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arXiv:0912.1239 · Physical Review A

Macroscopic self-trapping in Bose-Einstein condensates: Analysis of a dynamical quantum phase transition

Bruno Juliá-Díaz, Daniel Dagnino, Maciej Lewenstein, Joan Martorell, A. Polls

Published Feb 18, 2010• 86 citations
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Abstract

We consider a Bose-Einstein condensate in a double-well potential undergoing a dynamical transition from the regime of Josephson oscillations to the regime of self-trapping. We analyze the statistical properties of the ground state (or the highest excited state) of the Hamiltonian in these two regimes for attractive (repulsive) interactions. We demonstrate that it is impossible to describe the transition within the mean-field theory. In contrast, the transition proceeds through a strongly correlated delocalized state, with large quantum fluctuations, and spontaneous breaking of the symmetry.

DOI
10.1103/physreva.81.023615

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We consider a Bose-Einstein condensate in a double-well potential undergoing a dynamical transition from the regime of Josephson oscillations to the regime of self-trapping. We analyze the statistical properties of the ground state (or the highest excited state) of the Hamiltonian in these two regimes for attractive (repulsive) interactions. We demonstrate that it is impossible to describe the transition within the mean-field theory. In contrast, the transition proceeds through a strongly correlated delocalized state, with large quantum fluctuations, and spontaneous breaking of the symmetry.
An analytical insight into the symmetry-breaking mechanisms underlying the transition from Josephson to self-trapping regimes in Bose-Einstein condensates is presented. We obtain expressions for the ground-state properties of the system of a gas of attractive bosons modeled by a two-site Bose-Hubbard Hamiltonian with an external bias. Simple formulas are found relating the appearance of fragmentation in the condensate with the large quantum fluctuations of the population imbalance occurring in the transition from the Josephson to the self-trapped regime.
Dynamics of the repulsive Bose–Einstein condensate (BEC) in a double-well trap is explored within the 3D time-dependent Gross–Pitaevskii equation. The model avoids numerous common approximations (two-mode treatment, time-space factorization, fixed values of the chemical potential and barrier penetrability, etc) and thus may provide a realistic description of BEC dynamics, including both weak-coupling (sub-barrier) and strong-coupling (above-barrier) regimes and their crossover. The strong coupling regime is achieved by increasing the number N of BEC atoms and thus the chemical potential. The evolution with N of Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST) is examined and the crucial impact of the BEC interaction is demonstrated. At weak coupling, the calculations well reproduce the JO/MQST experimental data. At strong coupling with a significant overlap of the left and right BECs, we observe a remarkable persistence of the Josephson-like dynamics: the JO and MQST converge to a high-frequency JO-like mode where both population imbalance and phase difference oscillate around the zero averages.
The quantum self-trapping phenomenon of a Bose-Einstein condensate (BEC) represents a remarkable nonlinear effect of wide interest. By considering a purely dipolar BEC in a double-well potential, we study how the dipole orientation affects the ground-state structure and the transition between self-trapping and Josephson oscillations in dynamics. Three-dimensional numerical results and an effective two-mode model demonstrate that the onset of self-trapping of a dipolar BEC can be radically modified by the dipole orientation. We also analyze the failure of the two-mode model in predicting the rate of Josephson oscillations. We hope that our results can motivate experimental work as well as future studies of self-trapping of ultracold dipolar gases in optical lattices.
We consider a Bose gas with an attractive interaction in a symmetric double-well potential. In the mean-field approximation, the ground-state solution spontaneously breaks the symmetry of the trapping potential above a certain value of the interaction strength. We demonstrate how the Landau-Ginzburg scheme of the second-order phase transition emerges from the quantum model and show the link to the spontaneous symmetry breaking mentioned above. We identify the order parameter, the critical point and analyze quantum fluctuations around it.
Macroscopic quantum tunneling of Bose-Einstein Condensate in a double-well potential is discussed in the frame of an experimentally realizable Hamiltonian with ϕ 6 interaction under the two-mode approximation. Concentrating on a special case Δ E = Λ = 0, we analyze the stationary states and dynamics of the system to find that such system possesses very small phase spaces for Josephson oscillation and usually lies in the macroscopic quantum self-trapping state (MQST). Importantly we obtain three kinds of MQSTs and in terms of them show the phase diagrams of the system.
We develop a Fock-space WKB method for a Bose-Einstein condensate (BEC) in a double-well trap, using an analogy with a single particle of either positive or negative mass in a quantum potential. The usual mean-field approach is the classical limit, and the inverse number of atoms plays the role of Planck constant. The ground state of positive-mass particles corresponds to the mean-field fixed point of lower energy, while that of negative mass to the excited fixed point. In the case of attractive BECs above the threshold for symmetry breaking, the ground state is the Schr\"odinger cat state and we relate this to the double-well shape of the potential in Fock space. In the repulsive case, we identify the quantum states corresponding to the mean-field macroscopic self-trapping phenomenon. In particular, the phase-locked ($\ensuremath{\pi}$ phase) macroscopic quantum self-trapping of BECs is related to the double-well shape of the potential in Fock space. The running-phase macroscopic quantum self-trapping state is found to be subject to quantum collapses and revivals. The phase dispersion grows exponentially, reaching the absolute maximum just before the first collapse of the running phase, which may explain the growth of the phase fluctuations seen in the experiment on macroscopic quantum self-trapping.