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Annalen der Physik

On the averaged quantum dynamics by white‐noise hamiltonians with and without dissipation

Werner Fischer, Hajo Leschke, Peter Mülle

Published Jan 1, 1998• 9 citations
Publisher

Abstract

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.

DOI
10.1002/andp.19985100203

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On the averaged quantum dynamics by white-noise hamiltonians with and without dissipation

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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.
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.
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.
: The Brownian dynamics of the density operator for a quantum system interacting with a classical heat bath is described using a stochastic, non-linear Liouville equation obtained from a variational principle. The rate of the noise-induced transitions is expressed as a function of the environmental spectral density, and is discussed for the case of white noise and blackbody radiation. The time-scale of decoherence and dissipation is obtained numerically for a system of two quantum states. These are the ground and first excited state of the center of mass vibrations for a single ion confined in a harmonic trap. PACS numbers: 42.50.Lc,05.40.+j,32.80.Pj 1 Introduction The evolution of a quantum system surrounded by a thermal environment represents a fundamental problem of the quantum theory, increasingly important for applications. In the classical mechanics, a particle interacting with a heat bath has a Brownian evolution towards thermalization, which reduces to a purely dissipative moti...
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.
We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.
We study a classical system-bath model in which a system particle is linearly coupled to a bath of harmonic oscillators. In a system subject to white and correlated Gaussian noises, we derive an expression for the joint probability density, and the mean squared values of the system-bath particles are calculated. For white noise, the mean squared values of the system-bath particles exhibit an anomalous time dependence, which is different from that of normal diffusion. In particular, for a correlated Gaussian noise, the mean squared displacement and mean squared velocity of the bath particle show superspreading growths of t5t^5t5 and t3t^3t3 in t≪τt{\ll}τt≪τ, respectively. When τ=0τ=0τ=0, the mean squared velocity of a quantum particle under random noise is proportional to ttt, while the mean squared velocity of a bath particle in the presence of correlated Gaussian noise increases in proportion to t2t^2t2. This anomalous transport phenomenon results from the mixed derivative structure of the master equation, which couples with the transport coordinates in the diffusion dynamics of the relative coordinates. This result shows that the removal of dissipation in the Caldeira-Leggett framework leads to fundamentally different transport mechanisms characterized by non-diffusive quantum diffusion.