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 photodetection. 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.