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Quantum trajectories for a system interacting with environment in N-photon state

We derive stochastic master equation for a quantum system interacting with an environment prepared in a continuous-mode $N$-photon state. To determine the conditional evolution of the quantum system depending on continuous in time measurements of the output field the model of repeated interactions and measurements is applied. The environment is defined as an infinite chain of harmonic oscillators which do not interact between themselves and they are prepared initially in an entangled state being a discrete analogue of a continuous-mode $N$-photon state. We provide not only the quantum trajectories but also the analytical formulae for the whole statistics of the output photons and the solution to the master equation. The solution in the continuous case is represented by a simple diagrammatic technique with very transparent "Feynmann rules". This technique considerably simplifies the structure of the solution and enables one to find physical interpretation for the solution in terms of a few elementary processes.