In our group we study many-body quantum systems out-of-equilibrium. In particular, we focus on gases of ultracold atoms (Bose-Einstein condensate, optical lattices); quantum optics (optical amplifiers, resonators, lasers), andcondensed matter (high-temperature superconductors, Josephson junctions). The technical details of these systems are very different. For example, their temperature varies from room temperature (~300K) to below one-millionth of a degree above the absolute zero (<1uK). The number of atoms involved goes from a few hundreds, to ten-to-the-twenty. However, the theoretical questions that we ask follow a common thread.

**Thermalization.**When a system is not perturbed by any external force, we can always assume it to be found at thermal equilibrium and to be described by a Boltzmann distribution. What happens when a system is driven by a time-dependent force? How is Boltzmann (classical) statistics recovered in a system obeying quantum mechanics? In some cases, if the system is observed for long enough time, it is still possible to define an effective Boltzmann distribution or better, a “low-frequency effective temperature” (LET) (See for example hereor here). When does this concept apply and when does it not?**Universality.**At equilibrium, the concept of universality is intimately related to the integration over “imaginary time”. Can we achieve a similar degree of universality by integrating over “real-time”? By considering small perturbations around a quantum critical point, it is possible to show that, in non-equilibrium systems, the time coordinate follows the same scaling laws as the other dimensions. (See here and here). How do time and space coordinates differ?**Minimal model.**Sometimes, complicate phenomena such as “exceptional points” (see here) and “non-equilibrium quantum phase transitions” (see here) can be understood by mapping them to a harmonic oscillator. Similarly, competing orders in cuprates, can be explained by considering the effects of a single impurity inside a superconductor (see here). What are the limitations of this minimalistic approach? When do we need more complicate models?