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Journal of Physics A Mathematical and Theoretical

Damping of Bogoliubov excitations at finite temperatures

V. Pastukhov

Published Sep 14, 2015• 12 citations
Publisher

Abstract

We present a simple and efficient method to calculate the damping for the excitation spectrum of a uniform D-dimensional Bose gas. Starting from the original Popov's hydrodynamic description and integrating out phase variables, we obtained the effective action of amplitude fluctuations. Within this approach, the lifetime of quasi-particles with a finite momentum is calculated at a wide temperature range. It is shown that the correct use of the hydrodynamic approach leads to the damping rate, which coincides with results obtained by means of the perturbation theory.

DOI
10.1088/1751-8113/48/40/405002

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We present a simple and efficient method to calculate the damping for the excitation spectrum of a uniform D-dimensional Bose gas. Starting from the original Popov's hydrodynamic description and integrating out phase variables, we obtained the effective action of amplitude fluctuations. Within this approach, the lifetime of quasi-particles with a finite momentum is calculated at a wide temperature range. It is shown that the correct use of the hydrodynamic approach leads to the damping rate, which coincides with results obtained by means of the perturbation theory.
We develop a finite-temperature perturbation theory (beyond the mean field) for a Bose-condensed gas and calculate temperature-dependent damping rates and energy shifts for Bogolyubov excitations of any energy. The theory is generalized for the case of excitations in a spatially inhomogeneous (trapped) Bose-condensed gas, where we emphasize the principal importance of inhomogeneity of the condensate density profile and develop the method of calculating the self-energy functions. The use of the theory is demonstrated by calculating the damping rates and energy shifts of low-energy quasiclassical excitations, i.e., the quasiclassical excitations with energies much smaller than the mean-field interaction between particles. In this case the boundary region of the condensate plays a crucial role and the result for the damping rates and energy shifts is completely different from that in spatially homogeneous gases. We also analyze the frequency shifts and damping of sound waves in cylindrical Bose condensates and discuss the role of damping in the recent MIT experiment on the sound propagation.
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic theory that enables one to systematically study both the excitations and their decay. At zero temperature, the leading mechanism of decay of a quasiparticle is disintegration into three others. We find that low-energy quasiparticles (phonons) decay with the rate that scales with the seventh power of momentum, whereas the rate of decay of the high-energy quasiparticles does not depend on momentum. In addition, our approach allows us to study analytically the quasiparticle decay in the whole crossover region between the two limiting cases. When applied to integrable models, including the Lieb-Liniger model of bosons with contact repulsion, our theory confirms the absence of the decay of quasiparticle excitations. We account for two types of integrability-breaking perturbations that enable finite decay: three-body interaction between the bosons and two-body interaction of finite range.
We study the Landau damping of Bogoliubov excitations in two- and three-dimensional optical lattices at finite temperatures, extending our recent work on one-dimensional (1D) optical lattices. We use a Bose-Hubbard tight-binding model and the Popov approximation to calculate the temperature dependence of the number of condensate atoms nc0(T){n}^{\mathrm{c}0}(T)nc0(T) in each lattice well. As with 1D optical lattices, damping only occurs if the Bogoliubov excitations exhibit anomalous dispersion (i.e., the excitation energy bends upward at low momentum), analogous to the case of phonons in superfluid 4He^{4}\mathrm{He}4He. This leads to the disappearance of all damping processes in a DDD-dimensional simple cubic optical lattice when $U{n}^{\mathrm{c}0}\ensuremath{\geqslant}6DJ$, where UUU is the on-site interaction, and JJJ is the hopping matrix element.
We present the theory for low energy collective excitations of a trapped Bose-Einstein condensate which includes the dynamics of thermal quasiparticles. A linearized hydrodynamic description yields the equations expressed in terms of the small density fluctuation δn and velocity potential fluctuation δΦ , which is the generalization of the one given by Morii et al. to the non-uniform case. A expression of the damping rate of low energy collective excitations has been obtained for the higher temperature regime.
We study damping of the dipole oscillation in a Bose-condensed gas in a combined cigar-shaped harmonic trap and one-dimensional (1D) optical lattice potential at finite temperatures. In order to include the effect of thermal excitations in the radial direction, we derive a quasi-1D model of the Gross-Pitaevskii equation and the Bogoliubov equations. We use the Popov approximation to calculate the temperature dependence of the condensate fraction with varying lattice depth. We then calculate the Landau damping rate of the dipole oscillation as a function of the lattice depth and temperature. The damping rate increases with increasing lattice depth, which is consistent with experimental observations. The magnitude of the damping rate is in reasonable agreement with experimental data. We also find that the damping rate has a strong temperature dependence, showing a sharp increase with increasing temperature. Finally, we emphasize the importance of the radial thermal excitations in both equilibrium properties and the Landau damping.
We investigate the small frequency and momentum structure of the weakly interacting Bose gas in two dimensions using a functional renormalization group approach. The flow equations are derived within a derivative approximation of the effective action up to second order in spatial and temporal variables and investigated numerically. The truncation we employ is based on the perturbative structure of the theory and is well described as a renormalization group enhanced perturbation theory. It allows to calculate corrections to the Bogoliubov spectrum and to investigate the damping of quasiparticles. Our approach allows to circumvent the divergences which plague the usual perturbative approach.