Strongly correlated systems in atomic and condensed matter physics …cmt.harvard.edu/demler/TEACHING/Physics284/chapter11.pdf · Strongly correlated systems in atomic and condensed

Strongly correlated systems

in atomic and condensed matter physics

Lecture notes for Physics 284

by Eugene Demler

Harvard University

March 26, 2018

2

Chapter 11

Fermion pairing close toFeshbach resonance. TheBCS-BEC crossover.

11.1 BCS model in electron systems

We start by reviewing the BCS model of electron pairing in electron systems[13]. We consider a reduced Hamiltonian that includes terms which are decisivefor pairing of fermions

H =∑kσ

ξk c†kσ ckσ +

∑kl

Vkl c†k↑ c†−k↓ c−l↓ cl↑ (11.1)

Here ξk = k2

2m − µ ≡ εk − µ. The BCS wavefunction describes a many-bodystate with pairs of states (k ↑,−k ↓) occupied or unoccupied as units in coherentmanner. In the mean-field approximation we can describe such state by assign-ing finite expectation values to operators ck↑ c−k↓. The mean-field Hamiltonianis given by

H =∑kσ

ξk c†kσ ckσ +

∑kl

Vkl ( c†k↑ c†−k↓ bl + b∗k c−l↓ cl↑ − b∗kbl ) (11.2)

Here bk are numbers, not operators, which need to be determined self-consistently.We define

∆k =∑l

Vkl bl =∑l

Vkl 〈c−l↓ cl↑ 〉 (11.3)

Then

HMF =∑kσ

ξk c†kσ ckσ +

∑k

( ∆k c†k↑ c†−k↓ + ∆∗k c−k↓ ck↑ −∆k b

∗k ) (11.4)

3

4CHAPTER 11. FERMION PAIRING CLOSE TO FESHBACH RESONANCE. THE BCS-BEC CROSSOVER.

Fermionic part of the Hamiltonian can be diagonalized by the Bogoliubov trans-formation. In simple s-wave superconductors ∆k should have the same phasefor all k, although the actual value of the phase is not important. For simplicitywe set ∆k to be real

ck↑ = ukγk↑ + vkγ†−k↓

c−k↓ = −vkγ†k↑ + ukγ−k↓ (11.5)

When coefficients uk and vk satisfy

2ξkukvk + ∆ku2k −∆kv

2k = 0

u2k + v2

k = 1 (11.6)

we find that off-diagonal terms in the fermionic Hamitlonian vanish and weobtain

HMF =∑k

Ek ( γ†k↑ γk↑ + γ†k↓ γk↓ ) (11.7)

with

Ek =√ξ2k + ∆2

k (11.8)

From equation (11.3) we obtain self-consistency condition

∆k = −∑k′

Vkk′ 〈c−k′↓ ck′↑ 〉 = −∑k′

Vkk′ uk′ vk′ 〈 ( 1 − γ†k′↑ γk′↑ − γ†k′↓ γk′↓ ) 〉

(11.9)

At finite temperature we have 〈 γ†k′σ γk′σ 〉 = f(Ek′) ≡ 1/(1 + eβEk′ ) so

∆k = −∑k′

Vkk′∆k′

2Ek′tanh(

βEk′

2) (11.10)

There is also self-consistency equation on the chemical potential

n =∑kσ

〈 c†kσ ckσ 〉 = 2∑k

v2k −

∑kσ

(v2k − u2

k) 〈 γ†kσ γkσ 〉 (11.11)

In the BCS model of superconductivity in metals Vkk′ is constant but thereis energy cut-off |ξk| ≤ ωD, where ωD is the phonon Debye frequency. Then

− 1

V=

1

2

∑|ξk|≤ωD

tanh(βEk′2 )

Ek(11.12)

At T = 0 we find

− 1

V=

1

2

∑|ξk|≤ωD

1

Ek(11.13)

11.2. BCS FOR ULTRACOLD ATOMS 5

Assuming ωD << EF so variations of the density of states with energy can beneglected, one finds

∆ =ωD

sinh[

1N(0)V

] ' 2ωD e− 1N(0)V

Tc = 1.14ωD e− 1N(0)V (11.14)

Here N(0) is the density of states at the Fermi energy.

