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Physical Review A

Dark solitons and their head-on collisions in Bose-Einstein condensates

Guoxiang Huang, Manuel G. Velárde, Valeri A. Makarov

Published Jun 8, 2001• 137 citations
Publisher

Abstract

The evolution and collision of dark solitary waves (solitons) appearing in cigar-shaped Bose-Einstein condensates with repulsive atom-atom interaction are here considered using a Boussinesq-Korteweg--de Vries description. We provide theoretical predictions and computer experiment evidence about their phase shifts or change of trajectories, in the space-time plot, corresponding upon collisions. Details are also given about a suggested experiment that could assess their genuine solitonic nature.

DOI
10.1103/physreva.64.013617

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The evolution and collision of dark solitary waves (solitons) appearing in cigar-shaped Bose-Einstein condensates with repulsive atom-atom interaction are here considered using a Boussinesq-Korteweg--de Vries description. We provide theoretical predictions and computer experiment evidence about their phase shifts or change of trajectories, in the space-time plot, corresponding upon collisions. Details are also given about a suggested experiment that could assess their genuine solitonic nature.
We investigate the dynamics of pulses in a cigar-shaped Bose-Einstein condensate with repulsive atom-atom interactions without using Thomas-Fermi approximation. In the linear level our results give the Bogoliubov excitation spectrum for sound propagation with speed c = c 0 /(2) 1/2 , where c 0 is the speed for the case without a trap. We develop a Korteweg-de Vries (KdV) description for dark soliton propagation in the system and show that it is the quantum pressure that contributes the dispersion necessary for the formation of the dark solitons.
We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states are dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally, we present experimental realizations of multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.
The ring dark solitons and their head-on collisions in a Bose–Einstein condensates with thin disc-shaped potential are studied. It is shown that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by the two cylindrical KdV equations in the respective reference frames. By using the extended Poincare–Lighthill–Kuo perturbation method, the analytical phase shifts following the head-on collisions between two ring dark solitary waves are derived. It is shown that the phase shifts decrease with the radial coordinate r according to the r−1/3 law and depend on the initial soliton amplitude and radius.
We present experimental data showing the head-on collision of dark solitons generated in an elongated Bose-Einstein condensate. No discernable interaction can be recorded, in full agreement with the fundamental theoretical concepts of solitons as mutually transparent quasiparticles. Our soliton generation technique allows for the creation of solitons with different depths; hence, they can be distinguished and their trajectories be followed. Simulations of the 1D-Gross-Pitaevskii equation have been performed to compare the experiment with a mean-field description.
The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg--de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.