Nonlinear modes of a macroscopic quantum oscillator

We consider the Bose–Einstein condensate in a parabolic trap as a macroscopic quantum oscillator and describe, analytically and numerically, its collective modes — a nonlinear generalisation of the (symmetric and antisymmetric) Hermite–Gauss eigenmodes of a harmonic quantum oscillator.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.75.F; 03.75.-b; 03.50.-z; 73.20.Dx The recent observation of different types of Bose– Einstein condensation (BEC) in atomic clouds [1] led to the foundation of a new direction in the study of macroscopic quantum phenomena. From a general point of view, the dynamics of gases of cooled atoms confined in a magnetic trap at very low temperatures can be described by an effective equation for the condensate wave function known as the Gross–Pitaevskii (GP) equation [2]. This is a classical nonlinear equation that takes into account the effects of the particle interaction through an effective mean field, and therefore it can be treated as a nonlinear generalization of a text-book problem of quantum mechanics, i.e., as a macroscopic quantum oscillator. Similar models of the confined dynamics of macroscopic quantum systems appear in other fields, e.g., in the case of an electron gas confined in a quantum well [3], or optical modes in a photonic microcav* Corresponding author. E-mail address: [email protected] (Yu.S. Kivshar). ity [4]. In all such systems, confined single-particle states are restricted to a set of discrete energies that form a set of eigenmodes. A classical and probably most familiar example of such a system is a harmonic quantum oscillator with equally spaced energy levels [5]. When, instead of single-particle states, we describe quasiclassically a system of interacting bosons in a macroscopic ground state confined in an external potential, a standard application of the mean-field theory allows us to introduce a macroscopic wave function as a classical field Ψ (R, t) having the meaning of the order parameter. The equation for the function Ψ (R, t) looks similar to that of a single-particle oscillator, but it also includes the effect of interparticle interaction, taken into account as a mean-field nonlinear term. Then, the important questions are: Does the physical picture of eigenmodes remain valid in the nonlinear case, and what is the effect of nonlinearity on the modes? In this Letter we analyse nonlinear eigenmodes of a macroscopic quantum oscillator as a set of nonlinear stationary states that extend the 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0375-9601(00) 00 77 4X 226 Yu.S. Kivshar et al. / Physics Letters A 278 (2001) 225–230 well-known Hermite–Gauss eigenfunctions. We also make a link between seemingly different approximations, the well-known Thomas–Fermi approximation and the perturbation theory developed here for the case of weak nonlinearity. For both attractive and repulsive interaction, we demonstrate a close connection between the nonlinear modes and (bright and dark) multi-soliton stationary states. We would like to emphasise that the nonlinear modes we analyse here are not a generalisation of the linear-response excitations of the ground state discussed in the literature [7]; they are the macroscopic modes of the confined condensate as the whole. We consider the macroscopic dynamics of condensed atomic clouds in a three-dimensional, strongly anisotropic, external parabolic potential created by a magnetic trap. The BEC collective dynamics can be described by the GP equation