  We compare two different approaches to the control of the dynamics of a
continuously monitored open quantum system. The first is Markovian feedback as
introduced in quantum optics by Wiseman and Milburn [Phys. Rev. Lett. {\bf 70},
548 (1993)]. The second is feedback based on an estimate of the system state,
developed recently by Doherty {\em et al.} [Phys. Rev. A {\bf 62}, 012105
(2000)]. Here we choose to call it, for brevity, {\em Bayesian feedback}. For
systems with nonlinear dynamics, we expect these two methods of feedback
control to give markedly different results. The simplest possible nonlinear
system is a driven and damped two-level atom, so we choose this as our model
system. The monitoring is taken to be homodyne detection of the atomic
fluorescence, and the control is by modulating the driving. The aim of the
feedback in both cases is to stabilize the internal state of the atom as close
as possible to an arbitrarily chosen pure state, in the presence of inefficient
detection and other forms of decoherence. Our results (obtain without recourse
to stochastic simulations) prove that Bayesian feedback is never inferior, and
is usually superior, to Markovian feedback. However it would be far more
difficult to implement than Markovian feedback and it loses its superiority
when obvious simplifying approximations are made. It is thus not clear which
form of feedback would be better in the face of inevitable experimental
imperfections.
