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arXiv:cond-mat/0404272 · Physical Review A

Propagation of sound in a Bose-Einstein condensate in an optical lattice

C. Menotti, Meret Krämer, Augusto Smerzi, Лев П. Питаевский, S. Stringari

Published Aug 26, 2004• 30 citations
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Abstract

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.

DOI
10.1103/physreva.70.023609

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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.
The speed of sound of a Bose-Einstein condensate in an optical lattice is studied both analytically and numerically in all spatial three dimensions. Our investigation shows that the sound speed depends strongly on the strength of the lattice. In the one-dimensional case, the speed of sound falls monotonically with increasing lattice strength. The dependence on lattice strength becomes much richer in two and three dimensions. In the two-dimensional case, when the interaction is weak, the sound speed first increases and then decreases as the lattice strength increases. For the three-dimensional lattice, the sound speed can even oscillate with the lattice strength. These rich behaviors can be understood in terms of compressibility and effective mass. Our analytical results in the limit of weak lattices also offer an interesting perspective to help with our understanding: they show that the lattice component perpendicular to the sound propagation increases the sound speed while the lattice component parallel to the propagation decreases the sound speed. The various dependences of the sound speed on the lattice strength are the result of this competition.
We consider propagation of density wave packets in a Bose-Einstein condensate. We show that the shape of initially broad, laser-induced, density perturbation changes in the course of free time evolution so that a shock wave front finally forms. Our results are well beyond predictions of commonly used zero-amplitude approach, so they can be useful in extraction of a speed of sound from experimental data. We discuss a simple experimental setup for shock propagation and point out possible limitations of the mean-field approach for description of shock phenomena in a Bose-Einstein condensate.
The dynamics of Bose-Einstein condensates trapped in a deep optical lattice is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity and the intersite hopping rates are retrieved from a nonlinear tight-binding approximation taking into account the effective dimensionality of each condensate. We derive analytically the Bloch and the Bogoliubov excitation spectra and the velocity of sound waves emitted by a traveling condensate. Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of localized wave packets. We calculate the ground-state atomic distribution in the presence of a harmonic confining potential, the frequencies of small amplitude dipole, and quadrupole oscillations. We finally quantize the DNL, recovering an extended Bose-Hubbard model.
A paper presents a theoretical study of shock waves in a trapped Bose-Einstein condensate (BEC). The mathematical model of the BEC in this study is a nonlinear Schroedinger equation (NLSE) in which (1) the role of the wave function of a single particle in the traditional Schroedinger equation is played by a space- and time-dependent complex order parameter (x,t) proportional to the square root of the density of atoms and (2) the atoms engage in a repulsive interaction characterized by a potential proportional to | (x,t)|2. Equations that describe macroscopic perturbations of the BEC at zero temperature are derived from the NLSE and simplifying assumptions are made, leading to equations for the propagation of sound waves and the transformation of sound waves into shock waves. Equations for the speeds of shock waves and the relationships between jumps of velocity and density across shock fronts are derived. Similarities and differences between this theory and the classical theory of sound waves and shocks in ordinary gases are noted. The present theory is illustrated by solving the equations for the example of a shock wave propagating in a cigar-shaped BEC.