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Physical Review A

Virial expansion around the s -wave Feshbach resonance in mass-imbalanced Fermi gases

Ting Xie, Xue‐Jin Hu, Yin Hui Huang, Gao‐Ren Wang, Shu‐Lin Cong

Published Mar 11, 2014• 0 citations
Publisher

Abstract

We derive the expressions for the virial coefficients up to the third order for two-species fermion mixtures, and investigate the fugacity of the species and the interaction energy density around the $s$-wave Feshbach resonance in $^{6}\mathrm{Li}$-$^{40}\mathrm{K}$ mixtures. The theoretical results show that the magnitude of the virial coefficients depends on the mass ratio in a mass-imbalanced system. We find that the contribution of two-body bound states to high-order virial coefficients is very small at temperatures of microkelvin order. With the decrease of $T/{T}_{F}$, the positions of the maxima of fugacity and interaction energy density shift away from the resonance position in the third-order virial expansion calculation. The mass ratio determines the relevance between the fugacity and number density of heavy species. Our conclusion is also effective for other mass-imbalanced systems.

DOI
10.1103/physreva.89.032704

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We derive the expressions for the virial coefficients up to the third order for two-species fermion mixtures, and investigate the fugacity of the species and the interaction energy density around the sss-wave Feshbach resonance in 6Li^{6}\mathrm{Li}6Li-40K^{40}\mathrm{K}40K mixtures. The theoretical results show that the magnitude of the virial coefficients depends on the mass ratio in a mass-imbalanced system. We find that the contribution of two-body bound states to high-order virial coefficients is very small at temperatures of microkelvin order. With the decrease of T/TFT/{T}_{F}T/TF​, the positions of the maxima of fugacity and interaction energy density shift away from the resonance position in the third-order virial expansion calculation. The mass ratio determines the relevance between the fugacity and number density of heavy species. Our conclusion is also effective for other mass-imbalanced systems.
We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging sss-wave scattering length as{a}_{s}as​, that is, with 1/as=01/{a}_{s}=01/as​=0. The high-temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio $\ensuremath{\kappa}$ increases and diverges when Efimov physics sets in at $\ensuremath{\kappa}=13.61$. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for $\ensuremath{\kappa}=1$, but finite for $\ensuremath{\kappa}\ensuremath{\ne}1$ (with a majority of heavy particles). Compared to the equilibrium polarization of the noninteracting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.
We study near a Feshbach resonance, as a function of the mass ratio, the fermion-dimer scattering amplitude in fermionic mixtures of two fermion species. When masses are equal the physical situation is known to be quite simple. We show that, when the mass ratio is increased, the situation becomes much more complex. For the sss-wave contribution we obtain an analytical solution in the asymptotic limit of very large mass ratio. In this regime the sss-wave scattering amplitude displays a large number of zeros, essentially linked to the known large value of the fermion-dimer scattering length in this regime. We find by an exact numerical calculation that a zero is still present for a mass ratio of 15. For the ppp-wave contribution we make our study below the mass ratio of 8.17, where a fermion-dimer bound state appears. We find that a strong ppp-wave resonance is present at low energy, due to a virtual bound state, in the fermion-dimer system, which is a forerunner of the real bound state. This resonance becomes prominent in the mass ratio range around the one corresponding to the 40{}^{40}40K -6{}^{6}6Li mixtures, much studied experimentally. This resonance should affect a number of physical properties. These include the equation of state of unbalanced mixtures at very low temperature but also the equation of state of balanced mixtures at moderate or high temperature. The frequency and the damping of collective modes should also provide a convenient way to evidence this resonance. Finally it should be possible to modify the effective mass of one of the fermionic species by making use of an optical lattice. This would allow one to study the strong dependence of the resonance as a function of the mass ratio of the two fermionic elements. In particular one could check if the virtual bound state is relevant for the instabilities of these mixtures.
We give exact integral expressions of the third cluster or virial coefficients of binary mixtures of ideal Bose or Fermi gases, with interspecies interactions of zero range and infinite s-wave scattering length. In general the result depends on three-body parameters Rt appearing in three-body contact conditions, because an Efimov effect is present or because the mixture is in a preefimovian regime with a mass ratio close to an Efimov effect threshold. We give a new, exact integral expression of Rt for the microscopic narrow Feshbach resonance model. A divergence of Rt in the preefimovian regime at a scaling exponent s = 1/2 is predicted and physically discussed. The analytical results are applied to typical species used in cold atom experiments.
We consider a homogeneous Bose-Fermi mixture, with the boson-fermion interaction tuned by a Fano-Feshbach resonance, in the presence of mass and density imbalance between the two species. By using many-body diagrammatic methods, we first study the finite-temperature phase diagram for the specific case of the mass-imbalanced mixture 87{}^{87}87Rb-40{}^{40}40K for different values of the density imbalance. We then analyze the quantum phase transition associated with the disappearance at zero temperature of the boson condensate above a critical boson-fermion coupling. We find a pronounced dependence of the critical coupling on the mass ratio and a weak dependence on the density imbalance. For a vanishingly small boson density, we derive, within our approximation, the asymptotic expressions for the critical coupling in the limits of small and large mass ratios. These expressions are relevant also for the polaron-molecule transition in a Fermi mixture at small and large mass ratios. The analysis of the momentum distribution functions at sufficiently large density imbalances shows an interesting effect in the bosonic momentum distribution due to the simultaneous presence of composite fermions and unpaired fermions.
We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either sss-wave or ppp-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either sss-wave or ppp-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting sss-wave Fermi gas has already been proved very useful. In the case of ppp-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a ppp-wave Feshbach resonance.
We show that, near a Feshbach resonance, a strong p-wave resonance is present at low energy in atom-dimer scattering for 6^66Li-40^{40}40K fermionic mixtures. This resonance is due to a virtual bound state, in the atom-dimer system, which is present at this low energy. When the mass ratio between the two fermionic elements is increased, this virtual bound state goes to a known real bound state which appears when the mass ratio reaches 8.17. This resonance should affect a number of physical properties. These include the equation of state of unbalanced mixtures at very low temperature but also the equation of state of balanced mixtures at moderate or high temperature. The frequency and the damping of collective modes should also provide a convenient way to evidence this resonance. Finally it should be possible to modify the effective mass of one the fermionic species by making use of an optical lattice. This would allow to study the strong dependence of the resonance as a function of the mass ratio of the two fermionic elements.