  Atomic wave packets in optical lattices which are both spatially finite and
time-dependent exhibit many striking similarities with light pulses in photonic
crystals. We analytically characterize the transmission properties of such a
potential geometry for an ideal gas in terms of a position-dependent band
structure. In particular, we find that at specific energies, wave packets at
the center of the finite lattice may be enclosed by pairs of band gaps. These
act as mirrors between which the atomic wave packet is reflected, thereby
effectively yielding a matter wave cavity. We show that long trapping times may
be obtained in such a resonator and investigate the collapse and revival
dynamics of the atomic wave packet by numerical evaluation of the Schr\"odinger
equation.
