Nonlinear Schrödinger equation for a superfluid Fermi gas in the BCS-BEC crossover

Abstract

We introduce a quasianalytic nonlinear Schrödinger equation with beyond mean-field corrections to describe the dynamics of a zero-temperature dilute superfluid Fermi gas in the crossover from the weak-coupling Bardeen-Cooper-Schrieffer ͑BCS͒ regime, where k F ͉a͉ Ӷ 1 with a the s-wave scattering length and k F the Fermi momentum, through the unitarity limit k F a → Ϯϱ to the Bose-Einstein condensation ͑BEC͒ regime where k F a Ͼ 0. The energy of our model is parametrized using the known asymptotic behavior in the BCS, BEC, and the unitarity limits and is in excellent agreement with accurate Green's- The crossover from a Bardeen-Cooper-Schrieffer ͑BCS͒ Fermi superfluid at zero temperature for weak coupling to a Bose-Einstein condensation ͑BEC͒ of dimers ͓1͔ has been an intense area of research ͑both experimental ͓2,3͔ and theoretical ͓4-10͔͒ after the realization of a BCS to BEC crossover ͑BBC͒ in a trapped dilute Fermi superfluid near a Feshbach resonance. This allows one to change the system from the BCS regime, with small negative a ͑k F ͉a͉ Ӷ 1͒, through the unitarity regime of divergent a ͑k F a → Ϯϱ͒, to the BEC regime of dimers with positive a, with k F the Fermi momentum. The limiting behavior of the system in the weakcoupling ͑k F ͉a͉ Ӷ 1͒ ͓11-15͔ and unitarity limits for both positive and negative a is well known ͓12,13,16͔. Accurate information about the BBC dynamics has recently been available from numerical fixed-node Green's-function Monte Carlo ͑GFMC͒ calculations ͓5,7͔ at zero temperature. There is also a mean-field BCS ͑MFBCS͒ calculation of the same ͓6͔.A quasianalytical model for the BBC problem, generating a Ginzburg-Landau-͑GL-͒ type equation for fermions for a Ͻ 0 and a Gross-Pitaevskii-͑GP-͒ type equation for dimers for a Ͼ 0, both including beyond mean-field effects, so as to be valid in the unitarity limit of divergent a, should be useful. We propose such a model in three dimensions, called the BBC model ͑BBM͒, for the crossover problem of a dilute trapped Fermi superfluid at zero temperature using the known theoretical solution in the weak-coupling and unitarity limits in both the BCS and BEC regimes.We consider a dilute superfluid Fermi gas of N spin-1/2 atoms of mass m, atomic scattering length a, and density n with singlet pairing due to an attractive atomic interaction. We assume, for a dilute gas, that the results are universal, determined solely by the scattering length a, and independent of the details of the interaction potential and its range. We present an analytical model for energy and bulk chemical potentials in the entire crossover region and hence derive a nonlinear equation for the superfluid Fermi gas. We present results for radial and axial frequencies of collective oscillation in a cigar-shaped trap and compare with GFMC and MFBCS calculations as well as experimental data.At low densities, in the BCS regime ͑k F ͉a͉ Ӷ 1͒, gaps are negligible ͓4,12͔ and the total energy E per particle of the superfluid Fermi gas is given by ͓11,13͔ where2m ͑3 2 n͒ 2/3 is the Fermi energy...

Finite-temperature effective field theory for dark solitons in superfluid Fermi gases

“…The total phase change throughout the soliton is determined as the difference (26) and results in the integral,…”

“…This inspires attempts to develop complementary approaches for ultracold Fermi gases exploiting only a macroscopic wave function. Prominent examples of such attempts are the modifications of the Ginzburg-Landau (GL) approach [24,25] for cold Fermi gases in the BCS-BEC crossover, and the GP and nonlinear Schrödinger equations [26]. The GL method is valid in a rather narrow temperature region close to the critical temperature T c .…”

Faraday waves in quasi-one-dimensional superfluid Fermi-Bose mixtures

“…In general, Bose and Fermi subsystems are described by the Lieb-Liniger and Gaudin-Yang theories, respectively. Here we are interested in weak Bose-Bose interactions (we consider small positive g b ) and attractive Fermi-Fermi interactions and the superfluid Fermi-Bose system is described by the nonlinear Schrödinger-like equation (1) [29][30][31][32][33]. In the BCS weak attractive coupling limit the fermionic subsystem coefficient is κ = 1/4, while in the molecular unitarity limit it is κ = 1/16 [26].…”

Structure and dynamics of a rotating superfluid Bose-Fermi mixture

“…Here we choose (α − λ) = 32/(3 √ π), β = 4πα/η with η = 22.22, α = 32ξ/(3 √ π ), and λ = 32(ξ − 1)/(3 √ π) with ξ = 1.1 [10]. The bulk chemical potential of the Fermi superfluid is expressed as [10,16,17] μ p (n p ,a f ) = 2 2 m p (6π 2 n p ) 2/3 g 2 1/3 n 1/3 p a f ,…”

“…Thus the function (5) with Eq. (6) provides a smooth interpolation between the bulk chemical potential of a Fermi superfluid in the weak-coupling limit and that in the unitary limit for both the uniform case and the trapped case [10,17,18]. It was shown that the results with the choice of the fitting parameters δ = 20π/(3π 2 ) 2/3 , k = δ/(1 − ζ ), and ζ = 0.44 [10] agreed well with the corresponding Monte Carlo data [23,24].…”