19 research outputs found
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