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Control theory and applications in Quantum Mechanics

Course Type: 
PhD Course
Academic Year: 
20 h

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 different scales: averaging. Systems, with an unknown parameter.
The (finite-dimensional) Schroedinger equation for the wave function and for the propagator.

Lecture 2.
Families of vector fields. Lie groups and left invariant control systems. Lie brackets. Frobenious theorem. Non-integrable vector distributions.

Lecture 3.
Controllability 1. The Krener theorem, The Chow theorem.

Lecture 4.
Controllability 2. Convexification. Killing the drift. The recurrent drift theorem. Applications to finite dimensional quantum systems: the
Lie Algebraic Rank Condition. Controlling a Spin 1/2 particle on the Bloch sphere.

Lecture 5. Optimal control. The Pontryagin Maximum Principle

Lecture 6. Minimum time for a 2-level system.

Lecture 7. Minimum time for a 3-level system.

Lecture 8. The adiabatic theorem: averaging. Population transfer for systems presenting conical intersections.

Lecture 9. Systems with an unknown parameter. Two level systems: chirp pulses. Three level systems: the STIRAP Process. 


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