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Physical Review E

Dynamics of matter-wave condensates with time-dependent two- and three-body interactions trapped by a linear potential in the presence of atom gain or loss

D. Belobo Belobo, G. H. Ben-Bolie, Timoléon Crépin Kofané

Published Apr 21, 2014• 14 citations
Publisher

Abstract

Bose-Einstein condensates with time varying two- and three-body interatomic interactions, confined in a linear potential and exchanging atoms with the thermal cloud are investigated. Using the extended tanh-function method with an auxiliary equation, i.e., the Lenard equation, many exact solutions describing the dynamics of matter-wave condensates are derived. An important issue is the time management of the cubic and the quintic nonlinearities by tuning the rate of exchange of atoms between the condensate and the thermal background. In addition, adjusting the strength of the linear potential, the rate of exchange of atoms, and many other free parameters allow one to control many features of the condensate such as its height, width, position, velocity, acceleration, and its direction, respectively. Full numerical solutions corroborate the analytical predictions.

DOI
10.1103/physreve.89.042913

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Bose-Einstein condensates with time varying two- and three-body interatomic interactions, confined in a linear potential and exchanging atoms with the thermal cloud are investigated. Using the extended tanh-function method with an auxiliary equation, i.e., the Lenard equation, many exact solutions describing the dynamics of matter-wave condensates are derived. An important issue is the time management of the cubic and the quintic nonlinearities by tuning the rate of exchange of atoms between the condensate and the thermal background. In addition, adjusting the strength of the linear potential, the rate of exchange of atoms, and many other free parameters allow one to control many features of the condensate such as its height, width, position, velocity, acceleration, and its direction, respectively. Full numerical solutions corroborate the analytical predictions.
By using the F-expansion method associated with four auxiliary equations, i.e., the Bernoulli equation, the Riccati equation, the Lenard equation, and the hyperbolic equation, we present exact explicit solutions describing the dynamics of matter-wave condensates with time-varying two- and three-body nonlinearities. Condensates are trapped in a harmonic potential and they exchange atoms with the thermal cloud. These solutions include the generalized Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. In addition, we have also found rational function solutions. Solutions constructed here have many free parameters that can be used to manipulate and control some important features of the condensate, such as the position, width, velocity, acceleration, and homogeneous phase. The stability of the solutions is confirmed by their long-time numerical behavior.
To analytically investigate the matter-wave solitons of Bose–Einstein condensates (BECs) in time-dependent complex potential, we consider a cubic-quintic Gross–Pitaevskii (GP) equation with distributed coefficients and a dissipative term. By introducing a suitable ansatz, we establish the criterion of the modulational instability (MI) of the system and present an explicit expression for the growth rate of a purely growing MI. Effects of the parabolic background potential, as well as of the linear potential, the gain/loss parameter, and the two- and three-body interatomic interactions on the MI are investigated. We show how the feeding/loss parameter can be well used to control the instability of the system. The analytical resolution of the considered GP equation leads to exact bright, dark and kink solitary wave solutions which are used to investigate analytically the dynamics of matter-wave solitons in BECs under consideration. These analytical investigations show that the amplitude and the motion of bright, dark and kink solitary waves depend on the strengths of the two- and three-body interatomic interactions, as well as on the strengths of the external trapping potential and the parameter of the gain/loss of atoms in the condensate.
Using the F-expansion method we present analytical matter-wave solutions to Bose—Einstein condensates with two- and three-body interactions through the generalized three-dimensional Gross—Pitaevskii equation with time-dependent coefficients, for the periodically time-varying interactions and quadratic potential strength. Such solutions exist under certain conditions, and impose constraints on the functions describing potential strength, nonlinearities, and gain (loss). Various shapes of analytical matter-wave solutions which have important applications of physical interest are studied in details.
In this paper, we consider a Gross-Pitaevskii (GP) equation with a time-dependent nonlinearity and a spatiotemporal complex linear term which describes the dynamics of matter-wave solitons in Bose-Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms. We establish the integrability conditions under which analytical solutions describing the modulational instability and the propagation of both bright and dark solitary waves on a continuous wave background are obtained. The obtained integrability conditions also appear as the conditions under which the solitary waves of the BECs can be managed by controlling the functional gain or loss parameter. For specific BECs, the dynamics of bright and dark solitons are investigated analytically through the found exact solutions of the GP equation. Our results show that under the integrability conditions, the gain or loss parameter of the GP equation can be used to manage the motion of both bright and dark solitons. We show that for BECs with loss (gain) of atoms, the bright and dark solitons during their propagation have a compression (broadening) in their width. Furthermore, under a safe range of parameters and under the integrability conditions, it is possible to squeeze a bright soliton of BECs with loss of atoms into the assumed peak matter density, which can provide an experimental tool for investigating the range of validity of the 1D GP equation. Our results also reveal that under the conditions of the solitary wave management, neither the injection or the ejection of atoms from the condensate affects the soliton peak during its propagation.
We construct, through a further extension of the tanh-function method, the matter-wave solutions of Bose-Einstein condensates (BECs) with a three-body interaction. The BECs are trapped in a potential comprising the linear magnetic and the time-dependent laser fields. The exact solutions obtained include soliton solutions, such as kink and antikink as well as bright, dark, multisolitonic modulated waves. We realize that the motion and the shape of the solitary wave can be manipulated by controlling the strengths of the fields.
Using the F-expansion method, we present analytical matter-wave solutions to Bose-Einstein condensates with two- and three-body interactions through the generalized onedimensional Gross-Pitaevskii equation with time-dependent coefficients, for the experimentally relevant quadratic potential strength. Various shapes of the analytical matter-wave solutions which have important applications of physical interest are studied in detail. In particular, the matter-wave solitons are obtained in the quintic Gross-Pitaevskii equation model, which may be produced by tuning the cubic nonlinearity to zero via the Feschbach resonance technique. Direct numerical simulation has been performed to show the stable region of the matter-wave soliton.