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Physical Review A

Quantum localization and the rate of exploration of phase space

Eric J. Heller

Published Feb 1, 1987• 187 citations
Publisher

Abstract

Nonstationary phase-space distributions \ensuremath{\rho}(t) are shown to sweep out phase space at a rate scrR which can be determined from the autocorrelation function, P(t)=${h}^{N}$Tr[\ensuremath{\rho}\ensuremath{\rho}(t)]. It is demonstrated that, after the initial decay of P(t), subsequent recurrences (increases) of P(t) cause permanent slowdowns in the rate scrR. Such slowdowns may cause localization of quantum eigenstates, whereas no such implications apply to the classical stationary distributions. The localization is discussed in terms of the fraction of the available phase space which the dynamics accesses, and the phase-space distribution of individual eigenfunctions.

DOI
10.1103/physreva.35.1360

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Nonstationary phase-space distributions \ensuremath{\rho}(t) are shown to sweep out phase space at a rate scrR which can be determined from the autocorrelation function, P(t)=hN{h}^{N}hNTr[\ensuremath{\rho}\ensuremath{\rho}(t)]. It is demonstrated that, after the initial decay of P(t), subsequent recurrences (increases) of P(t) cause permanent slowdowns in the rate scrR. Such slowdowns may cause localization of quantum eigenstates, whereas no such implications apply to the classical stationary distributions. The localization is discussed in terms of the fraction of the available phase space which the dynamics accesses, and the phase-space distribution of individual eigenfunctions.
The rate and extent of the exploration of the available phase space of a bound quantum mechanical system are shown to depend on the repulsion of energy eigenstates. Central to the argument is the Fourier transform relating the survival probability (in time) of an initially prepared nonstationary state and the (frequency) autocorrelation function of the excitation spectrum. Strong repulsion of states, as in the Wigner surmise, leads to a rapid dephasing of the initially coherently prepared state. The rate and extent of sampling of phase space depend not only on the spacing distribution but also on the intensity fluctuations. The rate of dephasing is equal to that inferred from the width of the spectral autocorrelation function.
In a wide-open quantum system, the effect of the Hamiltonian is negligible by comparison with the effect of the environment. For open systems, this is the opposite limit to closed or isolated systems. The quantum state diffusion model provides equations for the localization or reduction of quantum states of wide-open systems. The ensemble localization of an operator is defined, and it is proved that the rate of selflocalization of a selfadjoint environment operator towards one of its eigenstates is no less than 2. A bound is also obtained for the rate of selflocalization of some non-selfadjoint operators, which localize to minimum indeterminacy wave packets. The theory is presented for quasiclassical systems. For a sufficient number of independent environment operators, the states localize asymptotically to wave packets with Heisenberg indeterminacy products close to the minimum, which look to classical eyes like phase space points. To zeroth order in h(cross), the time-dependent WKB theory of quantum state diffusion due to a single operator shows localization or reduction within and between fixed classical sheets or Lagrangian manifolds. To first order, the sheets themselves diffuse. The rate of localization in an ensemble is determined by commutation terms with either sign and by correlation terms which always increase the localization. For the quasiclassical case the latter dominate, and this leads to a purely classical theory of localization, with a picture based on the diffusion of phase space densities. This means that state diffusion dynamics, like Hamiltonian dynamics, has a purely classical form, in which Planck's constant plays no role.
This work presents the complete dynamical formulation of the potential layerwithin the Theory of Potential framework. The potential layer is described by a density operator ρ_P(t) acting on aHilbert space of realizable configurations. We derive a generalized masterequation governing the evolution of ρ_P(t), combining: • unitary evolution generated by the potential Hamiltonian H_P,• local Lindblad-type decoherence generating effectively classical branches,• a deterministic ontological reduction term selecting the realized sector. We further show that the full dynamics arises from a variational principle inoperator space, based on the functionalF = E_P − Θ S_P − Λ[…],linking potential-layer energy, entropy, and constraints from geometry and therealized layer R(t). The work derives the continuity equation for the Q-density field Q(x,t),identifies entropy production and exchange terms (σ_loc, σ_red), and analyzesthe characteristic timescales of decoherence and reduction. This paper establishes the dynamical core of the potential layer as aself-contained component of the Theory of Potential and provides thefoundation for subsequent developments involving realization dynamics,geometric coupling, and cosmology.
The recent pioneering experiments of the [Nature 412, 52 (2001)] and [Science, 293, 274 (2001)] groups have demonstrated the dynamical tunneling of cold atoms interacting with standing electromagnetic waves. It has been shown [Phys. Rev. Lett. 89, 253201 (2002)], that the tunneling oscillations observed in these experiments correspondingly stems from two- and three-Floquet quantum state mechanism and can be controlled by varying the experimental parameters. The question where are the fingerprints of the classical chaotic dynamics in a quantum dynamical process which is controlled by 2 or 3 quantum states remains open. Our calculations show that although the effective $\ensuremath{\hbar}$ associated with the two experiments is large, and the quantum system is far from its semiclassical limit, the quantum Floquet-Bloch quasienergy states still can be classified as regular and chaotic states. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. It is also shown that as the wave packet which is initially localized at one of the two inner regular islands oscillates between them through the chaotic sea, it accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum, ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩, as measured in both experiments. The extremely high sensitivity of the rate of decay of the oscillations of ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩ to the very small changes in the population of different Floquet-Bloch states, is another type of fingerprint of chaos in quantum dynamics that presumably was measured in the NIST and AUSTIN experiments for the first time.