11.2 BCS for ultracold atoms

When extending BCS approach to ultracold atoms (see Refs. [2, 11, 12] forreview), we should remember that there is no energy cut-of such as ωD. Thereis only ultraviolet cut-off in equation (11.13) when |k| ≤ R−1. Our approach isagain to trade bare interaction for the scattering length. We have

m

4πas=

1

V+

∑|k|≤R−1

m

k2(11.15)

We can combine equation (11.15) with

− 1

V=

1

2

∑|k|≤R−1

1

Ek(11.16)

to obtain

− m

4πas=

1

2

∑k

(1

Ek− m

k2) (11.17)

The last integral over k converges and so we sent the cut-off R−1 to infinity.

11.3 Mathematical details

When analyzing equations (11.11), (11.17) it is convenient to define dimension-less quantities [7]

x2 =k2

2m

1

∆

x0 =µ

∆

ξx =ξk∆

= x2 − x0

Ex =Ek∆

=√ξ2x + 1 (11.18)


We can rewrite (11.11), (11.17) as

− 1

as=

2

π(2m∆)1/2 I1(x0)

n =1

2π2(2m∆)3/2 I2(x0) (11.19)

Here

I1(x0) =

∫ ∞0

dxx2(1

Ex− 1

x2)

I2(x0) =

∫ ∞0

dxx2(1− ξxEx

) (11.20)

The last two integrals can be written as

I1(x0) = 2(x0I6(x0)− I5(x0))

I2(x0) =2

3(x0I5(x0) + I6(x0)) (11.21)

where

I5(x0) = (1 + x20)1/4E(

π

2, κ)− 1

4x21(1 + x2

0)1/4Fπ

2, κ)

I6(x0) =1

2(1 + x20)1/4

F (π

2, κ)

x21 =

(1 + x20)1/2 + x0

2

κ2 =x2

1

(1 + x20)1/2

(11.22)

and we used elliptic functions

F (α, κ) =

∫ α

0

dφ1√

1− κ2 sin2 φ

E(α, κ) =

∫ α

0

dφ

√1− κ2 sin2 φ (11.23)

11.3.1 BCS limit

When scattering length is small and negative we have x0 >> 1. In this limit wecan approximate

I5(x0) ≈√x0

I6(x0) ≈ lnx0/2√x0 (11.24)

11.3. MATHEMATICAL DETAILS 7

Then we find

∆0

EF=

1

x0µ

EF= 1

1

kFas= − 2

πlnx0 (11.25)

So we find the BCS like expression

∆gap = ∆ =8

e2EF e

π2kF a (11.26)

11.3.2 BEC Regime

When scattering length is positive and large we have essentially a weakly inter-acting gas of molecules. In this regime x0 < 0 and |x0| >> 1. Then

κ2 =1

4x20

I5(x0) ≈ π

16|x0|3/2

I6(x0) ≈ π

4|x0|1/2(11.27)

This gives us

∆

EF=

(16

3π

)2/3

|x0|1/3

µ

EF= −

(16

3π

)2/3

|x0|4/3

1

kFas=

(16

3π

)2/3

|x0|2/3 (11.28)

We find precisely what we expect

µ = − ~2

2ma2. (11.29)

For the quasiparticle gap we have ∆gap =√

∆2 + µ2. Since ∆ << |µ| we find

∆gap ≈ ~2

2ma2 .

11.3.3 Evolution of quasiparticles in the BCS-BEC crossover

Nontrivial change in the dispersion of quasiparticles across the BCS-BEC crossoveris shown in fig. 11.1. The minimum in the quasiparticle dispersion is at kF inthe BCS regime and at k = 0 on the BEC side.


Figure 11.1: Evolution of fermionic quasiparticles across the BCS-BECcrossover. Figure taken from [12].

Figure 11.2: Transition temperature as a function of the scattering length acrossthe BCS-BEC crossover, calculated using mean-field theory. The diamond cor-responds to the Monte Carlo simulation at unitarity by Burovski et al. Figuretaken from [11].

11.3.4 Phase Diagram

One can also use this approach to calculate ∆(T ). It is natural to define Tc asthe highest temperature at which 〈∆〉 6= 0.

TBCSc =

8eγE

πe2EF e

−π21

kF |as| (11.30)

with γE = 0.57772...

TBECc = 3.31

~2n2/3B

mB= 0.218EF (11.31)

Figure 11.2 shows Tc across the entire Feshbach resonance region. Note a smallincrease of Tc as the system approaches unitarity from the BEC side. This canbe understood as a result of repulsive ineraction between molecules. One knownartifact of the mean-field approach is the prediction of the first order transitioninto the superfluid state on the BEC side of the phase diagram, see [8] and fig.11.3.

