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arXiv:quant-ph/9503008 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum state diffusion, density matrix diagonalization, and decoherent histories: A model

J. J. Halliwell, Andreas Zoupas

Published Dec 15, 1995• 127 citations
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Abstract

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.

DOI
10.1103/physrevd.52.7294

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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.
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ(L2(R+)⊗(Cn⊕Cn)) and the Hilbert space L2(μ), where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B(t),t≥0}, we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.
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).
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.
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.
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
In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system (S∪E)(S\cup E)(S∪E) is isolated from the rest of the universe. So we have two independent Hamiltonians H^S\hat{H}_SH^S​, H^E\hat{H}_EH^E​ and an interaction V^\hat{V}V^ between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction V^\hat{V}V^ with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time t0=0t_0=0t0​=0, with a Gaussian kernel, for 0≤t<tc0 \le t < t_c0≤t<tc​ where tc∈]0,∞]t_c\in ]0, \infty ]tc​∈]0,∞] is a critical time such that reversibility is lost for t>tct > t_ct>tc​ if tc<+∞t_c<+\infty tc​<+∞. Some manifestations of decoherence, for t>0t>0t>0 small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.