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arXiv:0907.4314 · Journal of Physics A Mathematical and Theoretical

Control of unstable macroscopic oscillations in the dynamics of three coupled Bose condensates

Pierfrancesco Buonsante, Roberto Franzosi, Vittorio Penna

Published Jun 24, 2009• 19 citations
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Abstract

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.

DOI
10.1088/1751-8113/42/28/285307

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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.
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.
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 study an atomic Josephson junction (AJJ) in the presence of two interacting Bose–Einstein condensates (BECs) confined in a double-well trap. We assume that bosons of different species interact with each other. The macroscopic wavefunctions of the two components obey a system of two 3D coupled Gross–Pitaevskii equations (GPEs). We write the Lagrangian of the system, and from this we derive a system of coupled ordinary differential equations (ODEs), for which the coupled pendula represent the mechanical analogues. These differential equations control the dynamical behaviour of the fractional imbalance and of the relative phase of each bosonic component. We perform the stability analysis around the points which preserve the symmetry and get an analytical formula for the oscillation frequency around the stable points. Such a formula could be used as an indirect measure of the inter-species s-wave scattering length. We also study the oscillations of each fractional imbalance around zero and nonzero—the macroscopic quantum self-trapping (MQST)—time averaged values. For different values of the inter-species interaction amplitude, we carry out this study both by directly solving the two GPEs and by solving the corresponding coupled pendula equations. We show that, under certain conditions, the predictions of these two approaches are in good agreement. Moreover, we calculate the crossover value of the inter-species interaction amplitude which signals the onset of MQST.
We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The boson Josephson junction (BJJ) dynamics is described by the two-mode nonlinear Gross-Pitaevskii equation that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of dynamical regimes for the phase difference across the junction and for the population imbalance that are not accessible with superconductor Josephson junctions (SJJ's). These include oscillations with either or both of the following properties: (i) the time-averaged value of the phase is equal to $\ensuremath{\pi} (\ensuremath{\pi}$-phase oscillations); (ii) the average population imbalance is nonzero, in states with macroscopic quantum self-trapping. The (nonsinusoidal) generalization of the SJJ ac and plasma oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and the total number of condensate atoms) onto a single universal curve for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of ${}^{3}\mathrm{He}\ensuremath{-}B$ and the internal Josephson effect in ${}^{3}\mathrm{He}\ensuremath{-}A$ are also discussed.
We explore the interplay between tunneling and interatomic interactions in the dynamics of a bosonic Josephson junction. We tune the scattering length of an atomic ^{39}K Bose-Einstein condensate confined in a double-well trap to investigate regimes inaccessible to other superconducting or superfluid systems. In the limit of small-amplitude oscillations, we study the transition from Rabi to plasma oscillations by crossing over from attractive to repulsive interatomic interactions. We observe a critical slowing down in the oscillation frequency by increasing the strength of an attractive interaction up to the point of a quantum phase transition. With sufficiently large initial oscillation amplitude and repulsive interactions, the system enters the macroscopic quantum self-trapping regime, where we observe coherent undamped oscillations with a self-sustained average imbalance of the relative well population. The exquisite agreement between theory and experiments enables the observation of a broad range of many body coherent dynamical regimes driven by tunable tunneling energy, interactions and external forces, with applications spanning from atomtronics to quantum metrology.