  In this paper, the non-Markovian dynamics of a microcavity coupled to a
waveguide in photonic crystals is studied based on Fano-type tight binding
model. Using the exact master equation, we solve analytically and numerically
the temporal evolution of the cavity coherent state and the associated physical
observables. A critical transition is revealed when the coupling increase
between the cavity and the waveguide. In particular, the cavity field becomes
dissipationless when the coupling strength goes beyond a critical value, as a
manifestation of strong non-Markovian memory effect. The result also indicates
that the cavity can maintain in a coherent state with arbitrary small number of
photons when it strongly couples to the waveguide at very low temperature.
These properties can be measured experimentally through the photon current
flowing over the waveguide in photonic crystals.
