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arXiv:cond-mat/0306372 · Physical Review A

How do sound waves in a Bose-Einstein condensate move so fast?

P. M. Stevenson

Published Nov 3, 2003• 4 citations
arXivHTMLPDF

Abstract

Low-momentum excitations of a dilute Bose-Einstein condensate behave as phonons and move at a finite velocity ${v}_{s}.$ Yet the atoms making up the phonon excitation each move very slowly; ${v}_{a}=p/\stackrel{\ensuremath{\rightarrow}}{m}0.$ A simple ``cartoon picture'' is suggested to understand this phenomenon intuitively. It implies a relation ${v}_{s}{/v}_{a}\ensuremath{\approx}{N}_{\mathrm{ex}},$ where ${N}_{\mathrm{ex}}$ is the number of excited atoms making up the phonon. This relation does indeed follow from the standard Bogoliubov theory.

DOI
10.1103/physreva.68.055601

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Low-momentum excitations of a dilute Bose-Einstein condensate behave as phonons and move at a finite velocity vs.{v}_{s}.vs​. Yet the atoms making up the phonon excitation each move very slowly; va=p/m\ensuremath→0.{v}_{a}=p/\stackrel{\ensuremath{\rightarrow}}{m}0.va​=p/m\ensuremath→0. A simple ``cartoon picture'' is suggested to understand this phenomenon intuitively. It implies a relation vs/va\ensuremath≈Nex,{v}_{s}{/v}_{a}\ensuremath{\approx}{N}_{\mathrm{ex}},vs​/va​\ensuremath≈Nex​, where Nex{N}_{\mathrm{ex}}Nex​ is the number of excited atoms making up the phonon. This relation does indeed follow from the standard Bogoliubov theory.
We calculate the quantum Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ and butterfly velocity vB{v}_{B}vB​ in the dilute Bose gas at temperature TTT deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time-ordered correlators, from which ${\ensuremath{\lambda}}_{L}$ and vB{v}_{B}vB​ are extracted. At very low temperature where elementary excitations are phonon-like, we find ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{T}^{5}$ and ${v}_{B}\ensuremath{\sim}c$, the sound velocity. At relatively high temperature, we have ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}T$ and ${v}_{B}\ensuremath{\sim}c{(T/{T}_{*})}^{0.23}$. We find that ${\ensuremath{\lambda}}_{L}$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on TTT. The chaos diffusion constant ${D}_{L}={v}_{B}^{2}/{\ensuremath{\lambda}}_{L}$, on the other hand, differs from the energy diffusion constant DE{D}_{E}DE​. We find ${D}_{E}\ensuremath{\ll}{D}_{L}$ at very low temperature and ${D}_{E}\ensuremath{\gg}{D}_{L}$ otherwise.
Collective excitations of a dilute Bose condensate have been observed. These excitations are analogous to phonons in superfluid helium. Bose condensates were created by evaporatively cooling magnetically trapped sodium atoms. Excitations were induced by a modulation of the trapping potential, and detected as shape oscillations in the freely expanding condensates. The frequencies of the lowest modes agreed well with theoretical predictions based on mean-field theory. Before the onset of Bose-Einstein condensation, we observed sound waves in a dense ultracold gas.
A paper presents a theoretical investigation of subsonic and supersonic effects in a Bose-Einstein condensate (BEC). The BEC is represented by a time-dependent, nonlinear Schroedinger equation that includes terms for an external confining potential term and a weak interatomic repulsive potential proportional to the number density of atoms. From this model are derived Madelung equations, which relate the quantum phase with the number density, and which are used to represent excitations propagating through the BEC. These equations are shown to be analogous to the classical equations of flow of an inviscid, compressible fluid characterized by a speed of sound (g/Po)1/2, where g is the coefficient of the repulsive potential and Po is the unperturbed mass density of the BEC. The equations are used to study the effects of a region of perturbation moving through the BEC. The excitations created by a perturbation moving at subsonic speed are found to be described by a Laplace equation and to propagate at infinite speed. For a supersonically moving perturbation, the excitations are found to be described by a wave equation and to propagate at finite speed inside a Mach cone.
We study the propagation of sound waves in a Bose-Einstein condensate trapped in a one-dimensional optical lattice. We find that the velocity of the propagation of sound wave packets decreases with increasing optical lattice depth, as predicted by the Bogoliubov theory. The strong interplay between nonlinearities and the periodicity of the external potential generates phenomena that are not present in the uniform case. Shock waves, for instance, can propagate slower than sound waves, due to the negative curvature of the dispersion relation. Moreover, nonlinear corrections to the Bogoliubov theory appear to be important even with very small density perturbations, inducing a saturation of the amplitude of the sound signal.
We point out that the velocity of propagation of sound wavepackets in a Bose-Einstein condensate filling a three-dimensional cubic optical lattice undergoes a maximum with increasing lattice depth. For a realistic choice of parameters, the maximum sound velocity in a lattice condensate can exceed the sound velocity in a homogeneous condensate with the same average density by 30%. The maximum falls into the superfluid regime, and should be observable under currently achievable laboratory conditions.