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Theoretical and Mathematical Physics

Theory of stochastic processes in quantum dynamical systems

Yu. N. Barabanenkov, Vladimir D. Ozrin, A. V. Shelest

Published Feb 1, 1980• 1 citation
Publisher
DOI
10.1007/bf01032118

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In this paper we first introduce the Fock-Guichardet formalism for the quantum stochastic integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures of the QS integration over a space-time. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative is proved. Finally, Q-adapted dynamics is discussed, including Bosonic Q=1, Fermionic Q=-1, and monotone Q=0 quantum dynamics. These may be of particular interest to quantum field theory, quantum open systems, and quantum theory of stochastic processes.
Stochastic theories have been used for a long time to describe relaxation as well as line-shape functions. Since the pioneering work of Kubo, these theories have gained considerable interest in the application to optical processes. While previous theories focused on stochastic descriptions of the relaxation, our emphasis is on the dynamics induced by stochastic interactions in a system undergoing relaxation and dephasing processes, including pure dephasing. All the theories based either on the Hilbert-space representation, or on the Liouvillian-space representation, but requiring spectral decomposition methods, are unable to describe these effects simultaneously. In this paper, the implications of the statistical properties of the stochastic variables on the dynamics are elucidated, and the general time evolution of a stochastic N-level system is established in the particular Markovian or weakly colored noise cases.
Stochastic processes are shown to emerge from the time evolution of complex quantum systems. Using parametric, banded random matrix ensembles to describe a quantum chaotic environment, we show that the dynamical evolution of a particle coupled to such environments displays a variety of stochastic behaviors, ranging from turbulent diffusion to Lévy processes and Brownian motion. Dissipation and diffusion emerge naturally in the stochastic interpretation of the dynamics. This approach provides a derivation of a fractional kinetic theory in the classical limit and leads to classical Lévy dynamics.