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arXiv:0709.2004 · Physical Review A

Bright solitary-matter-wave collisions in a harmonic trap: Regimes of solitonlike behavior

Alan D. Martin, Charles S. Adams, S. A. Gardiner

Published Jan 25, 2008• 47 citations
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Abstract

Systems of solitary waves in the one-dimensional Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyze the soliton nature of these solitary waves, a particle analogy for the solitary waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particlelike manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the collision time of the solitons. We find that the collision time of the solitons is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This phase preservation can be used to find regimes where there is agreement between the wave and particle models. This also implies that soliton regimes may be found in three-dimensional geometries where solitary waves can be made to repeatedly collide out of phase, stabilizing the condensate against collapse. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carries over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes.

DOI
10.1103/physreva.77.013620

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Systems of solitary waves in the one-dimensional Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyze the soliton nature of these solitary waves, a particle analogy for the solitary waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particlelike manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the collision time of the solitons. We find that the collision time of the solitons is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This phase preservation can be used to find regimes where there is agreement between the wave and particle models. This also implies that soliton regimes may be found in three-dimensional geometries where solitary waves can be made to repeatedly collide out of phase, stabilizing the condensate against collapse. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carries over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes.
Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated and shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, presenting a paradigm for chaos in wave mechanics.
Bright solitary-waves may be created in dilute Bose-Einstein condensates of at tractively interacting atoms in one dimensional regimes. In integrable systems, such solitary waves are particle-like objects called solitons. We investigate the consequences of non-integrability on the solitary waves in trapped Bose-Einstein condensates caused by an axial harmonic trap, and non-integrability caused by three dimensional effects. To analyse the soliton-like nature of the solitary-waves in an axial harmonic trap, a particle analogy for the solitary-waves is formulated. Exact Soliton solutions exist in the absence of an external trapping potential, which behave in a particle-like manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the, collision time of the solitons. We find that the collision time of the solions is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carrier over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes. We predict that these chaotic regimes will be an indicator of rapid depletion of the condensate due to quantum transitions of the condensate particles into non-condensate modes. To analyse the residual effects of the three dimensional nature of the solitary waves, we use a nonlinear Schrödinger equation with an additional quintic term. We perform variational calculations, and confirm the collapse of a soliton when the number of particles contained therein is increased past a critical number. We investigate the effects of varying the axial trap frequency and scattering length on the critical number. We propose a method to model particle exchange between solitons by extending the variational treatment to two solitons.
Bose–Einstein condensates with attractive interactions have stable 1D solutions in the form of bright solitary-waves. These solitary waves behave, in the absence of external potentials, like macroscopic quantum particles. This opens up a wide array of applications for the testing of quantum mechanical behaviours and precision measurement. Here we investigate these applications with particular focus on the interactions of bright solitary-waves with narrow potential barriers. We first study bright solitons in the Gross–Pitaevskii equation as they are split on Gaussian and δ-function barriers, and then on Gaussian barriers in a low energy system. We present analytic and numerical results determining the general region in which a soliton may not be split on a finite width potential barrier. Furthermore, we test the sensitivity of the system to quantum fluctuations. We then study fast-moving bright solitons colliding at a narrow Gaussian potential barrier. In the limiting case of a δ-function barrier, we show analytically that the relative norms of the outgoing waves depends sinusoidally on the relative phase of the incoming waves, and determine whether the outgoing waves are bright solitons. We use numerical simulations to show that outside the high velocity limit nonlinear effects introduce a skew to the phase-dependence. Finally, we use these results to analyse the process of soliton interferometry. We develop analyses of both toroidal and harmonic trapping geometries for Mach–Zehnder interferometry, and then two implementations of a toroidal Sagnac inter- ferometer, also giving the analytical determination of the Sagnac phase in such systems. These results are again verified numerically. In the Mach–Zehnder case, we again probe the systems sensitivity to quantum fluctuations.
The dynamics of nonlinear solitary waves in studied by using the model of nonlinear Schrodinger equations(NSE) with an external harmonic potential. The model allows one to analyze on the general basis a variety of nonlinear phenomena appearing both in Bose-Einstein condensate in a magnetic trap, whose profile is describe by a quadratic function of coordinates, and an nonlinear optics, physics of lasers, and biophysics. It is shown that exact solutions for a quantum-mechanical particle in a harmonic potential and solutions obtained within the framework of the adiabatic perturbation theory of bright solitons in a parabolic trap are completely identical. This fact not only proves once more that solitons behave like particles but also that they can preserve such properties in different traps for the which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase soliton in a harmonic potential. The interaction dynamics of solitons in nonstationary potentials is studied and the possibility of the appearance of a soliton parametric resonance at with the amplitude of soliton oscillation in a trap exponentially increases with time is shown.
We consider the one-dimensional Gross–Pitaevskii (GP) equation, which governs the dynamics of Bose–Einstein condensate (BEC) matter waves with time-dependent scattering length and a harmonic trapping potential. We present the integrable condition for the one-dimensional GP equation and obtain the exact analytical solution which describes the modulational instability and the propagation of a bright solitary wave on a continuous wave (cw) background. Moreover, by employing the adiabatic perturbation theory for a bright soliton, we obtain approximative bright solitary wave solutions under near-integrable conditions. Both the exact and approximative solutions show that the amplitude of a bright solitary wave with zero boundary condition depends on the scattering length while its motion depends on the external trapping potential.
We investigate dynamical effects of a bright soliton in Bose–Einstein condensed (BEC) systems with local and smooth space variations of the two-body atomic scattering length. It includes a discussion about the possible observation of a new type of standing nonlinear atomic matter wave in cigar-type traps. A rich dynamics is observed in the interaction between the soliton and an inhomogeneity. By considering an analytical time-dependent variational approach and also full numerical simulation of one-dimensional and three-dimensional Gross–Pitaevskii equations, we study processes such as trapping, reflection and transmission of the bright matter soliton due to the impurity. We also derive conditions for the collapse of the bright solitary wave, considering a quasi-one-dimensional BEC with attractive local inhomogeneity.