GEOMETRIC CONTROL AND APPLICATIONS TO QUANTUM MECHANICS - UGO BOSCAIN - INRIA - CNRS - UNIVERSITE PARIS SORBONNE

DISMA (3rd floor)

The purpose of this course is to introduce the basic concepts in geometric control, from controllability
to optimal control. These concepts will be then applied to the problem of controlling simple quantum
mechanical systems that appear often in quantum technologies as Nuclear Magnetic Resonance and for
the realization of q-bits for quantum computers.
Lecture 1. The \control theory problem": controllability, stabilizability, optimal control. Example
of problems arising in quantum mechanics: Nuclear Magnetic Resonance, Stimulated Raman Adiabatic
Passages. Systems evolving on two di erent scales: averaging. Systems, with an unknown parameter.
The ( nite-dimensional) Schroedinger equation for the wave function and for the propagator.
Lecture 2. Families of vector elds. Lie groups and left invariant control systems. Lie brackets.
Frobenius theorem. Non-integrable vector distributions.
Lecture 3. Controllability 1. The Krener theorem, The Chow theorem.
Lecture 4. Controllability 2. Convexi cation. Killing the drift. The recurrent drift theorem. Applications to nite dimensional quantum systems: the Lie Algebraic Rank Condition. Controlling a Spin 1/2
particle on the Bloch sphere.
Lecture 5. Optimal control 1. Formulation of the problem. Existence.
Lecture 6. Optimal control 2. The Pontryagin Maximum Principle (proof for minimal energy for affine
systems).
Lecture 7. Minimal energy for a 2-level system.
Lecture 8. Minimum time for a 3-level system.
Lecture 9. The adiabatic theorem: averaging. Population transfer for systems presenting conical
intersections.
Lecture 10. Systems with an unknown parameter. Two level systems: chirp pulses. Three level sys-
tems: the STIRAP process.
[1] Jurdjevic, Velimir. Geometric control theory. Cambridge Studies in Advanced Mathematics, 52. Cam-
bridge University Press, Cambridge, 1997.
[2] D'Alessandro, Domenico Introduction to quantum control and dynamics. Chapman & Hall/CRC Ap-
plied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2008.
[3] A. Agrachev, D. Barilari, and U. Boscain. A Comprehensive Introduction to sub-Riemannian Ge-
ometry, volume 181 of Cam- bridge Studies in Advanced Mathematics. Cambridge University Press,
Cambridge, 2020. http://people.sissa.it/?barilari/Notes.html. xviii+746 pp.

Agenda:
Tuesday, January 25, h 14-18 in Auletta Seminari DISMA (3rd floor)
Wednesday, January 26, h 14-18 in Auletta Seminari DISMA (3rd floor)
Thursday, January 27, h 9-13 in Aula Buzano DISMA (3rd floor)