Photon counting probabilities of the output field for a single-photon input

TL;DR

This paper derives photon counting statistics for a single-photon interacting with a quantum system, providing analytical formulas for output field probabilities, quantum trajectories, and detection times, enhancing understanding of quantum measurement processes.

Contribution

It introduces new analytical expressions for photon counting probabilities, quantum trajectories, and detection times in single-photon quantum interactions, using quantum filtering theory.

Findings

Analytical formulas for output photon counting probabilities.

Quantum trajectories for continuous measurement are explicitly derived.

Probability densities for detection times are provided for arbitrary photon profiles.

Abstract

We derive photon counting statistics for an output field of a single-photon wave packet interacting with a quantum system (e.g. a quantum harmonic oscillator or a two-level atom). We determine the exclusive probability densities for the output field by making use of quantum filtering theory. The quantum trajectories for continuous in time measurements of the output field (reflected and transmitted), are determined starting from a collision model and difference filtering equations. We provide analytical formulae for quantum trajectories associated with a two-dimensional stochastic counting process, we describe their structure, and give a physical interpretation to them. Moreover, we provide analytical expressions for probability densities for the times of successive photon detections for a single-photon field scattered on a two-level atom, for an arbitrary photon profile and any initial state of the atom.