Accurate Number Densities of Ideal Photons in a One-Dimensional Barrel Cavity

Our experimental scheme is based on a barrel optical microcavity filled with a dye solution. It is found that the number of non-condensed photons is characterized by an analytical function, which involves a q-digamma function in mathematics. We employ the q-digamma function to calculate the spatial and momentum distributions of ideal photons in a one-dimensional barrel cavity. The first main finding in this paper is that the spatial and momentum distributions possess a similar profile. The second main finding is that when photons are in the normal state, the density profile exhibits Friedel oscillations. The third main finding is that when photons are in the BEC state, the density profile exhibits a sharp peak with extremely narrow width. The fourth main finding is that the central peak of the density distribution is a monotonically increasing function of the photon number N but is a monotonically decreasing function of the temperature T.