Starting point is a given semigroup of completely positive maps on the 2 times 2 matrices. This semigroup describes the irreversible evolution of a decaying 2-level atom. Using the integral-sum kernel approach to quantum stochastic calculus we couple the 2-level atom to an environment, which in our case will be interpreted as the electromagnetic field. The irreversible time evolution of the 2-level atom then stems from the reversible time evolution of atom and field together. Mathematically speaking, we have constructed a Markov dilation of the semigroup. The next step is to drive the atom by a laser and to count the photons emitted into the field by the decaying 2-level atom. For every possible sequence of photon counts we construct a map that gives the time evolution of the 2-level atom inferred by that sequence. The family of maps that we obtain in this way forms a so-called Davies process. In his book Davies describes the structure of these processes, which brings us into the field of quantum trajectories. Within our model we calculate the jump operators and we briefly describe the resulting counting process.
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