# Open Quantum Systems

### Goal of the program

The goals of the program are to stimulate progress in the theory of open quantum systems, both in the conceptual foundations and in the analysis of paradigmatic models, as well as to disseminate recent advances in the field. Topics to be featured include:

• Non-equilibrium steady states and transport,
• Large open systems,
• Driven systems at equilibrium,
• Reduced or effective dynamics,
• Scattering in quantum field theory.

### Structure of the program

We plan a series of seminars and lectures about the main topics of the program.

Two workshops devoted to open quantum systems will be organized, that will feature mini-courses by D. Ruelle, L. Accardi and C. A. Pillet. The program will overlap with a course given by one of the organizers, J. Derezinski, on to applications of operator algebras to quantum physics.

Applications to the program should be addressed to one of the organizers: yngvason@ap.univie.ac.at , gmgraf@itp.phys.ethz.ch or jan.derezinski@fuw.edu.pl .

### Tentative list of participants (unconfirmed marked with *)

 L. Accardi, 30 Jan. -- 10 Feb.; R. Alicki, 30 Jan. -- 18 Feb.; H. Araki, 30 Jan. - 31 March; J. Avron; 30 Jan. -- 6 Feb.; J. Barata, 30 Jan. -- 12 Feb.; J. Bellissard, 28 Jan. -- 7 Feb.; F. Benatti, 31 Jan. -- 6 Feb.; P. Blanchard, 6.Feb. -- 13.Feb.; J. Clark, 30 Jan. -- 05. Feb.; H. Cornean, 10 March -- 20 March; J. Derezinski, 30 Jan.-12 Feb., 15-18 Feb., 28 Feb.-4 March, 8-29 March, 4 April-6 May, 9-28 May; W. De Roeck, 31 Jan. -- 9 Feb.; A. Elgart, 20 Jan. -- 3 Feb.; F. Fagnola, 30 Jan. -- 6 Feb.; R. Floreanini, 29 Jan. -- 4 Feb.; J. Froehlich, 14 March - 18 March; R. Fruboes, 13 March - 18 March; C. Gerard 14 - 18 March S. Golenia, 13 April - 15 April; G. M. Graf, 27 Jan. -- 11 Feb., 7 -- 18 March; M. Griesemer, 6 -- 26 March; H. Grundling, 2 -- 9 March; C. Hainzl, 1 -- 20 March; M. Hellmich, 6 -- 13 Feb.; M. Hirokawa, 7 -- 18. March; F. Hiroshima, 7 -- 18. March; K. Ito, 7 -- 17 March; V. Jaksic, 13 -- 19 March; C. Jaekel, 28 Feb. -- 19 March; G. Jona-Lasinio, 13 -- 24 March; I. Klich, 30 Jan. -- 10 Feb.; A. Kossakowski, 13 -- 19 March; J. Lorinczi, 6 -- 13 March; W.A. Majewski, 28 Jan. -- 12 Feb.; T. Matsui, 9 March -- 19 March; K. Meissner, 14 -- 24 Apr.; M. Merkli, 4 -- 25 March; H. Moriya, 30 Jan -- 9 Feb.; K. Netocny, 6 -- 18 March; H. Narnhofer, local person; R. Olkiewicz, 30 Jan -- 12 Feb.; G. Panati, 30 Jan. -- 9 Feb; Y. Pautrat, 20 Jan. -- 4 Feb.; C. A. Pillet, 13 -- 18 March; R. Roschin, 1 -- 31 March; D. Ruelle, 30 Jan. -- 7 Feb.; J. Schenker, 30.Jan - 5. Feb. and 14 -- 18 March; B. Schlein, 20 Jan. -- 3 Feb.; R. Seiringer, 23 -- 31 Jan. and 11 -- 20 March; G. Sewell, 10 -- 31 March; H. Siedentop, 12 March -- 19 March; H. Skibsted, 7. March -- 19 March; J. P. Solovej, 12 March - 18 March; M. Travaglia, 31 Jan -- 8 Feb.; J. Yngvason, local person; H. Zenk, 13 -- 18 March; J. Schach-Moeller*, around 2nd workshop

### Seminars and Workshops.

• Thursday 26.01
• 16:00 Robert Seiringer, Princeton University

The Thermodynamic Pressure of a Dilute Fermi Gas

Abstract: We consider a gas of fermions with non-zero spin at temperature $T$ and chemical potential $\mu$. We show that if the range of the interparticle interaction is small compared to the mean particle distance, the thermodynamic pressure differs to leading order from the corresponding expression for non-interacting particles by a term proportional to the scattering length of the interparticle interaction. This is true for any repulsive interaction, including hard cores. The result is uniform in the temperature as long as $T$ is of the same order as the Fermi temperature, or smaller.

