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arXiv:quant-ph/0604158 · Physical Review A

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

S Mossmann, Christof Jung

Published Sep 1, 2006• 68 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.74.033601

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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.
A semiclassical Bohr-Sommerfeld approximation is derived for an NNN-particle, two-mode Bose-Hubbard system modeling a Bose-Einstein condensate in a double-well potential. This semiclassical description is based on the ``classical'' dynamics of the mean-field Gross-Pitaevskii equation and is expected to be valid for large NNN. We demonstrate the possibility to reconstruct quantum properties of the NNN-particle system from the mean-field dynamics. The resulting semiclassical eigenvalues and eigenstates are found to be in very good agreement with the exact ones, even for small values of NNN, both for subcritical and supercritical particle interaction strength where tunneling has to be taken into account.
An atomic Bose-Einstein condensate (BEC) is often described as a macroscopic object which can be approximated by a coherent state. This, on the surface, would appear to indicate that its behavior should be close to being classical. In this paper, we clarify the extent of how "classical" a BEC is by exploring the semiclassical equations for BECs under the mean field Gaussian approximation. Such equations describe the dynamics of a condensate in the classical limit in terms of the variables < x > and < p > as well as their respective variances. We compare the semiclassical solution with the full quantum solution based on the Gross-Pitaevskii Equation (GPE) and find that the interatomic interactions which generate nonlinearity make the system less "classical." On the other hand, many qualitative features are captured by the semiclassical equations, and the equations to be solved are far less computationally intensive than solving the GPE which make them ideal for providing quick diagnostics, and for obtaining new intuitive insight.
The simplest model of three coupled Bose-Einstein condensates is investigated using a group theoretical method. The stationary solutions are determined using the SU(3) group under the mean-field approximation. This semiclassical analysis, using system symmetries, shows a transition in the dynamics of the system from self trapping to delocalization at a critical value for the coupling between the condensates. The global dynamics are investigated by examination of the stable points, and our analysis shows that the structure of the stable points depends on the ratio of the condensate coupling to the particle-particle interaction, and undergoes bifurcations as this ratio is varied. This semiclassical model is compared to a full quantum treatment, which also displays a dynamical transition. The quantum case has collapse and revival sequences superimposed on the semiclassical dynamics, reflecting the underlying discreteness of the spectrum. Nonzero circular current states are also demonstrated as one of the higher-dimensional effects displayed in this system.
Summary form only given. We present a quantum and semiclassical analysis of the dynamics of three coupled atomic Bose-Einstein condensates (BEC). The quantum description is most conveniently analyzed in terms of the operator algebra of SU(3). The space of pure states corresponds to a real manifold of five dimensions corresponding to the parameters of the coset SU(3)/U(2). This is the higher dimensional analogue of the Bloch sphere which has been used to describe the dynamics of two coupled BECs. The equations of motion in the Heisenberg picture can be mapped onto equations in this real manifold. The semiclassical description also involves five real parameters, and corresponds to the mean values of the Heisenberg equations with a factorisation assumption for higher order moments. The symmetries of the problem may be used to construct the ground states of the quantum system for both positive and negative scattering lengths. The structure of the ground states change as the elastic scattering strength changes with respect to the linear tunnelling coupling between the condensate components.
We study the equilibrium dynamics of a weakly interacting Bose-Einstein condensate trapped in a box. In our approach we use a semiclassical approximation similar to the description of a multi-mode laser. In dynamical equations derived from a full N-body quantum Hamiltonian we substitute all creation (and annihilation) operators (of a particle in a given box state) by appropriate c-number amplitudes. The set of nonlinear equations obtained in this way is solved numerically. We show that on the time scale of a few miliseconds the system exhibits relaxation -- reaches an equilibrium with populations of different eigenstates fluctuating around their mean values.
We study nonlinear dynamics of Rydberg-dressed Bose-Einstein condensates (BECs) trapped in a triple-well potential in the semiclassical limit. The Rydberg-dressed BECs experience a long-range soft-core interaction, giving rise to strong nearest- and next-nearest-neighbor interactions in the triple-well system. Using mean-field Gross-Pitaevskii (GP) equations, we show that lower branches of the eigenspectra exhibit loops and level crossings when the soft-core interaction is strong. The direct level crossings eliminate the possibility of adiabatic Landau-Zener transitions when tilting of the triple-well potential. We demonstrate that the long-range interaction allows for self-trapping in one, two, or three wells, in a far more controllable manor than BECs with short-range or dipolar interactions. Exact quantum simulations of the three-well Bose-Hubbard model indicate that self-trapping and nonadiabatic transition can be observed with less than a dozen bosons. Our study is relevant to current research into collective excitation and nonlinear dynamics of Rydberg-dressed atoms.