  Using a microscopic master equation to account for the environmental effects,
we compute the decoherence culminated during the propagation of a microwave
pulse of arbitrary shape through a superconducting qubit. It is shown that the
qubit decoherence vanishes and the pulse shape remains absorption-free when the
latter adopts a soliton shape with $n\pi$ area. Otherwise, the environmental
feedback decelerates the velocity of the soliton envelop and induces an
monotonic increase of phase in the microwave. A pulse of non-$n\pi$ area thus
ramifies into a transparent part that travels decoherence-free at incident
velocity and a slowing part that decays through space. The ramification
explains the environmental origin of pulse splitting observed in self-induced
transparency.
