  This article presents a general master equation formalism for the interaction
between travelling pulses of quantum radiation and localized quantum systems.
Traveling fields populate a continuum of free space radiation modes and the
Jaynes-Cummings model, valid for a discrete eigenmode of a cavity, does not
apply. We develop a complete input-output theory to describe the driving of
quantum systems by arbitrary incident pulses of radiation and the quantum state
of the field emitted into any desired outgoing temporal mode. Our theory is
applicable to the transformation and interaction of pulses of radiation by
their coupling to a wide class of material quantum systems. We discuss the most
essential differences between quantum interactions with pulses and with
discrete radiative eigenmodes and present examples relevant to quantum
information protocols with optical, microwave and acoustic waves.