11.4. BCS WAVEFUNCION 9

Figure 11.3: Order parameter as a function of the interaction strength and tem-perature. On the BEC side the order parameter exhibits multivalued behaviornear Tc characteristic of the first order transition. Figure taken from [8].

11.4 BCS Wavefuncion

Mean-field solution (11.4) is equivalent to the BCS wavefunction (at T = 0)

|ΨBCS〉 =∏k

(uk + vkc†k↑ c†−k↓)|0〉 (11.32)

The BCS ground state derived in the last section should be the vacuum ofthe Bogoliubov quasiparticles γkσ. Using equations (11.5) we can verify thatγkσ |Ψ〉 = 0.

We can also write equation (11.32) as

|ΨBCS〉 =∏k

uk e∑k

vkuk

c†k↑ c†−k↓ |0〉 (11.33)

The part of |ΨBCS〉 that contains precisely N particles is[∑k

vkuk

c†k↑ c†−k↓

]N/2|0〉 (11.34)

This is a condensate of Cooper pairs with the wavefunction

φ(r) =

∫d3r ei

~k~r v~ku~k

(11.35)


Figure 11.4: Observation of Bose-Einstein condensation of molecules. Thiscollection of images shows bimodal density distributions in both in-situ profiles(Innsbruck and Rice) and after expansion (JILA, MIR, ENS). Figure taken from[12].

Far on the BEC side

φ(r) ∼ 1√2πar

e−r/a (11.36)

11.5 Experimental observation of superfluidityin fermionic systems

On the BEC side of the Feshbach resonance the superfluid state of tightly boundmolecules should be very similar to a condensate of Bosonic atoms. Hence onecan look for standard signatures of superfluidity: bimodal density distributionsboth inside the trap and after some expansion. Observation of vortex lattice isanother important signature of superfluidity.

11.5.1 Projection experiments

On the BCS side of the Feshbach resonance Cooper pairs exist only as a many-body effect. In the dilute limit there are only inidividual atoms so TOF expan-sion unbinds the molecules. To demonstrate pairing in this regime one shouldconvert Cooper pairs into molecules first and expand after that. This can beachieved by quickly sweeping the magnetic field to the BEC side of the resonance[12, 3]. We will discuss only the simplest model of such experiments, in which weassume that the sweep rate is very large, and we can treat these experiments as a

11.5. EXPERIMENTAL OBSERVATIONOF SUPERFLUIDITY IN FERMIONIC SYSTEMS11

Figure 11.5: Experimental demonstration of fermion pair condensates. Bimodaldistribution after short expansion. This figure shows axial density of the atomiccloud after switching off the optical trap, a rapid ramp to zero field, furthershort expansion, and dissociation of the resulting molecules by ramping backacross resonance. Ramping to zero field across the BEC regime converts Cooperpairs from the BCS/unitary regimes into Feshbach molecules (see section 11.5.1).These molecules then expand without dissociating. Figure taken from [12].

projection of the many-body wavefunction. In analyzing real experiments theremay be important corrections due to finite rate of the magnetic field sweep[1].

We can express the operator that creates a molecule in the final point of theprojection experiment as

b†q =

∫dk φf (k) c†q

2 +k↑ c†q2−k↓

(11.37)

Here q is the momentum of the molecule and φf (k) is the molecular wavefunc-tion. The number of molecules with momentum q is nm(q) = b†qbq. For aprojection type experiment we can calculate the number of molecules by takingthe expectation value of nm(q)in the initial state

nm(q) =

∫dk dk′ φ∗f (k)φf (k′) 〈c†q

2 +k↑ c†q2−k↓

c q2−k′↓ c

q2 +k′↑ 〉 (11.38)

We take the initial state to be of the type (11.32). Direct calculation gives

nm(q) = δ(q) |∫

dk φ(k) 〈 c†k↑ c†−k↓ 〉 |

2 +

∫dk|φ(k)|2 〈n q

2 +k↑ 〉〈n q2−k↓ 〉(11.39)

The first term in (11.39) measures the number of molecules created in the con-densate, i.e. in the q = 0 state. This contribution is present only when there iscoherent pairing in the initial state and 〈 c†k↑ c

†−k↓ 〉 6= 0. Not surprisingly this

term is proportional to the overlap of the wavefunctions for Cooper pairs andthe final state molecules. The second term in (11.39) gives the number of non-condensate molecules produced after the sweep. This contribution is presenteven when the initial state is not paired. It reflects a finite probability of atomsto be close to each other in the initial state, so that the magnetic field sweepcan turn them into molecules. In the simplest approximation one can take thewavefunction of the final state molecules to be constant for k < 1/a∗ and zero


Figure 11.6: Observation of vortices in a strongly interacting Fermi gas. Super-fluidity, coherence, and vortex lattice were established at different value of themagnetic field. magnetic field was ramped to the BEC side for imaging. Figuretaken from [12].

