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arXiv:0805.4036 · Journal of Physics A Mathematical and Theoretical

A diagrammatic calculation of the energy spectrum of quantum impurity in degenerate Bose–Einstein condensate

Alexey Novikov, Mikhail Ovchinnikov

Published Mar 4, 2009• 17 citations
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Abstract

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.

DOI
10.1088/1751-8113/42/13/135301

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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.
The motion of single quantum particle through Bose-Einstein condensate (BEC) is considered within perturbation theory with respect to the particle-BEC interaction. The Hamiltonian of BEC is diagonalized by means of Bogoliubov's method. The process of dissipation due to the creation of excitation in BEC is analyzed and the dissipation rate is calculated in the lowest order of perturbation theory. The Landau's criterion for energy dissipation in BEC is then recovered. The energy spectrum of the impurity particle due to the interaction with BEC and its effective mass are evaluated.
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.
The exact density of states for one particle in the field of an attractive $\ensuremath{\delta}$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The "Bose-Einstein phase transition" is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature Tc{T}_{c}Tc​. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, Eb{E}_{b}Eb​. The transition temperature and the discontinuity of the specific heat at Tc{T}_{c}Tc​ are also obtained as a function of Eb{E}_{b}Eb​.
In this paper we develop a gapless theory of Bose-Einstein condensation (BEC) which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number-conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher-order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first- and second-order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infrared divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultraviolet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T -matrix leads naturally to a high-energy renormalization. The theory developed in this paper is therefore well defined at both low and high energy and provides a systematic description of BEC in dilute gases. It can therefore be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition.
We derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.
Abstract The properties of an impurity immersed in a dilute D -dimensional Bose gas at temperatures close to its second-order phase transition point are considered. Particularly by means of the 1/ N -expansion, we calculate the leading-order polaron energy and the damping rate in the limit of vanishing boson–boson interaction. It is shown that the perturbative effective mass and the quasiparticle residue diverge logarithmically in the long-length limit, signalling the non-analytic behavior of the impurity spectrum and pole-free structure of the polaron Green’s function in the infrared region, respectively.