Mathematical Methods in the Control of Quantum Mechanical Systems

Thursday, October 15, 2015 - 11:00am - 12:00pm
Lind 305
Domenico D'Alessandro (Iowa State University)
In the last decades, advances in pulse shaping techniques have opened
up the possibility of manipulation of systems whose evolution follows
the laws of quantum mechanics. Moreover, novel applications, such as in
quantum information processing, have offered further motivation for
this study.

From a mathematical point of view, the field which is now known as'Quantum Control' is a combination of different mathematical techniques borrowed from a wide variety of mathematical areas. Different tools apply to different models which correspond to different approximations
of the physical system at hand. The simplest case is the one of a closed
system, i.e., a system non interacting with the environment in any way
other than through the external controls, controlled in open loop, and
whose state can be modeled as a vector in a finite dimensional Hilbert
space. In this case, the operator describing the evolution belongs to a
Lie group and the control system is determined by a family of right
invariant vector fields on such a Lie group. Techniques of geometric
control are therefore appropriate. As some of the above assumptions on
the physical model are relaxed, different tools have to be used. The
consideration of 'open' systems, which also allow for a continuous
measurement of the state and feedback, requires the introduction of
techniques of dynamical semigroups as well as stochastic calculus. The
study of infinite dimensional quantum control systems is often done
using tools of functional analysis and control of partial differential

This talk is a brief survey of the field from the point of view of the
mathematics that is used and needs to be developed. After introducing
basic notions of quantum mechanics and the relevant models used in applications I will indicate a number of open mathematical problems.
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