Correction to the quantum phase operator for photons

Abstract

The vector potential operator, $\hat{\mathbf{A}}$, is transformed and rewritten in terms of cosine and sine functions in order to get a clear picture of how the photon states relate to the $\mathbf{A}$ field. The phase operator, defined by $\hat{E}=\exp (-i\hat{\varphi })$, is derived from this picture. The result has a close resemblance with the known Susskind-Glogower (SG) operator, which is given by ${\hat{E}}_{SG}=({\hat{a}}_{\mathbf{k}}{\hat{a}}_{\mathbf{k}}^{†}{)}^{-1/2}{\hat{a}}_{\mathbf{k}}$. It will be shown that ${\hat{a}}_{\mathbf{k}}$ should be replaced by $({\hat{a}}_{\mathbf{k}}+{\hat{a}}_{-\mathbf{k}}^{†})$ instead to yield $\hat{E}=(({\hat{a}}_{\mathbf{k}}+{\hat{a}}_{-\mathbf{k}}^{†})({\hat{a}}_{\mathbf{k}}^{†}+{\hat{a}}_{-\mathbf{k}}){)}^{-1/2}({\hat{a}}_{\mathbf{k}}+{\hat{a}}_{-\mathbf{k}}^{†})$, which makes the operator unitary. $\hat{E}$ will also be analyzed when restricted to the space of only forward moving photons with wave vector $\mathbf{k}$. The resulting phase operator, ${\hat{E}}_{+}$, will turn out to resemble the SG operator as well, but with a small correction: Whereas $E_{SG}$ can be equivalently written as ${\hat{E}}_{SG}={\sum }_{n=0}^{\infty }|n⟩⟨n+1|$, the operator, ${\hat{E}}_{+}$, is instead given by ${\hat{E}}_{+}={\sum }_{n=0}^{\infty }{a}_n|n⟩⟨n+1|$, where $a_n=(n+1/2)!/(n!\sqrt{n+1})$. The sequence, $(a_n{)}_{n\in \{0,1,2,\dots \}}$, converges to $1$ from below for $n$ going to infinity.

10 pages, 2 figures, second version