Nearly Hamiltonian dynamics of laser systems

The Arecchi-Bonifacio (or Maxwell-Bloch) model is the benchmark for the description of active optical media. However, in the presence of a fast relaxation of the atomic polarization, its implementation is a challenging task even in the simple ring-laser configuration, due to the presence of multiple timescales. In this paper we show that the dynamics is nearly Hamiltonian over timescales much longer than those of the cavity losses. More precisely, we prove that it can be represented as a pseudo spatiotemporal pattern generated by a nonlinear wave equation equipped with a Toda potential. The existence of two constants of motion (identified as pseudo energies), thereby elucidates the reason why it is so hard to simplify the original model: the adiabatic elimination of the polarization must be accurate enough to describe the dynamics correctly over unexpectedly long timescales. Finally, since the nonlinear wave equation with Toda potential can be simulated on much longer times than the previous models, this opens up the route to the numerical (and theoretical) investigation of realistic setups.