v 2 1 0 N ov 2 00 5 STABILIZING FEEDBACK CONTROLS FOR QUANTUM SYSTEMS ∗

No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is neccessarily one with partial observations, and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator. 1. Introduction. Though they are both probabilistic theories, probability theory and quantum mechanics have historically developed along very different lines. Nonetheless the two theories are remarkably close, and indeed a rigorous development of quantum probability [18] contains classical probability theory as a special case. The embedding of classical into quantum probability has a natural interpretation that is central to the idea of a quantum measurement: any set of commuting quantum observables can be represented as random variables on some probability space, and conversely any set of random variables can be encoded as commuting ob-servables in a quantum model. The quantum probability model then describes the statistics of any set of measurements that we are allowed to make, whereas the sets of random variables obtained from commuting observables describe measurements that can be performed in a single realization of an experiment. As we are not allowed to make noncommuting observations in a single realization, any quantum measurement yields even in principle only partial information about the system. The situation in quantum feedback control [10, 11] is thus very close to classical stochastic control with partial observations [3]. A typical quantum control scenario, representative of experiments in quantum optics, is shown in Fig. 1.1. We wish to control the state of a cloud of atoms, e.g. we could be interested in controlling their collective angular momentum. To observe the atoms, we scatter a laser probe field off the atoms and measure the scattered light using a homodyne detector (a cavity can be used to increase the interaction strength between the light and the atoms). The observation process is fed into a controller which can feed back a control signal to the atoms through some actuator, e.g. a time-varying magnetic field. The entire setup can be described by a Schrödinger equation for the atoms and the probe field, …