  We present a wave atom optics theory of the Collective Atomic-Recoil Laser,
where the atomic center-of-mass motion is treated quantum mechanically. It
extends the previous ray atom optics theory, which treated the center-of-mass
atomic motion classically, to the realm of ultracold atoms. For the case of a
far off resonant pump laser we derive an analytical solution which gives the
linear response of the CARL system for both the quantum and classical regimes.
A linear stability analysis reveals significant qualitative differences between
these two regimes, which arise from the effects of diffraction on the atomic
center-of-mass motion.
