Fluctuation-Dissipation relation of a grand-canonical photon Bose-Einstein condensate

In thermal equilibrium, fluctuations of the particle number are related to the isothermal compressibility and the thermal energy k_BT. This relation has been observed in a variety of systems ranging from Brownian particles to quantum gases, but not yet in Bose-Einstein condensates despite these systems are intrinsically in thermal equilibrium. Here, we experimentally investigate the fluctuation-dissipation relation in a Bose-Einstein condensate of photons realized in a dye-microcavity. The photon gas is coupled to a reservoir of dye molecular excitations, which serve both as a heat bath and a particle reservoir. Our measurements demonstrate a good agreement with the expected ratio of statistical number fluctuations and compressibility for a thermalized photon gas system following the flucutation-dissipation theorem.