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International Journal of Modern Physics B

Variational approach for two-component condensates dynamics with two- and three-body interactions and external feeding

Maïk Delon Mboumba, A. B. Moubissi, Thierry Blanchard Ekogo, G. H. Ben-Bolie, Timoléon Crépin Kofané

Published Oct 13, 2015• 7 citations
Publisher

Abstract

We study theoretically and numerically the dynamical behavior of two-component condensates with two- and three-body interactions in variable shape optical lattices and external feeding. By means of the variational approach, the evolution of the condensate amplitudes, widths and number of particles are investigated. The stability of stationary two-component solitons are derived through the Vakhitov–Kolokolov criterion which depends on the cubic and quintic nonlinearities. Direct numerical results of the two coupled Gross–Piteavskii equations (GPEs) which describe the dynamics of the two-component condensates are found to be in good agreement with the analytical predictions.

DOI
10.1142/s0217979215502021

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We study theoretically and numerically the dynamical behavior of two-component condensates with two- and three-body interactions in variable shape optical lattices and external feeding. By means of the variational approach, the evolution of the condensate amplitudes, widths and number of particles are investigated. The stability of stationary two-component solitons are derived through the Vakhitov–Kolokolov criterion which depends on the cubic and quintic nonlinearities. Direct numerical results of the two coupled Gross–Piteavskii equations (GPEs) which describe the dynamics of the two-component condensates are found to be in good agreement with the analytical predictions.
By means of variational solution and direct numerical simulation of the Gross-Pitaevskii equation (GPE), we have studied the stability of matter-wave solitons in two-dimensional (2D) Bose-Einstein condensations (BECs), with 2D linear and nonlinear optical lattices (OLs). Using the static variational approach and Vakhitov-Kolokolov criterion necessary for stability, we obtain the stability condition for solitons in different combinations of OL's parameters. We show that the 2D linear and nonlinear optical lattices allow us to stabilize 2D solitons for both attractive and repulsive interactions. We also study the time-evolution problems of 2D BECs, using the time-dependent variational approach and numerical solution of GPE for 2D linear and nonlinear OLs. Very good agreement between the results corresponding to both treatments is observed.
The stability and collective excitations of binary Bose–Einstein condensates with cubic and quintic nonlinearities in variable anharmonic optical lattices are investigated. By using the variational approach, the influences of the quintic nonlinearities and the shape of the external potential on the stability are discussed in details. It is found that the quintic intraspecies and interspecies interatomic interactions profoundly affect the stability criterion and collective excitations of the system. The shape dependent potential form that characterizes the optical lattice deeply alters the stability regions. Direct numerical simulations of the mean-field coupled Gross–Pitaevskii equation describing the system agree well with the analytical predictions.
The dynamics of dissipative matter wave bright solitons with three- body recombination and external feeding, in a highly asymmetric trap and a deformable shape optical lattice, is studied. Using the variational approach, we derive and analyze the variational equations for the time evolution of the width and amplitude of dissipative solitons, and effective potential of the system. We examine the effects of the optical lattice deformability on the stability of the system. Furthermore, we perform direct numerical experiments to confirm our analytical approach, and good agreement is observed.
Beyond the mean-field theory, a new model of the Gross–Pitaevskii equation (GPE) that describes the dynamics of Bose–Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the three-body interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.
Solitons in one-dimensional (1D) Bose-Einstein condensates with attractive interaction in variable anharmonic periodic potentials are presented. These variable periodic potentials allow us to consider a variety of potential shapes between the two special limits of the Kronig-Penney model and the inverse Kronig-Penney model, with the sinusoidal optical lattice as an intermediate case. The Gross-Pitaevskii equation in (1D) is taken as a model [53]. We have derived the stability conditions for BECs in variable anharmonic periodic potentials using the Vakhitov-Kolokolov criterion. We have found that the stability condition for solitons depends significantly on the shape parameter. In particular, depending on the values of the shape parameter, one and three localized states for the 1D case, respectively, have been found. We have shown that some of these localized states are stable or unstable. The stability conditions obtained are an extension of the existing results to: (1) the more general class of external potential, and (2) the Vakhitov-Kolokolov criterion has been generalized in the case of an external potential of arbitrary shape. A fully numerical simulation of the 1D Gross-Pitaevskii equation finally tests the results of the variational approximation. A good agreement between both the methods is observed.