Photons in a Spherical Cavity

TL;DR

This paper resolves conflicting results on photon modes in a spherical cavity by deriving the allowed frequencies for electric and magnetic multipole photons, and explores the potential for various entangled photon states.

Contribution

It provides a detailed solution to the photon mode problem in a spherical cavity, distinguishing electric and magnetic multipole conditions, and analyzes bipartite photon entanglement possibilities.

Findings

Allowed frequencies differ for electric and magnetic multipole photons.

Identified 40 potential types of entangled photon states.

Developed second-quantized vector potential in spherical geometry.

Abstract

The iconic problem of photon modes in a spherical cavity has been discussed in the literature; however, conflicting results have been reported \cite{Heitler,Davydov_QuantumMechanics}. For this reason, the solution of this problem is worked out in detail here, starting with the Maxwell equations and applying boundary conditions at the surface of the bounding perfect conductor. Contrary to the treatments in the literature \cite{Heitler,Davydov_QuantumMechanics}, the allowed frequencies for photons in the sphere are given by two different conditions, one for electric and one for magnetic multipole photons. After establishing the modes and their allowed frequencies, we write down the second-quantized vector potential in the spherical geometry. Based on these spherical modes, bipartite photon entanglement is investigated showing that there are in-principle 40 different types of entangled photon states. Finally, we include some appendices about photon plane-wave and spherical-wave helicity states, helicity spherical harmonic vectors, and rotation of the helicity states and 3-d vectors using the Wigner $D$-matrix.