On parametric resonance in the laser action

We consider the selfconsistent semiclassical Maxwell--Schr\"odinger system for the solid state laser which consists of the Maxwell equations coupled to $N\sim 10^{20}$ Schr\"odinger equations for active molecules. The system contains time-periodic pumping and a weak dissipation. We introduce the corresponding Poincar\'e map $P$ and consider the differential $DP(Y^0)$ at suitable stationary state $Y^0$. We conjecture that the {\it stable laser action} is due to the {\it parametric resonance} (PR) which means that the maximal absolute value of the corresponding multipliers is greater than one. The multipliers are defined as eigenvalues of $DP(Y^0)$. The PR makes the stationary state $Y^0$ highly unstable, and we suppose that this instability maintains the {\it coherent laser radiation}. We prove that the spectrum Spec$\,DP(Y^0)$ is approximately symmetric with respect to the unit circle $|\mu|=1$ if the dissipation is sufficiently small. More detailed results are obtained for the Maxwell--Bloch system. We calculate the corresponding Poincar\'e map $P$ by successive approximations. The key role in calculation of the multipliers is played by the sum of $N$ positive terms arising in the second-order approximation for the total current. This fact can be interpreted as the {\it synchronization of molecular currents} in all active molecules, which is provisionally in line with the role of {\it stimulated emission} in the laser action. The calculation of the sum relies on probabilistic arguments which is one of main novelties of our approach. Other main novelties are i) the calculation of the differential $DP(Y^0)$ in the "Hopf representation", ii) the block structure of the differential, and iii) the justification of the "rotating wave approximation" by a new estimate for the averaging of slow rotations.