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.
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.
The ground-state properties of a system with a small number of interacting bosons over a wide range of densities are investigated. The system is confined in a two-dimensional isotropic harmonic trap, where the interaction between bosons is treated as a hard-core potential. By using variational Monte Carlo method, we diagonalize the one-body density matrix of the system to obtain the ground-state energy, condensate wavefunction and the condensate fraction. We find that in the dilute limit the depletion of central condensate in the 2D system is larger than in a 3D system for the same interaction strength; however as the density increases, the depletion at the centre of 2D trap will be equal to or even lower than that at the centre of 3D trap, which is in agreement with the anticipated in Thomas–Fermi approximation. In addition, in the 2D system the total condensate depletion is still larger than in a 3D system for the same scattering length.
The Bose-Einstein condensation of correlated atoms in a trap is studied by examining the effect of inter-particle 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 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.
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.
We investigate two-species Bose gases in traps with various interactions using variational quantum Monte Carlo (VMC) techniques at zero temperature. The bosons are represented by hard spheres whose diameter is equivalent to the
s-wave scattering length in the low-energy and long-wavelength approximation. We explore the role of repulsive and attractive interspecies or intraspecies interactions on the condensate properties of the mixtures, particularly the condensate fraction of each species as compared to the case when each species is in a separate trap of its own. We model the repulsive interactions by a hard-core (HC) potential and the attractive interactions by a shallow model potential. The VMC density profiles and energies are evaluated at various interactions and two mass ratios of the species.