Open Quantum Systems I: The Hamiltonian Approach by Alain Joye (auth.), Stéphane Attal, Alain Joye, Claude-Alain

By Alain Joye (auth.), Stéphane Attal, Alain Joye, Claude-Alain Pillet (eds.)

Understanding dissipative dynamics of open quantum platforms is still a problem in mathematical physics. This challenge is suitable in a number of parts of primary and utilized physics. From a mathematical viewpoint, it includes a wide physique of data. major growth within the realizing of such structures has been made over the last decade. those books found in a self-contained means the mathematical theories considering the modeling of such phenomena. They describe bodily correct types, increase their mathematical research and derive their actual implications.

In quantity I the Hamiltonian description of quantum open platforms is mentioned. This contains an advent to quantum statistical mechanics and its operator algebraic formula, modular idea, spectral research and their functions to quantum dynamical systems.

Volume II is devoted to the Markovian formalism of classical and quantum open platforms. a whole exposition of noise conception, Markov methods and stochastic differential equations, either within the classical and the quantum context, is supplied. those mathematical instruments are positioned into viewpoint with actual motivations and applications.

Volume III is dedicated to contemporary advancements and functions. the subjects mentioned contain the non-equilibrium houses of open quantum structures, the Fermi Golden Rule and susceptible coupling restrict, quantum irreversibility and decoherence, qualitative behaviour of quantum Markov semigroups and continuous quantum measurements.

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We shall essentially only deal with systems governed by Hamiltonians that are time-independent. These evolution equations are also called canonical equations of motion. e. ∂ ∂ Q˙k = K(Q, P ), P˙k = − K(Q, P ), with (Q(0), P (0)) = (Q0 , P 0 ), ∂Pk ∂Qk (5) are called canonical transformations. The energy conservation property (4) is just a particular case of time dependence of a particular observable. Assuming the Hamiltonian is time independent, but not necessarily given by (2), the time evolution of any (smooth) observable B : Γ → R defined on phase space computed along a classical trajectory Bt (q, p) ≡ B(q(t), p(t)) is governed by the Liouville equation d Bt (q, p) = LH Bt (q, p), with B0 (q, p) = B(q, p), dt (6) where the linear operator LH acting on the vector space of observables is given by LH = ∇p H(q, p) · ∇q − ∇q H(q, p) · ∇p , (7) with the obvious notation.

1 Density Matrices . . . . . . . . . . . . . . . . . . . . . 54 Boltzmann Gibbs . . . . . . . . . . . . . . . . . . . . . . . 57 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2 3 This set of lectures is intended to provide a flavor of the physical ideas underlying some of the concepts of Quantum Statistical Mechanics that will be studied in this school devoted to Open Quantum Systems.

1) The space RdN of the coordinates (q1 , q2 , · · · , qN ), with qk,j ∈ R, j = 1 · · · , d which we shall sometimes denote collectively by q (and similarly for p), is called the configuration space and the space Γ = RdN × RdN = R2dN of the variables (q, p) is called the phase space. A point (q, p) in phase space characterizes the state of the system and the observables of the systems, which are the physical quantities one can measure on the system, are given by functions defined on the phase space.

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