The Bose-Einstein condensation of correlated atoms in a trap is studied by examining the effect of interparticle correlations to one-body properties of atomic systems at zero temperature using a simplified formula for the correlated two-body density distribution. Analytical expressions for the density distribution and rms radius of the atomic systems are derived using four different expressions of the Jastrow-type correlation function. In one case, in addition, the one-body density matrix, momentum distribution and kinetic energy are calculated analytically, while the natural orbitals and natural occupation numbers are also predicted in this case. Simple approximate expressions for the mean-square radius and kinetic energy are also given.
An overview of the Bose–Einstein condensation of correlated atoms in a trap is presented by examining the effect of interparticle correlations to one- and two-body properties of the above systems at zero temperature in the framework of the lowest order cluster expansion. Analytical expressions for the one- and two-body properties of the Bose gas are derived using Jastrow-type correlation function. In addition numerical calculations of the natural orbitals and natural occupation numbers are also carried out. Special effort is devoted for the calculation of various quantum information properties including Shannon entropy, Onicescu informational energy, Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. The above quantities are calculated for the trapped Bose gases by comparing the correlated and uncorrelated cases as a function of the strength of the short-range correlations. The Gross–Piatevskii equation is solved, giving the density distributions in position and momentum space, which are employed to calculate quantum information properties of the Bose gas.
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.
Several properties of trapped hard-sphere bosons are evaluated using variational Monte Carlo techniques. A trial wave function composed of a renormalized single-particle Gaussian and a hard-sphere Jastrow function for pair correlations is used to study the sensitivity of condensate and noncondensate properties to the hard-sphere radius and the number of particles. Special attention is given to diagonalizing the one-body density matrix 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 with the highest occupation. The effect of interaction on other quantities such as the ground-state energy, the mean radial displacement, and the momentum distribution is calculated as well. Results are compared with mean-field theory in the dilute limit.
We introduce a family of correlated trial wave functions for the
N-particle ground state of an interacting Bose gas in a harmonic trap. For large
N, the correlations lead to a relative energy decrease of a fraction
3/5N, compared to mean field Gross-Pitaevskii theory. The kinetic energy in the weakly confining direction turns out to be most sensitive to our correlations and, remarkably, is higher by as much as a few per cent for condensates with atom numbers of a few thousand. Thus, the predicted deviations from Gross-Pitaevskii theory originating from ground state correlations might be observed in momentum distribution measurements of small condensates.
We investigate the properties of hard-core bosons in harmonic traps over a wide range of densities. Bose-Einstein condensation is formulated using the one-body density matrix (OBDM) which is equally valid at low and high densities. The OBDM is calculated using diffusion Monte Carlo methods and it is diagonalized to obtain the ``natural'' single-particle orbitals and their occupation, including the condensate fraction. At low boson density, ${\mathrm{na}}^{3}<{10}^{\ensuremath{-}5},$ where
n=N/V and a is the hard-core diameter, the condensate is localized at the center of the trap. As
na3 increases, the condensate moves to the edges of the trap. At high density, it is localized at the edges of the trap. At ${\mathrm{na}}^{3}<~{10}^{\ensuremath{-}4},$ the Gross-Pitaevskii theory of the condensate describes the whole system within
1 At ${\mathrm{na}}^{3}\ensuremath{\approx}{10}^{\ensuremath{-}3},$ corrections are
3 to the Grass-Pitaevskii energy but
30 to the Bogoliubov prediction of the condensate depletion. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}{10}^{\ensuremath{-}2},$ mean-field theory fails. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}0.1,$ the bosons behave more like a liquid
4He droplet than a trapped boson gas.