  We present a novel cavity QED system in which a Bose-Einstein condensate
(BEC) is trapped within a high-finesse optical cavity whose length may be
adjusted to access both single-mode and multimode configurations. We
demonstrate the coupling of an atomic ensemble to the cavity in both
configurations. The atoms are confined either within an intracavity
far-off-resonance optical dipole trap (FORT) or a crossed optical dipole trap
via transversely oriented lasers. Multimode cavity QED provides fully emergent
and dynamical optical lattices for intracavity BECs. Such systems will enable
explorations of quantum soft matter, including superfluid smectics, superfluid
glasses, and spin glasses as well as neuromorphic associative memory.
