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arXiv:0801.1815 · Physical Review A

Superfluid pairing between fermions with unequal masses

M. A. Baranov, C. Lobo, Gora V. Shlyapnikov

Published Sep 29, 2008• 46 citations
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Abstract

We consider a superfluid state in a two-component gas of fermionic atoms with equal densities and unequal masses in the BCS limit. We develop a perturbation theory along the lines proposed by Gorkov and Melik-Barkhudarov and find that for a large difference in the masses of heavy $(M)$ and light $(m)$ atoms one has to take into account both the second-order and third-order contributions. The result for the critical temperature and order parameter is then quite different from the prediction of the simple BCS approach. Moreover, the small parameter of the theory turns out to be $({p}_{F}\ensuremath{\mid}a\ensuremath{\mid}∕\ensuremath{\hbar})\sqrt{M∕m}⪡1$, where ${p}_{F}$ is the Fermi momentum and $a$ the scattering length. Thus, for a large mass ratio $M∕m$ the conventional perturbation theory requires significantly smaller Fermi momenta (densities) or scattering lengths than in the case of $M\ensuremath{\sim}m$, where the small parameter is ${p}_{F}\ensuremath{\mid}a\ensuremath{\mid}∕\ensuremath{\hbar}⪡1$. We show that three-body scattering resonances appearing at a large mass ratio due to the presence of three-body bound Efimov states do not influence the result, which in this sense becomes universal.

