  We consider the theory of feedback control of a Bose-Einstein condensate
(BEC) confined in a harmonic trap under a continuous measurement constructed
via non-destructive imaging. A filtering theory approach is used to derive a
stochastic master equation (SME) for the system from a general Hamiltonian
based upon system-bath coupling. Numerical solutions for this SME in the limit
of a single atom show that the final steady state energy is dependent upon the
measurement strength, the ratio of photon kinetic energy to atomic kinetic
energy, and the feedback strength. Simulations indicate that for a weak
measurement strength, feedback can be used to overcome heating introduced by
the scattering of light, thereby allowing the atom to be driven towards the
ground state.
