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arXiv:1406.5536 · New Journal of Physics

Temporal non-equilibrium dynamics of a Bose–Josephson junction in presence of incoherent excitations

Mauricio Trujillo-Martinez, Anna Posazhennikova, Johann Kroha

Published Jan 9, 2015• 8 citations
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Abstract

The time-dependent non-equilibrium dynamics of a Bose–Einstein condensate (BEC) typically generates incoherent excitations out of the condensate due to the finite frequencies present in the time evolution. We present a detailed derivation of a general non-equilibrium Greenʼs function technique that describes the coupled time evolution of an interacting BEC and its single-particle excitations in a trap, based on an expansion in terms of the exact eigenstates of the trap potential. We analyze the dynamics of a Bose system in a small double-well potential with initially all particles in the condensate. When the trap frequency is larger than the Josephson frequency, , the dynamics changes at a characteristic time, , abruptly from the slow Josephson oscillations of the BEC to fast Rabi oscillations driven by quasiparticle excitations in the trap. For times , the Josephson oscillations are undamped, in agreement with the experiments. We analyze the physical origin of the finite scale as well as its dependence on the trap parameter Δ .

DOI
10.1088/1367-2630/17/1/013006

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The time-dependent non-equilibrium dynamics of a Bose–Einstein condensate (BEC) typically generates incoherent excitations out of the condensate due to the finite frequencies present in the time evolution. We present a detailed derivation of a general non-equilibrium Greenʼs function technique that describes the coupled time evolution of an interacting BEC and its single-particle excitations in a trap, based on an expansion in terms of the exact eigenstates of the trap potential. We analyze the dynamics of a Bose system in a small double-well potential with initially all particles in the condensate. When the trap frequency is larger than the Josephson frequency, , the dynamics changes at a characteristic time, , abruptly from the slow Josephson oscillations of the BEC to fast Rabi oscillations driven by quasiparticle excitations in the trap. For times , the Josephson oscillations are undamped, in agreement with the experiments. We analyze the physical origin of the finite scale as well as its dependence on the trap parameter Δ .
We perform a detailed quantum dynamical study of nonequilibrium Josephson oscillations between interacting Bose-Einstein condensates confined in a finite-size double-well trap. We find that the Josephson junction can sustain multiple undamped Josephson oscillations up to a characteristic time scale tau(c) without quasipartcles being excited in the system. This may explain recent related experiments. Beyond a characteristic time scale tau(c) the dynamics of the junction is governed by fast, quasiparticle-assisted Josephson tunneling as well as Rabi oscillations between the discrete quasiparticle levels. We predict that an initially self-trapped state of the Bose-Einstein condensates will be destroyed by these fast dynamics.
Experiments on the nonequilibrium dynamics of an isolated Bose-Einstein condensate (BEC) in a magnetic double-well trap exhibit a puzzling divergence: While some show dissipation-free Josephson oscillations, others find strong damping. Such damping in isolated BECs cannot be understood on the level of the coherent Gross-Pitaevskii dynamics. Using the Keldysh functional-integral formalism, we describe the time-dependent system dynamics by means of a multimode BEC coupled to fluctuations (single-particle excitations) beyond the Gross-Pitaevskii saddle point. We find that the Josephson oscillations excite an excess of fluctuations when the effective Josephson frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{J}$ is in resonance with the effective fluctuation energy ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}}_{m}$, where both ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{J}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}}_{m}$ are strongly renormalized with respect to their noninteracting values. By evaluating and using the model parameters for the respective experiments, we describe quantitatively the presence or absence of damping.
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 investigate the coherent dynamics of a noninteracting Bose-Einstein condensate in a system consisting of two bosonic reservoirs coupled via a spatially localized mode. We describe this system by a two-terminal Fano-Anderson model and investigate analytically the time evolution of observables such as the Josephson current. In doing so, we find that the Josephson current sensitively depends on the on-site energy of the localized mode. This facilitates using this setup as a transistor for a Bose-Einstein condensate. We identify two regimes. In one regime, the system exhibits well-behaved long-time dynamics with a slowly oscillating and undamped Josephson current. In a second regime, the Josephson current is a superposition of an extremely weakly damped slow oscillation and an undamped fast oscillation. Our results are confirmed by finite-size simulations.
A Bose gas in a double-well potential, exhibiting a true Bose-Einstein condensate (BEC) amplitude and initially performing Josephson oscillations, is a prototype of an isolated, nonequilibrium many-body system. We investigate the quasiparticle (QP) creation and thermalization dynamics of this system by solving the time-dependent Keldysh-Bogoliubov equations. We find avalanchelike QP creation due to a parametric resonance between BEC and QP oscillations, followed by slow, exponential relaxation to a thermal state at an elevated temperature, controlled by the initial excitation energy of the oscillating BEC above its ground state. The crossover between the two regimes occurs because of an effective decoupling of the QP and BEC oscillations. This dynamics is analogous to elementary particle creation in models of the early universe. The thermalization in our setup occurs because the BEC acts as a grand canonical reservoir for the quasiparticle system.
The dynamical properties of partially coherent Bose-Einstein condensates in double wells are investigated in three typical regimes. In the extreme Fock regime, the time evolution of the degree of coherence is shown to decay rapidly. In the Rabi regime, a relation between the amplitude of Rabi oscillation and the degree of coherence is obtained, which is expected to determine the degree of coherence by measuring the amplitude of Rabi oscillation. The study on the self-trapping phenomena in the Josephson regime exhibits that both the degree of coherence and the initial relative phase can affect the final particle distribution.