Spectral properties of the two-photon absorption and emission process

The quantum Markov semigroup of the two-photon absorption and emission process has two extremal normal invariant states. Starting from an arbitrary initial state it converges toward some convex combination of these states as time goes to infinity (approach to equilibrium). We compute the exact exponential rate of this convergence showing that it depends only on the emission rates. Moreover, we show that off-diagonal matrix elements of any initial state go to zero with an exponential rate which is smaller than the exponential rate of convergence of the diagonal part. In other words quantum features of a state survive longer than the relaxation time of its classical part.