Interacting Quantum and Classical Continuous Systems II. Asymptotic Behavior of the Quantum Subsystem

In the framework of event-enhanced quantum theory the dynamical equation for the reduced density matrix of a quantum system interacting with a continuous classical system is derived. The asymptotic behavior of the corresponding dynamical semigroup is discussed. The example of a quantum–classical coupling on Lobatchevski space is presented.     

The low density limit for anN-level system interacting with a free bose or fermi gas

It is proved that the reduced dynamics of anN-level system coupled to a free quantum gas converges to a quantum dynamical semigroup in the low density limit. The proof uses a perturbation series of the quantum BBGKY-hierarchy, and the analysis of this series is based on scattering theory. The limiting semigroup contains the full scattering cross section, but it does not depend on the statistics of the reservoir. The dynamics of the semigroup is discussed.     

Environment-Induced Superselection Rules in Markovian Regime

A generalization of the Jacobs–deLeeuw–Glicksberg splitting for a quantum dynamical semigroup is introduced. This generalization, called the isometric-sweeping decomposition, is used to derive effective superselection rules in the quantum system. Received: 1 February 1999 / Accepted: 2 June 1999     

Markov dilations and quantum detailed balance

We construct quantum stochastic processes whose multi-time correlation functions, with suitable time ordering, can be obtained from a quantum dynamical semigroup. We prove that such a process defines a stationary Markov dilation of the associated semigroup if and only if (up to technicalities) the semigroup satisfies the quantum detailed balance condition with respect to its stationary state.     

Nonlinear quantum dynamical semigroups for many-body open systems

The notion of a nonlinear quantum dynamical semigroup is introduced, and the existence and uniqueness of solutions of the corresponding nonlinear evolution equations are studied in a more abstract framework. The construction of nonlinear quantum dynamical semigroups is carried out for two different mean-field models. First a mean-field coupling between a system of noninteracting subsystems and the bath is investigated. As examples, a nonlinear frictional Schrödinger equation and a model for a quantum Boltzmann equation are discussed. Second, a many-body system with mean-field interaction coupled to a bath is considered. Here, again, the form of the generator is derived; however, it cannot be obtained rigorously, except for some particular examples. Finally, the quantum Ising-Weiss model is briefly studied.     

Dynamical violation of the superposition principle for open quantum systems

It is argued that the impossibility of observing coherent superpositions of certain macroscopically distinguishable quantum states is a combined effect of collective dissipation processes generated by an interaction of the N-particle system with external quantum fields and a “coarse-grained” character of real measurements. A simple model involving a quantum dynamical semigroup is discussed.     

On the mixing-enhancing dynamical maps and mixing-enhancing evolution of a quantum system

The mixing-enhancing (in the sense of Uhlmann) dynamical maps and dynamical evolution is studied. We give a necessary and sufficient condition for a dynamical map (and dynamical evolution) of a quantum system to be mixing-enhancing. In the case of a finite- dimensional Hilbert space this condition is equivalent to the condition that the dynamical map (dynamical evolution) preserve the most mixed state and the von Neumann entropy be non- decreasing. It is proved that, in contrast with the finite-dimensional case, increasing of the von Neumann entropy under a dynamical map (for any initial state) does not imply that the dynamical map is mixing-enhancing. We also give a necessary and sufficient condition for an infinitesimal generator of a norm-continuous dynamical semigroup to be mixing-enhancing.     

On the asymptotic behavior of dynamical maps for a finite quantum system

We give some remarks on the dynamical evolution (also nonlinear) of finite quantum system. We are interested int-asymptotic behavior of density matrices in the Liouville space formalism and we show that for nonlinear dynamical semigroups, as well as for the dynamical maps that do not form semigroups, the stationary time evolution may be attained for finite time in contrast to the motion generated by the linear dynamical semigroup. Recently the problem of constructing a nonlinear analog of quantum mechanics with nonlinear wave equation playing the role of the Schrödinger equation has been investigated by some authors; see for example Mielnik (1974), Bergmann (1968). Our work is related to this investigation and gives a characteristic feature of the nonlinear time evolution.     

Some remarks on dynamical semigroups

We deal with time evolution of a finite quantum system given by a dynamical semigroup Λ_t. For the semigroup we define and give some properties of the convex Λ_t-invariant subset of states “pathological” in some aspect evolving in strictly reversible manner independently of the stochastic surroundings of the system.     

Extension of quantum dynamical semigroups

We discuss the problem of extending a quantum dynamical semigroup to a group.     

Properties of quantum Markovian master equations

In this paper we give an essentially self-contained account of some general structural properties of the dynamics of quantum open Markovian systems. We review some recent results regarding the problem of the classification of quantum Markovian master equations and the limiting conditions under which the dynamical evolution of a quantum open system obeys an exact semigroup law (weak coupling limit and singular coupling limit). We discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.     

Classical Properties of Infinite Quantum Open Systems

Long time asymptotic properties of a class of environmentally induced dynamical semigroups on arbitrary von Neumann algebras are discussed. Such a semigroup selects observables, called effective observables, which are immune to the process of decoherence and so evolve in a reversible automorphic way. In particular, it is shown that effective observables of the quantum system in the thermodynamic limit, subjected to a specific interaction with another quantum system, obey classical dynamics.This work was supported by the KBN research grant no 5P03B 081 21     

Quantum dynamical semigroups and approach to equilibrium

For a quantum dynamical semigroup possessing a faithful normal stationary state, some conditions are discussed, which ensure the uniqueness of the equilibrium state and/or the approach to equilibrium for arbitrary initial condition.A fellowship from the Italian Ministry of Public Education is acknowledged.     

On the generators of quantum dynamical semigroups

The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB() is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.     

Quantum Evolution beyond the Markovian Semigroup — Generalizing the Stenholm–Barnett Approach

We provide conditions for the memory kernel governing the time-nonlocal quantum master equation which guarantee that the corresponding dynamical map is completely positive and trace-preserving. This approach gives rise to the new parametrization of dynamical maps in terms of two completely positive maps – so-called legitimate pair. In fact, these new parameterizations are a natural generalization of Markovian semigroup. Interestingly our class contains recently studied models like semi-Markov evolution and collision models.     

Irreversible evolution in quantum logics

The notion of dynamical semigroup is introduced in the quantum logic scheme on the set of the states. Under suitable nonempty mathematical assumptions it is shown that a Heisenberg picture exists equivalent to the Schrödinger one and having many aspects similar to those of the Hilbert case.     

Stochastic evolution of Yang-Mills connections on the noncommutative two-torus

We study quantum stochastic parallel transport processes where the noise terms arise from quantum Brownian motion in Fock space and the connection is chosen to minimize the Yang-Mills functional on a Heisenberg module over the smooth algebra of the noncommutative two-torus. Each such process yields a dilation of a quantum dynamical semigroup whose action on components of the connection induces a family of transformations of the moduli space. From a physical point of view, this describes a highly singular interaction between quantized Yang-Mills fields and the free boson field.