A phase operator for photons

TL;DR

This paper introduces a unitary phase operator for photons in a single momentum mode, enabling accurate phase measurements that incorporate polarization effects and satisfy canonical commutation relations.

Contribution

It defines a new unitary phase operator for photons that acts as a true ladder operator and satisfies canonical commutation relations, advancing quantum optics theory.

Findings

Operator is unitary and acts as a true ladder operator

Satisfies the canonical commutation relation [n, φ] = i

Eigenstates of the operator are explicitly found

Abstract

We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, ${\hat{E}} = e^{i \hat{\phi}}$, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation $[{\hat{E}}, {\hat{n}}] = {\hat{E}}$; consequently, the phase $\hat{\phi}$ satisfies the canonical commutation relation $[{\hat{n}}, {\hat{\phi}}] = i$. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.