# The time-dependent spectrum of a single photon and its POVM

## TL;DR

This paper develops a formalism to describe the quantum measurement process of a single photon's time-dependent spectrum using POVMs, exploring implications for heralding photons and observing interference effects.

## Contribution

It constructs the POVM for time-dependent spectral measurements of single photons, including special cases and applications like heralding and interference phenomena.

## Key findings

- Constructed POVM for frequency-filtered photon detection
- Analyzed the limit where filter bandwidth and timing resolution satisfy cc 1
- Identified a Hong-Ou-Mandel type interference effect with frequency-filtered photons

## Abstract

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 $\Gamma$), and then record at what time (with some finite precision $\Delta t$, and with some finite efficiency $\eta$) and after passing what filter the photon is detected. What is the POVM (Positive-Operator Valued Measure, 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 $\Gamma\Delta t\ll 1$ (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.