Continuous-mode multiphoton filtering

The purpose of this paper is to derive filters for an arbitrary open quantum system driven by a light wave packet prepared in a continuous-mode multiphoton state. A continuous-mode multiphoton state is a state of a traveling light wave packet that contains a definite number of photons and is characterized by a temporal (or, equivalently, spectral) profile. After the interaction with the system, the outgoing light can be monitored by means of homodyne detection or photo-detection. Filters for both measurement schemes are derived in this paper. Unlike the vacuum or the coherent state case, the annihilation operator of the light field acting on a multiphoton state changes the state by annihilating a photon, and this makes the traditional filtering techniques inapplicable. To circumvent this difficulty, we adopt a non-Markovian embedding technique proposed in [J. E. Gough, M. R. James, and H. I. Nurdin, Quantum Inf. Process., 12 (2013), pp. 1469-1499] for the study of the single-photon filtering problem. However, the multiphoton nature of the problem addressed in this paper makes the study much more mathematically involved. Moreover, as demonstrated by an example -a two-level system driven by a continuous-mode two-photon state -multiphoton filters can reveal interesting strong nonlinear optical phenomena absent in both the single-photon state case and the continuous-mode Fock state case.