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Journal of Physics A Mathematical and Theoretical

Bose–Einstein condensation in a disordered system: the Feynman path integral approach

Wattana Lim, Virulh Sa‐Yakanit, Wichit Sritrakool

Published Apr 18, 2008• 1 citation
Publisher

Abstract

A modeled Bose system consisting of N particles with two-body interaction confined within volume V under the inhomogeneity of the system is investigated using the Feynman path integral approach. The two-body interaction energy is assumed to be dependent on the two-parameter interacting strength a and the correlation length l . The inhomogeneity of the system or the porosity can be represented as density with interacting strength b and correlation length L . The mean-field approximation on the two-body interaction in the Feynman path integrals representation is performed to obtain the one-body interaction. This approximation is equivalent to the Hartree approximation in the many-body electron gas problem. This approximation has shown that the calculation can be reduced to the effective one-body propagator. Performing the variational calculations, we obtain analytical results of the ground-state energy and the condensate density which are in agreement with that from Bugoliubov's approach.

DOI
10.1088/1751-8113/41/18/185002

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A modeled Bose system consisting of N particles with two-body interaction confined within volume V under the inhomogeneity of the system is investigated using the Feynman path integral approach. The two-body interaction energy is assumed to be dependent on the two-parameter interacting strength a and the correlation length l . The inhomogeneity of the system or the porosity can be represented as density with interacting strength b and correlation length L . The mean-field approximation on the two-body interaction in the Feynman path integrals representation is performed to obtain the one-body interaction. This approximation is equivalent to the Hartree approximation in the many-body electron gas problem. This approximation has shown that the calculation can be reduced to the effective one-body propagator. Performing the variational calculations, we obtain analytical results of the ground-state energy and the condensate density which are in agreement with that from Bugoliubov's approach.
A modeled Bose system consisting of N particles with two-body interaction confined within volume V under inhomogeneity of the system is investigated using the Feynman path integral approach. The two-body interaction energy is assumed to be dependent on the two-parameter interacting strength a and the correlation length l. The inhomogeneity of the system or the porosity can be represented as density [Formula: see text] with interacting strength b and correlation length L. The mean field approximation on the two-body interaction in the Feynman path integrals representation is performed to obtain the one-body interaction. This approximation is equivalent to the Hartree approximation in the many-body electron gas problem. This approximation has shown that the calculation can be reduced to the effective one-body propagator. Performing the variational calculations, we obtain analytical results of the ground state energy which is in agreement with that from Bugoliubov's approach.
We apply Feynman's path integral theory to study Bose-Einstein condensates in a double well potential and in a random potential. The main idea of this thesis is to perform the mean field approximation in two body interaction by the path integral approach. Bose particles are confined in the double well which is taken as a harmonic potential with Gaussian or cosine barrier. Performing the variational calculations we obtain analytical results of the ground state energy, wave functions, and overlap integrals. The overlap integrals enable us to calculate the tunneling rates which are in good agreement with numerical values for the two-mode model of Gross-Pitaevskii's equation. We also consider the model of a Bose system consisting of N particles with finite interacting pair potentials under the influence of random potentials. The porosity or nonhomogeneity of the system can be represented by the density of porosity with amplitude of the fluctuation, and a correlation length. We obtain analytical results of the effective mass and the ground state energy which are in good agreement with that of Bugoliubov approach. The propagator of the system is used to calculate the partition function, specific heat, the critical temperature, condensate density, and the superfluid density of the system.
A Bose–Einstein condensate in a double-well potential is considered by using the Feynman path integral theory combined with a variational approach. The system consists of N interacting bosons confined in the double well which is taken as a harmonic potential with a Gaussian barrier. The calculation is carried out within the first cumulant approximation measured with respect to the harmonic action containing variational parameters. Assuming a separable expression for the trial action corresponds to the usual mean field approximation. Performing the variational calculations, we obtain analytical results for the ground state energy and its condensate wavefunction. Using a projection onto the odd contributions, the first excited state energy and condensate wavefunction are determined, too. We find very good agreement with a full numerical solution of the Gross–Pitaevskii equation over the full range of potential parameters.
Properties of Bose–Einstein condensates (BEC) are studied in the presence of a Gaussian random potential of arbitrary strength of disorder and with arbitrary correlation length. Using the stochastic mean field approximation, a system having arbitrary strength of interactions is treated at finite temperature. A special case, in which we assume that the second moment of the random potential is Lorentzian, is treated in more detail at zero temperature. The total density of particles consists of condensate density, glassy density and normal density components. Our results are in agreement with those of Huang and Meng (HM). The system is stable below the critical value of the disorder parameter and undergoes a first-order phase transition at the critical value. We find that the critical value of the disorder parameter changes with changing the correlation length of the random potential.
We investigate the thermodynamic properties of a dilute Bose gas in a correlated random potential using exact path integral Monte Carlo methods. The study is carried out in continuous space and disorder is produced in the simulations by a three-dimensional (3D) speckle pattern with tunable intensity and correlation length. We calculate the shift of the superfluid transition temperature due to disorder and we highlight the role of quantum localization by comparing the critical chemical potential with the classical percolation threshold. The equation of state of the gas is determined in the regime of strong disorder, where superfluidity is suppressed and the normal phase exists down to very low temperatures. We find a T 2 dependence of the energy in agreement with the expected behavior in the Bose glass phase. We also discuss the major role played by the disorder correlation length and we make contact with a Hartree–Fock mean-field approach that holds if the correlation length is very large. The density profiles are analyzed as a function of temperature and interaction strength. Effects of localization and the depletion of the order parameter are emphasized in the comparison between local condensate and total density. At very low temperature, we find that the energy and the particle distribution of the gas are very well described by the T =0 Gross–Pitaevskii theory, even in the regime of very strong disorder.