- Ulrich
Kuhl (Marburg)
"Studies of resonances
in open microwave cavities by the method of harmonic inversion"
From the measurement of a reflection spectrum of an open
microwave cavity the poles of the scattering matrix in the complex
plane have been determined [1]. The resonances have been extracted by
means of the harmonic inversion method [2]. By this it became possible
to resolve the resonances in a regime where the line widths exceed the
mean level spacing up to a factor of 10, a value inaccessible in
experiments up to now. For example the distributions of line widths
were studied and found to be in good agreement with predictions from
random matrix theory [3].
[1] UK, R. Höhmann, J. Main, and H.-J. Stöckmann, Phys. Rev.
Lett. 100, 254101 (2008)
[2] J. Main, Phys. Rep. 316,
233 (1999).
[3] H.~J. Sommers, Y.~V. Fyodorov, and M. Titov, J. Phys. A 32,
L77 (1999).
- Fabrice Mortessagne
(Nice)
" Wave Chaos in a
chaotic optical fiber: from ergodic modes to gain-controlled scar
modes"
For now a few years, we experimentally proved that a passive
optical fiber with a chaotic-billiard-shape transverse section
constitutes a powerful system for studying (and imaging) wave chaos
[1], and particularly when scars are involved [2]. Recently, we propose
an original mechanism to select scar modes through coherent gain
amplification in a multimode D-shaped fiber. More precisely, we
demonstrate the selective amplification of scar modes by positioning a
gain region in the vicinity of the self-focal point of the shortest
periodic orbit in the transverse motion.
[1] V. Doya, O. Legrand and F. Mortessagne, Phys. Rev. E 65,
056223 (2002).
[2] V. Doya, O. Legrand and F. Mortessagne, Phys. Rev. Lett.
88, 014102 (2002).
[3] C. Michel, V. Doya, O. Legrand and F. Mortessagne, Phys.
Rev. Lett. 99, 224101 (2007).
- Mélanie
Lebental (Paris-Nord)
"Organic micro-lasers
as test-beds for wave chaos in open systems" Slides
There is
currently a great deal of interest in flat dielectric micro-cavities
due to their numerous practical applications and their use to test
fundamental physics such as quantum chaos. Actually, due to a formal
analogy, the electromagnetic field plays the role of a quantum
particule. The classical limit then corresponds to the geometrical
optics limit. The original aspect of our study is based on the use of
organic materials to make flat micro-lasers with various shapes
(stadium, disk, polygons, etc...). In fact their low refractive
indexes facilitate the output coupling of the light confined inside
the resonator. The generic features of these open billiards are very
different from what is expected for equivalent closed systems. In
particular the trace formula must be modified to take into account
the losses. It can be derived analytically in two cases and we
proposed a general expression which can be tested experimentally. We
demonstrated that lasing can be used to monitor the weight
coefficients of the periodic orbits and get an insight into the
imaginary part of the passive resonator spectrum.
- Achim
Richter (Darmstadt)
"Chaotic Scattering in
Microwave Billiards: Isolated and Overlapping Resonances*" Slides
Measurements of transmission and reflection spectra of
two-dimensional microwave resonators coupled to two antennas are
discussed. Below a certain cut-off frequency the resonators simulate a
chaotic quantum system whose eigenvalues manifest themselves as
resonances with average spacing D and average width Γ. Experimental
observables are scattering amplitudes, i.e. S-matrix elements, and
their phases determined in the region of isolated and overlapping
resonances for time-reversal invariant (GOE) and non-invariant (GUE)
systems, respectively, as a function of frequency. Particular emphasis
is given to the region of overlapping resonances, i.e. Γ ≥ D, where two
S-matrix elements are correlated over a certain range of frequency. If,
however, the S-matrix elements are Fourier transformed from the
frequency into the time domain a sample of statistically independent
Fourier coefficients is obtained which can then be compared to
predictions from the theory of chaotic scattering developed in the
framework of nuclear reaction theory [1,2,3]. One of the results
discussed is: although a single resonance decays exponentially in time
the decay of a system of resonances with random parameters is in
general non-exponential.
* Supported by the DFG within the SFB634
[1] H. Alt, H.-D. Gräf, H.L. Harney, R. Hofferbert, H.
Lengeler, A. Richter, P. Schardt, H.A. Weidenmüller, Phys. Rev.
Lett. 74,62 (1995).
[2] B.Dietz, T. Friedrich, H.L. Harney, M. Miski-Oglu, A.
Richter, F. Schäfer, H.A. Weidenmüller, Phys. Rev. E, in
press.
[3] B.Dietz, T. Friedrich, H.L. Harney, M. Miski-Oglu, A.
Richter, F. Schäfer, J.J.M. Verbaarshoot, H.A. Weidenmüller,
to be published.
- S.
Gouëzel (Rennes)
"Spectral
gap for piecewise hyperbolic maps
"
In the last few years, several approaches have been devised
to study the spectral properties of smooth hyperbolic maps.
Unfortunately, none of them could possibly apply to billiards. As a
first step in this direction, we have developed with V. Baladi a new
framework to deal with hyperbolic maps with discontinuities, combining
analytic (anisotropic spaces of Triebel type) and geometric (a class of
foliations that are locally invariant under the map) tools. This
approach covers a large class of maps but requires a bunching
condition, which is probably only an artefact of the method but is
always satisfied when the stable or unstable direction is
one-dimensional.
- O. Bandtlow
(Queen Mary, London)
"Asymptotics of Ruelle
resonances"
Ruelle resonances are the complex exponential decay-rates of
time correlation functions of a dynamical system. If the system is
given by a real analytic expanding map and the observables are smooth
then it is known that the Ruelle resonances form a sequence converging
to zero. It is also known that in this case the Ruelle resonances
correspond to the eigenvalues λ_{n} of a certain
compact operator (known as
transfer operator in this context), which acts on spaces of
holomorphic functions.
The talk will be concerned with the problem of determining
the
asymptotics of
the eigenvalue sequence {λ_{n}}. In particular,
I will discuss a method that yields simple, explicitly computable
upper
bounds for each λ_{n}.
This is joint work with O. Jenkinson.
- Hans Henrik Rugh
(Cergy)
"Resonances,
intermittency and Möbius transformations"
Möbius transformations mapping the interval [0,1] into
itself
have the advantage of admitting nice analytic extensions to
the Riemann Sphere. Pertinent examples are the Farey map and
the Gauss map.
Associated with these transformations we have Ruelle
transfer operators
and their Fredholm determinants, and the latter are poised for
interesting and surprising analytic properties. We will discuss
results for such maps and some related open problems.
- Nicolas
Roy (Berlin)
"A microlocal approach
to Ruelle resonances of Anosov diffeomorphisms"
If f is an Anosov diffeomorphism on a compact manifold, the
decay of the dynamical correlation functions is governed by the
so-called Ruelle resonances. It follows from the works of Baladi &
al and Liverani & al, that these resonances can be obtained by a
suitable spectral analysis of the composition operator (or another one
related to it) called the "Transfer operator".
In this talk, we will show how these results can be obtained by a
systematic microlocal analysis. We will explain also why microlocal
analysis is natural in this problem, and mention possible perspectives
of the approach.
The talk is based on the recent paper :
[Faure, Roy, Sjoestrand] "Semi-classical approach for Anosov
diffeomorphisms and Ruelle resonances". Open Math. Journal (2008), vol.
1, 35--81.