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arXiv:0710.1179 · Physical Review A

Dark matter-wave solitons in the dimensionality crossover

Georgios Theocharis, P. G. Kevrekidis, Markus K. Oberthaler, D. J. Frantzeskakis

Published Oct 17, 2007• 34 citations
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Abstract

We consider the statics and dynamics of dark matter-wave solitons in the dimensionality crossover regime from three dimensions (3D) to one dimension (1D). There, using the nonpolynomial Schr\"odinger mean-field model, we find that the anomalous mode of the Bogoliubov spectrum has an eigenfrequency which coincides with the soliton oscillation frequency obtained by the 3D Gross-Pitaevskii model. We show that substantial deviations (of the order of 10% or more) from the characteristic frequency ${\ensuremath{\omega}}_{z}∕\sqrt{2}$ (${\ensuremath{\omega}}_{z}$ being the longitudinal trap frequency) are possible even in the purely 1D regime.

DOI
10.1103/physreva.76.045601

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We report on the generation, subsequent oscillation and interaction of a pair of matter-wave dark solitons. These are created by releasing a Bose-Einstein condensate from a double well potential into a harmonic trap in the crossover regime between one dimension and three dimensions. Multiple oscillations and collisions of the solitons are observed, in quantitative agreement with simulations of the Gross-Pitaevskii equation. An effective particle picture is developed and confirms that the deviation of the observed oscillation frequencies from the asymptotic prediction nu(z)/sqrt 2, where nu(z) is the longitudinal trapping frequency, results from the dimensionality of the system and the soliton interactions.
We study the statics and dynamics of dark solitons in a cigar-shaped Bose-Einstein condensate confined in a double-well potential. Using a mean-field model with a noncubic nonlinearity, appropriate to describe the dimensionality crossover regime from one- to three-dimensional, we obtain branches of solutions in the form of single and multiple dark soliton states, and study their bifurcations and stability. It is demonstrated that there exist dark soliton states which do not have a linear counterpart and we highlight the role of anomalous modes in the excitation spectra. Particularly, we show that anomalous mode eigenfrequencies are closely connected to the characteristic soliton frequencies as found from the solitons' equations of motion, and how anomalous modes are related to the emergence of instabilities. We also analyze in detail the role of the height of the barrier in the double-well setting, which may lead to instabilities or decouple multiple dark soliton states.
Abstract: In this thesis we experimentally observe the dynamics of dark solitons in elongated Bose-Einstein condensates (BEC) in the crossover regime between 1D and 3D. Dark solitons are dynamically stable in this regime. This allows the observation of the collisional and oscillational dynamics of two dark solitons in a harmonically confined BEC over several oscillation periods. We can detect the repulsive intersolitonic interaction as an increase of the oscillation frequency of the two solitons strongly depending on their mean distance. Thereby we give the first experimental evidence of the theoretically predicted repulsive interaction between the solitons. We compare our experimental findings to numerical simulations of the Gross-Pitaevskii equation and effective 1D equations. Additionally we develop a model in which the solitons are approximated by repulsively interacting classical particles. This model is based on a soliton interaction potential suggested by [Kiv1995] and can describe the essentials of the soliton dynamics. We observe well agreement between the soliton oscillation frequencies measured in the experiment, the numerical simulations and the results of the particle model. Besides we verify the theoretical prediction that the oscillation frequency of dark solitons in a harmonically confined BEC should be slower than the trap frequency.
The interaction of matter–wave solitons with a potential barrier is a fundamentally important problem, and the splitting and subsequent recombination of the soliton by the barrier is the essence of soliton matter–wave interferometry. We demonstrate the three-dimensional (3D) character of the interactions in the case relevant to ongoing experiments, where the number of atoms in the soliton is relatively close to the collapse threshold. We examine the soliton dynamics in the framework of the effectively one-dimensional (1D) nonpolynomial Schrödinger equation (NPSE), which admits the collapse in a modified form, and in parallel we use the full 3D Gross–Pitaevskii equation (GPE). Both approaches produce similar results, which are, however, quite different from those produced in recent work that used the 1D cubic GPE. Basic features, produced by the NPSE and the 3D GPE alike, include (a) an increase in the first reflection coefficient for increasing barrier height and decreasing atom number; (b) large variation of the secondary reflection/recombination probability versus barrier height; (c) pronounced asymmetry in the oscillation amplitudes of the transmitted and reflected fragments; and (d) enhancement of the transverse excitations as the number of atoms is increased. We also explore effects produced by variations of the barrier width and outcomes of the secondary collision upon phase imprinting on the fragment in one arm of the interferometer.
The interaction of matter-wave solitons with a potential barrier is a fundamentally important problem, and the splitting and subsequent recombination of the soliton by the barrier is the essence of soliton matter-wave interferometry. We demonstrate the three-dimensional (3D) character of the interactions in the case relevant to ongoing experiments, where the number of atoms in the soliton is sufficiently large that the soliton is close to collapse. The mean-field description is quite accurate, but the proximity to the collapse threshold makes the use of the 1D Gross-Pitaevskii equation (GPE) irrelevant. We examine the soliton dynamics in the framework of the effectively 1D nonpolynomial Schr{\\"{o}}dinger equation (NPSE), which admits the collapse in a modified form, and in parallel we use the full 3D GPE. Both approaches produce results which are very different from those produced in recent work which used the 1D cubic GPE. The features observed in both the NPSE and the 3D GPE include (a) an increase in the first reflection coefficient for increasing barrier height and decreasing atom number; (b) strong modulation in the secondary reflection/recombination probability vs. barrier height; (c) a pronounced asymmetry in the oscillation amplitudes of the transmitted and reflected fragments; (d) an enhancement of the transverse excitations as the number of atoms is increased. We also explore effects produced by variations of the barrier width and outcomes of the secondary collision upon phase imprinting on the fragment in one arm of the interferometer.
The mean-field dynamics of Bose-Einstein condensates loaded in an optical lattice, confined by a parabolic potential, and subjected to a change of scattering length by means of the Feshbach resonance is considered. The system is described by the Gross-Pitaevskii (GP) equation with varying nonlinearity, which in a number of cases is reduced to one-dimensional (1D) perturbed nonlinear Schr\"odinger (NLS) equations, the particular form of which depends on the relation among the parameters of the problem. We analytically describe the adiabatic dynamics of periodic solutions of the respective NLS equations, provide a numerical study of 1D models confined by a parabolic trap, and carry out numerical simulations of the matter-wave dynamics within the framework of the radially symmetric 3D GP equation. Special attention is paid to processes of generation of trains of bright and dark matter solitons from initially periodic waves. The results of the 1D approximation are compared with direct numerical simulation of the original 3D GP eqation, showing remarkable coincidence for definite regions of parameters.