The scaling of weak field phase-only control in Markovian dynamics

We consider population transfer in open quantum systems, which are described by quantum dynamical semigroups (QDS). Using second order perturbation theory of the Lindblad equation, we show that it depends on a weak external field only through the field's autocorrelation function, which is phase independent. Therefore, for leading order in perturbation, QDS cannot support dependence of the population transfer on the phase properties of weak fields. We examine an example of weak-field phase-dependent population transfer, and show that the phase-dependence comes from the next order in the perturbation.