Photon position measure

TL;DR

This paper derives the POVM for photon counting arrays, showing they measure photon flux density over pixels, and discusses the implications for photon localization and quantum measurement theory.

Contribution

The paper provides a theoretical derivation of the photon counting array POVM and clarifies its relation to photon position measurement in quantum optics.

Findings

POVM equals photon flux density integrated over pixel area and time

Photon counting arrays measure a coarse-grained photon position

Collapse occurs to vacuum, not localized photon state

Abstract

The positive operator valued measure (POVM) for a photon counting array detector is derived and found to equal photon flux density integrated over pixel area and measurement time. Since photon flux density equals number density multiplied by the speed of light, this justifies theoretically the observation that a photon counting array provides a coarse grained measurement of photon position. The POVM obtained here can be written as a set of projectors onto a basis of localized states, consistent with the description of photon position in a recent quantum imaging proposal [M. Tsang, Phys. Rev. Lett. \textbf{102}, 253601 (2009)]. The wave function that describes a photon counting experiment is the projection of the photon state vector onto this localized basis. Collapse is to the electromagnetic vacuum and not to a localized state, thus violating the text book rules of quantum mechanics but compatible with the theory of generalized observables and the nonlocalizability of an incoming photon.