Dynamics of Sound Waves in an Interacting Bose Gas

TL;DR

This paper derives an effective non-linear equation describing sound wave dynamics in a large, interacting Bose gas, confirming Bogolyubov's speed of sound and exploring instabilities for attractive interactions.

Contribution

It provides a rigorous derivation of the effective dynamics of excitations in an interacting Bose gas, including explicit error bounds and analysis of sound speed and instabilities.

Findings

Effective non-linear equation for excitations derived

Confirmed Bogolyubov's formula for sound speed

Identified dynamical instability for attractive potentials

Abstract

We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to a box of volume $\Lambda$ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, $\rho=\frac{N}{\Lambda}$, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume $\Lambda$ and small ratio $\frac{\Lambda}{\rho}$. The initial state of the gas is assumed to be close to a \textit{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 $\Lambda$. 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 $\frac{\Lambda}{\rho}$. 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.