Fig. 2: Thermalization of light in nonlinear waveguide arrays with different nonlinearities.

Linear and nonlinear couplings in three optical lattices when acted upon by three different nonlinearities: a a Kerr nonlinearity, c cascade χ⁽²⁾ process, and e optomechanical nonlinearities, as described by Eqs. (1), (3), and (4), respectively. Numerical simulations provide the modal occupancies after thermalization in all these three scenarios, in good agreement with the predicted Rayleigh–Jeans (RJ) distributions (black lines), as shown in b, d, and f. The insets display a monotonic increase in entropy S as well as the invariants of the motion U and \({{{{{{{\mathcal{P}}}}}}}}\). Note that in all cases, numerical simulations are performed over ensemble averages. The thermal fluctuations of quasi-invariants (when applicable) are indicated by gray lines, depicting the instantaneous values of U and \({{{{{{{\mathcal{P}}}}}}}}\) around their mean values. In all cases, the nonlinear array has M = 100 sites and the dashed lines represent the initial occupancies for the linear optical supermodes.