On the Scattering Process in Quantum Optics

The derivation of a quantum Markovian model for an optomechanical system consisting of a quantum mechanical mirror interacting with quantum optical input fields via radiation pressure is a difficult problem which ultimately involves the scattering process of quantum stochastic calculus. We show that, while the scattering process may be approximated in a singular limit by regular processes using different schemes, the limit model is highly sensitive to the mathematical interpretation of the approximation scheme. We find two main types of stochastic limits of regular models, and illustrate the origin of this difference at the level of one-particle scattering. As an alternative modeling scheme, we consider models of mirrors as nontrivial dielectric media with boundaries that are themselves quantized. Rather than treating the plane waves for the electromagnetic field, we take the actual physical modes and quantize these. The input-output formalism is then obtained in the far zone where the plane-wave approximation is valid. Several examples are considered, and the quantum stochastic model is derived. We also consider the quantum trajectories problem for continual measurement of the reflected output fields, and derive the stochastic master equations for homodyning and photon-counting detection to estimate the mirror observables.