630 research outputs found
Trace identities and their semiclassical implications
The compatibility of the semiclassical quantization of area-preserving maps
with some exact identities which follow from the unitarity of the quantum
evolution operator is discussed. The quantum identities involve relations
between traces of powers of the evolution operator. For classically {\it
integrable} maps, the semiclassical approximation is shown to be compatible
with the trace identities. This is done by the identification of stationary
phase manifolds which give the main contributions to the result. The same
technique is not applicable for {\it chaotic} maps, and the compatibility of
the semiclassical theory in this case remains unsettled. The compatibility of
the semiclassical quantization with the trace identities demonstrates the
crucial importance of non-diagonal contributions.Comment: LaTeX - IOP styl
Can One Hear the Shape of a Graph?
We show that the spectrum of the Schrodinger operator on a finite, metric
graph determines uniquely the connectivity matrix and the bond lengths,
provided that the lengths are non-commensurate and the connectivity is simple
(no parallel bonds between vertices and no loops connecting a vertex to
itself). That is, one can hear the shape of the graph! We also consider a
related inversion problem: A compact graph can be converted into a scattering
system by attaching to its vertices leads to infinity. We show that the
scattering phase determines uniquely the compact part of the graph, under
similar conditions as above.Comment: 9 pages, 1 figur
Spin-Boson Hamiltonian and Optical Absorption of Molecular Dimers
An analysis of the eigenstates of a symmetry-broken spin-boson Hamiltonian is
performed by computing Bloch and Husimi projections. The eigenstate analysis is
combined with the calculation of absorption bands of asymmetric dimer
configurations constituted by monomers with nonidentical excitation energies
and optical transition matrix elements. Absorption bands with regular and
irregular fine structures are obtained and related to the transition from the
coexistence to a mixing of adiabatic branches in the spectrum. It is shown that
correlations between spin states allow for an interpolation between absorption
bands for different optical asymmetries.Comment: 15 pages, revTeX, 8 figures, accepted for publication in Phys. Rev.
Spectral Statistics in Chaotic Systems with Two Identical Connected Cells
Chaotic systems that decompose into two cells connected only by a narrow
channel exhibit characteristic deviations of their quantum spectral statistics
from the canonical random-matrix ensembles. The equilibration between the cells
introduces an additional classical time scale that is manifest also in the
spectral form factor. If the two cells are related by a spatial symmetry, the
spectrum shows doublets, reflected in the form factor as a positive peak around
the Heisenberg time. We combine a semiclassical analysis with an independent
random-matrix approach to the doublet splittings to obtain the form factor on
all time (energy) scales. Its only free parameter is the characteristic time of
exchange between the cells in units of the Heisenberg time.Comment: 37 pages, 15 figures, changed content, additional autho
Transport and dynamics on open quantum graphs
We study the classical limit of quantum mechanics on graphs by introducing a
Wigner function for graphs. The classical dynamics is compared to the quantum
dynamics obtained from the propagator. In particular we consider extended open
graphs whose classical dynamics generate a diffusion process. The transport
properties of the classical system are revealed in the scattering resonances
and in the time evolution of the quantum system.Comment: 42 pages, 13 figures, submitted to PR
Eigenstate Structure in Graphs and Disordered Lattices
We study wave function structure for quantum graphs in the chaotic and
disordered regime, using measures such as the wave function intensity
distribution and the inverse participation ratio. The result is much less
ergodicity than expected from random matrix theory, even though the spectral
statistics are in agreement with random matrix predictions. Instead, analytical
calculations based on short-time semiclassical behavior correctly describe the
eigenstate structure.Comment: 4 pages, including 2 figure
Effective Coupling for Open Billiards
We derive an explicit expression for the coupling constants of individual
eigenstates of a closed billiard which is opened by attaching a waveguide. The
Wigner time delay and the resonance positions resulting from the coupling
constants are compared to an exact numerical calculation. Deviations can be
attributed to evanescent modes in the waveguide and to the finite number of
eigenstates taken into account. The influence of the shape of the billiard and
of the boundary conditions at the mouth of the waveguide are also discussed.
Finally we show that the mean value of the dimensionless coupling constants
tends to the critical value when the eigenstates of the billiard follow
random-matrix theory
Fidelity amplitude of the scattering matrix in microwave cavities
The concept of fidelity decay is discussed from the point of view of the
scattering matrix, and the scattering fidelity is introduced as the parametric
cross-correlation of a given S-matrix element, taken in the time domain,
normalized by the corresponding autocorrelation function. We show that for
chaotic systems, this quantity represents the usual fidelity amplitude, if
appropriate ensemble and/or energy averages are taken. We present a microwave
experiment where the scattering fidelity is measured for an ensemble of chaotic
systems. The results are in excellent agreement with random matrix theory for
the standard fidelity amplitude. The only parameter, namely the perturbation
strength could be determined independently from level dynamics of the system,
thus providing a parameter free agreement between theory and experiment
Periodic-Orbit Theory of Anderson Localization on Graphs
We present the first quantum system where Anderson localization is completely
described within periodic-orbit theory. The model is a quantum graph analogous
to an a-periodic Kronig-Penney model in one dimension. The exact expression for
the probability to return of an initially localized state is computed in terms
of classical trajectories. It saturates to a finite value due to localization,
while the diagonal approximation decays diffusively. Our theory is based on the
identification of families of isometric orbits. The coherent periodic-orbit
sums within these families, and the summation over all families are performed
analytically using advanced combinatorial methods.Comment: 4 pages, 3 figures, RevTe
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