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arXiv:gr-qc/9407039 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Generalized uncertainty relations and long-time limits for quantum Brownian motion models

Charis Anastopoulos, J. J. Halliwell

Published Jun 15, 1995• 72 citations
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Abstract

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${)}^{2}$(\ensuremath{\Delta}q${)}^{2}$ 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 ${\mathit{scrA}}^{2}$=U-${\mathit{C}}_{\mathit{p}\mathit{q}}^{2}$, where ${\mathit{C}}_{\mathit{p}\mathit{q}}$=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.

DOI
10.1103/physrevd.51.6870

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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−−−isessentiallytheareaenclosedbythe1−\ensuremathσcontouroftheWignerfunction.)Inbothcasestheminimizinginitialstateisacorrelatedcoherentstate(anonminimalGaussianpurestate),andinthefirstcasethelowerboundisonlyanenvelope.Thesegeneralizeduncertaintyrelationssupplyameasureofthecomparativesizeofquantumandthermalfluctuations.Weprovetwosimpleinequalities,relatinguncertaintytovonNeumannentropy,−Tr(\ensuremathρln\ensuremathρ),andthevonNeumannentropytolinearentropy,1−TrscrA--- 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-TrscrA−−−isessentiallytheareaenclosedbythe1−\ensuremathσcontouroftheWignerfunction.)Inbothcasestheminimizinginitialstateisacorrelatedcoherentstate(anonminimalGaussianpurestate),andinthefirstcasethelowerboundisonlyanenvelope.Thesegeneralizeduncertaintyrelationssupplyameasureofthecomparativesizeofquantumandthermalfluctuations.Weprovetwosimpleinequalities,relatinguncertaintytovonNeumannentropy,−Tr(\ensuremathρln\ensuremathρ),andthevonNeumannentropytolinearentropy,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 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.
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Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time ttt is connected with the reduced density at initial time t0t_0t0​ for $t_0 \leq t
We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength. As an illustration of these new results we solve the corresponding master equation and calculate, among other physical quantities, the uncertainty function whose late time behavior can be obtained fully. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that any initial state always settles down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. For more general spectra we show that the solution of the master equation can be reduced to solving for the motion of a classical parametric oscillator with parametric frequency determined by the unsolved for master equation coefficients. States in these systems experience evolution that is parametrically similar to the simpler evolution explicitly determined for in the case of Laurent-series spectra.
In the framework of the Lindblad theory for open quantum systems, we derive closed analytical expressions of the Heisenberg and Schr\"odinger generalized uncertainty functions for a particle moving in a harmonic-oscillator potential. The particle is initially in an arbitrary correlated coherent state, and interacts with an environment at finite temperature. We describe how the quantum and thermal fluctuations contribute to the uncertainties in the canonical variables of the system and analyze the relative importance of these fluctuations in the evolution of the system. We show that upon contact with the bath, the system evolves from a quantum-dominated to a thermal-dominated state in a time that is of the same order as the decoherence time calculated in other models in the context of transitions from quantum to classical physics.