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arXiv:0801.2747 · Physical Review A

Energetics and structural properties of trapped two-component Fermi gases

J. von Stecher, Chris H. Greene, D. Blume

Published Apr 16, 2008• 84 citations
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Abstract

Using two different numerical methods, we study the behavior of two-component Fermi gases interacting through short-range $s$-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 $N$, as functions of the $s$-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 $L=0$ 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 $N=30$ atoms. We focus on the odd-even oscillations of the ground-state energy of the equal-mass unitary system having up to $N=30$ particles, which are related to the excitation gap of the system. Furthermore, we present a detailed analysis of the structural properties of these systems.

DOI
10.1103/physreva.77.043619

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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 \ensuremathκ\ensuremath{\kappa}\ensuremathκ 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 \ensuremathκ\ensuremath{\kappa}\ensuremathκ. 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.
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 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.
We theoretically investigate equal-mass spin-balanced two-component Fermi gases in which pairs of atoms with opposite spins interact via a short-range isotropic model potential. We probe the distinction between two-dimensional and quasi-two-dimensional harmonic confinement by tuning the effective range parameter within two-dimensional scattering theory. Our approach, which yields numerically exact energetic and structural properties, combines a correlated Gaussian basis-set expansion with the stochastic variational method. For systems containing up to six particles, we: 1) Present the ground- and excited-state energy spectra; 2) Study non-local correlations by analysing the one- and two-body density matrices, extracting from these the occupation numbers of natural orbitals, the momentum distributions of atoms and pairs, and the molecular `condensate fraction’; 3) Study local correlations by computing the radial and pair distribution functions. This paper extends current theoretical knowledge on the properties of trapped few-fermion systems as realised in state-of-the-art cold-atom experiments.
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.
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.
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.