## Open Quantum Systems: An IntroductionIn this volume the fundamental theory of open quantum systems is revised in the light of modern developments in the field. A unified approach to the quantum evolution of open systems is presented by merging concepts and methods traditionally employed by different communities, such as quantum optics, condensed matter, chemical physics and mathematical physics. The mathematical structure and the general properties of the dynamical maps underlying open system dynamics are explained in detail. The microscopic derivation of dynamical equations, including both Markovian and non-Markovian evolutions, is also discussed. Because of the step-by-step explanations, this work is a useful reference to novices in this field. However, experienced researches can also benefit from the presentation of recent results. |

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### Contents

1 | |

2 Time Evolution in Closed Quantum Systems | 15 |

3 Time Evolution in Open Quantum Systems | 18 |

4 Quantum Markov Process Mathematical Structure | 33 |

5 Microscopic Description Markovian Case | 49 |

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assume Banach space Benatti Berlin Breuer commutes completely positive map completely positive semigroup contraction semigroup convergent correlation functions define Definition density matrix differential equation differential problem dynamics of open E.C.G. Sudarshan eigenoperator eigenvalue environment eventually contractive evolution family evolution operator exponential finite dimensional given initial condition integral integro-differential interaction picture kernel Kossakowski Lett linear operator Markov process Markovian master equation Math mathematical method non-Markovian obtain one-parameter semigroup open quantum systems pA(t partial trace perturbative positive operator positive semidefinite Proof properties pure result Rivas and S. F. S. F. Huelga Schr鐰inger equation Schr鐰inger picture Sect secular approximation self-adjoint operators singular coupling limit solution spectral decomposition Spohn Springer SpringerBriefs in Physics steady subsystem term Theorem theory time-dependent To(t Tr(P trace preserving unitary operator weak coupling limit