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Physical Review A

Exact quantum phase model for mesoscopic Josephson junctions

J. R. Anglin, P. D. Drummond, Augusto Smerzi

Published Nov 13, 2001• 91 citations
Publisher

Abstract

Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional $\mathrm{cos}2\ensuremath{\varphi}$ term in the potential. We also find that the inner product in this representation is nonlocal in $\ensuremath{\varphi}.$ Our model exhibits phenomena, such as $\ensuremath{\pi}$ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.

DOI
10.1103/physreva.64.063605

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Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional cos2\ensuremathφ\mathrm{cos}2\ensuremath{\varphi}cos2\ensuremathφ term in the potential. We also find that the inner product in this representation is nonlocal in \ensuremathφ.\ensuremath{\varphi}.\ensuremathφ. Our model exhibits phenomena, such as \ensuremathπ\ensuremath{\pi}\ensuremathπ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.
We investigate the fully quantum evolution of the population imbalance in a perfectly symmetric Bose-Josephson junction modeled by a two-mode Bose-Hubbard Hamiltonian, focusing on the validity of macroscopic quantum self-trapping beyond the mean-field theory. We show that for any finite number of particles, the exact quantum dynamics leads to the breakdown of macroscopic quantum self-trapping after a finite time, regardless of the initial state. Using the symmetries of the Bose-Hubbard Hamiltonian, we provide a mathematical demonstration of this result and analyze the spectral properties governing the dynamics. We identify a branching behavior in the eigenvalue differences and a nontrivial structure of the population-imbalance amplitudes. These features allow us to distinguish two clearly different dynamical regimes and to elucidate the mechanism leading to the emergence of a quasi–macroscopic quantum self-trapping regime for large particle numbers. These findings bridge the gap between mean-field predictions and exact quantum dynamics and provide insight into the emergence of classical nonlinear behavior from finite quantum many-body systems.
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.
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 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.
The quantum Josephson Hamiltonian of two weakly linked Bose-Einstein condensates is written in an overcomplete phase representation, thus avoiding the problem of defining a Hermitian phase operator. We discuss the limit of validity of the standard, non-rigorous Mathieu equation, due to the onset of a higher order cos⁡2ϕ\cos 2 ϕcos2ϕ term in the Josephson potential, and also to the overcompleteness of the representation (the phase ϕϕϕ being the relative phase between the two condensates). We thereby unify the Boson Hubbard and Quantum Phase models.
There exists an increasing evidence supporting the picture of the Josephson junction (JJ) as a "macroscopic quantum system". On the other hand the interpretation of experimental data strongly depends on the assumed theoretical model. We analyse the possible states of a Cooper pair box ("charge qubit") for the two types of models : two-mode Bose-Hubbard model with its large NNN aproximations and the many-body description within the mean-field approximation (Gross-Pitaevski equation). While the first class of models supports the picture of JJ being a quantum subsystem of a single degree of freedom, the second approach yields an essentially classical structure of accessible quantum states which, in particular, implies the absence of entanglement for two coupled JJ's. The arguments in favor of the mean-field theory are presented and different experimental tests including a new proposal are briefly discussed.