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Effective Josephson dynamics in resonantly driven Bose-Einstein condensates

Heimsoth, Martin, Hochstuhl, D., C. E. Creffield, Lincoln D. Carr, Fernando Sols

Published Oct 7, 2013• 20 citations
OpenAlex

Abstract

We show that the orbital Josephson effect appears in a wide range of driven atomic Bose-Einstein condensed systems, including quantum ratchets, double wells and box potentials. We use three separate numerical methods: the Gross-Pitaevskii equation, exact diagonalization of the few-mode problem and the multi-configurational time-dependent Hartree for bosons algorithm. We establish the limits of mean-field and few-mode descriptions, demonstrating that the few-mode approximation represents the full many-body dynamics to high accuracy in the weak driving limit. Among other quantum measures, we compute the instantaneous particle current and the occupation of natural orbitals. We explore four separate dynamical regimes, the Rabi limit, chaos, the critical point and self-trapping; a favorable comparison is found even in the regimes of dynamical instabilities or macroscopic quantum self-trapping. Finally, we present an extension of the (t, t')-formalism to general time-periodic equations of motion, which permits a systematic description of the long-time dynamics of resonantly driven many-body systems, including those relevant to the orbital Josephson effect.

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We show that the orbital Josephson effect appears in a wide range of driven atomic Bose-Einstein condensed systems, including quantum ratchets, double wells and box potentials. We use three separate numerical methods: the Gross-Pitaevskii equation, exact diagonalization of the few-mode problem and the multi-configurational time-dependent Hartree for bosons algorithm. We establish the limits of mean-field and few-mode descriptions, demonstrating that the few-mode approximation represents the full many-body dynamics to high accuracy in the weak driving limit. Among other quantum measures, we compute the instantaneous particle current and the occupation of natural orbitals. We explore four separate dynamical regimes, the Rabi limit, chaos, the critical point and self-trapping; a favorable comparison is found even in the regimes of dynamical instabilities or macroscopic quantum self-trapping. Finally, we present an extension of the (t, t')-formalism to general time-periodic equations of motion, which permits a systematic description of the long-time dynamics of resonantly driven many-body systems, including those relevant to the orbital Josephson effect.
We show that the orbital Josephson effect appears in a wide range of driven atomic Bose–Einstein condensed systems, including quantum ratchets, double wells and box potentials. We use three separate numerical methods: the Gross–Pitaevskii equation, exact diagonalization of the few-mode problem and the multi-configurational time-dependent Hartree for bosons algorithm. We establish the limits of mean-field and few-mode descriptions, demonstrating that the few-mode approximation represents the full many-body dynamics to high accuracy in the weak driving limit. Among other quantum measures, we compute the instantaneous particle current and the occupation of natural orbitals. We explore four separate dynamical regimes, the Rabi limit, chaos, the critical point and self-trapping; a favorable comparison is found even in the regimes of dynamical instabilities or macroscopic quantum self-trapping. Finally, we present an extension of the ( t , t ')-formalism to general time-periodic equations of motion, which permits a systematic description of the long-time dynamics of resonantly driven many-body systems, including those relevant to the orbital Josephson effect.
In a system of ac-driven condensed bosons we study a new type of Josephson effect occurring between states sharing the same region of space and the same internal atom structure. We first develop a technique to calculate the long-time dynamics of a driven interacting many-body system. For resonant frequencies, this dynamics can be shown to derive from an effective time-independent Hamiltonian which is expressed in terms of standard creation and annihilation operators. Within the subspace of resonant states, and if the undriven states are plane waves, a locally repulsive interaction between bosons translates into an effective attraction. We apply the method to study the effect of interactions on the coherent ratchet current of an asymmetrically driven boson system. We find a wealth of dynamical regimes which includes Rabi oscillations, self-trapping and chaotic behavior. In the latter case, a full many-body calculation deviates from the mean-field results by predicting large quantum fluctuations of the relative particle number.
In a system of ac-driven condensed bosons we study a new type of Josephson effect occurring between states sharing the same region of space and the same internal atom structure. We first develop a technique to calculate the long-time dynamics of a driven interacting many body system. For resonant frequencies, this dynamics can be shown to derive from an effective time-independent Hamiltonian which is expressed in terms of standard creation and annihilation operators. Within the subspace of resonant states, and if the undriven states are plane waves, a locally repulsive interaction between bosons translates into an effective attraction. We apply the method to study the effect of interactions on the coherent ratchet current of an asymmetrically driven boson system. We find a wealth of dynamical regimes which includes Rabi oscillations, self trapping and chaotic behavior. In the latter case, a full many-body calculation deviates from the mean-field results by predicting large quantum fluctuations of the relative particle number.
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.
We study atomic Josephson junctions (AJJs) with one and two bosonic species confined by a double-well potential. Proceeding from the second quantized Hamiltonian, we show that it is possible to describe the zero-temperature AJJ microscopic dynamics by means of extended Bose–Hubbard (EBH) models, which include usually neglected nonlinear terms. Within the mean-field approximation, the Heisenberg equations derived from such two-mode models provide a description of AJJ macroscopic dynamics in terms of ordinary differential equations (ODEs). We discuss the possibility of distinguishing the Rabi, Josephson and Fock regimes in terms of the macroscopic parameters which appear in the EBH Hamiltonians, and then in the ODEs. We compare the predictions for the relative populations of the Bose gas atoms in the two wells obtained from the numerical solutions of the two-mode ODEs, with those deriving from the direct numerical integration of the Gross–Pitaevskii equations (GPEs). Our investigations show that the nonlinear terms of the ODEs are crucial to achieve a good agreement between the ODE and GPE approaches, and in particular to give quantitative predictions of the self-trapping regime.
We study the dynamics of three-dimensional weakly linked Bose-Einstein condensates using a multimode model with an effective interaction parameter. The system is confined by a ring-shaped four-well trapping potential. By constructing a two-mode Hamiltonian in a reduced highly symmetric phase space, we examine the periodic orbits and calculate their time periods both in the self-trapping and Josephson regimes. The dynamics in the vicinity of the reduced phase space is investigated by means of a Floquet multiplier analysis, finding regions of different linear stability and analyzing their implications on the exact dynamics. The numerical exploration in an extended region of the phase space demonstrates that two-mode tools can also be useful for performing a partition of the space in different regimes. Comparisons with Gross-Pitaevskii simulations confirm these findings and emphasize the importance of properly determining the effective on-site interaction parameter governing the multimode dynamics.