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Ground-state properties of an anisotropically trapped alkali gas using Feynman path integration

Natthapon Nakpathomkun, Virulh Sa-yakanit

Published Jan 1, 2001• 0 citations
Publisher

Abstract

The dilute weakly interacting alkali gas was shown to exhibit the phase transition called Bose-Einstein condensation for the first time in the experiment of Wieman's group in 1995. in this thesis, the ground state properties of the alkali gas trapped in an anisotropic magnetic field are investigated by using the variational method in path integral formalism. The interatomic interaction is approximated by the delta function which is short-range potential. The approximated density matrix is derived. Consequently the ground state energy and wavefunction are obtained. The ground state energy is then compared to the result obtained by the mean field approach both analytically and numerically. The result is shown to equivalent to the variational mean field approach. In addition, a comparison between path integral approach and the numerical mean field approach is shown to give no significant difference. The ground state wavefunction is extracted from the weighting function. Substitute the variational parameters into the expression of wavefunction one can see that the result is close to the numerical method.

DOI
10.58837/chula.the.2001.1064

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The dilute weakly interacting alkali gas was shown to exhibit the phase transition called Bose-Einstein condensation for the first time in the experiment of Wieman's group in 1995. in this thesis, the ground state properties of the alkali gas trapped in an anisotropic magnetic field are investigated by using the variational method in path integral formalism. The interatomic interaction is approximated by the delta function which is short-range potential. The approximated density matrix is derived. Consequently the ground state energy and wavefunction are obtained. The ground state energy is then compared to the result obtained by the mean field approach both analytically and numerically. The result is shown to equivalent to the variational mean field approach. In addition, a comparison between path integral approach and the numerical mean field approach is shown to give no significant difference. The ground state wavefunction is extracted from the weighting function. Substitute the variational parameters into the expression of wavefunction one can see that the result is close to the numerical method.
Based on the energy functional and variational method, we present a new method to investigate the ground state properties for a weakly interacting Bose-condensed gas in an anisotropic harmonic trap at zero temperature. With this method we are able to find the analytic expression of the ground-state wavefunction and to explore the relevant quantities, such as energy, chemical potential, and the aspect ratio of the velocity distribution. These results agree well with previous ground state numerical solutions of the Gross–Pitaevskii equation given by Dalfovo et al. [Phys. Rev. A 53 (1996) 2477] This new method is simple compared to other methods used to solve numerically the Gross–Pitaevskii equation, and one can obtain analytic and reliable results.
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.
In this thesis, the case of charged bosons confined in an isotropic magnetic trap is studied by using Feynman path integration. We show that the upper bound of the ground state energy can be evaluated by applying the variational path integral technique. Consequently, the approximated density matrix and wavefunction are obtained. The ground state energy is then compared with the result obtained by Ginzburg-Pitaevskii-Gross approach. it is shown that our result agrees with such an approach. Furthermore, from our result, when we neglect the effect of screened Coulomb potential in the ground state energy of the screened Coulomb case, the ground state energy of the Coulomb potential arises as expected.
The ground and excited states of a weakly interacting and dilute Bose-Einstein condensed gas, confined in a completely anisotropic harmonic oscillator potential, are determined at zero temperature within the Bogoliubov approximation. The numerical calculations employ a computationally efficient procedure based on a discrete variable representation (DVR) of the Hamiltonian. The DVR is efficient for problems where the interaction potential may be expressed as a local function of interparticle coordinates. In order to address condensates that are both very large ($\ensuremath{\sim}{10}^{6}$ atoms) and fully anisotropic, the ground state is found using a self-consistent field approach. Experience has demonstrated, however, that standard iterative techniques applied to the solution of the nonlinear partial differential equation for the condensate are nonconvergent. This limitation is overcome using the method of direct inversion in the iterated subspace (DIIS). In addition, the sparse structure of the DVR enables the efficient application of iterative techniques such as the Davidson and/or Lanczos methods, to extract the eigenvalues of physical interest. The results are compared with recent experimental data obtained for Bose-Einstein condensed alkali-metal vapors confined in magnetic traps.
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.