Threshold detection statistics of bosonic states

TL;DR

This paper introduces new matrix functions, the Bristolian and loop Torontonian, to accurately compute measurement probabilities for threshold detectors in quantum photonics, enabling better analysis of bosonic states.

Contribution

It develops the Bristolian and loop Torontonian functions, filling a gap in calculating threshold detection probabilities for various quantum states of light.

Findings

Introduces the Bristolian and loop Torontonian functions.

Provides a unified framework for bosonic measurement statistics.

Enhances tools for photonic quantum technology analysis.

Abstract

In quantum photonics, threshold detectors, distinguishing between vacuum and one or more photons, such as superconducting nanowires and avalanche photodiodes, are routinely used to measure Fock and Gaussian states of light. Despite being the standard measurement scheme, there is no general closed form expression for measurement probabilities with threshold detectors, unless accepting coarse approximations or combinatorially scaling summations. Here, we present new matrix functions to fill this gap. We develop the Bristolian and the loop Torontonian functions for threshold detection of Fock and displaced Gaussian states, respectively, and connect them to each other and to existing matrix functions. By providing a unified picture of bosonic statistics for most quantum states of light, we provide novel tools for the design and analysis of photonic quantum technologies.