Abstract:
There will be a short introduction to many-body physics with light containing a presentation of the driven-dissipative Bose-Hubbard model. One of the main numerical challenges for the description of these photonic lattice systems will be highlighted: the exponentially large Hilbert space. 

In the first part a recently developed numerical approach will be presented which circumvents this problem by truncating the higher order many-body correlations. This method is then applied to the antibunching to bunching crossover in extended 2D lattice structures. These results are benchmarked with the corner-space renormalisation method. 

Then, in the second part, the physics of optical bistability for a single driven-dissipative nonlinear cavity in the quantum regime is revisited. The dynamical nature of the associated hysteresis is revealed with a rich double power law behaviour of the hysteresis area. 

Finally, a scaling analysis based on the Liouvillian gap is presented which qualitatively explains the double power law.