Axiomatic construction of quantum Langevin equations

Abstract

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation-dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a markovian semi-classical approach, due to Bedeaux and Mazur, from a non-markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation-dissipation theorem is seen to be incompatible with a markovian dynamics. Possible applications to the quantum spherical model are discussed.

1 + 33 pages, 4 figures. Major revision. J. Stat. Mech at press

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• Fokker-Planck equations for a trapped particle in a quantum-thermal Ohmic bath: general theory and applications to Josephson junctions
• Continuous spontaneous localization as the white-noise limit of spontaneous unitarity violation
• Perturbation approach in Heisenberg equations for lasers
• Non-equilibrium dynamics of the open quantum $O(n)$-model with non-Markovian noise: exact results
• A Resolution of the Ito-Stratonovich Debate in Quantum Stochastic Processes