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arXiv:0807.1777 · Physical Review Letters

Mean-Field Dynamics of a Non-Hermitian Bose-Hubbard Dimer

Eva-Maria Graefe, H. J. Korsch, Astrid Elisa Niederle

Published Oct 9, 2008• 215 citations
arXivHTMLPDF

Abstract

We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.

DOI
10.1103/physrevlett.101.150408

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We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.
We investigate the many-particle and mean-field correspondence for a non-Hermitian NNN-particle Bose-Hubbard dimer where a complex on-site energy describes an effective decay from one of the modes. Recently a generalized mean-field approximation for this non-Hermitian many-particle system yielding an alternative complex nonlinear Schr\"odinger equation was introduced. Here we give details of this mean-field approximation and show that the resulting dynamics can be expressed in a generalized canonical form that includes a metric gradient flow. The interplay of nonlinearity and non-Hermiticity introduces a qualitatively new behavior to the mean-field dynamics: The presence of the non-Hermiticity promotes the self-trapping transition, while damping the self-trapping oscillations, and the nonlinearity introduces a strong sensitivity to the initial conditions in the decay of the normalization. Here we present a complete characterization of the mean-field dynamics and the fixed point structure. We also investigate the full many-particle dynamics, which shows a rich variety of breakdown and revival as well as tunneling phenomena on top of the mean-field structure.
In the limit of large particle numbers and low densities systems of cold atoms can be effectively described as macroscopic single-particle systems in a mean-field approximation. In the case of a Bose–Hubbard system, modelling bosons on a discrete lattice with on-site interactions, this yields a discrete nonlinear Schrödinger equation of Gross–Pitaevskii type. It has been recently shown that the correspondence between the Gross–Pitaevskii equation and the Bose–Hubbard system breaks down for complex extensions. In particular, for a Bose–Hubbard dimer with complex on-site energy the mean-field approximation yields a generalized complex nonlinear Schrödinger equation. Conversely, a Gross–Pitaevskii equation with complex on-site energies arises as the mean-field approximation of many-particle Lindblad dynamics rather than a complex extension of the Bose–Hubbard system. Here we address the question of how the mean-field description is modified in the presence of a complex-valued particle interaction term for a Bose–Hubbard dimer. We derive the mean-field equations of motion leading to nonlinear dissipative Bloch dynamics, related to a nontrivial complex generalization of the nonlinear Schrödinger equation. The resulting dynamics are analysed in detail. It is shown that depending on the parameter values there can be up to six stationary states, and for small values of the interaction strength there are limit cycles. Furthermore, we show how a Gross–Pitaevskii equation with a complex interaction term can be derived as the mean-field approximation of a Bose–Hubbard dimer with an additional Lindblad term modelling two-particle losses.
We discuss the dynamics of an open two-mode Bose–Hubbard system subject to phase noise and particle dissipation. Starting from the full many-body dynamics described by a master equation the mean-field limit is derived resulting in an effective non-Hermitian (discrete) Gross–Pitaevskii equation which has been introduced only phenomenologically up to now. The familiar mean-field phase-space structure is substantially altered by the dissipation. In particular, the character of the fixed points shows an abrupt transition from elliptic or hyperbolic to attractive or repulsive, respectively. This reflects the metastable behaviour of the corresponding many-body system which surprisingly also leads to a significant increase of the purity of the condensate. A comparison of the mean-field approximation to simulations of the full master equation using the Monte Carlo wavefunction method shows an excellent agreement for wide parameter ranges.
We study the dynamical properties in a non-Hermitian Bose-Hubbard dimer and show that the steady state consists mostly of the eigenstate whose eigenvalue has the largest imaginary part. A sharp phase transition occurs in the steady state and the phase transition occurs for a finite particle number, not for infinite ones, in contrast to the Hermitian system. We also investigate the quantum Fisher information and entanglement in two different phases. The results show that the steady state in the Josephson oscillation regime is fully N -particle entangled and the corresponding parameter sensitivity approaches the Heisenberg limit. In the self-trapping regime, the parameter sensitivity just scales as the shot-noise limit. Moreover, quantum Fisher information of the steady state is robust to the initial state, which indicates that the non-Hermitian dynamics takes more advantage than the Hermitian one for quantum metrology.
The Bose-Hubbard system has been studied extensively both theoretically and experimentally, in particular in the context of ultracold atomic gases in optical lattices. Even in the two-mode case the many-particle dynamics display complex interference effects resulting in revival and breakdown phenomena as well as tunnelling. The most basic theoretical description is the mean-field approximation, which can be derived from a time-dependent variational principle assuming the many-particle wave function is an SU(2) coherent state. Here we build on this to construct a simple initial-value coherent state propagator, summing over mean-field trajectories and keeping track of their phases, given by the corresponding mean-field actions. This yields an approximation to the full time-dependent many-particle state, and is able to reproduce breakdown and revival dynamics. Applying a time-slicing procedure on top of this, we are able to accurately capture many-particle tunnelling effects. While in this paper we focus our analysis on the Bose-Hubbard dimer, the methods developed can be applied to more general SU(2) Hamiltonians, and can be extended to SU(M) systems.
Abstract This paper is a first step toward rigorously establishing the validity of mean-field approximations for the dynamics of quantum systems in a large dimensional limit. Our result is a derivation of the large d mean-field limit of the Bose–Hubbard model on a square lattice. The Bose–Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator phase transition. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions.