V. A. Brazhnyi

THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length.

Evolution of periodic matter waves in one-dimensional Bose-Einstein condensates with time-dependent scattering length is described. It is shown that variation of the effective nonlinearity is a powerful tool for controlled generation of bright and dark solitons starting with periodic waves.

On classification of intrinsic localized modes for the discrete nonlinear Schrödinger equation

Abstract We consider localized modes (discrete breathers) of the discrete nonlinear Schrodinger equation i(dψn/dt)=ψn+1+ψn−1−2ψn+σ|ψn|2ψn, σ=±1, n∈ Z . We study the diversity of the steady-state solutions of the form ψn(t)=eiωtvn and the intervals of the frequency, ω, of their existence. The base for the analysis is provided by the anticontinuous limit (ω negative and large enough) where all the solutions can be coded by the sequences of three symbols “−”, “0” and “+”. Using dynamical systems approach we show that this coding is valid for ω ∗ ≈−3.4533 and the point ω ∗ is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for ω>ω ∗ and give the complete table of them for the solutions with codes consisting of less than four symbols.

Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations: Application to two-component Bose-Einstein condensates

The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schrödinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.

Dissipation-Induced Coherent Structures in Bose-Einstein Condensates

We discuss how to engineer the phase and amplitude of a complex order parameter using localized dissipative perturbations. Our results are applied to generate and control various types of atomic nonlinear matter waves (solitons) by means of localized dissipative defects.

Dark solitons as quasiparticles in trapped condensates

We present a theory of dark soliton dynamics in trapped quasi-one-dimensional Bose-Einstein condensates, which is based on the local-density approximation. The approach is applicable for arbitrary polynomial nonlinearities of the mean-field equation governing the system as well as to arbitrary polynomial traps. In particular, we derive a general formula for the frequency of the soliton oscillations in confining potentials. A special attention is dedicated to the study of the soliton dynamics in adiabatically varying traps. It is shown that the dependence of the amplitude of oscillations vs the trap frequency (strength) is given by the scaling law X{sub 0}{proportional_to}{omega}{sup -{gamma}} where the exponent {gamma} depends on the type of the two-body interactions, on the exponent of the polynomial confining potential, on the density of the condensate, and on the initial soliton velocity. Analytical results obtained within the framework of the local-density approximation are compared with the direct numerical simulations of the dynamics, showing a remarkable match. Various limiting cases are addressed. In particular for the slow solitons we computed a general formula for the effective mass and for the frequency of oscillations.

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

It is shown that inhomogeneous nonlinear interactions in a Bose-Einstein condensate loaded in an optical lattice can result in a delocalizing transition in one dimension, which sharply contrasts to the known behavior of discrete and periodic systems with homogeneous nonlinearity. The transition can be originated either by decreasing the amplitude of the linear periodic potential or by the change of the mean value of the periodic nonlinearity. The dynamics of the delocalizing transition is studied.

Driving Defect Modes of Bose-Einstein Condensates in Optical Lattices

We present an approximate analytical theory and direct numerical computation of defect modes of a Bose-Einstein condensate loaded in an optical lattice and subject to an additional localized (defect) potential. Some of the modes are found to be remarkably stable and can be driven along the lattice by means of a defect moving following a steplike function defined by the period of Josephson oscillations and the macroscopic stability of the atoms.

Dynamics of matter solitons in weakly modulated optical lattices

It was shown that matter solitons can be effectively managed by smooth variation of parameters of optical lattices in which the atomic condensate is embedded. The phenomenon is based on the effect of the lattice modulation of the carrier wave that transports the soliton and thus can be well described in terms of the effective mass approach, where a specific form of the band structure is of primary importance. We considered linear and parabolic modulation to demonstrate the possibilities of management of matter solitons.

Hydrodynamic flow of expanding Bose-Einstein condensates

We study expansion of quasi-one-dimensional (1D) Bose-Einstein condensate (BEC) after switching off the confining harmonic potential. Exact solution of dynamical equations is obtained in the framework of the hydrodynamic approximation and it is compared with the direct numerical simulation of the full problem, showing excellent agreement at realistic values of physical parameters. We analyze the maximum of the current density and estimate the velocity of expansion. The results of the 1D analysis provides also qualitative understanding of some properties of BEC expansion observed in experiments.