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Annales De L Institut Henri Poincare-physique Theorique

The classical limit of a class of quantum dynamical semigroups

Fabio Benatti

Published Jan 1, 1993• 0 citations
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Abstract

A class of quantum dynamical semigroups is proved to possess a residual phase-space stochasticity, if a weak-coupling limit is performed, whereas the usual limit in which h→0 is completely deterministic. As a consequence, solutions of the corresponding Chapman-Kolmogorov differential equation, with linear Liouville term, are constructed

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A note on a formula of the Lévy-Khinchin type in quantum probability

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Quantum Stochastic Processes

Luigi Accardi, Alberto Frigerio, John T. Lewis

A class of non-commutative stochastic processes is defined.These processes are defined up to equivalence by their multi-time correlation kernels.A reconstruction theorem, generalizing the Kolmogorov theorem for classical processes, is proved.Markov processes and their associated semigroups are studied, and some non-quasi free examples are constructed on the Clifford algebra, with the use of a perturbation theory of Markov processes.The connection with the Hepp-Lieb models is discussed.

Publications of the Research Institute for Mathematical Sciences•Apr 30, 1982•265 citations
A class of quantum dynamical semigroups is proved to possess a residual phase-space stochasticity, if a weak-coupling limit is performed, whereas the usual limit in which h→0 is completely deterministic. As a consequence, solutions of the corresponding Chapman-Kolmogorov differential equation, with linear Liouville term, are constructed
The classical limit of the infinitesimal generators of quantum dynamical semigroups obtained from quantum diffusions are shown to be second order semi-elliptic differential operators and thus (Markov) generators of classical diffusions in phase space. Necessary and sufficient conditions are established, in terms of the quantum generators, under which (a) the Markov generator is strictly elliptic, b) the classical diffusion is deterministic, and c) the classical flow is symplectic On montre que la limite classique du generateur infinitisimal d'un semigroupe dynamique quantique, obtenu a partir d'une diffusion quantique, et un operateur differentiel semi-elliptique d'ordre 2 et donc, le generateur (Markov) d'une diffusion classique dans l'espace de phase. On etablit des conditions necessaires et suffisantes pour que: a) le generateur soit (positivement) elliptique; b) la diffusion classique soit deterministe; c) le flot classique soit symplectique
In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to W∗W^*W∗-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deserves to be referred as "the" Quantum Fokker-Planck Equation. We then provide important examples of the limit dynamics, one of which constitutes a new Quantum generalization of the celebrated Fermi Golden Rule.
It has recently been shown [7, 8, 4] that Markov dilations of quantum dynamical semigroups may be constructed by solving corresponding quantum stochastic differential equations. These equations may be interpreted as describing the evolution of a quantum system with a singular coupling to a Boson reservoir, moreover, when solutions are combined with the free evolution of the reservoir, a reversible evolution results which may be interpreted as that of the system plus reservoir. It is the purpose of the present paper to characterize the above stochastic evolutions essentially by a condition of with respect to the free evolution.
We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one – a “reservoir”. The operator, which we call a “Friedrichs Hamiltonian”, has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before [AFL] in more complicated models and they are usually referred to as ”stochastic Limit”.
In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.