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arXiv:0911.5723 · Journal of Physics B Atomic Molecular and Optical Physics

Variational approach to the ground state of an impurity in a Bose–Einstein condensate

Alexey Novikov, Mikhail Ovchinnikov

Published Apr 26, 2010• 24 citations
arXivHTMLPDF

Abstract

In this paper, we consider the effect of self-localization of a quantum impurity in a Bose–Einstein condensate. The space correlation function of the impurity is evaluated with the help of the imaginary-time path integral approach. Employing Feynman's variational method, we calculate the impurity correlation function as well as the energy of the system associated with the impurity. The effect of self-localization predicted before within the Gross–Pitaevskii approach is recovered by our treatment. The strong coupling regime with negative ground state energy is reached by the variational method, and the corresponding correlation function is calculated.

DOI
10.1088/0953-4075/43/10/105301

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In this paper, we consider the effect of self-localization of a quantum impurity in a Bose–Einstein condensate. The space correlation function of the impurity is evaluated with the help of the imaginary-time path integral approach. Employing Feynman's variational method, we calculate the impurity correlation function as well as the energy of the system associated with the impurity. The effect of self-localization predicted before within the Gross–Pitaevskii approach is recovered by our treatment. The strong coupling regime with negative ground state energy is reached by the variational method, and the corresponding correlation function is calculated.
In this paper we consider a quantum particle moving through dilute Bose–Einstein condensate at zero temperature. In our formulation the impurity particle interacts with the gas of uncoupled Bogoliubov's excitations. The perturbation theory for the Green's function of the impurity particle with respect to the impurity–condensate interaction is constructed by employing the coherent-state path integral approach. The perturbative expansion for the Green's function is resumed into the expansion for its self-energy part with the help of the diagrammatic technique developed in this work. The dispersion relation for the impurity clothed by condensate excitations is obtained and effective mass is evaluated beyond the Golden rule approximation.
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.
Bose-Einstein condensates have been, by now, observed in as many as nine different atomic assemblies of bosons. Such a condensate is quantum mechanical interacting system whose ground state properties can be studied theoretically by solving the appropriate non-linear Gross-Pitaevskii-Ginzburg GPG equation. One can now study the change in the behavior of Bose-Einstein condensate by introducing a localized impurity which interacts with the condensate as a function of position of impurity in the condensate. The introduction of such an impurity can be mimicked by simply allowing an intensely focused laser beam to interact with the condensate. This would lead to alteration of ground state properties of the condensate as it would now interact with a potential of type V Sech2(r/w) where, V and w are amplitude and width of the impurity potential, respectively. The modified GPG equation in the presence of localized impurity potential as function of position in the condensate, has been numerically solved to obtain its various ground state properties as function of position, such as total energy per particle, chemical potential, kinetic, harmonic trap potential and two-body interaction energies per particle in addition to energy associated with impurity potential, correlation length, healing length etc. We have studied the behavior of the above-mentioned ground state properties as the position of localized impurity is changed in the condensate from core to peripheral position. While the total, harmonic oscillator potential and impurity energies decrease as the position of localized impurity is displaced from core of the condensate to its periphery, the value of two-body inter-particle interaction energy increases. Further, the values of chemical potential and total energy per particle shows decrease by ~ 9% and ~ 17% respectively, leading to the inference that the stability of condensate increases as the localized impurity is moved away from the core of the condensate.
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.
Abstract We consider two pointlike static impurities without direct interaction immersed in a three-dimensional Bose–Einstein condensate (BEC) at zero temperature. By solving the Gross–Pitaevskii (GP) equation in a perturbative manner, we calculate the ground state energy in the region where the atom–impurity interaction is assumed to be weak. We obtain an analytical expression for the spatial distribution of atom number density and the effective force between these two impurities. The effective force is found to be closely related to the strength of the atom–impurity interaction and the relative distance between these two impurities. Two critical relative distances are found between the two impurities. The first one corresponds to the vanishing of the perturbed energy with impurities, although the effective force between the two impurities still exists. At the second critical value, the energy of the impurities changes linearly with the atom–impurity interaction; otherwise, it changes quadratically with the atom–impurity interaction.
An impurity immersed in a Bose-Einstein condensate is no longer accurately described by the Fr\"ohlich Hamiltonian as the coupling between the impurity and the boson bath gets stronger. We study the dominant effects of the two-phonon terms beyond the Fr\"ohlich model on the ground-state properties of the polaron using Feynman's variational path-integral approach. The previously reported discrepancy in the effective mass between the renormalization group approach and this theory is shown to be absent in the beyond-Fr\"ohlich model on the positive side of the Feshbach resonance. Self-trapping, characterized by a sharp and dramatic increase of the effective mass, is no longer observed for the repulsive polaron once the two-phonon interactions are included. For the attractive polaron we find a divergence of the ground-state energy and effective mass at weaker couplings than previously observed within the Fr\"ohlich model.