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SUST Journal of Science and Technology (SUST JST)

Monte Carlo simulation of Bose-Einstein condensation in a harmonic potential trap

Dr Md Enamul Hoque

Published Dec 31, 2021• 0 citations
Publisher

Abstract

Bose-Einstein condensation has been attained using Monte Carlo simulation on non-interacting ideal bosons confined in a harmonic potential trap at near-zero temperature (~0K). A connection has been established between the cyclic paths originated by the interchange of identical bosons and condensation. At low temperatures, long cycles appear and condensation is obtained consequently. The simulated result is in perfect agreement with the corresponding theoretical findings.

DOI
10.63512/sustjst.2021.33

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Bose-Einstein condensation has been attained using Monte Carlo simulation on non-interacting ideal bosons confined in a harmonic potential trap at near-zero temperature (~0K). A connection has been established between the cyclic paths originated by the interchange of identical bosons and condensation. At low temperatures, long cycles appear and condensation is obtained consequently. The simulated result is in perfect agreement with the corresponding theoretical findings.
We present an analysis of Bose-Einstein condensation for a system of non-interacting spin-0 particles in a harmonic oscillator confining potential trap. We discuss why a confined system of particles differs both qualitatively and quantitatively from an identical system which is not confined. One crucial difference is that a confined system is not characterized by a critical temperature in the same way as an unconfined system such as the free boson gas. We present the results of both a numerical and analytic analysis of the problem of Bose-Einstein condensation in a general anisotropic harmonic oscillator confining potential.
Bose-Einstein condensation is known to occur in one-dimensional power-law potentials $V(x)\ensuremath{\propto}|x{|}^{\ensuremath{\eta}}$ as a true phase transition for \ensuremath{\eta}2, but only if there are no interactions between the particles. We show, via a path-integral quantum Monte Carlo scheme, that for both \ensuremath{\eta}2 and \ensuremath{\eta}=2 the spatial distribution of finite numbers of hard-core bosons suddenly becomes bimodal below a certain temperature, with a central condensate of particles distributed according to the lowest single-particle eigenstate. At still lower temperatures, the hard-core interactions cause the single-particle ground state to become a poor description for the interacting gas. If \ensuremath{\eta}2, the energy per particle undergoes a sudden decrease near the same temperature at which the bimodal distribution appears. It is found that this drop in energy disappears if the range of the interaction potential is sufficiently large.
We present a Monte Carlo method for the direct simulation of evaporative cooling and the formation of the Bose-Einstein condensate of magnetically trapped atoms. The motion of the atoms in the trap is treated classically, while quantum-mechanical probability is used to describe binary elastic collisions between bosons. Two implementations of Bose statistics in the frame of a classical Monte Carlo simulation have been developed. The one involves the evaluation of the number of quantum states in single-atom phase space and the other makes use of the quantum degeneracy of energy bands. Both techniques give results compatible with the analytic results for an ideal trapped Bose gas and with the theory of a weakly interacting Bose gas, as concerns the equilibrium condensate fraction. Our simulation method also makes it possible to study the dynamic formation of the condensate, giving characteristic condensation times consistent with the experiments.
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.
Bose-Einstein condensation has been experimentally found to take place in finite trapped systems when one of the confining frequencies is increased until the gas becomes effectively two-dimensional (2D). We confirm the plausibility of this result by performing path-integral Monte Carlo (PIMC) simulations of trapped Bose gases of increasing anisotropy and comparing them to the predictions of finite-temperature many-body theory. PIMC simulations provide an essentially exact description of these systems; they yield the density profile directly and provide two different estimates for the condensate fraction. For the ideal gas, we find that the PIMC column density of the squeezed gas corresponds quite accurately to that of the exact analytic solution and, moreover, is well mimicked by the density of a 2D gas at the same temperature; the two estimates for the condensate fraction bracket the exact result. For the interacting case, we find 2D Hartree-Fock solutions whose density profiles coincide quite well with the PIMC column densities and whose predictions for the condensate fraction are again bracketed by the PIMC estimates.
A path-integral quantum Monte Carlo method is used to calculate finite temperature properties of up to 1000 hard-core bosons in a two-dimensional isotropic harmonic-oscillator potential. If the interatomic repulsions are sufficiently short range, an abrupt increase in the condensate fraction and a hump in the specific heat occur close to the critical temperature of ideal bosons. The critical temperature and the condensate fraction are in general lowered by an increase in the hard-core radius a. If a is decreased below a certain level, the condensate fraction becomes indistinguishable from the corresponding value of the ideal bosons. For up to 1000 particles, this occurs when ${\mathrm{ln}}^{\ensuremath{-}1}{(1/na}^{2})\ensuremath{\lesssim}0.1,$ where n is the average particle density.