Quantum-state transformation by dispersive and absorbing four-port devices

Abstract

The recently derived input-output relations for radiation at a dispersive and absorbing four-port device [T. Gruner and D.-G. Welsch, Phys. Rev. A 54, 1661 (1996)] are used to derive the unitary transformation that relates the output quantum state to the input quantum state, including radiation and matter and without placing frequency restrictions. It is shown that for each frequency component the transformation can be regarded as a U(4) group transformation, which can directly be calculated from the underlying complex refractive-index profile of the device without additional postulates. If for narrow-bandwidth radiation far from the medium resonances the absorption matrix of the four-port device can be disregarded, the well-known U(2) group transformation for a lossless device is recognized. Explicit formulas for the transformation of Fock states, coherent states, and phase-space functions are given.