Coherence and Control of Quantum System

The aim of this course is to present the theoretical tools needed for the study and the description of quantum control strategies. We will focus on protocols designed to generate and/or preserve quantum states that are relevant for applications in quantum technologies and quantum information.
Particular attention will be given to the mathematical description of continuous-variable quantum systems (quantum harmonic oscillators), as for example quantum optomechanical systems, and to protocols based on continuous-measurements and feedback.

At the end of the course the student will have acquired the following competencies:
1. he will be able to mathematically describe continuous-variable quantum systems, with a particular attention on the description of Gaussian states, Gaussian evolutions and Gaussian measurements.
2. he will be able to derive the master equation in Lindblad form, describing the evolution of a quantum systems interacting with a Markovian environment.
3. he will be able to derive stochastic master equations, corresponding to the evolution of a quantum systems interacting with a Markovian environemnt that is continuously measured. In particular he will have focused on both continuous photo-detection and homodyne detection, and he will have considered both the description in terms of density operator in the Hilbert space, and in terms of first and second moments in the Gaussian formalism (for continuous-variable systems).
4. he will be able to describe control strategies based on continuous measurements and feedback (Markovian feedback vs Bayesian feedback).
5. he will be able to derive the Hamiltonian describing a quantum optomechanical systems, both in the non-linear and linearized regime.
6. he will be able to describe control protocols for quantum optomechanical systems, with the aim of, either cooling the mechanical oscillator towards its motional ground state, or generating non-classical states (squeezed states).