermions 501

quench the superﬂuid state. This is the reason why we have a partially-

polarized normal phase near resonance even at zero temperature. With

stronger coupling, the critical polarization σ

of a partially-polarized nor-

mal phase increases, and becomes unity at a critical interaction strength of

1/k

F ↑

a ≈ 0.7.

This means that beyond the critical coupling all minority

atoms pair up with majority atoms and form a Bose condensate. This is

the regime where a polarized two-component Fermi gas can be eﬀectively

described as a Bose-Fermi mixture.

In the limit of a BF mixture,

we have observed repulsive interactions

between the fermion dimers and unpaired fermions. They are parameterized

by an eﬀective dimer-fermion scattering length of a

= 1.23(3)a. This

value is in reasonable agreement with the exact value a

= 1.18a which has

been predicted over 50 years ago for the three fermion problem,

but has

never been experimentally conﬁrmed. Our ﬁnding excludes the mean-ﬁeld

prediction a

= (8/3)a. The boson-boson interactions were found to be

stronger than the mean-ﬁeld prediction in agreement with the Lee-Huang-

Yang prediction.

Including the LHY correction, the eﬀective dimer-dimer

scattering length was determined to be a

= 0.55(1)a, which is close to the

exact value for weakly bound dimers a

= 0.6a.

8. Tomographic RF Spectroscopy with a New Superﬂuid

RF spectroscopy of a two-component Fermi gas measures a single-particle

excitation spectrum by ﬂipping the spin state of an atom to a third spin

state. Since a fermion pair can be dissociated via spin ﬂip, RF spectroscopy

provides valuable information about the pair such as binding energy and

size. In early experiments,

58,59

a spectral shift has been observed in a Fermi

gas at low temperature and interpreted as a manifestation of pairing. How-

ever, it turned out that the spectral line shape is severely aﬀected by the

strong interactions of the third, ﬁnal spin state and broadened due to the

inhomogeneous density distribution of a trapped sample, preventing clear

comparison of the experimental results to theory. Recently, we have devel-

oped several experimental techniques to overcome these problems. In order

to minimize ﬁnal state eﬀects we have exploited a new spin mixture of states

|1i and |3i of

Li atoms

(corresponding to |F = 1/2, m

= 1/2i and

|F = 3/2, m

= −3/2i at low ﬁeld), and using a tomographic technique, we

have obtained local RF spectra from an inhomogeneous sample.

Figure 4 shows the RF spectra of the various phases in a trapped sample

with population imbalance. For a balanced superﬂuid, the majority and the

October 4, 2010 10:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch19

502 W. Ketterle, Y. Shin, A. Schirotzek and C. H. Schunk

spectral response / a.u.

3210

RF offset /

↑

3210 43210-1

a) b)

Fig. 4. Tomographic RF spectroscopy of strongly interacting Fermi mixtures. A

trapped, inhomogeneous sample has various phases in spatially diﬀerent regions.

The spectra of each region (red: majority, blue: minority) reveals the nature of

pairing correlation of the corresponding phase. (a) Balanced superﬂuid. (b) Polar-

ized superﬂuid. The additional peak in the majority spectrum is the contribution of

the excess fermions, which can be identiﬁed as fermionic quasiparticles in a super-

ﬂuid. From the separation of the two peaks, the pairing gap energy of a resonantly

interacting superﬂuid has been determined.

minority peak no longer overlaps with the majority spectrum, indicating the tran-

sition to polaronic correlations.

minority spectra completely overlap, showing the characteristic behavior of

pair dissociation, i.e. a sharp threshold and a slow high-energy tail. From the

spectral width, we have determined the pair size to be 2.6(2)/k

at unitarity,

about 20% smaller than the interparticle spacing.

These are the smallest

pairs so far observed in fermionic superﬂuids, highlighting the importance

of small fermion pairs for superﬂuidity at high critical temperature.

