Theory of multiplexed photon number discrimination

TL;DR

This paper derives an analytical formula for the probabilities in multiplexed photon number detection, enabling more accurate photon number retrodiction considering efficiency and noise, advancing quantum optics measurement techniques.

Contribution

It provides the first analytical expression for detection probabilities with arbitrary input photons and detectors, including efficiency and false counts, improving photon number retrodiction methods.

Findings

Derived analytical detection probability formula for multiplexed detectors.

Enabled unbiased photon number retrodiction considering efficiency and noise.

Demonstrated examples illustrating the method's application.

Abstract

Although some non-trivial photon number resolving detectors exist, it may still be convenient to discriminate photon number states with the method of multiplexed detection. Multiplexing can be performed with paths in real space, with paths in time, and in principle with any degree of freedom that has a sufficient number of eigenstates and that can be coupled to the photon number. Previous works have addressed the probabilities involved in these measurements with Monte Carlo simulations, or by restricting the number of detectors to powers of 2, or without including quantum efficiency or noise. In this work we find an analytical expression of the detection probabilities for any number of input photons and any number of on/off photon detectors with a quantum efficiency $0\%\leq\eta\leq100\%$ and a false count probability $\varepsilon\geq0$. This allows us to retrodict the number of photons that we had at the input in the least unbiased way possible. We conclude our work with some examples.