### Refine

Based on the Lindblad master equation approach we obtain a detailed microscopic model of photons in a dye-filled cavity, which features condensation of light. To this end we generalise a recent non-equilibrium approach of Kirton and Keeling such that the dye-mediated contribution to the photon-photon interaction in the light condensate is accessible due to an interplay of coherent and dissipative dynamics. We describe the steady-state properties of the system by analysing the resulting equations of motion of both photonic and matter degrees of freedom. In particular, we discuss the existence of two limiting cases for steady states: photon Bose-Einstein condensate and laser-like. In the former case, we determine the corresponding dimensionless photon-photon interaction strength by relying on realistic experimental data and find a good agreement with previous theoretical estimates. Furthermore, we investigate how the dimensionless interaction strength depends on the respective system parameters.

We investigate, both experimentally and theoretically, the static geometric properties of a harmonically trapped Bose–Einstein condensate of 6Li2 molecules in laser speckle potentials. Experimentally, we measure the in situ column density profiles and the corresponding transverse cloud widths over many laser speckle realizations. We compare the measured widths with a theory that is non-perturbative with respect to the disorder and includes quantum fluctuations. Importantly, for small disorder strengths we find quantitative agreement with the perturbative approach of Huang and Meng, which is based on Bogoliubov theory. For strong disorder our theory perfectly reproduces the geometric mean of the measured transverse widths. However, we also observe a systematic deviation of the individual measured widths from the theoretically predicted ones. In fact, the measured cloud aspect ratio monotonously decreases with increasing disorder strength, while the theory yields a constant ratio. We attribute this discrepancy to the utilized local density approximation, whose possible failure for strong disorder suggests a potential future improvement.