HAM-MARK
 PROJET A.N.R. BLANC
N° ANR-09-BLAN-0098-01

Starting 01/09/2009
Ending 31/08/2013

Headed by Stéphane ATTAL

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Rencontre "Systèmes Quantiques Ouverts"

Cergy-Pontoise, 17-19 Novembre 2009
  


Voici la liste des participants à cette rencontre avec les titres des exposés.


-- Walter Aschbacher : "Entropy density of quasifree states supported by left/right movers"

In quasifree fermionic systems over the discrete line whose states are supported by left/right movers, we study the correlation of a finite subsystem with its environment by means of the von Neumann entropy. Using Toeplitz theory, we show that the left/right mover structure reappears in a generic way in the entropy density in the limit of large subsystem size. We discuss this property for the XY chain out of equilibrium, in thermal equilibrium, and in the ground state. 


-- Stéphane Attal : "Dissipative Quantum Random Walks"


-- Jean-Marie Barbaroux : "Non-analyticity of the binding energy and the ground state energy for Hydrogen in non-relativistic QED"

The question of nonanalyticity in the fine structure constant for the ground state energy for atoms in NRQED was posed some years ago by Bach, Frohlich and Pizzo. I shall present here some recent results in collaboration with T. Chen, V. Vougalter and S. Wugalter, where we prove that both the binding energy and the ground state energy for hydrogen atom in NRQED contain a logarithmic divergent term in their "expansion".


-- Alberto Barchielli : "Quantum measurements in continuous time and non-Markovian evolutions"

In the last years there was a growing interest in quantum open systems with non-Markovian dynamics. Also some stochastic unravellings of such a kind of dynamics have been proposed because of their usefulness for numerical simulations. In the Markov case, the stochastic
Schrödinger equation, which unravel the usual Lindblad-type master equation, is also the starting point for the quantum theory of measurements in continuous time. However, in the non-Markov case it is not clear if the proposed unravellings can have a measurement interpretation. In this talk I want to present some proposals of unravellings which admit a physical measurement interpretation compatible
with the axiomatic formulation of quantum mechanics. Our unravellings are related to two classes of non-Markovian dynamics, the so called Lindblad rate equation and a dynamics representing a quantum system interacting with a coloured bath and possibly controlled by a measurement based feedback. The main part of the talk will be on the construction of unravelling and measurement interpretation of the Lindblad rate equation and on the presentation of a physical model (we study the heterodyne spectrum of a two-level system in a structured thermal-like bath). From joint works with C. Pellegrini, F. Petruccione, P. Di Tella, M. Gregoratti.


-- Nils Berglund : "Metastability in a class of parabolic stochastic PDEs"

The evolution of magnetisation in a ferromagnet can be described by a stochastic partial differential equation, in which thermal fluctuations are modelled by space-time white noise. This equation shows a metastable behaviour, in the sense that for certain initial conditions, the system needs an exponentially long time to reach equilibrium. Using potential-theoretic tools, we derive precise asymptotics of the relaxation time, going beyond traditional large-deviation estimates. Joint work with Florent Barret (Ecole Polytechnique) and Barbara Gentz (University of Bielefeld).


-- Laurent Bruneau : "Transport for the 1D Schrödinger equation via quasi-free systems"

We consider a finite sample of length N submitted to a potential V, coupled to two reservoirs (one at each end of the sample) and in the independent electron approximation. As time goes to infinity the system reaches a non-equilibrium steady state (NESS) which carries currents (charge and heat). The purpose of this talk is to relate the behavior of these currents as the size N of the sample grows in terms of spectral properties of the corresponding 1D Schrödinger operator $-\Delta+V$.


-- Horia Cornean : "A critical view on the Keldysh formalism applied to mesoscopic quantum systems"

This talk is intended as a "vulgarisation" of the Keldysh formalism towards mathematicians. We show in particular that the Keldysh contour-ordering is not necessary when calculating transient or steady-state currents through noninteracting mesoscopic systems. The Langreth rules can also be derived using elementary operations, only.


-- Stephan de Bievre : "Equilibration and generalized equipartition in dynamical Lorentz gases"

We prove approach to thermal equilibrium for the fully Hamiltonian dynamics of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a $d$-dimensional array of fixed soft scatterers that each possess an internal harmonic or anharmonic degree of freedom to which moving particles locally couple. We establish that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature $T$, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics--they need not be in a state of thermal equilibrium. The temperature $T$ to which the particles equilibrate obeys a generalized equipartition relation, in which the associated thermal energy $k_{\mathrm B}T$ is equal to an appropriately defined average of the scatterers' kinetic energy. In the equilibrated state, particle motion is diffusive.
Joint work with P. Parris (Missouri)


 -- Jan Derezinski : "Feynman diagrams - a beautiful and powerful formalism"

The diagrammatic method has a beautiful formal structure and many applications in physics. It involves a number of nontrivial combinatorial and analytic facts. Unfortunately, it is usually described in an ad hoc manner, which hides its beauty and versatility. I will try to give an introduction to this method, concentrating on the so-called Friedrichs diagrams. I will follow my joint work with C.Gerard.


