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Determination of ground state energy and effective mass of bose systems in random potentials by path integration method

Wattana Lim, Wichit Sritrakool, Virulh Sa-yakanit

Published Jan 1, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.58837/chula.the.2007.988

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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 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.
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.
We apply a path integral variational approach to obtain analytical expressions for condensate wave functions of an ultracold, interacting trapped Bose gases. As in many recent experiments, the particles are confined in a 1D or 3D harmonic oscillator trap which is superimposed by a periodic potential in one direction. Based on the first order cumulant expansion with respect to a harmonic trial action, and emplyoing a mean-field approximation, optimal variational parameters are obtained by minimizing an analytical expression for the ground state energy. Our largely analytical results for energy and condensate wave function are in good agreement with fully numerical calculations based on the Gross-Pitaevskii equation.
The diffusion Monte Carlo method is applied to describe a trapped atomic Bose-Einstein condensate at zero temperature, fully quantum mechanically and nonperturbatively. For low densities, ${n(0)a}^{3}<~2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ [n(0),[n(0),[n(0), peak density; a, s-wave scattering length], our calculations confirm that the exact ground-state energy for a sum of two-body interactions depends to a good approximation on only the atomic-physics parameter a, and no other details of the two-body model potential. Corrections to the mean-field Gross-Pitaevskii energy range from being essentially negligible to about 20% for N=2−−50N=2--50N=2−−50 particles in the trap with positive s-wave scattering length a=100−−10000a.u.a=100--10000\mathrm{a}.\mathrm{u}.a=100−−10000a.u. Our numerical calculations confirm that inclusion of an additional effective potential term in the mean-field equation, which accounts for quantum fluctuations [see, e.g., E. Braaten and A. Nieto, Phys. Rev. B 56, 14 745 (1997)], leads to a greatly improved description of trapped Bose gases.
By using exact path-integral Monte Carlo methods we calculate the equation of state of an interacting Bose gas as a function of temperature both below and above the superfluid transition. The universal character of the equation of state for dilute systems and low temperatures is investigated by modeling the interatomic interactions using different repulsive potentials corresponding to the same sss-wave scattering length. The results obtained for the energy and the pressure are compared to the virial expansion for temperatures larger than the critical temperature. At very low temperatures we find agreement with the ground-state energy calculated using the diffusion Monte Carlo method.