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

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.

expanded version, 8 pages

References in corpus (3)

• Heralded Generation of Ultrafast Single Photons in Pure Quantum States
• Limits on the heralding efficiencies and spectral purities of spectrally-filtered single photons from photon pair sources
• Single-photon Spectra in Quantum Optomechanics

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