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arXiv:0905.3908 · Physical Review A

Quantum linear Boltzmann equation with finite intercollision time

Lajos Diósi

Published Dec 17, 2009• 5 citations
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Abstract

Inconsistencies are pointed out in the usual quantum versions of the classical linear Boltzmann equation constructed for a quantized test particle in a gas. These are related to the incorrect formal treatment of momentum decoherence. We prove that ideal collisions with the molecules would result in complete momentum decoherence, the persistence of coherence is only due to the finite intercollision time. A corresponding quantum linear Boltzmann equation is proposed.

DOI
10.1103/physreva.80.064104

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Comment on “Quantum linear Boltzmann equation with finite intercollision time”

Klaus Hornberger, Bassano Vacchini

Inconsistencies are pointed out in a recent proposal [L.Diósi, Phys. Rev. A 80, 064104 (2009)] for a quantum version of the classical linear Boltzmann equation.

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Reply to “Comment on ‘Quantum linear Boltzmann equation with finite intercollision time’ ”

Lajos Diósi

Hornberger and Vacchini [Phys. Rev. A 82, 036101 (2010)] claim that the specific collisional momentum decoherence, pointed out in my recent work [Phys. Rev. A 80, 064104 (2009)], is already described by their theory. However, I have performed a calculation whereby I disprove the authors' claim and refute their conclusion that my recent work had no advantage over theirs.

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Inconsistencies are pointed out in the usual quantum versions of the classical linear Boltzmann equation constructed for a quantized test particle in a gas. These are related to the incorrect formal treatment of momentum decoherence. We prove that ideal collisions with the molecules would result in complete momentum decoherence, the persistence of coherence is only due to the finite intercollision time. A corresponding quantum linear Boltzmann equation is proposed.
Recently the general form of a translation-covariant quantum Boltzmann equation has been derived, which describes the dynamics of a tracer particle in a quantum gas. We develop a stochastic wave function algorithm that enables full three-dimensional Monte Carlo simulations of this equation. The simulation method is used to study the approach to equilibrium for various scattering cross sections and to determine dynamical deviations from Gaussian statistics through an investigation of higher-order cumulants. Moreover, we examine the loss of coherence of superpositions of momentum eigenstates and determine the corresponding decoherence time scales to quantify the transition from quantum to classical behavior of the state of the test particle.
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced by Hornberger [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a nonperturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum.
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.