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arXiv:cond-mat/0501641 · Physical Review Letters

Four-Body Problem and BEC-BCS Crossover in a Quasi-One-Dimensional Cold Fermion Gas

Christophe Mora, A. Komnik, Reinhold Egger, Alexander O. Gogolin

Published Aug 19, 2005• 38 citations
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Abstract

The four-body problem for an interacting two-species Fermi gas is solved analytically in a confined quasi-one-dimensional geometry, where the two-body atom-atom scattering length a(aa) displays a confinement-induced resonance. We compute the dimer-dimer scattering length a(dd) and show that this quantity completely determines the many-body solution of the associated BEC-BCS crossover phenomenon in terms of bosonic dimers.

DOI
10.1103/physrevlett.95.080403

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The four-body problem for an interacting two-species Fermi gas is solved analytically in a confined quasi-one-dimensional geometry, where the two-body atom-atom scattering length a(aa) displays a confinement-induced resonance. We compute the dimer-dimer scattering length a(dd) and show that this quantity completely determines the many-body solution of the associated BEC-BCS crossover phenomenon in terms of bosonic dimers.
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We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<<-1, however, the potential has long range, with a(ad)>0 and b(ad)>>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.
We consider a two-component gas of fermionic atoms confined to a quasi-two-dimensional (quasi-2D) geometry by a harmonic trapping potential in the transverse direction. We construct a mean-field theory of the BCS-BEC crossover at zero temperature that allows us to extrapolate to an infinite number of transverse harmonic oscillator levels. In the extreme BEC limit, where the confinement length exceeds the dimer size, we recover 3D dimers confined to two dimensions with weak repulsive interactions. However, even when the interactions are weak and the Fermi energy is less than the confinement frequency, we find that the higher transverse levels can substantially modify fermion pairing. We argue that recent experiments on pairing in quasi-2D Fermi gases [Y. Zhang et al., Phys. Rev. Lett. 108, 235302 (2012)] have already observed the effects of higher transverse levels.
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We determine the energetically lowest-lying states in the BEC-BCS crossover regime of sss-wave interacting two-component Fermi gases under harmonic confinement by solving the many-body Schr\"odinger equation using two distinct approaches. Essentially exact basis set expansion techniques are applied to determine the energy spectrum of systems with N=4N=4N=4 fermions. Fixed-node diffusion Monte Carlo methods are applied to systems with up to N=20N=20N=20 fermions, and a discussion of different guiding functions used in the Monte Carlo approach to impose the proper symmetry of the fermionic system is presented. The energies are calculated as a function of the sss-wave scattering length as{a}_{s}as​ for N=2−−20N=2--20N=2−−20 fermions and different mass ratios $\ensuremath{\kappa}$ of the two species. On the Bose-Einstein condensate (BEC) and BCS sides, our energies agree with analytically determined first-order correction terms. We extract the scattering length and the effective range of the dimer-dimer system up to $\ensuremath{\kappa}=20$. Our energies for the strongly interacting trapped system in the unitarity regime show essentially no shell structure, and are well described by a simple expression whose functional form can be derived using the local density approximation, with one or two parameters. The universal parameter $\ensuremath{\xi}$ for the trapped system for various $\ensuremath{\kappa}$ is determined, and comparisons with results for the homogeneous system are presented.
We investigate the crossover from BCS pairing to molecular Bose-Einstein condensation (BEC) in an atomic gas with two fermion species with masses ${m}_{\ensuremath{\uparrow}}\ensuremath{\ne}{m}_{\ensuremath{\downarrow}}$ tuned through a Feshbach resonance. We present results for the T=0T=0T=0 equation of state as a function of the scattering length including the effects of Gaussian fluctuations about the mean field ground state. We compute the ground state energy as a function of ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}$ at unitarity and find excellent agreement with the quantum Monte Carlo result for ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}=6.67$ for a 40K^{40}\mathrm{K}40K-6Li^{6}\mathrm{Li}6Li mixture. We show that the dimer scattering length in the BEC limit as a function of ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}$ compares well with the exact four-body results of Petrov et al. [J. Phys. B 38, S645 (2005)]. We also derive the condition for trapping frequencies to obtain an unpolarized gas in a harmonic trap.