Semiclassical approach to Bose-Einstein condensates in a triple well potential

Abstract

We present an approach for the analysis of Bose-Einstein condensates in a few mode approximation. This method has already been used to successfully analyze the vibrational modes in various molecular systems and offers a perspective on the dynamics in many particle bosonic systems. We discuss a system consisting of a Bose-Einstein condensate in a triple well potential. Such systems correspond to classical Hamiltonian systems with three degrees of freedom. The semiclassical approach allows a simple visualization of the eigenstates of the quantum system referring to the underlying classical dynamics. From this classification we can read off the dynamical properties of the eigenstates such as particle exchange between the wells and entanglement without further calculations. In addition, this approach offers insights into the validity of the mean-field description of the many particle system by the Gross-Pitaevskii equation, since we make use of exactly this correspondence in our semiclassical analysis. We choose a three mode system in order to visualize it easily and, moreover, to have a sufficiently interesting structure, although the method can also be extended to higher dimensional systems.⌽ ͑r ជ͒ = ͚ n,m n,m ͑r ជ͒â n,m , ͑2͒where we assume that the basis functions ͕ n,m ͖ of the oneparticle Hilbert space are exponentially localized in space and real, as is the case for the Wannier functions ͓22͔. The index n describes basis functions in different wells and we *Electronic address: mossmann@fis.unam.mx PHYSICAL REVIEW A 74, 033601 ͑2006͒

Semiclassical quantization of anN-particle Bose-Hubbard model

“…For three and more modes, the classical dynamics is chaotic (see, e.g., the studies of the three-mode system [4,31] or tilted optical lattices [32]). Chaoticity also appears in periodically driven two-mode systems [20,33] or the related kicked tops [24].…”

“…To avoid ambiguities one has to replace symmetrized products of the operators by the corresponding products of c-numbers. Therefore we will start on the many particle side with a symmetrized Bose-Hubbard Hamiltonian in the following, where then j are replaced byn s j = (â † jâ j +â jâ † j )/2 (see also [4]). This symmetrization affects only the nonlinear term in (1) and the symmetrizedĤ is related to (1) by an additive constant term depending only onN .…”

“…The classical-quantum correspondence has been studied in terms of phase space (Husimi) distributions [3] for such systems. Mossmann and Jung [4] demonstrate for a three-mode Bose-Hubbard model that the organization of the N -particle eigenstates follows the underlying classical, i.e. mean-field, dynamics.…”

Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer

“…(see definition (23)) that correspond to approaching 1) the curve g(v) from the right, at a given v, and 2) the straight line v = −1/2 from below, at a given τ , respectively. Hence, the approach to the regions where the classical instability crops up is driven by either η − 2 → 0 or |v| → 1/2.…”

“…This issue has been discussed in a series of papers examining the spectral properties of quantum trimer [20]- [22], its description within the phase-variable [23] and the Husimidistribution [24] pictures, and the inclusion of higherorder quantum correlations within the multiconfigurational Hartree method [25]. Quantum trimer has been also used to model coherent transport with weak interaction [26] and thermalization effects within the FokkerPlanck theory [35].…”