**Seminari e Convegni**

**Archiviata**

**25-27 gennaio 2022**

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

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)