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Stochastic Dynamics of Reduced Wave Functions and Continuous Measurement in Quantum Optics
Author(s) -
Breuer HeinzPeter,
Petruccione Francesco
Publication year - 1997
Publication title -
fortschritte der physik/progress of physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.2190450104
Subject(s) - stochastic differential equation , mathematics , path integral formulation , quantum probability , hilbert space , stochastic process , continuous time stochastic process , stochastic partial differential equation , master equation , probability amplitude , povm , quantum process , rigged hilbert space , mathematical analysis , statistical physics , differential equation , quantum , quantum dynamics , quantum mechanics , reproducing kernel hilbert space , physics , statistics
The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman‐Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman‐Kolmogorov equation is constructed by means of the path integral representation.