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arXiv:0710.1344 · Journal of Mathematical Physics

Decoherence rates for Galilean covariant dynamics

Jeremy Clark

Published May 1, 2008• 2 citations
arXivHTMLPDF

Abstract

We introduce a measure of decoherence for a class of density operators. For Gaussian density operators in dimension one, it coincides with an index used by Morikawa [“Quantum decoherence and classical correlation in quantum mechanics,” Phys. Rev. D 42, 2929–2931 (1990)]. Spatial decoherence rates are derived for three large classes of the Galilean covariant quantum semigroups introduced by Holevo [“On conservativity of covariant dynamical semigroups,” Rep. Math. Phys. 33, 95–110 (1993)]. We also characterize the relaxation to a Gaussian state for these dynamics and give a theorem for the convergence of the Wigner function to the probability distribution of the classical analog of the process.

DOI
10.1063/1.2912232

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We introduce a measure of decoherence for a class of density operators. For Gaussian density operators in dimension one, it coincides with an index used by Morikawa [“Quantum decoherence and classical correlation in quantum mechanics,” Phys. Rev. D 42, 2929–2931 (1990)]. Spatial decoherence rates are derived for three large classes of the Galilean covariant quantum semigroups introduced by Holevo [“On conservativity of covariant dynamical semigroups,” Rep. Math. Phys. 33, 95–110 (1993)]. We also characterize the relaxation to a Gaussian state for these dynamics and give a theorem for the convergence of the Wigner function to the probability distribution of the classical analog of the process.
Translation-covariant Markovian master equations used in the description of decoherence and dissipation are considered in the general framework of Holevo's results on the characterization of generators of covariant quantum dynamical semigroups. A general connection between the characteristic function of classical Lévy processes and loss of coherence of the statistical operator describing the center of mass degrees of freedom of a quantum system interacting through momentum transfer events with an environment is established. The relationship with both microphysical models and experimental realizations is considered, focusing in particular on recent interferometric experiments exploring the boundaries between classical and quantum world.
Translation-covariant Markovian master equations used in the description of decoherence and dissipation are considered in the general framework of Holevo’s results on the characterization of generators of covariant quantum dynamical semigroups. A general connection between the characteristic function of classical Lévy processes and loss of coherence of the statistical operator describing the center of mass degrees of freedom of a quantum system interacting through momentum transfer events with an environment is established. The relationship with both microphysical models and experimental realizations is considered, focusing in particular on recent interferometric experiments exploring the boundaries between classical and quantum world.
Abstract Relaxation rates define an important characteristics of quantum dynamics. They determine how quickly a quantum system interacting with an environment thermalizes, decoheres, and dissipates. Moreover, relaxation rates are also often directly accessible through experimental measurements. An analysis of relaxation rates of Markovian semigroups covariant w.r.t. maximally commuting subgroup of the unitary group is provided. It is shown by a direct proof that these rates satisfy the recently proved a universal constraint that each single rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. Furthermore, we introduce longitudinal and transversal relaxation rates for a general d -level system and derive nontrivial relations between them in the covariant case. These relations generalize the well-known inequality 2Γ T ≥ Γ L between the longitudinal relaxation rate Γ L and the transversal relaxation rate Γ T for two-level systems. Interestingly, for d ≥ 3, our generalization turns out to be much tighter than those valid for general quantum dynamical semigroups, thereby providing a potential means for the experimental verification of covariant generators.
Several quantum open-system models have been examined recently that dynamically reduce the off-diagonal terms of the density operator in the position representation by incorporating the effects of interaction with external degrees of freedom. A recent model [M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38 (1990)] includes explicit dependence on the spatial correlations of the decoherence rate of the off-diagonal elements (in the position representation) of the density operator. In this paper, we use influence-functional techniques to examine the effect of an external potential that randomly varies as function of position and time, which leads to a natural characterization of the decay rate of the coherence between spatially separated states in terms of the spatial correlations of the potential, and the corresponding fluctuating forces. We obtain the effective master equations for short environment relaxation time scales, and discuss the similarities and differences between the effects of random external potentials versus the effects of interactions with external degrees of freedom, such as that recently investigated by Stern, Aharonov, and Imry [Phys. Rev. A 41, 3436 (1990)].
We characterize general non-Markovian Gaussian maps which are covariant under Galilean transformations. In particular, we consider translational and Galilean covariant maps and show that they reduce to the known Holevo result in the Markovian limit. We apply the results to discuss measures of macroscopicity based on classicalization maps, specifically addressing dissipation, Galilean covariance and non-Markovianity. We further suggest a possible generalization of the macroscopicity measure defined by Nimmrichter and Hornberger [Phys. Rev. Lett. 110, 16 (2013)PRLTAO0031-9007].
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 TTT 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).