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arXiv:1102.3565 · Physical Review A

Condensate fraction of cold gases in a nonuniform external potential

G. E. Astrakharchik, Konstantin V. Krutitsky

Published Sep 22, 2011• 14 citations
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Abstract

Exact calculation of the condensate fraction in multidimensional inhomogeneous interacting Bose systems in a confining potential of arbitrary shape is a difficult computational problem. We have developed an iterative procedure which allows us to calculate the condensate fraction as well as the corresponding eigenfunction of the one-body density matrix. We successfully validate this procedure in diffusion Monte Carlo simulations of a Bose gas in an optical lattice at zero temperature. We also discuss the relation between different criteria used for testing coherence in cold Bose systems, such as the fraction of particles that are superfluid, condensed, or in the zero-momentum state.

DOI
10.1103/physreva.84.031604

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•Jan 1, 2002•1 citation
Exact calculation of the condensate fraction in multidimensional inhomogeneous interacting Bose systems in a confining potential of arbitrary shape is a difficult computational problem. We have developed an iterative procedure which allows us to calculate the condensate fraction as well as the corresponding eigenfunction of the one-body density matrix. We successfully validate this procedure in diffusion Monte Carlo simulations of a Bose gas in an optical lattice at zero temperature. We also discuss the relation between different criteria used for testing coherence in cold Bose systems, such as the fraction of particles that are superfluid, condensed, or in the zero-momentum state.
We evaluate the Bose-Einstein condensate density and the superfluid fraction of bosons in a periodic external potential using path-integral Monte Carlo methods. The periodic lattice consists of a cubic cell containing a potential well that is replicated along one dimension using periodic boundary conditions. The aim is to describe bosons in a one-dimensional optical lattice or helium confined in a periodic porous medium. The one-body density matrix is evaluated and diagonalized numerically to obtain the single boson natural orbitals and the occupation of these orbitals. The condensate fraction is obtained as the fraction of bosons in the orbital that has the highest occupation. The superfluid density is obtained from the winding number. From the condensate orbital and superfluid fraction, we investigate (1) the impact of the periodic external potential on the spatial distribution of the condensate, and (2) the correlation of localizing the condensate into separated parts and the loss of superflow along the lattice. For high-density systems, as the well depth increases, the condensate becomes depleted in the wells and confined to the plateaus between successive wells, as in pores between necks in a porous medium. For low-density systems, as the well depth increases the BEC is localized at the center of the wells (tight binding) and depleted between the wells. In both cases, the localization of the condensate suppresses superflow leading to a superfluid-insulator crossover. The impact of the external potential on the temperature dependence of the superfluidity is also investigated. The external potential suppresses the superfluid fraction at all temperatures, with a superfluid fraction significantly less than one at low temperature. The addition of an external potential does not, however, significantly reduce the transition temperature.
We investigate the zero-temperature properties of a diluted homogeneous Bose gas made of NNN particles interacting via a two-body square-well potential by performing Monte Carlo simulations. We tune the interaction strength to achieve arbitrary positive values of the scattering length and compute by Monte Carlo quadrature the energy per particle E/NE/NE/N and the condensate fraction N0/N{N}_{0}/NN0​/N of this system by using a Jastrow ansatz for the many-body wave function, which avoids the formation of the self-bound ground state and describes instead a (metastable) gaseous state with uniform density. In the unitarity limit, where the scattering length diverges while the range of the interatomic potential is much smaller than the average distance between atoms, we find a finite energy per particle [$E/N=0.70\phantom{\rule{4pt}{0ex}}{\ensuremath{\hbar}}^{2}{(6{\ensuremath{\pi}}^{2}n)}^{2/3}/2m$, with nnn the number density] and a quite large condensate fraction (N0/N=0.83{N}_{0}/N=0.83N0​/N=0.83).
We investigate the phenomenon of Bose-Einstein condensation and superfluidity in a Bose gas at zero temperature with disorder. By using the diffusion Monte Carlo method, we calculate the superfluid and the condensate fraction of the system as a function of density and strength of disorder. In the regime of weak disorder we find agreement with the analytical results obtained within the Bogoliubov model. For strong disorder the system enters an unusual regime where the superfluid fraction is smaller than the condensate fraction.
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By using the diffusion Monte Carlo method we calculate the one- and two-body density matrix of an interacting Fermi gas at T = 0 in the BCS to Bose-Einstein condensate (BEC) crossover. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.