  We explore the quantum dynamics of photoassociation of Bose-Einstein
condensed atoms into molecules using an optical cavity field. Inside of an
optical resonator, photoassociation of quantum degenerate atoms involves the
interaction of three coupled quantum fields for the atoms, molecules, and the
photons. The feedback created by a high-Q optical cavity causes the cavity
field to become a dynamical quantity whose behavior is linked in a nonlinear
manner to the atoms inside and where vacuum fluctuations have a more important
role than in free space. We develop and compare several methods for calculating
the dynamics of the atom-molecule conversion process with a coherently driven
cavity field. We first introduce an alternate operator representation for the
Hamiltonian from which we derive an improved form of mean field theory and an
approximate solution of the Heisenberg-Langevin (HL) equations that properly
accounts for quantum noise in the cavity field. It is shown that our improved
mean field theory corrects several deficiencies in traditional mean field
theory based on expectation values of annihilation/creation operators. Also, we
show by direct comparison to numerical solutions of the density matrix
equations that our approximate quantum solution of HL equations gives an
accurate description of weakly or undriven cavities where mean field theories
break down.
