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arXiv:1006.0769 · Physical Review A

Energy spectrum of harmonically trapped two-component Fermi gases: Three- and four-particle problem

K. M. Daily, D. Blume

Published May 17, 2010• 46 citations
arXivHTMLPDF

Abstract

Trapped two-component Fermi gases allow the investigation of the so-called BCS-BEC crossover by tuning the interspecies atom-atom $s$-wave scattering length ${a}^{(\mathit{aa})}$ 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 ${a}^{(\mathit{aa})}$ 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.

DOI
10.1103/physreva.81.053615

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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.
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.
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 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.
Four identical spinless bosons with purely attractive two-body short-range interactions and repulsive three-body interactions under external spherically symmetric harmonic confinement are considered. The repulsive three-body potential prevents the formation of deeply bound states with molecular character. The low-energy spectrum with vanishing orbital angular momentum and positive parity for infinitely large two-body sss-wave scattering length is analyzed in detail. Using the three-body contact, states are classified as universal, quasiuniversal, or strongly nonuniversal. Connections with the zero-range interaction model are discussed. The energy spectrum is mapped out as a function of the two-body sss-wave scattering length as,as>0{a}_{s}, {a}_{s}>0as​,as​>0. In the weakly to medium-strongly interacting regime, one of the states approaches the energy obtained for a hard-core interaction model. This state is identified as the energetically lowest-lying ``BEC state.'' Structural properties are also presented.
This work studies zero-temperature Fermi gases with short-ranged and attractive interactions with focus on the dimensionality aspects by confining the system from three to two dimensions. Interacting Fermi gases are superfluids characterized as being strongly correlated if the scattering length is much larger than the other scales present. In the many-body context, this category of systems requires a careful treatment since mean-field approaches are not expected to hold under the conditions which define the strong pairing. For this reason, we employ quantum Monte Carlo methods to study the dimensionality effects in two-component Fermi gases in the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover. To be able to investigate such aspects, we consider a three-dimensional gas with periodic boundary conditions in the xy plane and a confining potential in the z direction capable of reducing the dimensionality of the system, which defines a dimensional BCS-BEC crossover. We calculate the equation of state as the energy per particle as a function of the dimensionality and as a function of the interaction parameter, which are characterized by the oscillator length in the z direction and the scattering length respectively. The greater confinement in the system lowers the interparticle distances, which increases the effects of the attractive interactions between antiparallel-spin fermions.We also calculate the density profile in the z direction over the crossover for different confinements and different interaction parameters to quantify their influence on the spatial distribution of the particles. This study connects the perspectives between theoretical and experimental works in low-dimensional ultracold Fermi gases by utilizing an approach that is similar to what it is realized in modern experiments.
The properties of two-component Fermi gases become universal if the interspecies sss-wave scattering length as{a}_{s}as​ and the average interparticle spacing are much larger than the range of the underlying two-body potential. Using an explicitly correlated Gaussian basis set expansion approach, we determine the eigenenergies of two-component Fermi gases in a cubic box with periodic boundary conditions as functions of the interspecies sss-wave scattering length and the effective range of the two-body potential. The universal properties of systems consisting of up to four particles are determined by extrapolating the finite-range energies to the zero-range limit. We determine the eigenenergies of states with vanishing and finite momenta. In the weakly attractive BCS regime, we analyze the energy spectra and degeneracies using first-order degenerate perturbation theory. Excellent agreement between the perturbative energy shifts and the numerically determined energies is obtained. For the infinitely large scattering length case, we compare our results---where available---with those presented in the literature.