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arXiv:cond-mat/9404044 · Physical Review Letters

Dynamics by White-Noise Hamiltonians

Werner Fischer, Hajo Leschke, Peter Müller

Published Sep 19, 1994• 17 citations
arXivHTMLPDF

Abstract

A new class of random quantum-dynamical systems in continuous space is introduced. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time independent, and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is $\ensuremath{\delta}$ correlated in time and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both averaged states and observables are obtained. If the deterministic part is that of a particle subject to a constant magnetic field, spatial variance of the averaged state grows diffusively for long times independent of the initial state.

DOI
10.1103/physrevlett.73.1578

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A new class of random quantum-dynamical systems in continuous space is introduced. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time independent, and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is \ensuremathδ\ensuremath{\delta}\ensuremathδ correlated in time and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both averaged states and observables are obtained. If the deterministic part is that of a particle subject to a constant magnetic field, spatial variance of the averaged state grows diffusively for long times independent of the initial state.
Exact results are derived on the averaged dynamics of a class of random quantum-dynamical systems in continuous space. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time-independent and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time, but smoothly correlated in position and momentum. The averaged dynamics of the resulting white-noise system is shown to be a monotone mixing increasing quantum-dynamical semigroup. Its generator is computed explicitly. Typically, in the course of time the mean energy of such a system grows linearly to infinity. In the second part of the paper an extended model is studied, which, in addition, accounts for dissipation by coupling the white-noise system linearly to a quantum-mechanical harmonic heat bath. It is demonstrated that, under suitable assumptions on the spectral density of the heat bath, the mean energy then saturates for long times.
Abstract Exact results are derived on the averaged dynamics of a class of random quantum‐dynamical systems in continuous space. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time‐independent and quadratic, the Weyl‐Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time, but smoothly correlated in position and momentum. The averaged dynamics of the resulting white‐noise system is shown to be a monotone mixing increasing quantum‐dynamical semigroup. Its generator is computed explicitly. Typically, in the course of time the mean energy of such a system grows linearly to infinity. In the second part of the paper an extended model is studied, which, in addition, accounts for dissipation by coupling the white‐noise system linearly to a quantum‐mechanical harmonic heat bath. It is demonstrated that, under suitable assumptions on the spectral density of the heat bath, the mean energy then saturates for long times.
The stochastic dynamics of a quantum system driven by NNN statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with increasing NNN the system approaches a deterministic limit, indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite-dimensional quantum system which, according to an asymptotic analysis, decreases at least as 1/N1/N1/N. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite NNN the distance (according to the Frobenius norm) between the averaged evolution unitary operator generated by the Hamiltonian HHH and the unitary evolution operator generated by the averaged Hamiltonian $\ensuremath{\langle}H\ensuremath{\rangle}$ scales as 1/N1/N1/N. Numerical simulations enlarge this result to a broader class of noncommuting protocols.
This thesis studies randomness in quantum dissipative and chaotic dynamics. We first focus on a detailed study of the effect of noise in the Hamiltonian of a quantum system, going beyond the noise-average and characterizing higher moments. The main quantity we introduce is the Stochastic Operator Variance (SOV), which is an observable characterizing the spread of trajectories around the average evolution. Interestingly, this quantity fulfills different types of uncertainty relations and is related to quantum information scrambling through out-of-time-order correlators (OTOC). We illustrate the SOV-OTOC connection in a stochastic version of the Lipkin-Meshkov-Glick (LMG) model, which shows a positive Lyapunov exponent from an unstable saddle point. We find that under the action of the noise, this Lyapunov exponent can change sign, thus stabilizing the unstable phase of the model while destabilizing the stable one. We then study the interplay between noise and decay in a non-Hermitian Hamiltonian. The noise-average evolution follows an antidephasing master equation beyond Lindblad form. We characterize the purity dynamics and the steady states of this master equation, and study this new evolution in a stochastically driven version of the Dissipative Qubit. By characterizing its spectral and steady state properties, we find that there are three phases: the PT unbroken, PT broken, and a novel Noise Induced (NI) phase where the qubit converges to the lossy state. We further investigate the validity of our model to explain experimental data, such as the residual damping rate of the PT unbroken phase. In the last Chapter, we study randomness as a model for chaotic dynamics. In particular, leveraging a Wigner-like surmise for the k-th neighbor level spacing distribution, we compute analytically the k-th neighbor Spectral Form Factor (knSFF), which characterizes the contribution of the k-th neighbor spacings to the Spectral Form Factor (SFF). We study the properties of the individual knSFF and characterize their role in building the universal ramp of the SFF. Interestingly, we find that the very short-range and very long-range spacings are the ones that contribute the most to the extent of the ramp. We finish by discussing our results and possible pathways for further research.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.
For a special class of potentials, the dynamical evolution of any quantum wavepacket is entirely determined by the laws of classical mechanics. Here, the properties of this class are investigated both from the viewpoint of the Ehrenfest theorem (which provides the evolution of the average position and momentum), and the Wigner representation (which expresses quantum mechanics in a phase space formalism). Finally, these results are extended to the case of a charged particle in a uniform magnetic field.