Laurent Bruneau

Repeated interactions in open quantum systems

Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.

Bogoliubov Hamiltonians and one-parameter groups of Bogoliubov transformations

On the bosonic Fock space, a family of Bogoliubov automorphisms corresponding to a strongly continuous one-parameter group of symplectic maps (R(t))t∊R is considered. We give conditions that guarantee it to be implemented by a strongly continuous one-parameter group U(t) of unitary operators. The generator of such U(t) will be called a Bogoliubov Hamiltonian. Given (R(t))t∊R, a Bogoliubov Hamiltonian is defined up to an additive constant. We introduce two kinds of Bogoliubov Hamiltonians: type I, characterized by vanishing of the expectation value at the vacuum, and type II, characterized by the fact that its infimum equals zero. We give conditions so that they are well defined. We show that there exist cases when only HI is well defined, even though the classical Hamiltonian is positive (which may be interpreted as a kind of an infrared catastrophe), and when only HII is well defined (which means that one needs to introduce an infinite counterterm in the formula for the Hamiltonian).

PAULI-FIERZ HAMILTONIANS DEFINED AS QUADRATIC FORMS

Abstract We study Pauli-Fierz Hamiltonians-self-adjoint operators describing a small quantum system interacting with a bosonic field. Using quadratic form techniques, we extend the results of Derezinski-Gerard and Gerard about the self-adjointness, the location of the essential spectrum and the existence of a ground state to a large class of Pauli-Fierz Hamiltonians.

Infinite products of random matrices and repeated interaction dynamics

A flexible magnetic sheet for therapeutic use made of a rubbery-flexible synthetic material in which permanent-magnetic ferrite particles have been embedded, the surface facing the body site to be treated of said sheet having been magnetized with magnetic poles of alternating polarities, which poles are in the form of some geometrical shape such as concentrically arranged rings, sectors, quadrangles and the like. Such sheet may also be composed of a plurality of individual parts.

Landauer-Büttiker Formula and Schrödinger Conjecture

We study the entropy flux in the stationary state of a finite one-dimensional sample

Scattering induced current in a tight-binding band

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On the singularity of random matrices with independent entries

connected at its left and right ends to two infinitely extended reservoirs