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arXiv:quant-ph/0412153 · Physical Review A

Modulational instability of two-component Bose-Einstein condensates in an optical lattice

Guangri Jin, Chul Koo Kim, Kyun Nahm

Published Oct 25, 2005• 32 citations
arXivHTMLPDF

Abstract

We study modulational instability of two-component Bose-Einstein condensates in an optical lattice, which is modeled as a coupled discrete nonlinear Schr\"odinger (DNLS) equation. The excitation spectrum and the modulational instability condition of the total system are presented analytically. In the long-wavelength limit, our results agree with the homogeneous two-component Bose-Einstein condensates case. The discreteness effects result in the appearance of the modulational instability for the condensates in miscible region. The numerical calculations confirm our analytical results and show that the interspecies coupling can transfer the instability from one component to another.

DOI
10.1103/physreva.72.045601

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We study modulational instability of two-component Bose-Einstein condensates in an optical lattice, which is modeled as a coupled discrete nonlinear Schr"odinger (DNLS) equation. The excitation spectrum and the modulational instability condition of the total system are presented analytically. In the long-wavelength limit, our results agree with the homogeneous two-component Bose-Einstein condensates case. The discreteness effects result in the appearance of the modulational instability for the condensates in miscible region. The numerical calculations confirm our analytical results and show that the interspecies coupling can transfer the instability from one component to another.
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We present a comprehensive study of modulational instability (MI) in a binary Bose-Einstein condensate with spin-orbit coupling, confined to a deep optical lattice. The system is modeled by a set of discrete Gross-Pitaevskii equations. Using linear stability analysis, we derive the explicit MI conditions for the system, elucidating the critical and distinct roles played by spin-orbit coupling, inter-species nonlinearity, and intra-species nonlinearity. Our analysis, conducted for both unstaggered and staggered fundamental modes, reveals markedly different instability landscapes for these two configurations. The analytical predictions are confirmed by extensive numerical simulations of the full nonlinear dynamics, which vividly illustrate the spatiotemporal evolution of wave amplitudes, phase coherence, and energy localization during the instability process. The numerical results, obtained via a fourth-order Runge-Kutta method, show excellent agreement with the linear stability theory and provide a complete picture of the MI-induced pattern formation.
By means of the multiple-scale expansion method, the coupled nonlinear Schrödinger equations without an explicit external potential are obtained in two-dimensional geometry for a self-attractive Bose–Einstein condensate composed of different hyperfine states. The modulational instability of two-component condensate is investigated by using a simple technique. Based on the discussion about two typical cases, the explicit expression of the growth rate for a purely growing modulational instability and the optimum stable conditions are given and analysed analytically. The results show that the modulational instability of this two-dimensional system is quite different from that in a one-dimensional system.
We investigate the effect of higher-order interactions induced by shape-dependent confinement in the modulational instability (MI) of a binary mixture of Bose-Einstein condensates. For this, we present and compute both analytically and numerically a system of coupled Gross-Pitaevskii equations with residual nonlinearity that rule the dynamics of the mixture. Using the linear stability approach, we obtain the instability criteria of the mixture and find that the MI can be excited in miscible condensates and altered in immiscible condensates due to the effect of residual nonlinearity. Direct numerical calculations are performed to support the analytical predictions, and a good agreement is found. The space-time evolution of the condensate density is displayed in both cases when the mixture is miscible and immiscible, showing the generation of bright solitons for modes predicted to be unstable.
Modulational instability and discrete matter wave solitons in dipolar BECs, loaded into a deep optical lattice, are investigated analytically and numerically. The process of modulational instability of nonlinear plane matter waves in a dipolar nonlinear lattice is studied and the regions of instability are established. The existence and stability of bulk discrete solitons are analyzed analytically and confirmed by numerical simulations. In marked contrast with the usual discrete nonlinear Schr\"odinger behavior (no dipolar interactions), we found a region where the two fundamental modes are simultaneously unstable, allowing enhanced mobility across the lattice for large norm values. To study the existence and properties of surface discrete solitons, an analysis of the dimer configuration is performed. The properties of symmetric and antisymmetric modes including stability diagrams and bifurcations are investigated in closed form. For the case of a bulk medium, properties of fundamental on-site and intersite localized modes are analyzed. On-site and intersite surface localized modes are studied, and we find that they do not exist when nonlocal interactions predominate with respect to local ones.
From GP equation and discrete Schrdinger equation,the modulational instability of dipole Bose-Einstein Condensates(BEC) in an optical lattice is studied.The excitation spectrum of the dipole BEC in the optical lattice and the relation between the dipole-dipole interaction and the modulational instability are obtained.The results show that the influences of the dipoledipole interaction on the nonlinear excitation and stability of dipole BEC in the optical lattice are large.This can give us some useful information for manipulating dipole BECs in practice.