Time-evolution operator driven terms

From this viewpoint, which is the most fundamental, the line shape as a whole is the sum of the diagonal matrix elements of the time evolution operator of the driven damped quantum harmonic oscillator in the IP representation with respect to the diagonal part of the Hamiltonian of this oscillator. According to Eq. (120), each diagonal element is a sum of time-dependent terms... 

The aim of this appendix is to disentangle the driven term hbQ appearing in (G.l). That may be performed with the aid of the IP [54], which allows us to write the time-evolution operator (G.2) according to... 

In other words, we split the antisymmetric time evolution into two terms, one remaining proportional to the original equilibrium distribution, turned into an antisymmetric form, and the other corresponding to an antisymmetric form whose symmetric counterpart is supposed to be different from the equilibrium distribution. In fact, it were not so, this latter term would contribute to the former. In conclusion, we proceed as follows. First, we apply to the equilibrium distribution the operator This has the effect of turning the equilibrium distribution into an unstable antisymmetric form that has to be driven by the antisymmetric operator LA. At time t = t we apply to this unstable antisymmetric distribution the evolution operator exp(LAt) and we let this distribution evolve up to the time t2 > h. At time t = t2, we apply to this antisymmetric distribution the operator again, so that A(t2 — t )pe (y, 0) + PA°neq( y, f2 — h) becomes A(t2 — t )peq(y, 0) + Ps°neq(y, t2 — t ). The authors of Ref. 100 proved that A(t2 — t ) = (t2 — h), with (t2 — t ) given by Eq. (148). They also... 

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