Excess fermions in a low-temperature superﬂuid constitute quasiparticles

populating the minimum of the dispersion curve. The RF spectrum of a su-

perﬂuid with such quasiparticles shows two peaks, which, in the BCS limit,

would be split by the superﬂuid gap ∆. Therefore, RF spectroscopy of quasi-

particles is a direct way to observe the superﬂuid gap in close analogy with

tunneling experiments in superconductors. In a polarized superﬂuid near the

phase boundary, we have obtained a local majority spectrum with a double-

peak structure, from which the superﬂuid gap has been determined to be

∆ = 0.44(3)E

F ↑

at unitarity.

In addition, a Hartree term of −0.43(3)E

F ↑

is necessary to explain the observed spectral behavior. Evidence for a gap

was also obtained using momentum resolved RF spectroscopy.

The peak positions of the majority and the minority spectra become

diﬀerent in the partially-polarized normal phase, but still overlap in the

October 4, 2010 10:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch19

Superﬂuidity in a Gas of Strongly Interacting Fermions 503

high-energy tail. At large spin polarization, the limit of a single minority

immersed in a majority Fermi sea is approached, where several theoretical

studies suggest a polaron picture, associating the minority with weakly in-

teracting quasiparticles in a normal Fermi liquid.

63–65

More recently, this

picture has been experimentally conﬁrmed and the binding energy

and the

eﬀective mass

of these polarons have been determined. Figure 4 shows that

the diﬀerent kinds of pairing correlations are smoothly connected across the

superﬂuid-to-normal phase transition at ﬁnite temperature.

9. Summary and Discussion

In a series of experiments with population-imbalanced Fermi mixtures near

Feshbach resonances, we have established the phase diagram of a two-

component Fermi gas with strong interactions. This includes the identiﬁca-

tion of a tricritical point at which the critical lines for ﬁrst-order and second-

order phase transitions meet, and the veriﬁcation of a zero-temperature

quantum phase transition from a balanced superﬂuid to a partially-polarized

normal gas at unitarity. The observed critical points such as the critical po-

larization of a normal phase and the critical interaction strength of a compos-

ite boson in a Fermi sea provide quantitative tests for theoretical treatments

of strongly interacting fermions.

Our work can be summarized with the phase diagram of the system in a

3D parameter space (versus temperature, spin polarization and interaction

strength) shown in Fig. 5. For a complete understanding, this macroscopic

characterization of the diﬀerent phases should be complemented by an in-

vestigation of their microscopic properties. At low temperature, one can in-

terpret the observed polarized superﬂuid as a result of thermal population of

spin-polarized quasiparticles. However, the behavior at higher temperature

or/and in a stronger coupling regime is not yet completely understood and

could include a gapless region (h > ∆). Another open question is whether

the Fermi liquid description is still valid for high minority concentrations,

where Pauli blocking in the minority Fermi sea might play an important role.

Furthermore, it has been speculated that exotic pairing states might exist

in the partially-polarized phase.

So far, predicted exotic superﬂuid states

such as the breached-pair state in a stronger coupling regime and the FFLO

state in a weaker coupling regime have not been observed, but they may be

hidden in thin layers near the superﬂuid-normal boundary. This discussion

underlines that the population imbalanced Fermi system is the richest sys-

tem realized thus far with ultracold gases and therefore nicely illustrates the

October 4, 2010 10:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch19

504 W. Ketterle, Y. Shin, A. Schirotzek and C. H. Schunk

Normal

Bose Liquid

Fermi Liquid

Temperature

Interaction

Strength

Spin polarization

BCS-side

BEC-side

Superfluid

Phase separation

Bose-Fermi

mixture

Fig. 5. Various phases of a two-component Fermi gas. The structure of the phase

diagram is illustrated in the parameter space of temperature, interaction strength

and spin polarization.

novel approach to engineer interesting many-body systems using the tools

and methods of atomic physics.

Acknowledgments

This work was supported by NSF, ONR, MURI and ARO Award W911NF-

07-1-0493 (DARPA OLE Program).

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October 4, 2010 10:17 World Scientiﬁc Review Volume - 9.75in x 6.5in ch20

BCS FROM NUCLEI AND NEUTRON STARS TO

QUARK MATTER AND COLD ATOMS

Gordon Baym

Department of Physics, University of Illinois,

1110 W. Green St., Urbana, IL 61801, USA

This article describes extensions of BCS theory to atomic nuclei and neutron

stars, to high density quark matter and connections to pairing in ultracold

clouds of atomic fermions.

