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arXiv:1007.2329 · Modern Physics Letters B

DYNAMICS OF BISOLITONIC MATTER WAVES IN A BOSE–EINSTEIN CONDENSATE SUBJECTED TO AN ATOMIC BEAM SPLITTER AND GRAVITY

Alain M. Dikandé, I. Ndifon Ngek, Ebobenow Joseph

Published Oct 26, 2010• 1 citation
arXivHTMLPDF

Abstract

A theoretical scheme for an experimental implementation involving bisolitonic matter waves from an attractive Bose–Einstein condensate, is considered within the framework of a non-perturbative approach to the associate Gross–Pitaevskii equation. The model consists of a single condensate subjected to an expulsive harmonic potential creating a double-condensate structure, and a gravitational potential that induces atomic exchanges between the two overlapping post condensates. Using a non-isospectral scattering transform method, exact expressions for the bright-matter–wave bisolitons are found in terms of double-lump envelopes with the co-propagating pulses displaying more or less pronounced differences in their widths and tails depending on the mass of atoms composing the condensate.

DOI
10.1142/s0217984910025243

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A theoretical scheme for an experimental implementation involving bisolitonic matter waves from an attractive Bose–Einstein condensate, is considered within the framework of a non-perturbative approach to the associate Gross–Pitaevskii equation. The model consists of a single condensate subjected to an expulsive harmonic potential creating a double-condensate structure, and a gravitational potential that induces atomic exchanges between the two overlapping post condensates. Using a non-isospectral scattering transform method, exact expressions for the bright-matter–wave bisolitons are found in terms of double-lump envelopes with the co-propagating pulses displaying more or less pronounced differences in their widths and tails depending on the mass of atoms composing the condensate.
We theoretically investigate the dynamics of a matter-wave soliton created in a harmonic potential, which is attractive in the transverse direction but expulsive in the longitudinal direction. This Bose-Einstein-condensate (BEC) bright soliton made of 7Li^{7}\mathrm{Li}7Li atoms has been observed in a recent experiment [Khaykovich et al., Science 296, 1290 (2002)]. We show that the nonpolynomial Schr\"odinger equation, an effective one-dimensional equation we derived from the three-dimensional Gross-Pitaevskii equation, is able to reproduce the main experimental features of this BEC soliton in an expulsive potential.
We investigate the four-wave mixing of matter wave packets created from a Bose–Einstein condensate, realized experimentally by utilizing light pulses to create two high-momentum wave packets via Bragg diffraction from a stationary condensate. Based on the Gross–Pitaevskii equation, a set of nonlinearly coupled envelope equations including self- and cross-phase modulational effects are derived systematically using a method of multiple-scales. The exact and explicit analytical solutions are provided for the coupled envelope equations and the evolution of the wave packets after turning off trapping potential is discussed and compared with experiment.
We investigate the properties of modulational instability in the Salerno equation in quasi-one dimension in Bose-Einstein condensate (BEC). We analyzed the regions of modulational instability of nonlinear plane waves and determine the conditions of its existence in BEC. The existence and stability of strongly localized modes in discrete media is investigated with the framework of the Salerno model by using a linear analysis method. The regions of stability and instability are determined. Also the existence of localized modes for different values of parameters is shown numerically by homoclinic orbits intersection method. The response of Bose–Einstein condensates is studied with strong dipole–dipole atomic interactions to periodically varying perturbation. The dynamics is governed by the Gross–Pitaevskii equation with an additional nonlinear term, corresponding to nonlocal dipolar interactions. A mathematical model, based on the variational approximation, has been developed and applied to study parametric excitation of the condensates due to coefficient of nonlocal nonlinearity varying periodically. The model predicts the waveform of solitons in dipolar condensates and describes their small amplitude dynamics quite accurately. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation and a good agreement between them is found. The results can lead to better understanding of the properties of ultra-cold quantum gases, such as 52Cr, 164Dy and 168Er, where the long-range dipolar atomic interactions dominate the usual contact interactions. Dynamics of a matter wave soliton bouncing on the reflecting surface (atomic mirror) under the effect of gravity has been studied by analytical and numerical means. The iv analytical description is based on the variational approach. Resonant oscillations of the soliton's center of mass and width, induced by appropriate modulation of the atomic scattering length and the slope of the linear potential are analyzed. In numerical experiments, we observe the Fermi type acceleration of the soliton when the vertical position of the reflecting surface is periodically varied in time. Analytical predictions are compared with the results of numerical simulations of the Gross-Pitaevskii equation and a qualitative agreement between them is found.
