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arXiv:cond-mat/0010449 · Physical Review A

States without a linear counterpart in Bose-Einstein condensates

Roberto D’Agosta, Carlo Presilla A

Published Mar 27, 2002• 77 citations
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Abstract

We show the existence of stationary solutions of a one-dimensional Gross-Pitaevskii equation in the presence of a multiwell external potential that do not reduce to any of the eigenfunctions of the associated Schr\"odinger problem. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.

DOI
10.1103/physreva.65.043609

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We show the existence of stationary solutions of a one-dimensional Gross-Pitaevskii equation in the presence of a multiwell external potential that do not reduce to any of the eigenfunctions of the associated Schr"odinger problem. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.
The stationary solutions of the Gross-Pitaevskii equation can be divided in two classes: those which reduce, in the limit of vanishing nonlinearity, to the eigenfunctions of the associated Schrödinger equation and those which do not have linear counterpart. Analytical and numerical results support an existence condition for the solutions of the first class in terms of the ratio between their proper frequency and the corresponding linear eigenvalue. For one-dimensional confined systems, we show that solutions without linear counterpart do exist in presence of a multi-well external potential. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.
In the present contribution, we explore a host of different stationary states, namely dark-bright solitons and their lattices, that arise in the context of multicomponent atomic Bose-Einstein condensates. The latter are modeled by systems of coupled Gross-Pitaevskii equations with general interaction (nonlinearity) coefficients gij{g}_{ij}gij​. It is found that in some particular parameter ranges such solutions can be obtained in analytical form, however, numerically they are computed as existing in a far wider parametric range. Many features of the solutions under study, such as their analytical form without the trap or the stability and dynamical properties of one dark-bright soliton even in the presence of the trap are obtained analytically and corroborated numerically. Additional features, such as the stability of soliton lattice homogeneous states or their existence and stability in the presence of the trap, are examined numerically.
The analytical dark and bright soliton solutions of the one-dimensional Gross-Pitaevskii equation with a confining potential are obtained. For the bright soliton, the recent experimental finding is studied, and the particle number of the soliton and the window of the particle numbers for the bright soliton to occur are estimated analytically and in good agreement with the experimental data. The existence of dark soliton for the attractive interaction and bright soliton for the repulsive interaction is predicted under proper conditions.
The existence and stability of solitonic states in one-dimensional repulsive Bose-Einstein condensates is investigated within a fully many-body framework by considering the limit of infinite repulsion (Tonks-Girardeau gas). A class of stationary, shape-invariant states propagating at constant velocity are explicitly found and compared to the known solution of the Gross-Pitaevskii equation. The typical features attributed to nonlinearity are thus recovered in a purely linear theory, provided the full many-particle physics is correctly accounted for. However, the formation dynamics predicted by the Gross-Pitaevskii approximation considerably differs from the exact many-body evolution.
Exact solutions of the generalized stationary one-dimensional Gross-Pitaevski equation taking into account the nonlocality of interatomic interactions in a Bose-Einstein condensate were found. These solutions correspond to regimes of bright and dark solitons, as well as to complete and incomplete periodic modulation of the atomic concentration. A regime of complete purely harmonic modulation was revealed. The qualitative analysis of the phase plane of the corresponding equations was performed. The possibility of interpreting the experimentally observed soliton chains as a regime of complete or deep modulation was indicated.