By Ángel Rivas, Susana F. Huelga

ISBN-10: 3642233538

ISBN-13: 9783642233531

ISBN-10: 3642233546

ISBN-13: 9783642233548

In this quantity the elemental thought of open quantum structures is revised within the gentle of recent advancements within the box. A unified method of the quantum evolution of open structures is gifted by way of merging strategies and strategies ordinarily hired by means of varied groups, comparable to quantum optics, condensed subject, chemical physics and mathematical physics.

The mathematical constitution and the final homes of the dynamical maps underlying open approach dynamics are defined intimately. The microscopic derivation of dynamical equations, together with either Markovian and non-Markovian evolutions, is additionally mentioned. end result of the step by step motives, this paintings is an invaluable connection with rookies during this box. despite the fact that, skilled researches may also enjoy the presentation of contemporary results.

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**Example text**

Of course some skepticism may arise thinking that if the system B is out of our control, there is not warranty that this assumption is fulfilled. 1, that the global system is in a product state at the initial time. These two assumptions make to vanish the second and the third term in Eq. 16) 0 where K(t, u) = PV(t)G(t, u)QV(u). If this kernel is homogeneous K(t, u) = K(t − u), which is the case for stationary states ρ B (as exemplified in next section), this equation can be formally solved by a Laplace transformation [4], but this usually turns out into an intractable problem in practice.

F. 1007/978-3-642-23354-8_4, © The Author(s) 2012 33 34 4 Quantum Markov Process: Mathematical Structure p(x, t; x , t ) = p(x, t|x , t ) p(x , t ), where p(x, t; x , t ) is the joint probability that random variable X takes value x at time t and x at time t . By integrating with respect to x we find the relation between unconditional probabilities p(x, t) = d x p(x, t|x , t ) p(x , t ), and let us write it as p(x, t) = d x K (x, t|x , t ) p(x , t ) with K (x, t|x , t ) = p(x, t|x , t ). 2) The Markov process is called homogeneous if K (x, t|x , t ) is only a function of the difference between the two time parameters involved K (x, t|x , t ) = K t−t (x|x ).

X0 , t0 ) = p(xn , tn |xn−1 ), ∀tn ∈ I. 1) This property is sometimes expressed in the following way: a Markov process does not have memory of the history of past values of X. They are so-named after the Russian mathematician A. Markov. Consider now this process on a continuous interval I, then tn−1 is time infinitesimally close to tn . So for any t > t just from the definition of conditional probability Á. Rivas and S. F. 1007/978-3-642-23354-8_4, © The Author(s) 2012 33 34 4 Quantum Markov Process: Mathematical Structure p(x, t; x , t ) = p(x, t|x , t ) p(x , t ), where p(x, t; x , t ) is the joint probability that random variable X takes value x at time t and x at time t .

### Open Quantum Systems: An Introduction by Ángel Rivas, Susana F. Huelga

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