Figure 11.7: Condensate fraction as a function of interaction strength. Com-parison of theory and experiments. Figure taken from [12]. On the BEC sideheating due to vibrational relaxation leads to the rapid decay of the condensate.

otherwise. Here a∗ is the size of the molecule. Note that this is not the size ofthe closed channel bound state but the size of the Feshbach molecules includingthe open channel, which should be of the order of the scattering length. For thecoherent part we find

N0

V=

6a3∗

(2π)3

∫ a−1∗

0

dk k2 ∆/2√∆2 + ξ2

k

=9n

8

(∆

Ef

)2

kf a∗ (11.40)

Here n is the density of atoms. It is easy to understand why the final result isproportional to |∆|2 and a∗. The Cooper pair wavefunction goes as φc(r) ∼ ∆/r

at short distances. Molecular wavefunction is φm ∼ a−3/2∗ for r < a∗. Hence

|〈φc|φm〉|2 ∼ |∆|2 a∗. Figure 11.7 shows comparison of this simple model withthe experimental results.

11.5. EXPERIMENTAL OBSERVATIONOF SUPERFLUIDITY IN FERMIONIC SYSTEMS13

Figure 11.8: Condensate fraction as a function of magnetic field and temperaturein experiments on 6Li. Arrow marks the position of the Feshbach resonance.Figure taken from [12].

11.5.2 Observing pairing in noise correlations

Noise correlations analysis that we discussed in Chapter 8 can be also usedto probe pairing. We consider TOF experiments with fermionic atoms on theBCS side or at unitarity. Ballistic (collision free) expansion can be achieved byramping the magnetic field to the weakly interacting regime. After sufficientlylong expansion times we find real space distributions of atoms reflecting mo-mentum occupations nkσ = c†kσckσ. In the ideal case of zero temperature anduniform system one could argue that smearing of nkσ should be a signature ofCooper pairing. In Fermi liquid states one expects to find a discontinuity inthe occupation number at the Fermi momentum. However finite temperatureand inhomogeneous density can also smear the Fermi surface. A more strikingfeature of pairing is correlations in the occupation numbers. The BCS wave-function (11.32) demonstrates that occupation of the states k ↑ and −k ↓ areperfectly correlated. Each of these states can be either occupied or not (e.g. atthe Fermi momentum the occupation probability is 50%). However these statesare either occupied or empty simultaneously. If one detects an atom in statek ↑, there must be an atom in a state −k ↓. If there was no atom in the sate k ↑,there should be no atom in the state −k ↓. A proof of concept experiment wasdone by M. Greiner et. al. [6] and is summarized in Fig. 11.9. This experimentwas done on the BEC side with an RF pulse was applied to dissociate molecules.


Figure 11.9: Pairing correlations in a condensate of Fermi pairs. Correlationsin the TOF images reflect pairing of k and −k states. Figure taken from [12].

11.6 RF spectroscopy

11.6.1 Momentum unresolved RF spectroscopy

The idea of RF spectroscopy is to transfer a small number of atoms into anotherhyperfine state by applying an electro-magnetic pulse at radio frequencies. Typ-ically one measures the number of atoms transferred as a function of the pulsefrequency. If the number of transferred atoms is sufficiently small one can uselinear response theory to describe the process of the atom transfer.

LetHsyst be the Hamiltonian that describes two interacting species of fermionsckσ, which we assume to be given in the grand canonical ensemble. We take theRF pulse that couples ck↑ fermions to the third state fk.

HRF =∑k

(ω0 +k2

2m)f†kfk + ΩR

∑k

( e−iωtf†kck↑ + eiωtc†k↑fk ) (11.41)

Here ΩR is the Rabi frequency controlled by the intensity of the pulse, ω0 is theenergy of the f fermions relative to c↑. Note that the RF pulse does not changemomenta of fermions since photon momentum is negligible.