### 1st Workshop.

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

• Monday 7.02
• 11:00-12:00 Adam Majewski, Gdansk University

Positive quantum maps, states and entanglement

• Tuesday 8.02
• 11:00-12:00 Joao C. A. Barata, Univ. of Sao Paolo

Some stability results for quasi-periodically perturbed Hill's equations

Abstract: We show that quasi-periodic solutions of Hill's equations under quasiperiodic perturbations remain quasi-periodic if the strength of the perturbation belongs to a Cantor set of finite measure on a sufficiently small interval. One method is based on a resummation of formal Lindstedt series obtained as a solution of a generalized Ricatti equation. This is a joint work with G. Gentile (U, Roma III) and D. A. Cortez (IFUSP).

• Wednesday 9.02
• 11:00-12:00 Robert Olkiewicz, Wroclaw University

Decoherence in infinite quantum spin systems

• 16:00-17:00 Luigi Accardi, University of Rome

• Thursday 10.02.
• 11:00-12:00 Luigi Accardi, University of Rome

The emergence of the non crossing diagrams and of the QED. Hilbert module in the stochastic limit of non-relativistic QED.

• Wednesday, 9.03
• 15:30-16:30 Jozsef Lorinczi, Munich

Nelson's model: Ground state properties and infrared behaviour

• Thursday, 10.03
• 14:30-15:30 Roman Roschin, Moscow

Renormalized powers of boson fields

We define and study "powers" of creation and annihilation operators in boson Fock space by applying the standard renormalization procedure consists in introducing a cut-off and then trying to remove it by a limiting procedure. We show that, independently of the choice of the cut-off, one has the following symbolic relations: $(a^\dag_k)^{m} \to b^\dag_{m,k}$, $(a_k)^{m} \to b_{m,k}$, $(a^\dag_k)^{m-n} (a_k)^n \to 0$, where for each $m$ $b_{m,k}$, $b^\dag_{m,k}$ are new independent boson fields, $(a^\dag_k)^{m}$ is some reasonably defined sequence of (well-defined) operators in original boson Fock space, and the limit is understood in the sense of correlators. This oversimplified result can be considered as one more motivation for a new renormalization technique was introduced based on the idea of renormalizing a closed set of commutation relations and then finding a nontrivial representation for them.

• Friday, 11.03
• 14:30-15:30 H. Cornean, Aalborg

We revisit the theory of the Faraday effect in solids, which amounts for the computation of the conductivity tensor of noninteracting electrons, subjected to a periodic scalar potential and a constant magnetic field. Using the recently developed gauge invariant magnetic perturbation theory, we obtain a representation of the conductivity tensor which is free of the usual divergencies caused by the linear growth of the magnetic vector potential. An exact formula in the case of free electrons is also derived.

• 16:00-17:00 E. Skibsted Aarhus

Scattering for long-range magnetic fields

Two-dimensional magnetic systems with the field assumed to be homogeneous of degree $-1$ (to leading order) have a rich dynamical structure. We give an account of various recent results obtained with Horia Cornean and Ira Herbst. In particular if the field never vanishes for large $|x|$ there exists classically at high enough energies a globally attracting Lagrangian manifold given by logarithmic spirals. Based on this fact, we construct an approximate dynamics in quantum mechanics and prove asymptotic completeness. If the field has zeros all of them define classical channels. Only the stable ones exist in quantum mechanics, and for each of them we construct an approximate dynamics. Under certain conditions we prove completeness. For certain energies it may happen that the channel of logarithmic spirals and the channels of attracting half-lines coexist. There are also examples where none of them exist.

### 2nd Workshop.

Mon, 14 March -- Fri, 18 March, 2005.

One of the organizers, Jan Derezinski, will offer a course

during the period March 1 - May 30.