On single-frequency asymptotics for the Maxwell-Bloch equations: mixed states

Abstract

We consider damped driven Maxwell-Bloch equations which are finite-dimensional approximation of the damped driven Maxwell-Schr\"odinger equations. The equations describe a single-mode Maxwell field coupled to a two-level molecule. Our main result is the construction of solutions with single-frequency asymptotics of the Maxwell field in the case of quasiperiodic pumping. The asymptotics hold for solutions with harmonic initial values which are stationary states of averaged equations in the interaction picture. We calculate all harmonic states and analyse their stability. The calculations rely on the Bloch-Feynman gyroscopic representation of von Neumann equation for the density matrix. The asymptotics follow by application of the averaging theory of the Bogolyubov type. The key role in the application of the averaging theory is played by a special a priori estimate.