  In the spectrum of systems showing chaos-assisted tunneling, three-state
crossings are formed when a chaotic singlet intersects a tunnel doublet. We
study the dissipative quantum dynamics in the vicinity of such crossings. A
harmonically driven double well coupled to a bath serves as a model. Markov and
rotating-wave approximations are introduced with respect to the Floquet
spectrum of the time-dependent central system. The resulting master equation is
integrated numerically. We find various types of transient tunneling,
determined by the relation of the level width to the inherent energy scales of
the crossing. The decay of coherent tunneling can be significantly retarded or
accelerated. Modifications of the quantum asymptotic state by the crossing are
also studied. The comparison with a simple three-state model shows that in
contrast to the undamped case, the participation of states outside the crossing
cannot be neglected in the presence of dissipation.
