**Adiabatic quantum dynamics and applications to open systems**

Alain Joye

*Université Grenoble Alpes*

These lectures will present several versions of the adiabatic theorem of quantum mechanics, starting with the setup where the time dependent Hamiltonian has an isolated part in its spectrum for all times. The systematic construction of improved adiabatic approximations known as superadiabatic approximations will then be discussed. The methods allowing us to achieve exponential accuracy and to prove the famous Landau-Zener formula will also be introduced, and the adiabatic theorem without the familiar gap assumption will be presented. Then, we will turn to applications of these ideas and methods to effective models of time dependent open quantum systems in an adiabatic regime. In particular, the effect of a quantum environment on the transition probabilities between the states of a two-level system will be addressed.

Course Materials:

Lecture notes: [Joye 1/4] - [Joye 2/4] - [Joye 3/4] - [Joye 4/4] - [References]

Lecture notes (printable version): [Print 1/4] - [Print 2/4] - [Print 3/4] - [Print 4/4]

**Open quantum systems: structure and dynamics **

Marco Merkli

*University of Newfoundland*

The course will include the following topics:

- Recap of the postulates of quantum theory
- The dissipative Jaynes-Cummings model
- CPTP maps, Kraus representation, dilation
- Markovian semigroups and the GKSL theorem
- Non-Markovianity
- Heuristic derivation of the master equation
- Coupling an infinitely extended reservoir to a spin
- Proving the validity of the master equation

Course Materials:

Lecture notes: [Merkli 1-3/4] - [Merkli 4/4] - Exercises: [Exercises 1-6]

**Bogoliubov theory for many-body quantum systems**

Benjamin Schlein

*University of Zurich*

In many physically interesting limits, the behaviour of many-body quantum systems can be approximated by classical theories. Bogoliubov theory describes corrections to the classical limit, known as quantum fluctuations. In this mini-course, I am going to discuss recent applications of Bogoliubov theory to trapped Bose gases (first in the mean-field regime and then in the more challenging Gross-Pitaevskii limit), to systems of weakly interacting fermions and to the polaron in the strong coupling limit. We will show how Bogoliubov theory can be used to understand the low-energy spectrum and also to approximate the time-evolution of these systems.

Course Materials:

Lecture notes: [Schlein 1/4] - [Schlein 2/4] - [Schlein 3/4] - [Schlein 4/4] - [References]