# 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

equations.

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.

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

equations.

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.

MSC Code:

70Q05

Keywords: