The starting point is a given semigroup of completely positive maps on the 2 * 2 matrices. This semigroup describes the irreversible evolution of a decaying two-level atom. By using the integral-sum kernel approach to quantum stochastic calculus, the two-level atom is coupled to an environment, which in this case will be interpreted as the electromagnetic field. The irreversible time evolution of the two-level atom then stems from the reversible time evolution of the atom and the field together. Mathematically speaking, a Markov dilation of the semigroup has been constructed. The next step is to drive the atom by a laser and to count the photons emitted into the field by the decaying two-level atom. For every possible sequence of photon counts, a map is constructed that gives the time evolution of the two-level atom implied by that sequence. The family of maps obtained in this way forms a so-called Davies process. In this book, Davies describes the structure of these processes, which brings us into the field of quantum trajectories. Within the model presented in this paper, the jump operators are calculated and the resulting counting process is briefly described.
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