I am a member of the ANR “HAM-MARK” Project

It is focused on the description of Open Quantym Systems

Find out more “HAM-MARK“.

### Overview of the project

Development of rigorous statistical mechanics for quantum open systems.

There exist actually two different approaches, two different groups of researchers, in mathematics as well as in physics, which study the behaviour of dissipative quantum systems. They are: the Hamiltonian approach and the Markovian approach. ( = Ham–Mark )

Up to the last few years the connections and collaboration between these two groups of researchers have been very poor. Their techniques are very different but the problems they study are exactly the same. It has been one the successes of some of the members of this A.N.R. project to initiate, 7 years ago, discussions and collaborations between these two groups. Already, concrete results have been obtained (common publications, developments of new tools, collaborations with physicists, …).

The aim of this A.N.R. is to help this group of researchers to increase their cooperation in order to solve some of the most challenging problems connected to rigourous quantum statistical mechanics. In particular, we have in mind: the Fourier’s law for heat transport on quantum systems, the description of non-equilibrium steady states for simple spin models, the development of quantum Lanvegin equation techniques in order to describe the action of quantum heat baths.

### Current Research Project : Application of the Repeated Interaction Process to Quadratic Fermionic Systems

In collaboration with Dragi Karevski and Stephane Attal, I am currently working on describing Open Quadratic Ferminic Systems (of arbitrary geometry). The repeated interaction process is used to describe the interaction with the environment. Starting from a Gaussian initial state, and under quadratic interactions, we can show that the density matrix of the system remains Gaussian under its time evolution. It follows that the system state can be fully described by the matrix form of two points correlators.

### Another Project : Quantum Nonequilibrium Steady States Induced by Repeated Interactions

We study the steady state of a finite XX chain coupled at its boundaries to quantum reservoirs made of free spins that interact one after the other with the chain. The two-point correlations are calculated exactly, and it is shown that the steady state is completely characterized by the magnetization profile and the associated current. Except at the boundary sites, the magnetization is given by the average of the reservoirs’ magnetizations. The steady-state current, proportional to the difference in the reservoirs’ magnetizations, shows a nonmonotonic behavior with respect to the system-reservoir coupling strength, with an optimal current state for a finite value of the coupling. Moreover, we show that the steady state can be described by a generalized Gibbs state.