Coherence and control of quantum system

A.Y. 2020/2021
Overall hours
Learning objectives
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.
Expected learning outcomes
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).
Course syllabus and organization

Single session

Lesson period
First semester
Teaching methods
Classes will be live-streamed on ZOOM following the lessons timetable. Live-streamed classes will be recorded and will be available on the ARIEL website of the course. Relevant information about how to join the classes on ZOOM and other instructions will be available on the ARIEL website too. Please browse it regularly.
Students will be informed as soon as possible, through the course ARIEL website, also in case they will be able to attend classes at University. This possibility will depend upon the evolution of the emergency and the respect of the security directives.

Program and reference material
The course program and the reference material are unchanged.

Assessments methods and criteria
Assessments methods and criteria are unchanged: the oral exam will be taken either in class or through ZOOM, depending on the rules in force at that time.
Course syllabus
- Quantum harmonic oscillators and Gaussian formalism: Gaussian states; Gaussian operations; Gaussian measurements.
- Open quantum systems and time-continuous measurement: derivation of stochastic master equations (diffusive, quantum jumps).
- Quantum control via feedback strategies: measurement-based feedback (Markovian feedback and Bayesian feedback); coherent feedback.
- Quantum optomechanical systems: pressure-radiation Hamiltonian; sideband-cooling and feedback-cooling strategies; generation of mechanical squeezed states.
Prerequisites for admission
The course is structured to be self-consistent and assumes that students have the basics notions of quantum mechanics. However, it is recommended, but not necessary, that students have previously taken courses in "Quantum Optics" and "Quantum Information Theory".
Teaching methods
The adopted didactic method is based on theoretical lectures with the use of the blackboard.
Teaching Resources
- Lecture notes (available online on the online platform Ariel)
- A. Serafini, Quantum Continuous Variables (CRC Press, 2017).
- H.M. Wiseman and G.J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).
- W.P. Bowen and G.J. Milburn, Quantum Optomechanics (CRC Press, 2015).
Assessment methods and Criteria
The exam consists of an oral interview (lasting from 45 to 90 minutes) in which one will assess both the knowledge acquired during the lectures and the critical skills about problems related to the same topics discussed in class.
FIS/03 - PHYSICS OF MATTER - University credits: 6
Lessons: 42 hours