  We present a novel continuous-time control strategy to exponentially
stabilize an eigenstate of a Quantum Non-Demolition (QND) measurement operator.
In open-loop, the system converges to a random eigenstate of the measurement
operator. The role of the feedback is to prepare a prescribed QND eigenstate
with unit probability. To achieve this we introduce the use of Brownian motion
to drive the unitary control actions; the feedback loop just adapts the
amplitude of this Brownian noise input as a function of the system state.
Essentially, it "shakes" the system away from undesired eigenstates by applying
strong noise there, while relying on the open-loop dynamics to progressively
reach the target. We prove exponential convergence towards the target
eigenstate using standard stochastic Lyapunov methods. The feedback scheme and
its stability analysis suggest the use of an approximate filter which only
tracks the populations of the eigenstates of the measurement operator. Such
reduced filters should play an increasing role towards advanced quantum
technologies.

Quantum Physics