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Study of localized solutions of the nonlinear discrete model for dipolar Bose-Einstein condensate in an optical lattice by the homoclinic orbit method

Azeddine Messikh, Bakhram Umarov

Published Nov 10, 2012• 0 citations
OpenAlex

Abstract

It is well known that the dynamics of the Bose Einstein condensate (BEC) trapped in an optical lattice can be described in tight-binding approximation by the discrete Nonlinear Schrodinger Equation (DNLSE) [1,2]. This model opens the way to study different aspects of the BEC dynamics in an optical lattice, such as discrete solitons and nonlinear localized modes and their stability and dynamics, modulational instability, superfluid-insulator transition, etc.

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We use homoclinic orbits to find solutions of a dynamical system of the dipolar Bose Einstein Condensate (BEC) in a deep optical lattice. The equation of motion is transformed to a two-dimensional map and its homoclinic orbits are computed numerically. Each homoclinic orbit leads to a different solution. These different solutions lead to different types of solitons. We also analyse the stability of the solutions.

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The present work is devoted to the theoretical investigation of the dynamics of Bose –Einstein condensates (BEC) in optical lattice with the long range magnetic dipole-dipole interaction between atoms. The system is described by a nonlocal nonlinear Schrodinger equation (NLSE). We consider the case when the lattice depth is sufficiently large (tight binding approximation). In this case nonlocal NLSE can be reduced to discrete NLSE equation. We show that discrete NLSE equation for some range of parameters can be reduced further to integrable Ablowitz-Ladik equation which supports moving solitons. Taking the one-soliton solution of Ablowitz-Ladik equation as an initial condition we have performed detailed numerical study of a soliton dynamics in both nonlocal NLSE and discrete NLSE. The conditions of existence and stability of moving solitons in dipolar BEC in an optical lattice are numerically revealed. Also the applicability limits for the tight binding approximation for the experimentally achievable range of parameters are investigated.

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Santiago Rojas-Rojas, Rodrigo A. Vicencio, Mario I. Molina +1 more

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.

arXiv:1106.3499•Physical Review A•Sep 16, 2011•28 citations
It is well known that the dynamics of the Bose Einstein condensate (BEC) trapped in an optical lattice can be described in tight-binding approximation by the discrete Nonlinear Schrodinger Equation (DNLSE) [1,2]. This model opens the way to study different aspects of the BEC dynamics in an optical lattice, such as discrete solitons and nonlinear localized modes and their stability and dynamics, modulational instability, superfluid-insulator transition, etc.
The properties of Bose-Einstein condensates (BEC) can be studied and controlled effectively when trapped in optical lattices. In the limit of tight binding, BEC in optical lattices are well described by the discrete nonlinear Schrödinger equation (DNLSE). Spatially localised modes of excitations, known as discrete breathers or lattice solitons, can occur in the DNLSE. These have been shown to evolve from initial conditions of Gaussian wave-packets. Using this method with the addition of an initial momentum, mobile breathers that travel across the lattice with a constant velocity can also be formed.
The present work is devoted to the theoretical investigation of the dynamics of Bose –Einstein condensates (BEC) in optical lattice with the long range magnetic dipole-dipole interaction between atoms. The system is described by a nonlocal nonlinear Schrodinger equation (NLSE). We consider the case when the lattice depth is sufficiently large (tight binding approximation). In this case nonlocal NLSE can be reduced to discrete NLSE equation. We show that discrete NLSE equation for some range of parameters can be reduced further to integrable Ablowitz-Ladik equation which supports moving solitons. Taking the one-soliton solution of Ablowitz-Ladik equation as an initial condition we have performed detailed numerical study of a soliton dynamics in both nonlocal NLSE and discrete NLSE. The conditions of existence and stability of moving solitons in dipolar BEC in an optical lattice are numerically revealed. Also the applicability limits for the tight binding approximation for the experimentally achievable range of parameters are investigated.
We investigate the properties of discrete breathers in a Bose-Einstein condensate with two- and three-body interactions in optical lattice. In the tight-binding approximation the Gross-Pitaevskii equation with periodic potential for the condensate wavefunction is reduced to the cubic-quintic discrete nonlinear Schrödinger equation. We analyze the regions of modulational instabilities of nonlinear plane waves. This result is important to obtain the conditions for generation of discrete solitons(breathers) in optical lattice. Also using the Page approach, we derive the conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The numerical simulations conform with the analytical predictions.
We study localized nonlinear excitations of a dilute Bose-Einstein condensate (BEC) with spin-orbit coupling in a deep optical lattice (OL). We use Wannier functions to derive a tight-binding model that includes the spin-orbit coupling (SOC) at the discrete level in the form of a generalized discrete nonlinear Schödinger equation. Spectral properties are investigated and the existence and stability of discrete solitons and breathers with different symmetry properties with respect to the OL is demonstrated. We show that the symmetry of the modes can be changed from on-site to inter-site and to asymmetric modes simply by changing the interspecies interaction. Asymmetric modes appear to be novel modes intrinsic of the SOC.
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.