### Topical School

## Open quantum Systems

## Nancy, October 17th, 18th and 19th, 2007

#### Speakers

Stéphane Attal (Lyon) |
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Quantum Open Systems and Quantum Noises (1) |

In this course we present the Markovian approach for quantum open systems. In this approach we concentrate on the effective dissipative dynamics of a small quantum system in contact with a huge environment which is usually unknown. From this effective dynamics one can construct an environment and an explicit interaction system-environment which give rise to the observed restricted dynamics. This "dilation" of the small system is obtained with the help of quantum noises, by solving an adequate quantum Langevin equation. In this course we shall introduce the notions of quantum noises, quantum Langevin equation and show how this dilation can be explicitely realized. Our approach is based on a very simple physical model: repeated quantum interactions. This is an Hamiltonian model of a simple quantum system interacting repeatedly with some identical pieces of some environment. We shall show that quantum noises are spontaneously emerging from this Hamiltonian model. By the way this will give us a very intuitive and simple approach of what quantum noises are exactly. For example, we will understand why there are exactly 3 quantum noises, no more, no less. The three courses should be more or less divided as follows. 1st course: 1) Open Quantum System (density matrices, completely positive maps) 2) Repeated Quantum Interaction (physical model, mathematical setup, dilation of CP maps) |

Quantum Open Systems and Quantum Noises (2) |

2nd course: 3) Quantum Probability (observables, laws, connection with classical random variables, quantum Bernoulli) 4) Discrete Time Quantum Noises (universality, probabilistic interpretations, repeated quantum interactions and random walks) |

Quantum Open Systems and Quantum Noises (3) |

3rd course: 5) Continuous Spin Chain (heuristics, basic vectors, basic operators, the 3 quantum noises) 6) Quantum Langevin Equations (definition, unitarity, dilation of CP semigroups) 7) Approximation by the Spin Chain (embedding and approximation, convergence theorem) |

Wojciech de Roeck (Leuven) |

Fluctuations in nonequilibrium quantum systems (1) |

We review some elements of recent interest in nonequilibrium statistical mechanics; Gallavotti-Cohen fluctuation theorem and Jarzynsky relation, connection with Green-Kubo,... We discuss the relevance of large deviations in thermodynamics; equivalend of ensembles, variational principles |

Fluctuations in nonequilibrium quantum systems (2) |

In quantum mechanics, large deviations are still a field in construction. We point out what is the problem and the difference with classical mechanics. We also give some results that can be achieved alike in classical and quantum mechanics; e.g. H-theorems. |

Fluctuations in nonequilibrium quantum systems (3) |

We focus on the thermodynamics of quantum master equations. We discuss unravelings and fluctuation theorem. We also comment on how these master equation results emerge from a more fundamental microscopic description. |

Ernesto Medina (Caracas) |

Two-electron-entanglement enhancement by an inelastic scattering process |

In order to assess inelastic effects on two fermion entanglement production, we address an exactly solvable two-particle scattering problem where the target is an excitable scatterer. Useful entanglement, as measured by the two particle concurrence, is obtained from post-selection of oppositely scattered particle states. The S matrix formalism is generalized in order to address nonunitary evolution in the propagating channels. We find the striking result that inelasticity can actually increase concurrence as compared to the elastic case by increasing the uncertainty of the single particle subspace. Concurrence zeros are controlled by either single particle resonance energies or total reflection conditions that ascertain precisely one of the electron states. Concurrence minima also occur and are controlled by entangled resonance situations where the electron becomes entangled with the scatterer, and thus does not give up full information of its state. In this model, exciting the scatterer can never fully destroy phase coherence due to an intrinsic limit to the probability of inelastic events. |

Francesco Petruccione (Durban) |

Quantum decoherence (1) |

Decoherence denotes the environment-induced, dynamical destruction of quantum coherence. It leads to a dynamical selection of a distinguished set of pure states of the open system and counteracts the superposition principle in the Hilbert space of the open system. We will start with a general discussion of the dynamical structure which leads to ideal environment-induced decoherence, that is to the destruction of quantum coherence without damping. Basic concepts, such as dynamical selection of a preferred basis, decoherence time, and the emergence of coherent subspaces will be illustrated with the help of an analytically solvable example. |

Quantum decoherence (2) |

In the second part we will analyse and discuss the various fundamental physics mechanisms that lead to the space localization of composite quantum objects. In particular we will discuss high-temperature Brownian motion, decoherence through excitation and deexcitation of internal degrees of freedom and decoherence through scattering of particles. |

Quantum decoherence (3) |

In the last part, we will discuss the interplay between between decoherence and dissipation. If time permits, we will treat decoherence in the Caldeira-Leggett model. We will conclude, with a discussion of the role of decoherence in the theory of quantum measurement. |

Gunter Schütz (Jülich) |

Large scale dynamics of many-body quantum systems |

A fundamental question of statistical mechanics is how macroscopic equations of motion (such as e.g. the Navier-Stokes equation) arise from the microscopic interactions between the particles in the underlying many-body systems. There is a considerable body of work on classical systems, particularly stochastic interacting particles systems, that illuminates this phenomenon of emergence for many fundamentally important systems, but very little has been achieved for systems with genuine quantum mechanical dynamics, i.e., where no classical description is known or possible on intermediate scales. Taking the quantum XX-spin chain as an example for an open system kept at low temperatures we propose a hydrodynamic limit for the large-scale dynamics of the conserved local order parameter. We verify this description by investigating the same system in contact not with a surrounding heat bath, but with an infinite quantum XX-reservoir at its boundaries. |

#### Organizing comittee

Bertrand Berche |

Dragi Karevski |

#### Our partners

UFA - DFH | Nancy Université - UHP |