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arXiv:cond-mat/0107184

Stationary states of Bose-Einstein condensates in single- and multi-well trapping potentials

Roberto D’Agosta, Boris A. Malomed, Carlo Presilla A

Published Jul 9, 2001• 17 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0107184

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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.
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.
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.
We present analytical stationary solutions for the Gross-Pitaevskii equation (GPE) of a Bose-Einstein condensate (BECs) trapped in a double-well potential. These solutions are compared to those described by [Mahmud et al., Phys. Rev. A 66, 063607 (2002)]. In particular, we provide further evidence that symmetry preserving stationary solutions can be reduced to the eigenstates of the corresponding linear Schr\"odinger equation. Moreover, we have found that the symmetry-breaking solutions can emerge not only from bifurcations, but also from isolated points in the chemical potential--nonlinear interaction diagram. We also have found that there are some moving nodes in the symmetry-breaking solutions.
Bose-Einstein condensates (BECs) have been one of the most active areas of research since their experimental birth in 1995. The complicated nature of the experiments on BECs suggests to observe them in reduced dimensions. The dependence of the collective excitations of the systems on the spatial degrees of freedom allows the study in lower dimensions. In this thesis, we first study two effectively one-dimensional parallel linearly coupled BECs in the presence of external potentials. The system is modelled by linearly coupled Gross-Pitaevskii (GP) equations. In particular, we discuss the dark solitary waves and the grey-soliton-like solutions representing analogues of superconducting Josephson fluxons which we refer to as the fluxon analogue (FA) solutions. We analyze the existence, stability and time dynamics of FA solutions and coupled dark solitons in the presence of a harmonic trap. We observe that the presence of the harmonic trap destabilizes the FA solutions. However, stabilization is possible by controlling the effective linear coupling between the condensates. We also derive theoretical approximations based on variational formulations to study the dynamics of the solutions semi-analytically.\n\nWe then study multiple FA solutions and coupled dark solitons in the same settings. We examine the effects of trapping strength on the existence and stability of the localized solutions. We also consider the interactions of multiple FA solutions as well as coupled dark solitons. In addition, we determine the oscillation frequencies of the prototypical structures of two and three FA solutions using a variational approach.\n\nFinally, we consider two effectively two-dimensional parallel coupled BECs enclosed in a double well potential. The system is modelled by two GP equations coupled by linear and nonlinear cross-phase-modulations. We study a large set of radially symmetric nonlinear solutions of the system in the focusing and defocusing cases. The relevant three principal branches, i.e. the ground state and the first two excited states, are continued as a function of either linear or nonlinear couplings. We investigate the linear stability and time evolution of these solutions in the absence and presence of a topological charge. We notice that only the chargeless or charged ground states can be stabilized by adjusting the linear or nonlinear coupling between the condensates.
In the framework of coupled 1D Gross-Pitaevskii equations, we explore the dynamics of a binary Bose-Einstein condensate where the intra-component interaction is repulsive, while the inter-component one is attractive. The existence regimes of stable self-trapped localized states in the form of symbiotic solitons have been analyzed. Imbalanced mixtures, where the number of atoms in one component exceeds the number of atoms in the other component, are considered in parabolic potential and box-like trap. When all the intra-species and inter-species interactions are repulsive, we numerically find a new type of symbiotic solitons resembling dark-bright solitons. A variational approach has been developed which allows us to find the stationary state of the system and frequency of small amplitude dynamics near the equilibrium. It is shown that the strength of inter-component coupling can be retrieved from the frequency of the localized state's vibrations.