  We analyze the optomechanics-like properties of a Bose-Einstein condensate
(BEC) trapped inside an optical resonator and driven by both a classical and a
quantized light field. We find that this system exhibits a nature of role
reversal between the matter-wave field and the quantized light field. As a
result, the matter wave field now plays the role of the quantized light field,
and the quantized light field behaves like a movable mirror, in contrast to the
familiar situation in BEC-based cavity optomechanics [Brennecke et al., Science
322, 235 (2008); Murch et al., Nat. Phys. 4, 561 (2008)]. We demonstrate that
this system can lead to the creation of a variety of nonclassical matter-wave
fields, in particular cat states, and discuss several possible protocols to
measure their Wigner function.
