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Journal of Physics B Atomic Molecular and Optical Physics

Soliton dynamics in the collisions of Bose - Einstein condensates: an analogue of the Josephson effect

William P. Reinhardt, Charles W. Clark

Published Nov 28, 1997• 173 citations
Publisher

Abstract

The usual mean-field description of trapped atomic Bose - Einstein condensates predicts the existence of solitons in one-dimensional and quasi-one-dimensional confinement. If the net atomic pair interaction is attractive, these solitons appear as coherently propagating matter wavepackets; if it is repulsive, soliton behaviour is seen in the motion of `holes' in the condensate density, the velocity of which is determined by the phase difference between adjoining pieces of condensate, in analogy to the familiar Josephson effect. We explore the properties of these solitons via elementary analytic models and computer simulation, and make specific suggestions for their experimental observation.

DOI
10.1088/0953-4075/30/22/001

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The usual mean-field description of trapped atomic Bose - Einstein condensates predicts the existence of solitons in one-dimensional and quasi-one-dimensional confinement. If the net atomic pair interaction is attractive, these solitons appear as coherently propagating matter wavepackets; if it is repulsive, soliton behaviour is seen in the motion of `holes' in the condensate density, the velocity of which is determined by the phase difference between adjoining pieces of condensate, in analogy to the familiar Josephson effect. We explore the properties of these solitons via elementary analytic models and computer simulation, and make specific suggestions for their experimental observation.
Solitons are non-dispersive wave packets that balance the wave dispersion with a focusing nonlinearity. They maintain their shape and velocity after collisions with other solitons, owing to the integrability of the underlying nonlinear partial differential equations. They appear in various physical systems, such as water waves, light waves in optical fiber, and matter waves. In our laboratory, we study solitons and their interactions in a Bose-Einstein condensate with an attractive nonlinearity, trapped in a quasi-one-dimensional waveguide. We investigate the collisions of a pair of solitons and explore the relative phase implications on the boundary of integrability. We study soliton train formation with the quench of the nonlinearity strength, and interactions between neighboring solitons in the train. We form a composite soliton, known as a breather, where multiple solitons overlap and the wavefunction ``breathes'' in time. We also explore the response of an elongated condensate to an external modulation of the nonlinearity. These experiments characterize the mean-field descriptions of solitons and Bose-Einstein condensates and explore the boundaries between the mean-field theory and quantum many-body theory.
We review recent experimental and theoretical work on the creation of bright matter wave solitons in Bose–Einstein condensates. In two recent experiments, solitons are formed from Bose–Einstein condensates of 7 Li by utilizing a Feshbach resonance to switch from repulsive to attractive interactions. The solitons are made to propagate in a one-dimensional potential formed by a focused laser beam. For repulsive interactions, the wavepacket undergoes dispersive wavepacket spreading, while for attractive interactions, localized solitons are formed. In our experiment, a multi-soliton train containing up to ten solitons is observed to propagate without spreading for a duration of 2 s. Adjacent solitons are found to interact repulsively, in agreement with a calculation based on the nonlinear Schrödinger equation assuming that the soliton train is formed with an alternating phase structure. The origin of this phase structure is not entirely clear.
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.
Relativistic scalar field theories with a conserved global charge Q often possess (meta) stable spherically symmetric soliton solutions, called Q-balls. We elaborate on the perfect formal analogy which exists between Q-balls and spherically symmetric solitons in certain non-relativistic atomic Bose-Einstein condensates, for which the dominant interatomic interaction can be tuned attractive. In a harmonic trap, present in existing experiments, the Q-ball solution is modified in an essential way. If the trap is significantly prolongated in one direction, however, then genuine solitons do appear, and actual experimental data can be obtained for some of the Q-ball properties studied numerically in the relativistic cosmological context, such as their formation and collisions. We also suggest conditions under which the same cosmologically relevant analogies could be extended to the fully three-dimensional case.
For the first time, we find the complex solitons for a quasi-one-dimensional Bose-Einstein condensate with two-and three-body interactions. These localized solutions are characterized by a power law behaviour. Both dark and right solitons can be excited in the experimentally allowed parameter domain, when two-and three-body interactions are,respectively, repulsive and attractive. The dark solitons travel with a constant speed, which is quite different from the Lieb mode, where profiles with different speeds, bounded above by sound velocity, can exist for specified interaction strengths. We also study the properties of these solitons in the presence of harmonic confinement with time-dependent nonlinearity and loss. The modulational instability and the Vakhitov-Kolokolov criterion of stability are also studied.
We experimentally study the excitation modes of bright matter-wave solitons in a quasi-one-dimensional geometry. The solitons are created by quenching the interactions of a Bose-Einstein condensate of cesium atoms from repulsive to attractive in combination with a rapid reduction of the longitudinal confinement. A deliberate mismatch of quench parameters allows for the excitation of breathing modes of the emerging soliton and for the determination of its breathing frequency as a function of atom number and confinement. In addition, we observe signatures of higher-order solitons and the splitting of the wave packet after the quench. Our experimental results are compared to analytical predictions and to numerical simulations of the one-dimensional Gross-Pitaevskii equation.