1. Introduction

Starting very shortly after its initial discovery, BCS theory began to be ap-

plied to physical systems beyond laboratory superconductors, starting with

pairing of nucleons in atomic nuclei,

and its extension soon after to neutron

star matter.

2,3

Its impact on models of elementary particles and broken sym-

metry in high energy physics

has been profound. BCS formed the basis of

predicting a superﬂuid state

of degenerate

He dissolved in superﬂuid

He,

which would produce a novel mixture of a fermion superﬂuid dissolved in a

Bose superﬂuid.

Application to the helium liquids reached fruition with the

discovery

of the low temperature superﬂuid states of

He. Later BCS was

applied to pairing of quarks in degenerate quark-gluon plasmas,

and more

recently to ultracold atomic fermion clouds.

In view of the articles in this volume by Leggett on superﬂuid

He, Ket-

terle on cold atoms, and Weinberg and Wilcek on high energy applications,

this article will focus on BCS in atomic nuclei, and its extensions to neutron

star matter, and then high density quark matter, as well as touching on a

few connections of quark matter physics to ultracold atomic physics.

∗

Research discussed here has been supported in part by the U.S. National Science

Foundation over the years, most recently by NSF Grant PHY07-01611.

Unfortunately the BCS transition temperature was in the microkelvin range, well

beyond reach.

509

October 4, 2010 10:17 World Scientiﬁc Review Volume - 9.75in x 6.5in ch20

510 G. Baym

2. Application to Nuclei: Copenhagen, 1957

The ﬁrst application of BCS, to pairing of even numbers of neutrons or pro-

tons outside closed shells in atomic nuclei, had its inception before the ink

was even dry on the original BCS paper.

In the summer of 1957, David

Pines brought news of the BCS theory to Niels Bohr’s Institute in Copen-

hagen — the University Institute for Theoretical Physics — where he was

asked by Aage Bohr and Ben Mottelson to lecture on the theory. The lec-

ture, with Niels Bohr in the audience,

inspired discussion among A. Bohr,

Mottelson, and Pines of possible extensions of BCS to nuclei; they soon re-

alized that BCS would provide the solution to the outstanding puzzle of the

much larger single particle excitation energies, of order in MeV, in nuclei

that have an even number of neutrons and protons (even–even nuclei), com-

pared with the excitation energies in nuclei in which the neutron number or

proton number is odd (odd-A nuclei). The eﬀect — seen in the mass range

130 < A < 190 and A > 228, and strikingly in the high-resolution spectro-

scopic studies of

182

W versus

183

W — was as they realized the analog of the

excitation gap in metallic superconductors; the ground states of even–even

nuclei are eﬀectively BCS paired with a gap ∼ 12/

√

A MeV. Pines gave a

preliminary report of this ﬁrst application of BCS to nuclei in September at

the International Conference on Nuclear Structure at the Weizmann Insti-

tute,

and their paper, “Possible analogy between the excitation spectra of

nuclei and those of the superconducting state,” was submitted at the end of

the year.

In addition to pinning down the origin of the energy gap in nuclei, the

paper makes the crucial observation that, “The correlations giving rise to

the energy gap may also aﬀect many other nuclear properties; thus, they

appear to be responsible for the observed fact that the rotational moments

of inertia are appreciably smaller than the values corresponding to rigid

rotation.”

The rotational spectrum has the simple semi-classical form E =

I(I + 1)/2I, where ~I is the total angular momentum of the nucleus, and

I is the rotational moment of inertia; more widely spaced levels indicate

immediately that the moment of inertia is reduced from its classical value.

This reduction of the moment of inertia is in fact the analogue in neutral

Schrieﬀer, who was in Copenhagen in Fall 1957, recalls Niels Bohr commenting on

the BCS theory, “It’s an interesting idea but nature isn’t that simple.”

Bohr himself

had struggled with superconductivity; his paper, “Zur frage der Superleitung,”

submitted to Naturwissenschaften in June 1932, was withdrawn in proof, in response

to Felix Bloch’s strong criticisms.