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arXiv:quant-ph/9608046 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Post-decoherence density matrix propagator for quantum Brownian motion

J. J. Halliwell, Andreas Zoupas

Published Apr 15, 1997• 36 citations
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Abstract

Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the so-called localization time $(\ensuremath{\Elzxh}/\ensuremath{\gamma}{kT)}^{1/2},$ where \ensuremath{\gamma} is the dissipation and $T$ the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time ${\ensuremath{\gamma}}^{\ensuremath{-}1}.$ We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a completely different method, a result we previously obtained using the quantum state diffusion picture [Phys. Rev. D 52, 7294 (1995)]. We also go beyond this earlier result, in that we derive an explicit expression for the weighting of each phase space localized state in the approximately diagonal density matrix, as a function of the initial state. For sufficiently long times it is equal to the Wigner function, and we confirm that the Wigner function is positive for times greater than the localization time (multiplied by a number of order 1).

DOI
10.1103/physrevd.55.4697

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Using the path integral representation of the density matrix propagator of quantum Brownian motion, we derive its asymptotic form for times greater than the so-called localization time (\ensuremath\Elzxh/\ensuremathγkT)1/2,(\ensuremath{\Elzxh}/\ensuremath{\gamma}{kT)}^{1/2},(\ensuremath\Elzxh/\ensuremathγkT)1/2, where \ensuremath{\gamma} is the dissipation and TTT the temperature of the thermal environment. The localization time is typically greater than the decoherence time, but much shorter than the relaxation time \ensuremathγ\ensuremath−1.{\ensuremath{\gamma}}^{\ensuremath{-}1}.\ensuremathγ\ensuremath−1. We use this result to show that the reduced density operator rapidly evolves into a state which is approximately diagonal in a set of generalized coherent states. We thus reproduce, using a completely different method, a result we previously obtained using the quantum state diffusion picture [Phys. Rev. D 52, 7294 (1995)]. We also go beyond this earlier result, in that we derive an explicit expression for the weighting of each phase space localized state in the approximately diagonal density matrix, as a function of the initial state. For sufficiently long times it is equal to the Wigner function, and we confirm that the Wigner function is positive for times greater than the localization time (multiplied by a number of order 1).
We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.
We study the time evolution of the reduced density operator for a class of quantum Brownian motion models consisting of a particle moving in a potential V(x) and coupled to an environment of harmonic oscillators in a thermal state. Our principal tool is the Wigner function of the reduced density operator and for linear systems we derive an explicit expression for the Wigner function propagator. We use it to derive two generalized uncertainty relations. The first consists of a sharp lower bound on the uncertainty function U=(\ensuremath{\Delta}p)2{)}^{2})2(\ensuremath{\Delta}q)2{)}^{2})2 after evolution for time t in the presence of an environment. The second, a stronger and simpler result, consists of a lower bound at time t on the quantity scrA2{\mathit{scrA}}^{2}scrA2=U-Cpq2{\mathit{C}}_{\mathit{p}\mathit{q}}^{2}Cpq2​, where Cpq{\mathit{C}}_{\mathit{p}\mathit{q}}Cpq​=1/2〈\ensuremath{\Delta}p^\ensuremath{\Delta}q^+\ensuremath{\Delta}q^\ensuremath{\Delta}p^〉. ($scrA--- is essentially the area enclosed by the 1-\ensuremath{\sigma} contour of the Wigner function.) In both cases the minimizing initial state is a correlated coherent state (a nonminimal Gaussian pure state), and in the first case the lower bound is only an envelope. These generalized uncertainty relations supply a measure of the comparative size of quantum and thermal fluctuations. We prove two simple inequalities, relating uncertainty to von Neumann entropy, -Tr(\ensuremath{\rho} ln\ensuremath{\rho}), and the von Neumann entropy to linear entropy, 1-Tr${\mathrm{\ensuremath{\rho}}}^{2}$. We also prove some results on the long-time limit of the Wigner function for arbitrary initial states. For the harmonic oscillator the Wigner function for all initial states becomes a Gaussian at large times (often, but not always, a thermal state). We derive the explicit forms of the long-time limit for the free particle (which does not in general go to a Gaussian), and also for more general potentials in the approximation of high temperature. We discuss connections with previous work by Hu and Zhang and by Paz and Zurek.
We analyze the evolution of a quantum Brownian particle starting from an initial state that contains correlations between this system and its environment. Using a path-integral approach, we obtain a master equation for the reduced density matrix of the system finding relatively simple expressions for its time-dependent coefficients. We examine the evolution of delocalized initial states (Schr\"odinger cats) investigating the effectiveness of the decoherence process. Analytic results are obtained for an Ohmic environment (Drude's model) at zero temperature.
We consider the decoherence of phase-space histories in quantum Brownian motion models, consisting of a particle moving under a potential V(x)V(x)V(x) in contact with a heat bath of temperature TTT and dissipation constant $\ensuremath{\gamma}$ in the Markovian regime. The phase-space histories are described by quasiprojectors consisting of Gaussian density matrices smeared over large phase-space cells and are characterized by the size [$\ensuremath{\Gamma}$] of the phase-space cell together with the size [MMM] of the margin (the region at the boundary of $\ensuremath{\Gamma}$ in which the Weyl symbol of the projector goes from 1 to 0). We generalize earlier results of Hagedorn to show that an initial Gaussian density matrix remains approximately Gaussian under nonunitary time evolution deriving a bound giving the validity of this approximation. Following the earlier work of Omnes [J. Stat. Phys. 51, 351 (1989)] we use this result to compute the time evolution of the phase-space projectors under the master equation and show that histories of phase-space samplings approximately decohere, and that the probabilities for these histories are peaked about classical dissipative evolution, but with an element of unpredictability which is reflected in the increase of the size of the margin.
We study entanglement dynamics in quantum Brownian motion (QBM) models. Our main tool is the Wigner function propagator. Time evolution in the Wigner picture is physically intuitive and it leads to a simple derivation of a master equation for any number of system harmonic oscillators and spectral density of the environment. It also provides generalized uncertainty relations, valid for any initial state, that allow a characterization of the environment in terms of the modifications it causes to the system's dynamics. In particular, the uncertainty relations are very informative about the entanglement dynamics of Gaussian states, and to a lesser extent for other families of states. For concreteness, we apply these techniques to a bipartite QBM model, describing the processes of entanglement creation, disentanglement, and decoherence at all temperatures and time scales.
A model that generalizes the study of quantum Brownian motion (BM) is constructed. We consider disordered environment that may be either static (quenched), noisy or dynamical. The Zwanzig-Caldeira-Leggett BM model formally constitutes a special case where the disorder autocorrelation length is taken to be infinite. Alternatively, a localization problem is obtained if the noise autocorrelation time is taken to be infinite. Also the general case of weak nonlinear coupling to a thermal, possibly chaotic bath is handled by the same formalism. A general, Feynman-Vernon type path-integral expression for the propagator is introduced. A Wigner transformed version of this expression is utilized in order to facilitate comparison with the classical limit. It is demonstrated that nonstochastic genuine quantal manifestations are associated with the model. It is clarified that such effects are absent in the standard BM model, either the disorder or the chaotic nature of the bath are essential. Quantal correction to the classical diffusive behavior is found even in the limit of high temperatures. The suppression of interference due to dephasing is discussed, leading to the observation that due to the disorder the decay of coherence is exponential in time, and no longer depends on geometrical considerations. Fascinating non-Markovian effects due to time-correlated (colored) noise are explored. For this, a strategy is developed in order to handle the integration over paths. This strategy is extended in order to demonstrate how localization comes out from the path-integral expression.