Coupled-mode theory for Bose-Einstein condensates

Abstract

We apply the concepts of nonlinear guided-wave optics to a Bose-Einstein condensate (BEC) trapped in an external potential. As an example, we consider a parabolic double-well potential and derive coupled-mode equations for the complex amplitudes of the BEC macroscopic collective modes. Our equations describe different regimes of the condensate dynamics, including the nonlinear Josephson effect for any separation between the wells. We demonstrate macroscopic self-trapping for both repulsive and attractive interactions, and confirm our results by numerical simulations.A system of interacting bosons confined within an external potential at zero temperature can be described by a macroscopic wave function having the meaning of an order parameter and satisfying the Gross-Pitaevskii (GP) equation [1]. The GP equation is a nonlinear equation that takes into account the effects of the particle interactions through an effective mean field, and it describes the condensate dynamics in a confined geometry. Similar models of the confined dynamics of macroscopic systems appear in other fields, e.g. in the case of an electron gas confined in a quantum well, or optical modes of a photonic microcavity [2]. In all such systems, confined single-particle states are restricted to discrete energies that form a set of eigenmodes.The physical picture of eigenmodes remains valid in the nonlinear case [3], and nonlinear collective modes correspond to the ground and higher-order (excited) states of the Bose-Einstein condensate (BEC) [4]. Moreover, it is possible to observe at least the first excited (antisymmetric) collective mode experimentally [5], through the collapses and revivals in the dynamics of strongly coupled two-component BECs [6]. The interest in the nonground-state collective modes of BECs has grown dramatically with the study of vortex states, very recently successfully created in the experiment [7].The modal structure of the condensate macroscopic (ground and excited) states allows us to draw a deep analogy between BEC in a trap and guided-wave optics, where the concept of nonlinear guided modes is widely used [8]. The physical description of confined condensate dynamics in time is akin to that of stationary beam propagation along a nonlinear optical waveguide, with the BEC chemical potential playing the role of the beam propagation constant. As is well known from nonlinear optics, the guided waves become coupled in the presence of nonlinearity, and the mode coupling can lead to the nonlinear phase shifting between the modes, power exchange, and self-trapping.In this paper, we apply the concepts of nonlinear guided-wave optics to the analysis of mode coupling and intermodal population exchange in trapped BECs. As the most impressive (and also physically relevant) example of the applications of our theory, we consider the BEC dynamics in a harmonic double-well potential, recently discussed in the literature [9]. We study the coupling between the BEC ground-state mode and the first excited (antisymmetric) mode in such a poten...

Universality in nonadiabatic behavior of classical actions in nonlinear models of Bose-Einstein condensates

“…The phenomenon is very important for BEC physics: we consider here nonlinear two-mode models related to tunnelling between coupled BEC in a double well [23], nonlinear Landau-Zener tunnelling [24,25], Feshbach resonance passage in atom-molecule systems [26,27,28]. Nonlinear two-mode models were extensively studied previously (sometimes beyond the mean-field approxima- * Electronic address: alx˙it@yahoo.com tion, [23,24,25,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]), and destruction of adiabaticity was discussed already in [24,25,26], still there are regimes of motion that were not analyzed in these papers from the point of view of nonadiabatic behaviour, that is, when initial populations of both modes are not zero (or very small), but finite. We presented some of our results on that theme in [27,28]; nevertheless destruction of adiabatic invariance has not been studied systematically in BEC-related models yet.…”

Multiple Fluxon Analogues and Dark Solitons in Linearly Coupled Bose–Einstein Condensates

“…Such tunnelling has been observed experimentally where a single [5,6] and an array [7] of short Bose-Josephson junctions (BJJs) were realized. The dynamics of the phase difference between the wavefunctions of the condensates [2,3,4,8,9,10] resembles that of pointlike Josephson junctions [11].…”

Symmetry-Breaking Effects for Polariton Condensates in Double-Well Potentials

“…Importantly, the latter technique was used for the study of a ''polariton Josephson junction'' [38], in the spirit of earlier studies on ''bosonic Josephson junctions'' [39] in the context of atomic BECs. Importantly, a large volume of theoretical studies has accompanied these developments, first in the context of atomic BECs, through investigations related to finite-mode reductions and symmetry-breaking bifurcations [40][41][42][43][44][45][46][47][48][49], quantum effects [50], and nonlinear variants of the double-well potential [51], and more recently in the context of polariton condensates, especially as concerns Josephson oscillations therein [52][53][54][55]. It should be mentioned in passing that similar (spontaneous symmetry breaking) effects have been monitored in the realm of nonlinear optics: in this context, formation of asymmetric states in dual-core fibers [56][57][58][59][60][61][62], self-guided laser beams in Kerr media [63], and optically-induced dual-core waveguiding structures in photorefractive crystals [64] have been reported.…”