  We propose a feedback scheme for preparation of photon number states in a
microwave cavity. Quantum Non-Demolition (QND) measurements of the cavity field
and a control signal consisting of a microwave pulse injected into the cavity
are used to drive the system towards a desired target photon number state.
Unlike previous work, we do not use the Galerkin approximation of truncating
the infinite-dimensional system Hilbert space into a finite-dimensional
subspace. We use an (unbounded) strict Lyapunov function and prove that a
feedback scheme that minimizes the expectation value of the Lyapunov function
at each time step stabilizes the system at the desired photon number state with
(a pre-specified) arbitrarily high probability. Simulations of this scheme
demonstrate that we improve the performance of the controller by reducing
"leakage" to high photon numbers.