Within linear response theory one can use Fermi’s golden rule to calculatethe rate of the fermion transfer

I(ω)

2πΩ2R

=∑n

|〈n|A†|0〉 |2 δ(ω − (ωn0 − µ ) )

A† =∑k

f†kck (11.42)

Here ωn0 = En−E0 with En being the energy of state |n〉, in which one fermionhas been transferred into the f state, and E0 being the energy of the initial state.It may seem surprising that formula (11.42) has the chemical potential term inthe delta function for the energy. This is is because in calculating En and E0

we want to use the grand canonical representation of Hsyst, which includes the−µN term. The actual physical frequency measured in experiments needs to becalculated using the canonical form of the Hamiltonian. However we know thatstate |n〉 will have one fermion transferred from the c↑ to f state, hence we can

11.6. RF SPECTROSCOPY 15

simply add the −µ term in equation (11.42) to compensate for the use of thegrand canonical ensemble.

It may seem awkward that we choose to work in the grand canonical ensem-ble and then compensate for it by adding µ to the Fermi’s golden rule. Themotivation for doing this is that in the case of many-body pairing we know howto solve the problem in the grand canonical ensemble. For example, Bogoliubovoperators γ†kσ (see equation (11.5)) create excited states in the BCS Hamilto-nian. The energy of these excited states, which we obtained earlier, correspondsto the grand-canonical ensemble representation of the system.

Let us start with the simplest case of noninteracting fermions. Then state|n〉 corresponds to moving one of the fermions from the Fermi sea of ck↑ intofk state. The energy of this state is En = (ω0 + k2/2m) + (µ − k2/2m). Hereω0 + k2/2m comes from adding a fermion into state fk, and µ − k2/2m comesfrom creating a hole excitation in the Fermi sea of ck↑ (remember that we workin the grand canonical ensemble). Hence in this case we find ωn0 = ω0 + µregardless of the momentum of the fermion which has been transferred. Forthe RF spectrum we find I(ω) = 2πΩ2

Rnδ(ω − ω0), where n is the density offermions.

Let us now consider a paired state described by the BCS mean-field theory.Excited states |ek〉 should be of the form

|ek〉 = γ†−k↓f†k |ΨBCS 〉 (11.43)

The energy of this state (in the grand canonical ensemble again) is Ek+ω0+ k2

2m .Using Bogoliubov transformation formulas (11.5) we easily find that the matrixelement k〈n|A†|0〉 = vk, hence

I(ω)

2πΩ2R

=

∫k

v2kδ(ω − (Ek +

k2

2m+ ω0 − µ)) (11.44)

We recall that ξk = k2

2m − µ and using the usual rules of δ-functions we canrewrite the last formula as

I(ω)

2πΩ2R

=

∫ξk

ρ(ξk) δ(ω − (Ek + ξk + ω0)) = ρ(ξk)v2k

2u2k

|Ek+ k2

2m−µ+ω0=ω(11.45)

In writing the last equation we used 2u2k = (1 + ξk/Ek).

The smallest energy for which we find a nonzero I(ω) comes from k = 0 in(11.45). So I(ω) starts from the frequency ωth

ωth − ω0 =√µ2 + ∆2 − µ (11.46)

We have the following limiting cases

ωth − ω0 =

∆2

2EFin the BCS limit

0.31 ∆ on resonance

|EB| = ~2

ma2 in the BEC limit

(11.47)


Figure 11.10: Schematic representation of RF experiments in the paired states.The RF pulse changes the hyperfine state of all atoms without affecting theirmomenta.

The most important feature of RF spectra in (11.47) is the presence of thepairing gap. Finite energy is required to break pairs into individual atoms. Inthe BEC limit the value of ωth is simply the binding energy. The BCS valueof ωth is more surprising. Naively RF experiments appear similar to tunnelingexperiments in electron systems. However the gap observed in tunneling experi-ments (which are naturally in the BCS regime) is ∆. It is useful to discuss whatcauses the difference. In tunneling experiments momentum is not conserved.So the smallest energy required for taking an electron out corresponds to creat-ing a quasiparticles at momentum near kF and requires energy ∆. The crucialcomponent of RF spectroscopy is conservation of momentum. As a result RFspectra start with fermions at k = 0, which are only weakly affected by pairingin the BCS regime. This is shown schematically in fig. 11.10.

