Bose-Einstein Condensates in Standing Waves: The Cubic Nonlinear Schrödinger Equation with a Periodic Potential

Abstract

We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with a Jacobian elliptic function potential. In the limit of a sinusoidal potential our solutions model a dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided the ratio of the height of the variations of the condensate to its DC offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Numerical simulations suggest such stationary states are experimentally observable.The dilute-gas Bose-Einstein condesate (BEC) in the quasi-one-dimensional regime is modeled by the cubic nonlinear Schrödinger equation (NLS) with a potential [1][2][3]. The various traps which are used to contain the BEC have spurred the solution of the NLS with new potentials [4,5]. BECs trapped in a standing light wave have been used to study phase coherence [6] and matter-wave diffraction [7] and have been predicted to have applications in quantum logic [8] and matter-wave transport [9]. Exact solutions have been obtained for the Kronig-Penney potential [10] and some researchers have used a Bloch function description [11]. In this letter, we study new explicit solutions of the NLS with a Jacobian elliptic function potential.We consider the mean-field model of a quasi-onedimensional repulsive BEC trapped in an external potential which is given by the nonlinear Schrödinger equation [1]In experiments, the trapping potential is generated by a standing light wave [6]. As a model for such a potential we use the periodic potentialwhere sn(x, k) denotes the Jacobian elliptic sine function [12] with elliptic modulus 0 ≤ k ≤ 1. In the limit k → 1 − , V (x) becomes an array of well-separated hyperbolic secant potential barriers or wells, while in the limit k → 0 + it becomes purely sinusoidal. We note that for intermediate values (e.g. k = 1/2) the potential closely resembles the sinusoidal behavior and thus provides a good approximation to the standing wave potential generated experimentally [6]. We present stationary solutions in closed form and study their stability analytically and numerically. We begin by constructing solutions to Eq. (1) which have the formwherewhere B determines a mean amplitude and acts as a DC offset for the number of condensed atoms. The strength of the nonlinearity, which for the BEC is a function of both the atomic coupling and the number of condensed atoms, is determined by the parameters V 0 + k 2 and B, as is apparent in the amplitude of the solutions given by Eq. (3). Note that if x is scaled so that V (x) undergoes only a single oscillation on the ring (in the limit k → 1) the Jacobian elliptic potential provides a model of a single barrier or well [13]. For simplicity we focus on two special cases: (1) k arbitrary and trivial phase (c = 0), and (2) k = 0 with nontrivial phase (c = 0) so that the solutions are trigonometric functions. Trivial Phase Case -In the limit of c = 0, the solutions given in Eqs. (3)-(4) reduce tovalid for V 0 ≥ −k 2 , and ψ(x, t) = −(V 0 +k 2...

Nonlinear bifurcation diagrams of the current states of microcavity exciton-polaritons in the lattice

“…This approach is similar to the method used in [11] to determine the energy band structure of atomic BEC localized in an optical lattice. Unlike the latter case the exciton-polariton condensate is strongly affected by the spatial modulation of excitonic reservoir [12], and that is accounted by the second expansion in equation (4).…”

“…To describe polariton condensate properties we use an open driven-dissipative model that accounts for the particle exchange between the condensate and the excitonic reservoir formed by a non-resonant pump [12]. This approach is based on a mean field theory and assuming the spontaneous formation of the exciton-polariton condensate, we consider an open-dissipative Gross-Pitaevskii (GP) model, that describes incoherently pumped condensate coupled to an excitonic reservoir.…”

“…The existence of such nonlinear current-currying states was interpreted as a consequence of superfluid behaviour of a condensate in periodic potential [10]. The linear stability properties of these nonlinear Bloch waves have been intensively studied [12][13][14]. The regions of both the energetic and dynamical instabilities have been identified for such essentially conservative systems.…”

“…In such class of problems analytical methods are suitable only for special kinds of a potential such as elliptic [12] or Kronig-Penney [13,14] potentials. That is why in this paper we use the numerical methods of finding nonlinear Bloch waves which are appropriate for the strongly nonlinear case when perturbative approaches fail.…”

SpectrUW: A laboratory for the numerical exploration of spectra of linear operators

“…As a result of direct numerical simulations on (16), it was conjectured in [4,5] that this solution with V 0 = 1 is unstable for B < 1/2, but stable for B > 1. We reexamine this problem here, by considering (14) with V 0 = 1, but for a variety of B values, ranging from 0 to 1, using SpectrUW's animation feature.…”