  For equilibrium systems, the magnitude of thermal fluctuations is closely
linked to the dissipative response to external perturbations. This
fluctuation-dissipation relation has been described for material particles in a
wide range of fields. Here we experimentally probe the relation between the
number fluctuations and the response function for a Bose-Einstein condensate of
photons coupled to a dye reservoir, demonstrating the fluctuation-dissipation
relation for a quantum gas of light. The observed agreement of the scale factor
with the environment temperature both directly confirms the thermal nature of
the optical condensate and demonstrates the validity of the
fluctuation-dissipation theorem for a Bose-Einstein condensate.
