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The Theory of Open Quantum Systems$
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Heinz-Peter Breuer and Francesco Petruccione

Print publication date: 2007

Print ISBN-13: 9780199213900

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199213900.001.0001

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Stochastic Dynamics In Hilbert Space

Stochastic Dynamics In Hilbert Space

Chapter:
(p.303) 6 STOCHASTIC DYNAMICS IN HILBERT SPACE
Source:
The Theory of Open Quantum Systems
Author(s):

Heinz-Peter Breuer

Francesco Petruccione

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199213900.003.06

This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.

Keywords:   dynamical semigroups, piecewise deterministic processes, Hilbert space path, integral representation, continuous measurements, direct photodetection, homodyne photodetection, heterodyne photodetection, stochastic Schrödinger equation

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