In the discussion so far we assumed that state f does not interact with statesckσ. In many experimental systems, however, this interaction is not negligible.It often leads to strong modifications of the RF spectra. For example, let usassume that state f interacts with state c↓ precisely the same way as the statec↑. Then the RF pulse should rotate ck↑ fermions into fk, without affecting thenature of the paired state. So the RF spectrum should look like a δ-functionwith ω = ω0 [14]. For example, RF spectra measured originally by the Innsbruckgroup, see fig. ??, appear much sharper than theoretical predictions shown infig. 11.11. More recent experiments tried to minimize effects of finite stateinteractions, see fig. 11.13.

11.7. PROBLEMS TO CHAPTER ?? 17

Figure 11.11: RF spectra across the BCS-BEC crossover. Figure taken from[12].

11.6.2 Momentum resolved RF spectroscopy. Photoemis-sion

D. Jin and collaborators developed an interesting refinement of the RF spec-troscopy technique[10]. They went beyond measuring the total number of atomscreated by the RF pulse, by following the RF pulse with the TOF expansion ofthe atoms. This allowed them to measure the number of excited fk atoms as afunction of momentum k[10]. These experiments provide cold atoms analogueof photoemission spectroscopy in electron systems[4]. The simplest way to un-derstand such experiments is to think of energy conservation. For a given valueof momentum k we know the RF frequency ωk, which converts ck↑ into fk. We

also know the kinetic energy of fermions in the final state Efinal k = ω0 + k2

2m .Then from energy conservation we can calculate the energy of fermions in theinitial state, Einit k = Efinal k − ωk. We can also think abut it as finding theenergy of creating a hole excitation in the interacting many-body system. Fig.11.14 shows results of photoemission experiments across the BCS-BEC crosover.

11.7 Problems to Chapter 11

Problem 1.Consider mean-field equations (11.11) and (11.17) for the paired state in the

BCS-BEC crossover regime at T = 0. These equations need to be solved for thechemical potential µ and the order parameter ∆.

a) Show that in the BCS regime of negative small as we find equations(11.25), (11.26).


Figure 11.12: Experimentally measured RF spectra across the BCS-BECcrossover. Upper plots correspond to T/TF ≈ 6, middle plots to T/TF ≈ 0.5,bottom plots to T/TF < 0.2. FInal state interactions are important for inter-preting these results. Figure taken from [5].

b) Show that in the BEC regime of positive small as we find equation (11.29)and that ∆ << |µ|

11.7. PROBLEMS TO CHAPTER ?? 19

Figure 11.13: Experimentally measured RF spectra across the BCS-BECcrossover. Interaction strengths (for c↓ − f and c↓ − c↑ respectively), 1/kFai,are (a) 0.2, 0.4; (b) 0.1, 0; (c) 0.1, -0.3. Figure taken from [9].


Figure 11.14: Photoemission in ultracold fermions. Plot a is for weakly interact-ing gas, plot b is on resonance and T ≈ Tc, plot c is on the BEC side, kFa ≈ 1.The upper feature on plot c is attributed to unpaired fermions. Figure takenfrom [10].

Bibliography

[1] Ehud Altman and Ashvin Vishwanath. Dynamic projection on feshbachmolecules: A probe of pairing and phase fluctuations. Phys. Rev. Lett.,95(11):110404, Sep 2005.

[2] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physicswith ultracold gases. Rev. Mod. Phys., 80(3):885–964, Jul 2008.

[3] Diener and Ho. arXiv:cond-mat/0404517, 2004.

[4] A. Damascelli et al. Rev. Mod. Phys., 75:473, 2002.

[5] C. Chin et al. Science, 305:1128, 2004.

[6] M. Greiner et al. Phys. Rev. Lett., 94:110401, 2005.

[7] Marini et al. Eur.J. Phys. B, 1:151, 1998.

[8] R. Haussmann et al. Phys. Rev. A, 75:23610, 2007.

[9] Schunck et al. Nature, 454:739, 2008.

[10] Stewart et al. Nature, 454:744, 2008.

[11] Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari. Theory of ultra-cold atomic fermi gases. Rev. Mod. Phys., 80(4):1215–1274, Oct 2008.

[12] W. Ketterle and M. Zwierlein. Proceedings of the International School ofPhysics ”Enrico Fermi”, Course CLXIV, Varenna, arXiv:0801.2500, 2000.

[13] J.R. Schrieffer. Theory of Superconductivity. Addison-Wesley PublishingCompany, 1983.

[14] Z. Yu and G. Baym. Phys. Rev. A, 73:63601, 2006.

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