Lecture Notes in Mathematics: Open Quantum Systems II: The Markovian Approach (Volume 1881)

This volume is the second in a series of three volumes dedicated to the lecture notes of the summer school “Open Quantum Systems” which took place in the Institut Fourier in Grenoble, from June 16th to July 4th 2003. The contributions presented in thesevolumesarerevisedandexpandedversionsofthenotesprovidedtothestudents during the school. After the rst volume, developing the Hamiltonian approach of open quantum systems, this second volume is dedicated to the Markovian approach. The third volume presents both approaches, but at the recent research level. Open quantum systems A quantum open system is a quantum system which is interacting with another one. This is a general de nition, but in general, it is understood that one of the s- tems is rather “small” or “simple” compared to the other one which is supposed to be huge, to be the environment, a gas of particles, a beam of photons, a heat bath ... The aim of quantum open system theory is to study the behaviour of this coupled system and in particular the dissipation of the small system in favour of the large one.

Understanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. From a mathematical point of view, it involves a large body of knowledge. Significant progress in the understanding of such systems has been made during the last decade. These books present in a self-contained way the mathematical theories involved in the modeling of such phenomena. They describe physically relevant models, develop their mathematical analysis and derive their physical implications.

In Volume I the Hamiltonian description of quantum open systems is discussed. This includes an introduction to quantum statistical mechanics and its operator algebraic formulation, modular theory, spectral analysis and their applications to quantum dynamical systems.

Volume II is dedicated to the Markovian formalism of classical and quantum open systems. A complete exposition of noise theory,Markov processes and stochastic differential equations, both in the classical and the quantum context, is provided. These mathematical tools are put into perspective with physical motivations and applications.

Volume III is devoted to recent developments and applications. The topics discussed include the non-equilibrium properties of open quantum systems, the Fermi Golden Rule and weak coupling limit, quantum irreversibility and decoherence, qualitative behaviour of quantum Markov semigroups and continual quantum measurements.