Bellman equations for optimal feedback control of qubit states

Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.Comment: 10 pages, no figures, replaced the simpler model in section

On the long time behavior of Hilbert space diffusion

Stochastic differential equations in Hilbert space as random nonlinear modified Schroedinger equations have achieved great attention in recent years; of particular interest is the long time behavior of their solutions. In this note we discuss the long time behavior of the solutions of the stochastic differential equation describing the time evolution of a free quantum particle subject to spontaneous collapses in space. We explain why the problem is subtle and report on a recent rigorous result, which asserts that any initial state converges almost surely to a Gaussian state having a fixed spread both in position and momentum.Comment: 6 pages, EPL2-Te

On a general kinetic equation for many-particle systems with interaction, fragmentation and coagulation

We deduce the most general kinetic equation that describe the low density limit of general Feller processes for the systems of random number of particles with interaction, collisions, fragmentation and coagulation. This is done by studying the limiting as ε -> 0 evolution of Feller processes on ∪n∞ Xn with X = Rd or X = Zd described by the generators of the form ε-1 ∑K k=0 εkB(k), K ∈ N, where B(k) are the generators of k-arnary interaction, whose general structure is also described in the paper

Stochastic evolution as a quasiclassical limit of a boundary value problem for Schrödinger equations

We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those decribing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo Fock spaces. The latter are presented as the semi-classical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians

Continuous Quantum Measurement: Local and Global Approaches

In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schrödinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menski's formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schrödinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator. </jats:p