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arXiv:0907.3096 · Physical Review A

Four-mode Bose-Hubbard model with two greatly differing tunneling rates as a model for the Josephson oscillation of heat

Martin P. Strzys, J. R. Anglin

Published Apr 19, 2010• 23 citations
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Abstract

As a model for mesoscopic quantum systems in thermal contact, we consider a four-mode Bose-Hubbard model with two greatly differing tunneling rates. By a series of Holstein-Primakoff transformations, we show that the low-frequency dynamics of this system consists in general of two slow Josephson oscillations, rather than the single slow mode predicted by linear Bogoliubov theory. We identify the second slow Josephson oscillation as a heat exchange mode analogous to second sound.

DOI
10.1103/physreva.81.043616

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As a model for mesoscopic quantum systems in thermal contact, we consider a four-mode Bose-Hubbard model with two greatly differing tunneling rates. By a series of Holstein-Primakoff transformations, we show that the low-frequency dynamics of this system consists in general of two slow Josephson oscillations, rather than the single slow mode predicted by linear Bogoliubov theory. We identify the second slow Josephson oscillation as a heat exchange mode analogous to second sound.
A simple four-mode Bose–Hubbard model with intrinsic time-scale separation can be considered as a paradigm for mesoscopic quantum systems in thermal contact. In our previous work, we showed that in addition to coherent particle exchange, a novel slow collective excitation can be identified by a series of Holstein–Primakoff transformations. This resonant energy exchange mode is not predicted by the linear Bogoliubov theory, and its frequency is sensitive to interactions among Bogoliubov quasi-particles; it may be referred to as a second Josephson oscillation, in analogy to the second sound mode of liquid helium II. In this paper, we will explore this system beyond the Gross–Pitaevskii mean-field regime. We directly compare the classical mean-field dynamics to the exact full quantum many-particle dynamics and show good agreement over a wide range of system parameters. The second Josephson frequency becomes imaginary for stronger interactions however, indicating dynamical instability of the symmetric state. By means of a generalized quantum phase model for the full four-mode system, we then show that, in this regime, high-energy Bogoliubov quasiparticles tend to accumulate in one pair of sites, while the actual particles preferentially occupy the opposite pair. We interpret this as a simple model of thermal pressure.
We consider a theoretical model of a four-mode Bose-Hubbard model consisting of two pairs of wells coupled via two processes with two different rates. The model is naturally divided into two subsystems with strong intrasystem coupling and much weaker coupling between the two subsystems and has previously been introduced as a model for Josephson heat oscillations by Strzys and Anglin [Phys. Rev. A 81, 043616 (2010)]. We examine the quantum dynamics of this model for a range of different initial conditions, in terms of both the number distribution among the wells and the quantum statistics. We find that the time evolution is different to that predicted by a mean-field model and that this system exhibits a wide range of interesting behaviours. We find that the system equilibrates to a maximum entropy state and is thus a useful model for quantum thermalisation. As our model may be realized to a good approximation in the laboratory, it becomes a candidate for experimental investigation.
Adiabatic approximations are a powerful tool for simplifying nonlinear quantum dynamics, and are applicable whenever a system exhibits a hierarchy of time scales. Current interest in small nonlinear quantum systems, such as few-mode Bose-Hubbard models, warrants further development of adiabatic methods in the particular context of these models. Here we extend our recent work on a simple four-mode Bose-Hubbard model with two distinct dynamical time scales, in which we showed that among the perturbations around excited stationary states of the system is a slow collective excitation that is not present in the Bogoliubov spectrum. We characterized this mode as a resonant energy exchange with its frequency shifted by nonlinear effects, and referred to it as a second Josephson oscillation, in analogy with the second sound mode of liquid helium II. We now generalize our previous theory beyond the mean-field regime, and construct a general Bogoliubov free quasiparticle theory that explicitly respects the system's adiabatic invariant as well the exact conservation of particles. We compare this theory to the numerically exact quantum energy spectrum with up to 40 particles, and find good agreement over a significant range of parameter space.
We show that the Josephson plasma frequency for a condensate in a double-well potential, whose dynamics is described by the Gross-Pitaevskii (GP) equation, can be obtained with great precision by means of the usual Bogoliubov approach, whereas the two-mode model---commonly constructed by means of a linear combination of the low-lying states of the GP equation---generally provides accurate results only for weak interactions. A proper two-mode model in terms of the Bogoliubov functions is also discussed, revealing that in general a two-mode approach is formally justified only for not too large interactions, even in the limit of very small amplitude oscillations. Here we consider specifically the case of a one-dimensional system, but the results are expected to be valid in arbitrary dimensions.
We present an analytical description of the tunneling dynamics between two coupled Bose-Einstein condensates in the Josephson regime. The model relies on the classical analogy with a rigid pendulum and focuses on two dynamical modes of this system: Josephson oscillations and macroscopic quantum self-trapping. The analogy is extended to include an energy difference between the two superfluids caused by an asymmetry in the trapping potential. The model is compatible with the mean-field predictions of the two-mode Bose-Hubbard model. It gives insights on the mean-field model by involving experimentally measurable parameters with reduced correlations. For agreement with recent experimental observations, we establish heuristic formulas including a dissipation. We conclude with a convincing application of the model to several sets of experimental results.
We investigate the low-energy dynamics of two coupled anisotropic Bose-Einstein condensates forming a long Josephson junction. The theoretical study is performed in the framework of the two-dimensional Gross-Pitaevskii equation and the Bogoliubov--de Gennes formalism. We analyze the excitation spectrum of the coupled Bose condensates and show how low-energy excitations of the condensates lead to multiple-frequency oscillations of the atomic populations in the two wells. This analysis generalizes the standard bosonic Josephson equation approach. We also develop a one-dimensional hydrodynamic model of the coupled condensates, that is capable of reproducing the excitation spectrum and population dynamics of the system.