Time-dependent spectrum of a single photon and its positive-operator-valued measure

Suppose we measure the time-dependent spectrum of a single photon. That is, we first send the photon through a set of frequency filters (which we assume to have different filter frequencies but the same finite bandwidth $\mathrm{\Gamma }$) and then record at what time (with some finite precision $\mathrm{\Delta }t$ and some finite efficiency $\eta$) and after passing what filter the photon is detected. What is the positive-operator-valued measure (POVM), the most general description of a quantum measurement, corresponding to such a measurement? We show how to construct the POVM in various cases, with special interest in the case $\mathrm{\Gamma }\mathrm{\Delta }t\ll 1$ (time-frequency uncertainty still holds, even in that limit). One application of the formalism is to heralding single photons. We also find a Hong-Ou-Mandel type of interference effect with two photons entering a frequency filter.

Figure 1

Modes used to describe spectral filtering. There is one set of input modes of interest, which contains the light we are filtering, described by annihilation operators $\widehat{a}\left(\omega \right)$. There is a second set of auxiliary input modes, which are assumed to be in the vacuum state, described by $\widehat{b}\left(\omega \right)$. There are two sets of output modes: The operators ${\widehat{a}}^{\prime }\left(\omega \right)$ describe light transmitted through the filter, and that is the light we are going to perform measurements on with the aim to learn about the light in the input modes $\widehat{a}\left(\omega \right)$. The other “reflected” modes, described by ${\widehat{b}}^{\prime }\left(\omega \right)$, are available for further filtering operations (at different frequencies).