DOI
10.1103/physreva.78.033620

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We consider a superfluid state in a two-component gas of fermionic atoms with equal densities and unequal masses in the BCS limit. We develop a perturbation theory along the lines proposed by Gorkov and Melik-Barkhudarov and find that for a large difference in the masses of heavy (M)(M)(M) and light (m)(m)(m) atoms one has to take into account both the second-order and third-order contributions. The result for the critical temperature and order parameter is then quite different from the prediction of the simple BCS approach. Moreover, the small parameter of the theory turns out to be (pF\ensuremath∣a\ensuremath∣∕\ensuremathℏ)M∕m⪡1({p}_{F}\ensuremath{\mid}a\ensuremath{\mid}∕\ensuremath{\hbar})\sqrt{M∕m}⪡1(pF​\ensuremath∣a\ensuremath∣∕\ensuremathℏ)M∕m​⪡1, where pF{p}_{F}pF​ is the Fermi momentum and aaa the scattering length. Thus, for a large mass ratio M∕mM∕mM∕m the conventional perturbation theory requires significantly smaller Fermi momenta (densities) or scattering lengths than in the case of M\ensuremath∼mM\ensuremath{\sim}mM\ensuremath∼m, where the small parameter is pF\ensuremath∣a\ensuremath∣∕\ensuremathℏ⪡1{p}_{F}\ensuremath{\mid}a\ensuremath{\mid}∕\ensuremath{\hbar}⪡1pF​\ensuremath∣a\ensuremath∣∕\ensuremathℏ⪡1. We show that three-body scattering resonances appearing at a large mass ratio due to the presence of three-body bound Efimov states do not influence the result, which in this sense becomes universal.
We treat small trapped unequal-mass two-component Fermi gases at unitarity within a nonperturbative microscopic framework and investigate the system properties as functions of the mass ratio κ, and the numbers N1 and N2 of heavy and light fermions. While equal-mass Fermi gases with infinitely large interspecies s-wave scattering length a(s) are universal, we find that unequal-mass Fermi gases are, for sufficiently large κ and in the regime where Efimov physics is absent, not universal. In particular, the (N₁,N₂) = (2, 1) and (3, 1) systems exhibit three-body and four-body resonances at κ=12.314(2) and 10.4(2), respectively, as well as surprisingly large finite-range effects. These findings have profound implications for ongoing experimental efforts and quantum simulation proposals that utilize unequal-mass atomic Fermi gases.
Two-component Fermi and Bose gases with infinitely large interspecies sss-wave scattering length as{a}_{s}as​ exhibit a variety of intriguing properties. Among these are the scale invariance of two-component Fermi gases with equal masses, and the favorable scaling of Efimov features for two-component Bose gases and Bose-Fermi mixtures with unequal masses. This paper builds on our earlier work [Phys. Rev. Lett. 105, 170403 (2010)] and presents a detailed discussion of our studies of small unequal-mass two-component systems with infinite as{a}_{s}as​ in the regime where three-body Efimov physics is absent. We report on nonuniversal few-body resonances. Just like with two-body systems on resonance, few-body systems have a zero-energy bound state in free space and a diverging generalized scattering length. Our calculations are performed within a nonperturbative microscopic framework and investigate the energetics and structural properties of small unequal-mass two-component systems as functions of the mass ratio $\ensuremath{\kappa}$, and the numbers N1{N}_{1}N1​ and N2{N}_{2}N2​ of heavy and light atoms. For purely attractive Gaussian two-body interactions, we find that the (N1,N2)=(2,1)({N}_{1},{N}_{2})=(2,1)(N1​,N2​)=(2,1) and (3,1)(3,1)(3,1) systems exhibit three-body and four-body resonances at mass ratios $\ensuremath{\kappa}=12.314(2)$ and 10.4(2)10.4(2)10.4(2), respectively. The three- and four-particle systems on resonance are found to be large. It seems feasible that the features discussed in this paper can be probed experimentally with present-day technology.
We in this paper investigate the phase diagram associated with the Bardeen-Cooper-Schrieffer--Bose-Einstein-condensate (BCS-BEC) crossover of a three-component ultracold superfluid Fermi gas of different chemical potentials and equal masses in two dimensions. The gap-order parameter and number densities are found analytically by using the functional-path-integral method. The balance of pairing will be broken in the free space due to the unequal chemical potentials. We obtain the same particle-number density and condensed fraction in the BCS-superfluid phase as those in a recent paper [Phys. Rev. A 83, 033630 (2011)] while the Sarma phase of the coexistence of normal and superfluid Fermi gases is characteristic of an inhomogeneous system. The minimum ratio of the BCS-superfluid phase becomes 1/3 in the BCS limit corresponding to the zero ratio in the two-component system in which the critical point of phase separation is ${\ensuremath{\epsilon}}_{B}/{\ensuremath{\epsilon}}_{F}=2$, but it becomes 3 in the three-component case.
We consider a Fermi gas with two components of different masses, with the s-wave two-body interaction tuned to unitarity. In the range of mass ratio 8.62<M/m<13.6, it is possible for a short-range interaction between heavy fermions to produce a resonance in a three-body channel. The resulting system is scale invariant and has universal properties, and is very strongly interacting. When M/m is slightly above the lower limit 8.62, the ground state energy of a 2ratio1 mixture of heavy and light fermions is less than 2% of the energy of a noninteracting gas with the same number densities. We derive exact relationships between the pressures of the unitary Fermi gases with and without three-body resonance when the mass ratio is close to the critical values of 8.62 and 13.6.
The superfluidity and pairing phenomena in ultracold atomic Fermi gases have been of great interest in recent years, with multiple tunable parameters. Here we study the BCS-BEC crossover behavior of balanced two-component Fermi gases in a one-dimensional optical lattice, which is distinct from the simple three-dimensional (3D) continuum and a fully 3D lattice often found in a condensed matter system. We use a pairing fluctuation theory which includes self-consistent feedback effects at finite temperatures and find widespread pseudogap phenomena beyond the BCS regime. As a consequence of the lattice periodicity, the superfluid transition temperature Tc{T}_{\mathrm{c}}Tc​ decreases with pairing strength in the BEC regime, where it approaches asymptotically ${T}_{\mathrm{c}}=\ensuremath{\pi}an/2m$, with aaa being the sss-wave scattering length, and nnn (mmm) the fermion density (mass). In addition, the quasi-two dimensionality leads to fast-growing (absolute value of the) fermionic chemical potential $\ensuremath{\mu}$ and pairing gap $\mathrm{\ensuremath{\Delta}}$, which depends exponentially on the ratio d/ad/ad/a. Importantly, Tc{T}_{\mathrm{c}}Tc​ at unitarity increases with the lattice constant ddd and hopping integral ttt. The effect of the van Hove singularity on Tc{T}_{\mathrm{c}}Tc​ is identified. The superfluid density exhibits T3/2{T}^{3/2}T3/2 power laws at low TTT, away from the extreme BCS limit. These predictions can be tested in future experiments.
We consider pairing in a three-component gas of degenerate fermions. In particular, we solve the finite-temperature mean-field theory of an interacting gas for a system where both interaction strengths and fermion masses can be unequal. At zero temperature, we find the possibility of a quantum phase transition between states associated with pairing between different pairs of fermions. On the other hand, finite-temperature behavior of the three-component system reveals some qualitative differences from the two-component gas: for a range of parameters it is possible to have two different critical temperatures. The lower one corresponds to a transition between different pairing channels, while the higher one corresponds to the usual superfluid-normal transition. We discuss how these phase transitions could be observed in ultracold gases of fermionic atoms.