Open Quantum Systems II: The Markovian Approach
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.
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adjoint assume Attal bounded operators Brownian motion classical cocycle compact completely positive map computation condition consider constant convergence deﬁned Deﬁnition denote dilations domain elements entropy production ergodic evolution exists Fagnola finite ﬁrst ﬂows Fock space Gaussian given Hamiltonian Hilbert space inequality interactions isometry Ito formula Ito integral Lemma Liapunov function Markov process martingale Math Neumann algebra norm obtain operator process operators on h Parthasarathy probabilistic interpretations process xt Proof Proposition prove Pt(x quantum dynamical semigroup quantum Markov semigroup quantum noises quantum stochastic calculus quantum stochastic differential quantum stochastic integrals random variable representation reservoir resp satisﬁes satisfy separable Hilbert space solution space h spectral stationary distribution stochastic calculus stochastic differential equations stochastic integrals stochastic process subspace tensor product Theorem theory toy Fock space unique unitary vectors von Neumann algebra