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arXiv:0909.0673

Virial relations for ultracold trapped Fermi gases with finite range interactions through the BCS-BEC crossover

L. E. C. Rosales-Zarate R. Jauregui

Published Sep 3, 2009• 0 citations
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Abstract

We study the virial relations for ultracold trapped two component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the BCS-BEC crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many body energy with respect to the inverse of the scattering length and with respect to the interaction range. They are found to have extreme values at the unitary limit. The virial results are used to check the quality of the variational wave function involved in the calculations.

DOI
10.48550/arxiv.0909.0673

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We study the virial relations for ultracold trapped two component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the BCS-BEC crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many body energy with respect to the inverse of the scattering length and with respect to the interaction range. They are found to have extreme values at the unitary limit. The virial results are used to check the quality of the variational wave function involved in the calculations.
We study the virial relations for ultracold trapped two-component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the Bardeen–Cooper–Schrieffer (BCS) Bose–Einstein-condensate (BEC) crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many-body energy with respect to the inverse of the scattering length and with respect to the interaction range. Once the binding energy of the formed molecules in the BEC side is subtracted, the corresponding energy derivatives are found to have extreme values at the unitary limit. The value of the derivative with respect to the potential range in that limit is large enough to yield measurable differences between the total energy and twice the trapping energy unless the interacting system is described by extremely short potential ranges. The virial results are used to check the quality of the variational wavefunction involved in the calculations.
We perform a variational quantum Monte Carlo simulation of an interacting Fermi gas confined in a three dimensional harmonic potential. This gas is considered as the precursor system from which a molecular bosonic gas is formed. Based on the results of two-body calculations for trapped atoms, we propose a family of variational many-body wave functions that takes into account the qualitative different nature of the Bardeen-Cooper-Schieffer and Bose-Einstein condensate regimes as a function of the scattering length. Energies, densities, and correlation functions are calculated and compared with previous results for homogeneous gases. Universality tests at the unitarity limit are performed including the verification of the virial relation and the evaluation of the $\ensuremath{\beta}$ parameter.
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 extend a recently introduced separable interaction for the unitary trapped Fermi gas to all values of the scattering length. We derive closed expressions for the interaction matrix elements and the two-particle eigenvectors. At unitarity we analytically demonstrate the convergence of this interaction to the zero-range two-body pseudopotential for sss-wave scattering. We apply this effective interaction to the three- and four-particle systems along the BEC-BCS crossover and find that their low-lying energies exhibit convergence in the regularization parameter that is much faster than for the conventional renormalized contact interaction. We find similar convergence properties of the three-particle free energy at unitarity.
We investigate the non-universal behavior of the BCS-BEC crossover model at the normal to superfluid transition. By using a finite temperature quantum field theoretical approach due to Nozieres and Schmitt-Rink and by making the effective range expansion of the effective two-body interaction we numerically calculate the crossover transition temperature as a function of the scattering length, aF,a_{F},aF​, and the effective range parameter, re.r_{\mathrm{e}}.re​. In an ultracold two-component atomic Fermi gas BCS-BEC crossover physics is expected to appear near magnetic-field-induced Feshbach resonances. By matching two-body scattering properties near a Feshbach resonance to a simple renormalized model potential, a broad resonance is characterized by a small effective range and the gas displays a universal BCS-BEC crossover behavior. On the other hand, for a narrow resonance thermodynamic quantities depend the effective range which may be large and negative at resonance. For increasing values of −re-r_{\mathrm{e}}−re​ we find the transition temperature to be suppressed in the crossover region. We furthermore argue for existence of a lower bound of TcT_{c}Tc​ at fixed coupling for increasing −re-r_{\mathrm{e}}−re​ in the crossover region.
Properties of the ground state of an unpolarized ultracold Fermi gas along the BCS-BEC crossover are investigated by the variational and diffusion Monte Carlo methods. We apply the Wigner-Bethe-Peierls boundary condition in our calculations to avoid any bias from using an interatomic potential with finite effective range. Properties for several values of the scattering length are studied in the range $\ensuremath{-}8\ensuremath{\le}1/a{k}_{F}\ensuremath{\le}4$, including the unitary limit. The contact parameters as a function of scattering length are obtained by fitting the pair distribution functions for particles with different spins. The energies and contact parameters are in very good agreement with experimental data reported in the literature.