Calculation of excitations in Bose-Einstein condensates within second order theory

Mark Edwards

Tao Chen, Bo Yan

We apply the time-dependent Hartree-Fock-Bogoliubov (td-HFB) theory to describe the stimulated excitation driven by periodically modulating the interactions in a Bose-Einstein condensate (BEC). A comparison with the results calculated from the typical Bogoliubov approximation indicates that the additional interaction terms contributed by the excited modes play an important role to explain the dynamics of the stimulating process. The td-HFB model has not only painted a clear picture of the density wave propagation, but also partly explained the generation of the second order harmonic of the excited modes. The theoretical framework can be directly employed to study similar driven processes.

Pankaj Kumar

This paper present, using the gapless second order theory, the excitation energies and dynamics of the collective excitation for a partially condensed harmonic trapped quasi two dimensional bosonic gas. For numerical estimation we have taken a model of pancake shaped cloud consisting of N=200023 Na atom trapped with trapping frequency ωc = 25 x 350 Hz. We have shown the evaluated result of temperature dependent mean energies of the excitation using second order theory together with HFB-Popov theory and exact energy of Kohn modes. According to Kohn theorem a system of harmonically trapped interacting particle in any eigenstate of the Hamiltonian has an eigenstate with the amount ?ωi. In higher order theories, the dynamics of the thermal gas and its interaction with the condensate have to be taken into account accurately obtained results in agreement with Kohn theorem. In our calculation, it is shown that within the second order theory, the energies of the Kohn mode is very close to ?ω for temperature T< 0.8Tc.

Dániel Schumayer

Kunal K. Das

S. A. Morgan

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.