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New Journal of Physics

Wave-kinetic approach to the Schrödinger–Newton equation

José Tito Mendonça

Published Jan 22, 2019• 21 citations

Abstract

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.

DOI: 10.1088/1367-2630/ab0045

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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.

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.

Generalised Wigner and Weyl transformations of quantum operators are defined and their properties, as well as those of the algebraic structure induced on the phase-space are studied. Using such transformations, quantum linear evolution equations are given a phase-space representation. In particular this is done for the general kinetic equation of the Lindblad type. The resulting expressions are better suited for the passage to the classical limit and for a general comparison of classical and quantum systems. In this context a preliminary discussion of a number of problems of kinetic theory of open systems is given, whereas explicit applications are made in the next paper of the series.

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.

Using both fluid and kinetic descriptions, with repulsive forces between nearby atoms included, we discuss the basic oscillations and waves of a cloud of ultracold atoms confined in a magneto-optical trap. The existence of a hybrid mode, with properties similar to both plasma and acoustic waves, is described in detail. Tonks-Dattner resonances for confined hybrid modes in a spherical cloud are discussed, and the prediction of a nonlinear coupling between the dipole resonance and the hybrid modes is considered. Landau damping processes and quasilinear diffusion in velocity space are also discussed.