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arXiv:1106.5958 · Physical Review A

Natural and unnatural parity states of small trapped equal-mass two-component Fermi gases at unitarity and fourth-order virial coefficient

Debraj Rakshit, K. M. Daily, D. Blume

Published Mar 26, 2012• 61 citations
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Abstract

Equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement with large $s$-wave scattering length are considered. Using the stochastic variational approach, we determine the lowest 286 and 164 relative eigenenergies of the $(2,2)$ and $(3,1)$ systems at unitarity as a function of the range ${r}_{0}$ of the underlying two-body potential and extrapolate to the ${r}_{0}\ensuremath{\rightarrow}0$ limit. Our calculations include all states with vanishing and finite angular momentum $L$ (and natural and unnatural parity $\ensuremath{\Pi}$) with relative energy up to 10.5 $\ensuremath{\hbar}\ensuremath{\Omega}$, where $\ensuremath{\Omega}$ denotes the angular trapping frequency of the external confinement. Our extrapolated zero-range energies are estimated to have uncertainties of 0.1$%$ or smaller. The $(2,2)$ and $(3,1)$ energies are used to determine the fourth-order virial coefficient of the trapped unitary two-component Fermi gas in the low-temperature regime. Our results are compared with recent predictions for the fourth-order virial coefficient of the homogeneous system. We also calculate small portions of the energy spectra of the $(3,2)$ and $(4,1)$ systems at unitarity.

DOI
10.1103/physreva.85.033634

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Equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement with large sss-wave scattering length are considered. Using the stochastic variational approach, we determine the lowest 286 and 164 relative eigenenergies of the (2,2)(2,2)(2,2) and (3,1)(3,1)(3,1) systems at unitarity as a function of the range r0{r}_{0}r0​ of the underlying two-body potential and extrapolate to the r0\ensuremath→0{r}_{0}\ensuremath{\rightarrow}0r0​\ensuremath→0 limit. Our calculations include all states with vanishing and finite angular momentum LLL (and natural and unnatural parity \ensuremathΠ\ensuremath{\Pi}\ensuremathΠ) with relative energy up to 10.5 \ensuremathℏ\ensuremathΩ\ensuremath{\hbar}\ensuremath{\Omega}\ensuremathℏ\ensuremathΩ, where \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ denotes the angular trapping frequency of the external confinement. Our extrapolated zero-range energies are estimated to have uncertainties of 0.1% or smaller. The (2,2)(2,2)(2,2) and (3,1)(3,1)(3,1) energies are used to determine the fourth-order virial coefficient of the trapped unitary two-component Fermi gas in the low-temperature regime. Our results are compared with recent predictions for the fourth-order virial coefficient of the homogeneous system. We also calculate small portions of the energy spectra of the (3,2)(3,2)(3,2) and (4,1)(4,1)(4,1) systems at unitarity.
The properties of two-component Fermi gases with zero-range interactions are universal. We use an explicitly correlated Gaussian basis set expansion approach to investigate small equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement. At unitarity, we determine the ground-state energy for systems with up to 10 particles interacting through finite-range two-body potentials for both even and odd numbers of particles. We extrapolate the energies to the zero-range limit using a novel scheme that removes the linear and, in some cases, also the quadratic dependence of the ground-state energies on the two-body range. Our extrapolated zero-range energies are compared with results from the literature. We also calculate the two-body Tan contact and structural properties.
We investigate small equal-mass two-component Fermi gases under external spherically symmetric confinement in which atoms with opposite spins interact through a short-range two-body model potential. We employ a non-perturbative microscopic framework, the stochastic variational approach, and determine the system properties as functions of the interspecies s -wave scattering length a s , the orbital angular momentum L of the system, and the numbers N 1 and N 2 of spin-up and spin-down atoms (with N 1 − N 2 = 0 or 1 and N ⩽ 6 , where N = N 1 + N 2 ). At unitarity, we determine the energies of the five- and six-particle systems for various ranges r 0 of the underlying two-body model potential and extrapolate to the zero-range limit. These energies serve as benchmark results that can be used to validate and assess other numerical approaches. We also present structural properties such as the pair distribution function and the radial density. Furthermore, we analyze the one-body and two-body density matrices. A measure for the molecular condensate fraction is proposed and applied. Our calculations show explicitly that the natural orbitals and the momentum distributions of atomic Fermi gases approach those characteristic for a molecular Bose gas if the s -wave scattering length a s , a s > 0 , is sufficiently small.
A two-component four-fermion system with equal masses, interspecies sss-wave scattering length as{a}_{s}as​, and vanishing intraspecies interactions under external spherically symmetric harmonic confinement is considered. Using a correlated Gaussian basis set expansion approach, we determine the energies and various structural properties of the energetically lowest-lying gaslike state throughout the crossover for various ranges of the underlying two-body potential. Extrapolating to the zero-range limit, our numerical results show explicitly that the total energy, the trap energy, as well as certain aspects of the pair distribution function and of the momentum distribution are related through the so-called integrated contact intensity I(as)I({a}_{s})I(as​). Furthermore, it is shown explicitly that the total energy and the trap energy are related through a generalized virial theorem that accounts for a nonzero range.
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.
Using two different numerical methods, we study the behavior of two-component Fermi gases interacting through short-range sss-wave interactions in a harmonic trap. A correlated Gaussian basis-set expansion technique is used to determine the energies and structural properties, i.e., the radial one-body densities and pair distribution functions, for small systems with either even or odd NNN, as functions of the sss-wave scattering length and the mass ratio $\ensuremath{\kappa}$ of the two species. Particular emphasis is put on a discussion of the angular momentum of the system in the BEC-BCS crossover regime. At unitarity, the excitation spectrum of the four-particle system with total angular momentum L=0L=0L=0 is calculated as a function of the mass ratio $\ensuremath{\kappa}$. The results are analyzed from a hyperspherical perspective, which offers unique insight into the problem. Additionally, fixed-node diffusion Monte Carlo calculations are performed for equal-mass Fermi gases with up to N=30N=30N=30 atoms. We focus on the odd-even oscillations of the ground-state energy of the equal-mass unitary system having up to N=30N=30N=30 particles, which are related to the excitation gap of the system. Furthermore, we present a detailed analysis of the structural properties of these systems.
Trapped two-component Fermi gases allow the investigation of the so-called BCS-BEC crossover by tuning the interspecies atom-atom sss-wave scattering length a(aa){a}^{(\mathit{aa})}a(aa) from attractive to repulsive, including vanishing and infinitely large values. Here, we numerically determine the energy spectrum of the equal-mass spin-balanced four-fermion system---the smallest few-particle system that exhibits BCS-BEC crossoverlike behavior---as a function of a(aa){a}^{(\mathit{aa})}a(aa) using the stochastic variational approach. For comparative purposes, we also treat the two- and three-particle systems. States with vanishing and finite total angular momenta as well as with natural and unnatural parities are considered. In addition, the energy spectrum of weakly attractive and weakly repulsive gases is characterized by employing a perturbative framework that utilizes hyperspherical coordinates. The hyperspherical coordinate approach allows the straightforward assignment of quantum numbers and furthermore provides great insight into the strongly interacting unitary regime.