By Heinz-Peter Breuer
The physics of open quantum platforms performs a huge position in glossy experiments and theoretical advancements of quantum mechanics. Written for graduate scholars and readers with learn pursuits in open structures, this publication presents an advent into the most principles and ideas, as well as constructing analytical tools and desktop simulation strategies.
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Extra resources for The Theory of Open Quantum Systems
75) corresponding to the initial value 4) 0 (x) = z. 76). 77) This equation tells us that the probability density for the process to reach the point x at time t, under the condition that it was in x' at time t', is different from zero if and only if the phase flow carries x' to x in the time interval from t' to t, that is, if and only if x = t _ t , (x`). 59). 77) satisfies the Chapman-Kolmogorov equation. Thus we have constructed a solution of the Chapman-Kolmogorov equation and defined a simple Markov process.
Essentially, such a process is obtained by combining a deterministic time-evolution with a jump process. Piecewise deterministic processes have a large variety of applications in the natural sciences and technology. A number of examples and a mathematically rigorous treatment of PDPs may be found in the book by Davis (1993). In the theory of open quantum systems PDPs provide the appropriate mathematical concept to describe the evolution of the state of a continuously monitored system (see Chapter 6).
It is defined by some initial density p(x,t 0 ) and by a propagator which describes a deterministic time evolution corresponding to a system of ordinary differential equations d — x(t) = g(x(t)), x(t) E Rd . 75) 20 CLASSICAL PROBABILITY THEORY AND STOCHASTIC PROCESSES Here, g(x) denotes a d-dimensional vector field. e. that the vector field g(x) does not depend explicitly on time, such that the resulting process becomes homogeneous (see below). 75) represents the Hamiltonian equations of motion in phase space.
The Theory of Open Quantum Systems by Heinz-Peter Breuer