  A precise time-dependent control of a quantum system relies on an accurate
account of the quantum interference among the system, the control and the
environment. A diagrammatic technique has been recently developed to precisely
calculate this quantum correlation for a fast multimode coherent photon control
against slow relaxation, valid for both Markovian and non-Markovian systems. We
review this formalism in comparison with the existing approximate theories and
extend it to cases with controls by photon state other than the coherent state.
