Photon counting probabilities of the output field for a single-photon input

Abstract

In this paper, photon-counting statistics for an output field of a single-photon wave packet interacting with a quantum system was derived. The problem was formulated using a collision model with a bidirectional field approximated by two sequences of qubits. It was shown that the collision model formulated for the light with the temporal correlations allows deriving formulae for quantum trajectories in an effective and intuitive way. Starting from the discrete in-time evolution of the composed system, we determined the conditional and unconditional evolution of the quantum system, provided a physical interpretation of quantum trajectories, and determined the analytical formulae for the exclusive probability densities for the reflected and transmitted fields. Finally, our results were applied to a two-level atom, and we also determined analytical expressions for probability densities of arrival times of the successive counts.