Dynamics of sound waves in an interacting Bose gas

Abstract

We consider a non-relativistic quantum gas of N bosonic atoms confined to a box of volume Λ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, ρ = N Λ , of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume Λ and small ratio Λ ρ . The initial state of the gas is assumed to be close to a product state of one-particle wave functions that are approximately constant throughout the box. The initial one-particle wave function of an excitation is assumed to have a compact support independent of Λ. We derive an effective non-linear equation for the time evolution of the one-particle wave function of an excitation and establish an explicit error bound tracking the accuracy of the effective non-linear dynamics in terms of the ratio Λ ρ . We conclude with a discussion of the dispersion law of low-energy excitations, recovering Bogolyubov's well-known formula for the speed of sound in the gas, and a dynamical instability for attractive two-body potentials.

A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics

“…The functional was used in [51,39] for the derivation of the bosonic Hartree equation, and in [52,50] for the derivation of the Gross-Pitaevskii equation. A variant of it has been used in [21,20] to derive the effective dynamics of a tracer particle in a Bose gas and in [3] to derive the Hartree-von Neumann limit.…”

Bogoliubov corrections and trace norm convergence for the Hartree dynamics

“…Along with some modifications, the n = 1 case has been also used to derive the time-dependent Hartree equation for very singular potentials [22,Section 4], the Hartree equation in a large volume mean field limit [10], the NLS equation without positivity condition [36] as well as the Gross-Pitaevskii equation [38]. The idea is also applicable for fermions to derive the Hartree and Hartree-Fock equations in the corresponding mean field limit [35,2].…”

On coupling kinetic and Schrödinger equations

“…Since the coupling between the equation involving C 12 [f ] and the cubic nonlinear Schrödinger equation is difficult to study, let us replace C 12 [f ] by the linear quantum Boltzmann operator…”

“…For more discussions and references on this topic, we refer to the review paper [2] and the books [34,48]. Let us note that besides the kinetic theory point of view, there are other approaches to the study of BECs and excitations: the excitation spectrum [52], Fock space approach used to improve convergence rate in the analysis of Hepp, Rodnianski-Schlein [28,30,29], Fock space approach central limit theorem [5], Quasifree reduction [3], the time evolution of the one-particle wave function of an excitation [12,45], and cited references. Quantum kinetic theory, on the other hand, is both a genuine kinetic theory and a genuine quantum theory.…”