Bright solitons in a quasi-1D Bose–Einstein condensate can be used to enhance precision in matter-wave interferometry, due to their inherent robustness and support against dispersion. Such a soliton interferometer typically relies on a potential barrier used to split a single soliton into two smaller coherent solitons which can then be recombined on the same barrier. In this thesis we examine two extensions to this scheme. Firstly, we investigate a binary BEC system consisting of two bright solitons which are coupled through a mutual nonlinear interaction term. We derive a set of conditions under which the two components can be separated on a potential barrier and use numerical simulations to probe the regimes beyond which this mathematical treatment is applicable. We then use the numerical simulations to look at the effect of the nonlinear coupling on the dynamics of the binary solitons interacting with the barrier. We also look at the interference behaviour found by doubling the simulation time in either a ring trap or a harmonic trapping potential (to ensure recombination on the barrier); as well as the case where the solitons start spatially separated on either side of the barrier in order to find conditions under which the solitons will combine on the barrier. We find a good agreement between the analytical predictions and the results of simulations. Beyond the regions of parameter space where the predictions are expected to hold, we find complex transmission and interference behaviour as a result of nonlinear effects. The second part of this thesis consists of an examination of the prospect of using a subwavelength barrier scheme in a soliton interferometry experiment. This involves using two resonant coupling beams in a Λ-system with a spatially varying intensity. Under certain conditions, this can be used to form an effective potential barrier with a width which is not diffraction-limited. We look at suitable parameter regimes for such a barrier to split and recombine solitons in an interferometer and probe the effects of possible complications such as misalignment in the beams and different scattering lengths in the different states. We simulate the soliton interferometer using the full three component GPE as well as the single component analogue with the effective potential in order to characterise the soliton behaviour and the dependence of the interferometer sensitivity on the system parameters. We find a trend towards an idealised soliton interferometer with a decreasing value of the parameter, w, controlling the barrier characteristics. Also, we demonstrate an agreement between the relevant three component GPE and the analogous single component GPE, in the limit of strong coupling fields.
Using time dependent nonlinear (s-wave scattering length) coupling between the components of a weakly interacting two component Bose–Einstein condensate (BEC), we show the possibility of matter wave switching (fraction of atoms transfer) between the components via shape changing/intensity redistribution (matter redistribution) soliton interactions. We investigate the exact bright–bright N-soliton solution of an effective one-dimensional (1D) two component BEC by suitably tailoring the trap potential, atomic scattering length, and atom gain or loss. In particular, we show that the effective 1D coupled Gross–Pitaevskii equations with time dependent parameters can be transformed into the well known completely integrable Manakov model described by coupled nonlinear Schrödinger equations by effecting a change of variables of the coordinates and the wave functions under certain conditions related to the time dependent parameters. We obtain the one-soliton solution and demonstrate the shape changing/matter redistribution interactions of two and three-soliton solutions for the time-independent expulsive harmonic trap potential, periodically modulated harmonic trap potential, and kinklike modulated harmonic trap potential. The standard elastic collision of solitons occur only for a specific choice of soliton parameters.
We analyze the dynamics of a bright soliton in Bose–Einstein condensates (BECs) with time-dependent atomic scattering length in an expulsive parabolic potential. Under a safe range of parameters in which the Gross–Pitaevskii (GP) equation is effective in one dimension, our results show that, the dynamics of the bright soliton can be classed into two phases, depending on the value of the scattering length. Meanwhile, there exists a critical value of the absolute value of the atomic scattering length, below which, the dynamics of the bright soliton is very regular. Those phenomena can be useful for developing concrete applications of the nonlinear matter waves. We also obtain the orbital equation of the bright soliton and get some interesting data which may be useful for the experimental observation of the bright soliton and the application of the atom laser with manipulated intensity.