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Journal of Physics A Mathematical and Theoretical

A simple derivation of decoherence energies

G. Morchio

Published Jan 17, 2011• 0 citations
Publisher

Abstract

A long calculation on the basis of a general inequality by Gaveau and Schulman on decoherence energies is reduced to the application of few basic observations on product structures in a class of time evolutions of product states.
DOI
10.1088/1751-8113/44/6/068001

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The act of measurement on a quantum state is supposed to collapse the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schr\odinger's equation. There are two general approaches to the study of wave-function collapse: one called the consistent or decoherent histories approach and the other, the environmental decoherence approach, which studies the effect of the environment upon the quantum system, to explain wave-function collapse. In the environmental decoherence approach, one usually studies a Markovian-approximated Master equation to study the time-evolution of reduced density matrix and obtains the long-term dependence of the off-diagonal elements of this matrix. We do not make a Markovian assumption and study a particularly simple and calculable example. We find, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential, which is a new result (the long-term behavior is, of course, still exponential). This allows one to connect the Fermi-golden rule quadratic-in-time behavior of a transition probability to the exponential long-time behavior of a collapsing wave-function.
The self-consistent propagation of generalized D1 [coherent-product] states and of a class of Gaussian density-matrix generalizations is examined, at both zero and finite temperature, for arbitrary interactions between the localized lattice (electronic or vibronic) excitations and the phonon modes. It is shown that in all legitimate cases, the evolution of D1 states reduces to the disentangled evolution of the component D2 states. The self-consistency conditions for the latter amount to conditions for decoherence-free propagation, which complement the D2 Davydov soliton equations in such a way as to lift the nonlinearity of the evolution for the on-site degrees of freedom. Although it cannot support Davydov solitons, the coherent-product ansatz does provide a wide class of exact density-matrix solutions for the joint evolution of the lattice and phonon bath in compatible systems. Included are solutions for initial states given as a product of a [largely arbitrary] lattice state and a thermal equilibrium state of the phonons. It is also shown that external pumping can produce self-consistent Frohlich-like effects. A few sample cases of coherent, albeit not solitonic, propagation are briefly discussed.