Open Quantum Systems

International Erwin-Schrödinger-Institute for Mathematical Physics
January 20 to March 31, 2005

organized by J. Yngvason, G. M. Graf and J. Derezinski.


1st Workshop.

Mon, Jan. 31 - Fri, Feb. 4, 2005.

Schedule

Abstracts

D. Ruelle will give a 5 hours mini-course under the title:

AN INTRODUCTION TO NONEQUILIBRIUM STATISTICAL MECHANICS

(THE PHYSICS AND THE MATHEMATICS).

Statistical mechanics aims at explaining macroscopic properties of matter in terms of its microscopic interactions. There are two parts: equilibrium and nonequilibrium statistical mechanics. While equilibrium statistical mechanics is well understood, nonequilibrium (which deals with irreversibility, entropy creation, etc.) is still in a formative stage: different approaches exist, and the relations between these approaches has not been clarified. Here we shall give a broad overview of the subject, discuss some approaches in more detail, and emphasize the relations between the physics of nonequilibrium and the mathematical formalisms that try to explain the physics.

I. A broad overview (equilibrium, entropy and its increase, variety of nonequilibrium processes, macroscopic evolution equations, variety of mathematical idealizations, digression on life).

II. Infinite quantum spin systems.

III. Finite classical systems with isokinetic thermostats.

During the workshop and immediately after it L. Accardi will give a 5 hours mini-course under the title

THE STOCHASTIC LIMIT OF QUANTUM THEORY

Lecture I

Outline of the stochastic limit of quantum theory:

1. Notations and Statement of the Problem

2. Quantum Fields and their states

3. Those Kinds of Fields we call Noise

Lecture II

Dissipation, transport and coherence in quantum physics:

1. Discrete systems with dipole type interactions: canonical forms

2. The stochastic resonance principle

3. White noise Hamiltonian equations

4. Equivalence between normally ordered white noise Hamiltonian equations and quantum stochastic differential equations

Lecture III

1. The stochastic golden rule

2. The stochastic Heisenberg (Langevin) equation

3. The master equation

4. Classical subprocesses and generic systems

5. The "driving principle": generically the reservoir drives the system to a state which "resembles" the initial state of the reservoir

Lecture IV:

Electrical conductivity and the quantum Hall effect

Lecture V:

1. Non--equilibrium quantum field theory

2. Equivalence between dynamical detailed balance and local KMS condition

3. Comparison between the stochastic limit and the C*-algebraic approach in the conductivity problem.

Robert Alicki, University of Gdansk

A trap method in the theory of transport phenomena

A localised trap absorbing particles can be used to characterize transport properties of classical and quantum systems in terms of an asymptotic behavior of the particle current [1]. Classical random walks, diffusion models and quantum fermionic quasi-free systems are presented as illustrations. For the later the relations to quantum dynamical entropy are presented. The possible extensions of this approach to interacting models and to other transport phenomena are discussed.

[1] R. Alicki, M. Fannes, B. Haegeman and D. Vanpeteghem, J.Stat.Phys.{\bf 113}, 549 (2003)

Huzihiri Araki, RIMS Kyoto

Standard Potentials for the Non-Even Dynamics

In earlier works with Moriya, we have introduced the notion of standard potentials for each given dyanmics under the assumption of even dynamics and utilized it to prove equivalence of various charaterization of equilibrium states. In this talk, we generalize this notion to non-even dynamics. We show that this can be done with the same generality as the case of even dynamics. The time derivative of local observables is $i=\sqrt{-1}$ times the sum of the commutator with the potentials, the standardness condition for the potential is the same as the case of even dynamics, but the convergence condition takes somewhat different form as compared with the case of even dynamics. As for the Local Thermodynamical Stability condition, which is one of characterizations of equilibrium states, its formulation works as before, but there seems to be one more reason for favouring its mathematical version. The equivalence of various characterization of equilibrium states remains valid, except that the variational principle is out of the game because the translation invariance of the dynamics is incompatible with non-even dyanmics, as is already known.

Jean Bellissard, Georgia Institute of Technology

Dissipative transport and Kubo's formula in Aperiodic Solids.

Dissipative transport in solids can be described by a Markov semigroup of completely positive operators on the observable algebra of the charge carriers creation and annihilation operators. A model of generators of such semigroups, called the quantum jump model, will be presented. The linear response theory will be shown to provide the expression of transport coefficients through a Green-Kubo formula. This formula will be justified rigorously through the spectral property of the generator of the quantum jump model in various situations. The case of aperiodic solids, such as strongly disordered systems will be emphasized, in view of its relevance in the theory of the Quantum Hall effect. Title of paper: Dissipative transport and Kubo's formula in Aperiodic Solids.

