Ergodic Properties of the Spin - Boson System

We investigate the dynamics of a 2-level atom (or spin-1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared.Comment: 31 pages, postscrip

Resonances for a Hydrogenic System or a Harmonic Oscillator Strongly Coupled to a Field

We calculate resonances which are formed by a particle in a potential which is either Coulombian or quadratic when the particle is strongly coupled to a massless boson, taking only two energy levels into consideration. From these calculations we derive how the moving away of the particle from its attraction center goes together with the energy lowering of hybrid states that this particle forms with the field. We study the width of these states and we show that stable states may also appear in the coupling.Comment: 17 pages, 6 figure

A note on reflectionless Jacobi matrices

The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of [Breuer-Ryckman-Simon]. The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection. To appear in Commun. Math. Phys.Comment: 10 page

Entropic fluctuations in thermally driven harmonic networks

We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional which satisfies the Gallavotti--Cohen fluctuation theorem, i.e., a global large deviation principle with a rate function I(s) obeying the Gallavotti--Cohen fluctuation relation I(-s)-I(s)=s for all s. We also consider perturbations of our functional by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I(s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of our functional and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations

Entropic fluctuations in XY chains and reflectionless Jacobi matrices

We study the entropic fluctuations of a general XY spin chain where initially the left(x0) part of the chain is in thermal equilibrium at inverse temperature Tl/Tr. The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans-Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti-Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated to a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans-Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix h canonically associated to the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at 1 (i.e., the Kawasaki identity fails) and does not have the celebrated ES/GC symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the cases where h is a Schr\"odinger operator, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schr\"odinger operators

On the consequences of the fact that atomic levels have a certain width

This note presents two ideas. The first one is that quantum theory has a fundamentally perturbative basis but leads to nonperturbative states which it would seem natural to take into account in the foundation of a theory of quantum phenomena. The second one consists in questioning the validity of the present notion of time. Both matters are related to the fact that atomic levels have a certain width. This note is presented qualitatively so as to evidence its main points, independently of the models on which these have been tested.Comment: 8 page

What is absolutely continuous spectrum?

This note is an expanded version of the author's contribution to the Proceedings of the ICMP Santiago, 2015, and is based on a talk given by the second author at the same Congress. It concerns a research program devoted to the characterization of the absolutely continuous spectrum of a self-adjoint operator H in terms of the transport properties of a suitable class of open quantum systems canonically associated to H

Full statistics of erasure processes: Isothermal adiabatic theory and a statistical Landauer principle

We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauer's Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauer's bound.Comment: 24 pages, 4 figures; In the new version, Section 4 contains an extended discussion of the violation of Landauer's boun