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arXiv:0902.0366 · Physical Review Letters

Ground State Properties of Cold Bosonic Atoms at Large Scattering Lengths

Jun Liang Song, Fei Zhou

Published Jul 8, 2009• 47 citations
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Abstract

In this Letter, we study bosonic atoms at large scattering lengths using a variational method where the condensate amplitude is a variational parameter. We further examine momentum distribution functions, chemical potentials, the speed of sound, and spatial density profiles of cold bosonic atoms in a trap in this limit. The latter two properties turn out to bear similarities to those of Fermi gases. The estimates obtained here are applicable near Feshbach resonances, particularly when the fraction of atoms forming three-body structures is small and can be tested in future cold atom experiments.

DOI
10.1103/physrevlett.103.025302

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In this Letter, we study bosonic atoms at large scattering lengths using a variational method where the condensate amplitude is a variational parameter. We further examine momentum distribution functions, chemical potentials, the speed of sound, and spatial density profiles of cold bosonic atoms in a trap in this limit. The latter two properties turn out to bear similarities to those of Fermi gases. The estimates obtained here are applicable near Feshbach resonances, particularly when the fraction of atoms forming three-body structures is small and can be tested in future cold atom experiments.
Using a scattering length parametrization of the crossover from a BCS state to a Bose-Einstein condensate as well as the local density approximation for the density profile, we calculate the momentum distribution of a harmonically trapped atomic Fermi gas at zero temperature. Various interaction regimes are considered, including the BCS phase, the unitarity limit, and the molecular regime. We show that the relevant parameter which characterizes the crossover is given by the dimensionless combination N1/6a/aho,{N}^{1/6}{a/a}_{\mathrm{ho}},N1/6a/aho​, where N is the number of atoms, a is the scattering length, and aho{a}_{\mathrm{ho}}aho​ is the oscillator length. The width of the momentum distribution is shown to depend in a crucial way on the value and sign of this parameter. Our predictions can be relevant for experiments on ultracold atomic Fermi gases near a Feshbach resonance.
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 calculate the ground-state energy and the collective excitation frequency of trapped bosons at large scattering length interacting via the realistic two-body van der Waals potential. Our many-body method keeps two-body correlations produced by all interacting pairs. When the scattering length is small compared to the trap size and the number of bosons in the trap is of the order of a few thousands, the mean-field results are in good agreement with the many-body results. However for large particle numbers, even when the condensate is sufficiently dilute, the interatomic correlation comes into the picture. When the scattering length is quite large near the Feshbach resonance, the Bose gas becomes highly correlated. The many-body results are close to the Gross-Pitaevskii results for a small number of bosons, however, large deviations are noted in the large particle limit. We also calculate the lowest collective excitation and the interaction energy for large scattering lengths. The monopole excitation frequency exhibits a pronounced dependence on the scattering length. We also observe a universal behavior for the interaction energy at the limit of large scattering length.
Dilute Fermi systems with large sss-wave scattering length as{a}_{s}as​ exhibit universal properties if the interparticle spacing r0{r}_{0}r0​ greatly exceeds the range of the underlying two-body interaction potential. In this regime, r0{r}_{0}r0​ is the only relevant length scale and observables such as the energy per particle depend only on r0{r}_{0}r0​ (or, equivalently, the energy EFG{E}_{\mathrm{FG}}EFG​ of the free Fermi gas). This paper investigates Bose and Fermi systems with nonvanishing angular momentum lll using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of an lll-dependent length scale ${\ensuremath{\xi}}_{l}$, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite lll. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.
The ground state of an atomic Fermi gas near the Feshbach resonance for a negative scattering length is investigated using the variational method. The structure of the superfluid state is formed by two coherently coupled subsystems, viz., the quasimolecular subsystem in a closed channel and the subsystem of atomic pairs in an open channel. The derived system of equations makes it possible to describe the properties of the ground state for arbitrary values of the parameters (in particular, to find the gap in the single-particle Fermi excitation spectrum and the speed of sound characterizing the branch of collective Bose excitations).