Static and dynamic properties of self-bound droplets of light in hot vapours

The propagation of light in nonlinear media is well described by a $2$D nonlinear Schr\"odinger equation (NLSE) within the paraxial approximation, which is equivalent to the Gross-Pitaesvskii equation (GPE), the mean-field description for the dynamics of Bose-Einstein condensates (BECs). Due to this similarity, many theoretical and experimental investigations of phenomena which have already been studied and realized in BECs have been recently analysed in alternative experimental platforms such as hot atomic vapours. In this work, we study the formation of droplets of light in these media, attempting to establish a mapping between the experimental parameters normally used in BEC experiments and those needed to observe the analogous phenomenon in hot atomic vapours. We obtain the energy functional for the susceptibility of the medium in the $\chi^{(3)}$ , $\chi^{(3)}+\chi^{(5)}$ and saturating regimes for a two-level atomic configuration considering the focusing (attractive) regime. We apply a Gaussian variational approach and check its predictions through numerical simulations of the NLSE for each regime. Finally, we study the real-time dynamics of the system for both the $\chi^{(3)}+\chi^{(5)}$ and saturating nonlinearities, focusing our attention on the behaviour of the breathing mode and on the analysis of droplet formation for realistic experimental conditions.