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arXiv:nlin/0502059 · Journal of Experimental and Theoretical Physics

A kinetic approach to Bose-Einstein condensates: Self-phase modulation and Bogoliubov oscillations

J. T. Mendonça, R. Bingham, P. K. Shukla

Published Nov 1, 2005• 6 citations
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Abstract

A kinetic approach to Bose-Einstein condensates (BECs) is proposed based on the Wigner-Moyal equation (WME). In the semiclassical limit, the WME reduces to the particle-number conservation equation. Two examples of applications are (i) a self-phase modulation of a BE condensate beam, where we show that part of the beam is decelerated and eventually stops as a result of the gradient of the effective self-potential, and (ii) the derivation of a kinetic dispersion relation for sound waves in BECs, including collisionless Landau damping.

DOI
10.1134/1.2149073

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A kinetic approach to Bose-Einstein condensates (BECs) is proposed based on the Wigner-Moyal equation (WME). In the semiclassical limit, the WME reduces to the particle-number conservation equation. Two examples of applications are (i) a self-phase modulation of a BE condensate beam, where we show that part of the beam is decelerated and eventually stops as a result of the gradient of the effective self-potential, and (ii) the derivation of a kinetic dispersion relation for sound waves in BECs, including collisionless Landau damping.
We consider Landau damping of elementary excitations in Bose-Einstein condensates (BECs) with dipolar interactions. We discuss quantum and quasiclassical regimes of Landau damping. We use a generalized wave-kinetic description of BECs which, apart from the long-range dipolar interactions, also takes into account the quantum fluctuations and the finite-energy corrections to short-range interactions. Such a description is therefore more general than the usual mean-field approximation. The present wave-kinetic approach is well suited for the study of kinetic effects in BECs, such as those associated with Landau damping, atom trapping, and turbulent diffusion. The inclusion of quantum fluctuations and energy corrections changes the dispersion relation and the damping rates, leading to possible experimental signatures of these effects. Quantum Landau damping is described with generality, and particular examples of dipolar condensates in two and three dimensions are studied. The occurrence of roton-maxon excitations, and their relevance to Landau damping, are also considered in detail. The present approach is mainly based on a linear perturbative procedure, but the nonlinear regime of Landau damping, which includes atom trapping and atom diffusion, is also briefly discussed.
We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate.It is based on the coupled evolution equations of the parameters describing the system.These equations have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal Bose gas with a separated condensate.The dispersion equations for sound oscillations were obtained by linearization of the mentioned evolution equations in the collisionless approximation.They were analyzed both analytically and numerically.The expressions for sound speed and decrement rate were obtained in high and low temperature limiting cases.We have shown that at low temperature the dependence of the obtained quantities on temperature significantly differs from those obtained by other authors in the semi-phenomenological approaches.Possible effects connected with non-analytic temperature dependence of dispersion characteristics of the system were also indicated.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
We discuss the Schrödinger–Newton (SN) equation in the context of cold atom physics. For that purpose, we establish the wave-kinetic equation equivalent to the SN equation, which stays valid in different physical scenarios relevant to cold atoms. They include: (1) the usual scenario of matter confined in a self-gravitating field, (2) atomic molasses, confined and cooled by laser beams in a magneto-optical trap (MOT), (3) Bose–Einstein condensates, with or without long range dipolar interactions, and (4) electron states in a quantum plasma. We show that these different systems can be described by a formally identical equation, and they also manifest similar elementary excitations. The wave-kinetic equation is obtained by following the well-known Wigner–Moyal procedure, allowing the representation of quantum states in a classical phase-space. It is particularly well suited to discuss kinetic properties associated with Landau damping, as shown. We also consider generalisation of the SN equation onto the relativistic domain, and its impact on the proposed wave-kinetic description.
We show how to adapt the ideas of local energy and momentum conservation in order to derive modifications to the Gross-Pitaevskii equation which can be used phenomenologically to describe irreversible effects in a Bose-Einstein condensate. Our approach involves the derivation of a simplified quantum kinetic theory, in which all processes are treated locally. It is shown that this kinetic theory can then be transformed into a number of phase-space representations, of which the Wigner function description, although approximate, is shown to be the most advantageous. In this description, the quantum kinetic master equation takes the form of a Gross-Pitaevskii equation with noise and damping added according to a well defined prescription - an equation we call the stochastic Gross-Pitaevskii equation . From this, a very simplified description we call the phenomenological growth equation can be derived. We use this equation to study (i) the nucleation and growth of vortex lattices, and (ii) nonlinear losses in a hydrogen condensate, which it is shown can lead to a curious instability phenomenon.
We show how to adapt the ideas of local energy and momentum conservation in order to derive modifications to the Gross-Pitaevskii equation which can be used phenomenologically to describe irreversible effects in a Bose-Einstein condensate. Our approach involves the derivation of a simplified quantum kinetic theory, in which all processes are treated locally. It is shown that this kinetic theory can then be transformed into a number of phase-space representations, of which the Wigner function description, although approximate, is shown to be the most advantageous. In this description, the quantum kinetic master equation takes the form of a GPE with noise and damping added according to a well-defined prescription--an equation we call the stochastic GPE. From this, a very simplified description we call the phenomenological growth equation can be derived. We use this equation to study i) the nucleation and growth of vortex lattices, and ii) nonlinear losses in a hydrogen condensate, which it is shown can lead to a curious instability phenomenon.