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Physical Review A

Implications of the intrinsic decoherence in quantum mechanics to nonequilibrium statistical mechanics

A. K. Rajagopal

Published Aug 1, 1996• 5 citations
Publisher

Abstract

The discrete-time jump model for intrinsic decoherence in quantum mechanics proposed recently by Milburn is here generalized to explicitly time-dependent Hamiltonians. In this model, a time-dependent entropy is obtained and thus it admits of an interpretation that this is a model for an open system. Unlike in the Milburn model, we introduce a minimum uncertainty phase change dictated by the minimum ``time-energy uncertainty product'' to define the shortest time scale in the system. Implications of this model to quantum nanometric devices such as quantum dots or resonant tunneling devices are indicated. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1124

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The discrete-time jump model for intrinsic decoherence in quantum mechanics proposed recently by Milburn is here generalized to explicitly time-dependent Hamiltonians. In this model, a time-dependent entropy is obtained and thus it admits of an interpretation that this is a model for an open system. Unlike in the Milburn model, we introduce a minimum uncertainty phase change dictated by the minimum ``time-energy uncertainty product'' to define the shortest time scale in the system. Implications of this model to quantum nanometric devices such as quantum dots or resonant tunneling devices are indicated. \textcopyright{} 1996 The American Physical Society.
Four models of energy decoherence are discussed from the common perspective of intrinsic time-uncertainty. The four authors -- Milburn, Adler, Penrose, and myself -- have four different approaches. The present work concentrates on their common divisors at the level of the proposed equations rather than at the level of the interpretations. General relationships between time-uncertainty and energy-decoherence are presented in both global and local sense. Global and local master equations are derived. (The local concept is favored.)
Four models of energy decoherence are discussed from the common perspective of intrinsic time-uncertainty. The four authors - Milburn, Adler, Penrose, and myself - have four different approaches. The present work concentrates on their common divisors at the level of the proposed equations rather than at the level of the interpretations. General relationships between time-uncertainty and energy-decoherence are presented in both global and local sense. Global and local master equations are derived. (The local concept is favored.)
A model for intrinsic decoherence in quantum mechanics is proposed, based on a simple modification of unitary Schr\"odinger evolution. On sufficiently small time scales the system evolves by a random sequence of unitary phase changes generated by the Hamiltonian. The Schr\"odinger equation is obtained to zeroth order in the expansion parameter. Higher-order corrections lead to a loss of coherence in the energy basis. The rate of coherence loss becomes very large as the energy scale of the system is increased. The expansion parameter determines an uncertainty in the time step on very short times scales. A number of testable consequences are derived including anomalous dispersion of a free particle, decay of oscillatory systems, destruction of interference-fringe visibility, and a phase shift of interference fringes.
We propose a novel approach to intrinsic decoherence without adding new assumptions to standard Quantum Mechanics. We generalize the Liouville equation just by requiring the dynamical semigroup property of time evolution and dropping the unitarity requirement. With no approximations and specific statistical assumptions we find a generalized Liouville equation which depends on two characteristic time t1 and t2 and reduces to the usual equation in the limit t1 = t2 --> 0. However, for t1 and t2 arbitrarily small but finite, our equation can be written as a finite difference equation which predicts state reduction to the diagonal form in the energy representation. The rate of decoherence becomes faster at the macroscopic limit as the energy scale of the system increases. In our approach the evolution time appears, a posteriori, as a statistical variable as if time evolution would take place randomly at average intervals t2, each evolution having a time width t1. A generalized Tam Mandelstam inequality is derived. The relation with previous work by Milburn is discussed. The agreement with recent experiments on damped Rabi oscillations is described.