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arXiv:cond-mat/0610532 · Physical Review Letters

Dynamical Instability in a Trimeric Chain of Interacting Bose-Einstein Condensates

Pierfrancesco Buonsante, Roberto Franzosi, Vittorio Penna

Published Feb 5, 2003• 63 citations
arXivHTMLPDF

Abstract

We analyze thoroughly the mean-field dynamics of a linear chain of three coupled Bose-Einstein condensates, where both the tunneling and the central-well relative depth are adjustable parameters. Owing to its nonintegrability, entailing a complex dynamics with chaos occurrence, this system is a paradigm for longer arrays whose simplicity still allows a thorough analytical study. We identify the set of dynamics fixed points, along with the associated proper modes, and establish their stability character depending on the significant parameters. As an example of the remarkable operational value of our analysis, we point out some macroscopic effects that seem viable to experiments.

DOI
10.1103/physrevlett.90.050404

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We analyze thoroughly the mean-field dynamics of a linear chain of three coupled Bose-Einstein condensates, where both the tunneling and the central-well relative depth are adjustable parameters. Owing to its nonintegrability, entailing a complex dynamics with chaos occurrence, this system is a paradigm for longer arrays whose simplicity still allows a thorough analytical study. We identify the set of dynamics fixed points, along with the associated proper modes, and establish their stability character depending on the significant parameters. As an example of the remarkable operational value of our analysis, we point out some macroscopic effects that seem viable to experiments.
We report on the dynamics of three bosonic wells (trimer model) coupled cyclically and occupied by N bosons within the standard mean-field picture. Based on the identification of the fixed points of the dynamics, we find periodic solutions representing the trimer proper modes (the system behavior is discussed for initial conditions close to such states in some relevant cases) and the origin of dynamical instabilities. A diffused chaotic behavior and various macroscopic phenomena are detected numerically. Their dependence on macroscopic model parameters is evaluated as well. Finally, we consider the trimer dynamics for an open chain in the case when the central well has an adjustable depth.
We study the dynamics of two Bose–Einstein condensates (BECs) tunnel-coupled by a double-well potential. A real three-body interaction term is considered and a two-mode approximation is used to derive two coupled equations, which describe the relative population and relative phase. By solving the equations and analyzing the stability of the system, we find the stable stationary solutions for a constant atomic scattering length. When a periodically time-varying scattering length is applied, Melnikov analysis and numerical calculation demonstrate the existence of chaotic behavior and the dependence of chaos on the three-body interaction parameters.
The Bose-Hubbard Hamiltonian capturing the essential physics of the arrays of interacting Bose-Einstein condensates is addressed, focusing on arrays consisting of two (dimer) and three (trimer) sites.In the former case, some results concerning the persistence of mean-field features in the energy spectrum of the symmetric dimer are extended to the asymmetric version of the system, where the two sites are characterized by different on-site energies.Based on a previous systematic study of the mean-field limit of the trimer, where the dynamics is exhaustively described in terms of its fixed points for every choice of the significant parameters, an interesting mapping between the dimer and the trimer is emphasized and used as a guide in investigating the persistence of mean-field features in the rather complex energy spectrum of the trimer.These results form the basis for the systematic investigation of the purely quantum trimer extending and completing the existing mean-field analysis.In this respect we recall that, similar to larger arrays, the trimer is characterized by a non-integrable mean-field dynamics featuring chaotic trajectories.Hence, the correspondence between mean-field fixed points and quantum energy levels emphasized in the present work may provide a key to investigate the quantum counterpart of classical instability.
The dynamics of the three coupled bosonic wells (trimer) containing N bosons is investigated within a standard (mean-field) semiclassical picture based on the coherent-state method. Various periodic solutions (configured as pi-like, dimerlike, and vortex states) representing collective modes are obtained analytically when the fixed points of trimer dynamics are identified on the N=const submanifold in the phase space. Hyperbolic, maximum and minimum points are recognized in the fixed-point set by studying the Hessian signature of the trimer Hamiltonian. The system dynamics in the neighborhood of periodic orbits (associated with fixed points) is studied via numeric integration of trimer motion equations, thus revealing a diffused chaotic behavior (not excluding the presence of regular orbits), macroscopic effects of population inversion, and self-trapping. In particular, the behavior of orbits with initial conditions close to the dimerlike periodic orbits shows how the self-trapping effect of dimerlike integrable subregimes is destroyed by the presence of chaos.
We study the dynamical stability of the macroscopic quantum oscillations characterizing a system of three coupled Bose–Einstein condensates arranged into an open-chain geometry. The boson interaction, the hopping amplitude and the central-well relative depth are regarded as adjustable parameters. After deriving the stability diagrams of the system, we identify three mechanisms for realizing the transition from unstable to stable behaviour and analyse specific configurations that, by suitably tuning the model parameters, give rise to macroscopic effects which are expected to be accessible to experimental observation. Also, we pinpoint a system regime that realizes a Josephson-junction-like effect. In this regime, the system configuration does not depend on the model interaction parameters and the population oscillation amplitude is related to the condensate-phase difference. This fact makes estimating the latter quantity possible, since the measure of the oscillating amplitudes is experimentally accessible.
The dynamics of a mean-field model of three coupled condensates is studied for initial states characterized by twin condensates. Condensates occupy an open three-well chain, where the central potential-well depth is an independent parameter. Despite its diversity from the closed-chain symmetric model, such model is shown to have an integrable regime involving a predictable evolution for twin-condensate initial states. After establishing the dependence of the phase-space topology on the model parameters, the latter are shown to allow the control of various macroscopic effects such as rapid population inversions, pulsed phase changes, self-trapping and long-duration periodic oscillations. Finally, based on quantizing the system in terms of suitable ``microscopic'' canonical variables, the stability conditions of the integrable regime are related to the su(1,1) structure of the Hamiltonian spectrum.