-- Julien Deschamps : "Limite du continu pour des systèmes classiques d' interactions répétées"

Nous nous intéresserons à la limite en temps continu de l'évolution d'un système classique interagissant avec un environnement selon un schéma d' interactions répétées. Ce dernier consiste à considérer l'environnement comme constitué d'une infinité de parties identiques qui agissent les unes après les autres de manière indépendantes sur le système pendant un pas de temps h. Sous certaines hypothèses, la dynamique du système est alors donnée par une chaîne de Markov. Nous formaliserons tout ceci grâce à un parallèle entre système dynamiques déterministes et processus markoviens. Ensuite, nous étudierons la limite de ces interactions quand le pas de temps h tend vers 0. Nous verrons que dans certains cas la dynamique limite du système est donnée par une solution d' équation différentielle stochastique. De plus, j' illustrerai ces interactions répétées par des exemples simples de systèmes hamiltoniens.


-- Gian-Michele Graf : "Adiabatic evolution and dephasing"

Lindbladians are generators of the effective dynamics of open quantum systems. We focus on dephasing Lindbladians. Like Hamiltonians of isolated quantum mechanical system, but in contrast to generic Lindbladians, they exhibit several stationary states.

The adiabatic evolution of an isolated quantum mechanical system exhibits no irreversible transitions if its Hamiltonian undergoes a slow, transient time-dependence. By contrast, if a dephasing Lindbladian undergoes such a change, transitions are typically irreversible. I'll present a formulation of the adiabatic theorem which accounts for the different kinds of transition, though treating Hamiltonian and Lindbladian dynamics on equal footing. We'll then present two applications.

The first one is the solution of an optimization problem. Given are a path of dephasing Lindbladians, two states ("base" and "target") which are stationary w.r.t. the corresponding endpoint, and an amount of time to be spent. Sought is time schedule to be used on the path in order to bring the evolved "base" as close as possible to the "target".

In a second application we apply the result to transport and linear response theory. In the context of dephasing Lindbladians, the coefficients of dissipative conductance are determined by the Fubini-Study metric, while their non-dissipative counterparts are determined by the adiabatic curvature. If the metric and the (symplectic) curvature form are compatible, in the sense of defining a Kähler structure, then the non-dissipative resistance coefficients are immune to dephasing. We give some examples of compatible systems.

(Joint work with Y. Avron, M. Fraas, P. Grech, and O. Kenneth).
-- Ion Nechita : "Random quantum states"

The density operator in quantum mechanics is represented by a positive, unit trace matrix. We analyze various ensembles of random density operators of a fixed size and study constructive algorithms to generate them at random. A standard construction is based on generating random pure states on a composite Hilbert space according to the unique, unitarily invariant Fubini-Study measure and then partial tracing a part of the system. Other measures can be obtained from a procedure which associates to a given graph a random pure state such that the graph encodes the entanglement of the state. We develop techniques for studying statistics for the eigenvalues of these random density matrices such as the von Neumann entropy, the purity or the spectral density. This is joint work with B. Collins, K. Penson and K. Zyczkowski.


-- Francis Nier : "Adiabatic evolution of resonances in far from equilibrium systems"

After recalling the physical systems of resonant tunneling diodes and the functional framework for such far from equilibrium quantum system, the necessity of an asymptotic analysis and of a new approach for the adiabatic evolution of quantum resonant states will be presented. Possibly, some numerical issues which are as much challenging as the theoretical ones will be discussed.


-- Francesco Petruccione : "Engineering inverse power law decoherence of a qubit"

The exact dynamics of a Jaynes-Cummings model for a qubit interacting with a bath of bosons, characterized by a special form of the spectral density, is evaluated analytically. The special reservoir is designed to induce anomalous decoherence, resulting in an inverse power law relaxation, of power three half, on an evaluated long time scale. If compared to the exponential-like relaxation obtained from the original Jaynes-Cummings model for Lorentzian-type spectral density functions, decoherence is considerably hindered.


-- Claude-Alain Pillet : "Diffusion in a repeated interaction model"

In a joint work with S. De Bièvre and L. Bruneau we consider the transport properties of an electron in the tight binding approximation and subject to an homogeneous static electric field. We show that repeated interactions of the electron with two-level systems, in thermal equilibrium suppress the Bloch oscillations. We study the statistical properties of the induced steady current.


-- David Taj :  "Van Hove Limit for Infinitely Extended Open Quantum Systems"

We extend Davies celebrated markovian generator to open systems with arbitrary spectra (i.e. possibly infinitely extended). We present our new generator and prove that it matches the exact system evolution in the Van Hove Limit (weak coupling), under the same hypotheses of Davies approach. Our theory is then applied to the case of a free particle in the euclidean space coupled to a heat bath. Also, we discuss a possible application to Quantum Transport where we present preliminary results for a Wigner Function approach. Application to the study of diffusion is also envisaged.


 -- Sylvain Vogelsberger :  "Evolution of entanglement of two qubits in a model of repeated interactions"

We consider a system of two qubits in interaction with an environment. In the model of repeated interactions, we calculate the evolution of entanglement of two qubits. We focus our work on the initial states wich have an X-form. So in appendix we give a panorama of the set of X-states : ?x.

 -- Valentin Zagrebnov :  "Lieb-Robinson Bounds and Construction of Dynamics"

A W*-dynamical system is constructed for infinite anharmonic crystal with help of Lieb-Robinson estimates.