关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stochastic differential equations and diffusion processes

Probability and Measure

Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. An overview is given of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations, and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. The physical interpretation and derivation of the various approaches are emphasized, how they are connected is explored, and how different methods may be suitable for solving different problems is examined.

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Dynamics of non-Markovian open quantum systems

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Stochastic Differential Equations And Applications

Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.

Measurements continuous in time and a posteriori states in quantum mechanics

I General theory.- The Stochastic Schr#x00F6 dinger Equation.- The Stochastic Master Equation: Part I.- Continuous Measurements and Instruments.- The Stochastic Master Equation: Part II.- Mutual Entropies and Information Gain in Quantum Continuous Measurements.- II Physical applications.- Quantum Optical Systems.- A Two-Level Atom: General Setup.- A Two-Level Atom: Heterodyne and Homodyne Spectra.- Feedback.

Quantum Trajectories and Measurements in Continuous Time

Starting from the idea of generalized observables, related to effect-valued measures, as introduced by Ludwig, some examples oi continual observations in quantum mechanics are discussed. A functional probability distribution, on the set of the trajectories which are obtained as output of the continual observation, is constructed in the form of a Feynman integral. Interesting connections with the theory of dynamical semi-groups are pointed out. The examples refer to small systems, but they are interesting for the light they may shed on the problem of the connections between the quantum and the macroscopic levels of description for a large body; the idea of continuous trajectories indeed seems to be essential for the macroscopic level of description.

A model for the macroscopic description and continual observations in quantum mechanics

Continuous (in time) measurements can be introduced in quantum mechanics by using operation-valued measures and quantum stochastic calculus. In this paper quantum stochastic calculus is used for showing the connections between measurement theory and open-system theory. In particular, it is shown how continuous measurements are strictly related to the concept of output channels, introduced in the framework of quantum stochastic differential equations by Gardiner and Collet.

Measurement theory and stochastic differential equations in quantum mechanics.

Quantum stochastic calculus is an operator analogue of classical ItG's stochastic calculus which was originally developed for treating quantum noise. However, it turned out to be also useful in other cases of open systems, as in the case of measurement theory in quantum mechanics and in the treatment of quantum input and output channels. In the present paper, this calculus is used for developing a theory of direct, homodyne and heterodyne detection in quantum optics.

http://iopscience.iop.org/article/10.1088/0954-8998/2/6/002/pdf

Alberto Barchielli

Papers

Measurements continuous in time and a posteriori states in quantum mechanics

Quantum Trajectories and Measurements in Continuous Time

A model for the macroscopic description and continual observations in quantum mechanics

Measurement theory and stochastic differential equations in quantum mechanics.

Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics