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arXiv:0909.1660 · Physical Review Letters

Matter-Wave Dark Solitons: Stochastic versus Analytical Results

S. P. Cockburn, Hector E. Nistazakis, Theodoros P. Horikis, P. G. Kevrekidis, Nick P. Proukakis, D. J. Frantzeskakis

Published Apr 29, 2010• 72 citations
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Abstract

The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically derived equation for the soliton center based on perturbation theory for dark solitons.

DOI
10.1103/physrevlett.104.174101

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The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically derived equation for the soliton center based on perturbation theory for dark solitons.
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We study matter-wave dark solitons in atomic Bose-Einstein condensates (BECs) at finite temperatures, under the effect of linear and periodic potentials. Our model, namely, a dissipative Gross-Pitaevskii equation, is treated analytically by means of dark-soliton perturbation theory and the Landau dynamics approach, which result in a Newtonian equation of motion for the dark-soliton center. This reduced model, which incorporates an effective washboard potential and an antidamping term accounting for finite-temperature effects, constitutes an example of an antidamped Josephson junction. We perform a qualitative (local and global) analysis of the equation of motion. We present results of systematic numerical simulations for both zero- and finite-temperature BECs to highlight the differences between the Hamiltonian and dissipative settings. For sufficiently small wave numbers of the periodic potential and weak linear potentials, the analytical results are found to be in good agreement with pertinent ones obtained via a Bogoliubov--de Gennes analysis and direct numerical simulations.
We study analytically, as well as numerically, single- and multiple-dark matter-wave solitons in atomic Bose-Einstein condensates at finite temperatures. Our analysis is based on the study of the dissipative Gross-Pitaevskii equation, which incorporates a phenomenological damping term accounting for the interaction of the condensate with the thermal cloud. We illustrate how the negative Krein sign eigenmodes (associated with the the single- or multiple-dark soliton states) can give rise to Hopf bifurcations and oscillatory instabilities, whose ensuing dynamics is also elucidated. In all cases, the finite-temperature induced dynamics results in soliton decay, and the system eventually asymptotes to the ground state.
We consider the dynamics of dark matter wave solitons moving through non-uniform cigar-shaped Bose–Einstein condensates described by the mean field Gross–Pitaevskii equation with generalized nonlinearities, in the case when the condition for the modulation stability of the Bose–Einstein condensate is fulfilled. The analytical expression for the frequency of the oscillations of a deep dark soliton is derived for nonlinearities which are arbitrary functions of the density, while specific results are discussed for the physically relevant case of a cubic–quintic nonlinearity modelling two- and three-body interactions, respectively. Opposite to the usual (cubic) Gross–Pitaevskii equation for which the dark soliton effective mass is known to be constant (equal to 2), in the presence of a cubic–quintic nonlinearity we find that the effective mass depends on the product of the initial density background and the ratio between the coefficient of quintic and cubic nonlinearities, this leading to the interesting possibility of measuring three-body interactions directly from the dark soliton dynamics. A comparison between analytical results and direct numerical simulations of the cubic–quintic Gross–Pitaevskii equation shows good agreement between them which confirms the validity of our approach.
The dissipative dynamic stability is investigated of dark solitons in elongated Bose–Einstein condensates that can be described by the Gross–Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear Schrodinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons turn out to be thermodynamically unstable.
We consider the decay of a dark soliton in a homogeneous BEC at finite temperature, with a particular emphasis on the effect of thermal noise on the stability of the soliton. We present an analytical treatment of a damped soliton in an unbounded system, comparing the predicted dynamics with numerical stochastic simulations of a spatially confined system. By varying the system temperature we study the relative importance of noise on the motion of the soliton. In the regime of low temperature the soliton is largely immune to thermal fluctuations, and is well described by the damped Gross-Pitaevskii equation. In this regime our analytical treatment is in close agreement with numerical simulations. For sufficiently high temperature, the thermal fluctuations have the interesting effect of increasing the stability of the soliton, extending its lifetime beyond the predictions of damped Gross-Pitaevskii theory.