Franco Fagnola, Universita` di Genova

Quantum Markov Semigroups

Quantum Markov Semigroups (QMS) $({\mathcal{T}}_t)_{t\ge 0}$ on a von Neumann algebra ${\mathcal{A}}$ arose in the physical literature as the mathematical structure for modelling the irreversible evolution of quantum open systems. QMS can be viewed as a natural generalisation of classical Markov semigroups on a function space, which is replaced by a (non-commutative) operator algebra. This generalisation gives a rigorous basis to the study of the qualitative behaviour of evolution equations (master equations) on an operator algebra. In the physical literature these are often studied either by explicit solutions or numerical simulations. General tools for studying the behaviour of quantum master equations have been developed recently and fruitfully applied to several physical models. In the talk we shall discuss the following the problems on the behaviour of the evolution described by a QMS (master equation):

  1. irreducibility and structure,
  2. time spent in a projection (recurrence or transience),
  3. existence and uniqueness of stationary states,
  4. convergence to invariant states,
  5. exponential convergence to invariant states.
In a number of applications ${\mathcal{A}}$ is the von Neumann algebra ${{\mathcal{B}}}({\mathsf{h}})$ of all bounded operators on ${\mathsf{h}}$ and the generator of a QMS is represented in the form \[ {\mathcal{L}}(x) = i [H,x] - \frac{1}{2}\sum_{\ell\ge 1} \left(L^*_\ell L_\ell x -2 L^*_\ell x L_\ell + xL^*_\ell L_\ell\right), \] under suitable hypotheses on $H, L_\ell$. In all these cases we will provide simple conditions on $H, L_\ell$ allowing us to solve problems (i),...,(iv). We illustrate the general methods studying applications to the master equationsof a couple of quantum open systems: the two-photon absorption and emission process and the one-photon absorption and stimulated emission process.

Roberto Floreanini, INFN Trieste

Entanglement generation in the Unruh effect

A uniformly accelerated two-level system in weak interaction with a scalar field in the Minkowski vacuum can be treated as an open quantum system. Its time evolution is described by a master equation generating a semigroup of completely positive maps: it encodes the well known Unruh effect, leading to a purely thermal equilibrium state. Remarkably, this asymptotic state turns out to be entangled when the uniformly accelerating system is composed by two, independent two-level atoms.

Israel Klich, Caltech

Entanglement Entropy and Quantum Noise

We discuss two problems: The nonequillibrium quantum noise generated when quasi-free fermions are scattered through a time dependent contact between (possibly biased) metals, and the entanglement entropy of a Fermi sea which is split into two parts. We show that the two are related through a simple inequality. The mathematical tools involved are calculation and regularization of a certain operator determinant, representing charge measurements in the fermionic Fock space. In the linear dispersion regime the determinant may be rewritten as a solution to a Riemann Hilbert problem.

Gianluca Panati, TU München

Peierls substitution, Hofstadter butterfly and duality of bundles.

One of the central problems of solid state physics is to understand the dynamics of an electron in the periodic potential generated by the ionic cores, under the additional influence of external slowly-varying potentials. This problem exhibits a multiscale structure and can therefore fruitfully studied by space-adiabatic methods. With this approach we obtain an effective quantum Hamiltonian governing the intraband dynamics, whose leading order is the celebrated Peierls substitution. This result is used to prove that, in the limit of a very weak (resp. strong) constant magnetic field, the original Hamiltonian is asintotically unitarily equivalent to the Hofstadter (resp. Harper) model. Finally, the duality between the corresponding "colourful butterflies", pointed out by Avron and coworkers by a thermodynamic argument, is explained from a bundle-theoretic viewpoint. (based on joint work with H. Spohn, S. Teufel and F. Faure)

Yan Pautrat, Université Paris-Sud

Fluctuation algebra for fermionic systems

In a recent work with Aschbacher, Jaksic and Pillet, quantum systems consisting of a "small system" coupled to fermionic reservoirs were studied. The full thermodynamics of the joint system were derived from hypotheses at the microscopic level. In such systems, physical results such as the fluctuation-dissipation theorem raise interest in a mathematical investigation of time fluctuations of observables such as heat fluxes. Because different flux observables do not commute, a full central limit theorem for these observables has to describe the commutation relations satisfied by the limiting objects, which we view as fluctuation observables. In this talk we state such a theorem, which shows in particular that equilibrium situations (i.e. situations where all reservoirs are at the same temperature) are characterized by the commutativity of the fluctuation observables.

Wojciech de Roeck, K.U. Leuven

Fluctuations of the Dissipated Heat in a Quantum Stochastic Model

We introduce a quantum stochastic dynamics for heat conduction. A multi-level subsystem is coupled to reservoirs at different temperatures. Energy quanta are detected in the reservoirs allowing the study of steady state fluctuations of the entropy production. Our main result states a symmetry in its large deviation rate function.

Benjamin Schlein, Stanford

Derivation of the Gross-Pitaevskii Equation for the Dynamics of the Bose Einstein Condensate.

We consider a system of N bosons interacting through a pair potential with scattering length of order 1/N (Gross-Pitaevskii scaling). In the limit of large N, one expects that the time-evolution of the one particle marginal of the density matrix is described by the time-dependent Gross-Pitaevskii equation. I will report on some recent progress towards a proof of this conjecture. This is a joint work with L. Erdos and H.-T. Yau.

List of participants of the 1st workshop