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Communications in Theoretical Physics

Collective Excitations in Spin-2 Bose–Einstein Condensates

Jing-Min Hou, Li-Jun Tian

Published Dec 1, 2005• 0 citations
Publisher

Abstract

The Green's functions and the correlation functions in spin-2 Bose–Einstein condensates at finite temperature are defined and the generalized Dyson–Beliaev equations are introduced. We discuss the spin conservation in z direction and decouple the Green's functions and the generalized Dyson–Beliaev equations according to different spin conservations in z direction. The anomalous vertex functions are introduced and the self-energies are separated into the proper self-energies and the improper self-energies. The generalized Dyson–Beliaev equations are decoupled according to separation of the self-energies. We calculate the Green's functions step by step in the Bogoliubov approximation and discuss the collective excitations in spin-2 Bose–Einstein condensates in the polar, ferromagnetic, and cyclic cases, respectively.

DOI
10.1088/6102/44/6/1025

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The Green's functions and the correlation functions in spin-2 Bose-Einstein condensates at finite temperature are defined and the generalized Dyson-Beliaev equations are introduced. We discuss the spin conservation in z direction and decouple thre Green's functions and the generalized Dyson-Beliaev equations according to different spin conservations in z direction. The anomalous vertex functions are introduced and the self-energies are separated into the proper self-energies and the improper self-energies. The generalized Dyson-Beliaev equations are decoupled according to separation of the self-energies. We calculate the Green's functions step by step in the Bogoliubov approximation and discuss the collective excitations in spin-2 Bose-Einstein condensates in the polar, ferromagnetic, and cyclic cases,respectively.
Starting from the Dyson-Beliaev and generalized Gross-Pitaevskii equations with an extra nonlocal potential, we derive an exact expression of the two-particle Green's function $\underset{̱}{\mathcal{K}}$ for an interacting Bose-Einstein condensate in terms of unambiguously defined self-energies and vertices. The formula can be a convenient basis for approximate calculations of $\underset{̱}{\mathcal{K}}$. It also tells us that poles of $\underset{̱}{\mathcal{K}}$ are not shared with (i.e., shifted from) those of the single-particle Green's function, contrary to the conclusion of previous studies.
We give an overview of the current theory of collective modes in trapped atomic gases at finite temperatures, when the dynamics of the condensate and non-condensate must both be considered. A simple introduction is given to the quantum field formulation of the dynamics of an interacting Bose-condensed system, based on equations of motion for the condensate wavefunction and single-particle Green's functions for the non-condensate atoms. We discuss the nature of excitations in the mean-field collisionless region, including the Beliaev second-order approximation for the self-energies. We also sketch the derivation of coupled two-fluid hydrodynamic equations using a simple kinetic equation which includes collisions between condensate and non-condensate atoms.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
The finite temperature Green's function technique is used to calculate the energies and damping rates of the elementary excitations of homogeneous, dilute, spin-1 Bose gases below the Bose-Einstein condensation temperature in both the density and spin channels. For this purpose a self-consistent dynamical Hartree-Fock model is formulated, which takes into account the direct and exchange processes on equal footing by summing up certain classes of Feynman diagrams. The model is shown to satisfy the Goldstone theorem and to exhibit the hybridization of one-particle and collective excitations correctly. The results are applied to gases of 23Na{}^{23}\mathrm{Na}23Na and 87Rb{}^{87}\mathrm{Rb}87Rb atoms.
A homogeneous gas of atoms with hyperfine spin F=1F=1F=1 is considered below the critical temperature of the Bose-Einstein condensation. The general structure of the correlation functions (both for density and spin fluctuations) and their connection to the thermal Green's functions are worked out. The damping of the modes is calculated in the random-phase approximation.