Configuration-Space Photon Number Operators in Quantum Optics

Abstract

Some properties of the operator ${\hat{n}}_{V,t}$ representing the number of photons localized in a finite volume $V$ at time $t$ are investigated. To some extent these properties reflect the well-known difficulty of localizing photons in space-time. However, it is shown that, when the linear dimensions of $V$ are large compared with the wave-length of any occupied mode of the field, the ${\hat{n}}_{V,t}$ operator acquires some simple properties. The commutator of ${\hat{n}}_{V,t}$ and the detection operator $\hat{\mathrm{A}}(\mathrm{x}, t)$ is expressible in an interesting form. The commutator and relations derived from it become particularly simple for certain space-time regions which we label conjoint and disjoint. An orthogonal set of eigenstates of ${\hat{n}}_{V,t}$ is found, together with the corresponding eigenvalues, and it is shown that an arbitrary state is expressible in terms of these eigenstates. Some $N\mathrm{th}$-order correlations of the ${\hat{n}}_{V,t}$ operators are evaluated, and the results are used to calculate the probability distribution of eigenvalues of ${\hat{n}}_{V,t}$ for an arbitrary state of the field.