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arXiv:0712.3002 · Journal of Physics B Atomic Molecular and Optical Physics

Collisions of bright solitary matter waves

Nicholas G. Parker, A. Martin, Simon L. Cornish, Charles S. Adams

Published Feb 11, 2008• 60 citations
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Abstract

The collisions of three-dimensional bright solitary matter waves formed from atomic Bose–Einstein condensates are shown to exhibit rich behaviour. Collisions range from being elastic to completely destructive due to the onset of collapse during the interaction. Through a detailed quantitative analysis we map out the role of relative phase, impact speed and interaction strength. In particular, we identify the importance of the collapse time in the onset of unstable collisions and show how the relative phase controls a population transfer between the waves. Our analysis enables us to interpret recent experimental observations of bright solitary matter waves.

DOI
10.1088/0953-4075/41/4/045303

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The collisions of three-dimensional bright solitary matter waves formed from atomic Bose–Einstein condensates are shown to exhibit rich behaviour. Collisions range from being elastic to completely destructive due to the onset of collapse during the interaction. Through a detailed quantitative analysis we map out the role of relative phase, impact speed and interaction strength. In particular, we identify the importance of the collapse time in the onset of unstable collisions and show how the relative phase controls a population transfer between the waves. Our analysis enables us to interpret recent experimental observations of bright solitary matter waves.
We study bright solitary waves of three dimensional trapped Bose-Einstein condensates and their collisions. For a single solitary wave, in addition to an upper critical number, we also find a {\em lower} cut-off, below which no stable state can be found. Collisions between solitary waves can be elastic, inelastic with either reduced or increased outgoing speed, or completely unstable due to a collapse instability. A πππ-phase difference between the waves promotes elastic collisions, and gives excellent agreement with recent experimental results over long timescales.
In recent years, bright soliton-like structures composed of gaseous Bose-Einstein condensates have been generated at ultracold temperature. The experimental capacity to precisely engineer the nonlinearity and potential landscape experienced by these solitary waves offers an attractive platform for fundamental study of solitonic structures. The presence of three spatial dimensions and trapping implies that these are strictly distinct objects to the true soliton solutions. Working within the zero-temperature mean-field description, we explore the solutions and stability of bright solitary waves, as well as their interactions. Emphasis is placed on elucidating their similarities and differences to the true bright soliton. The rich behaviour introduced in the bright solitary waves includes the collapse instability and symmetry-breaking collisions. We review the experimental formation and observation of bright solitary matter waves to date, and compare to theoretical predictions. Finally we discuss the current state-of-the-art of this area, including beyond-mean-field descriptions, exotic bright solitary waves, and proposals to exploit bright solitary waves in interferometry and as surface probes.
We model the dynamics of formation of multiple, long-lived, bright solitary waves (BSWs) in the collapse of Bose–Einstein condensates with attractive interactions as studied in the experiment of Cornish et al (2006 Phys. Rev. Lett. 96 170401). We use both mean-field and approximate quantum field simulation techniques. While a number of separated wave packets form as observed in the experiment, they do not have a repulsive π phase difference as has been previously inferred. We observe that the inclusion of quantum fluctuations causes soliton dynamics to be predominantly repulsive in one-dimensional (1D) simulations independent of their initial relative phase. However, indicative 3D simulations do not show a similar effect. In contrast, in 3D quantum noise has a negative impact on BSW lifetimes. Finally, we show that condensate oscillations, after the collapse, may serve to deduce three-body recombination rates.
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.
Bright solitary waves in a Bose-Einstein condensate contain thousands of identical atoms held together despite their only weakly attractive contact interactions. They nonetheless behave like a compound object, staying whole in collisions, with their collision properties strongly affected by intersoliton quantum coherence. We show that separate solitary waves decohere due to phase diffusion, dependent on their effective ambient temperature, after which their initial mean-field relative phases are no longer well defined or relevant for collisions. In this situation, collisions occur predominantly repulsively and can no longer be described within mean-field theory. When considering the timescales involved in recent solitary wave experiments where nonequilibrium phenomena play an important role, these features could explain the predominantly repulsive collision dynamics observed in most condensate soliton train experiments.
We present a comprehensive analysis of the form and interaction of dipolar bright solitons across the full parameter space afforded by dipolar Bose–Einstein condensates, revealing the rich behavior introduced by the non-local nonlinearity. Working within an effective one-dimensional description, we map out the existence of the soliton solutions and show three collisional regimes: free collisions, bound state formation and soliton fusion. Finally, we examine the solitons in their full three-dimensional form through a variational approach; along with regimes of instability to collapse and runaway expansion, we identify regimes of stability which are accessible to current experiments.