Quantum-state input-output relations for absorbing cavities

The quantized electromagnetic field inside and outside an absorbing high-$Q$ cavity is studied, with special emphasis on the absorption losses in the coupling mirror and their influence on the outgoing field. Generalized operator input-output relations are derived, which are used to calculate the Wigner function of the outgoing field. To illustrate the theory, the preparation of the outgoing field in a Schr\"{o}dinger cat-like state is discussed.