P1254156884NvKqe

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Scattering echoes in a waveguide with a ripple
cavity Hoshik Lee and L. E. ReichlThe Center for
Complex Quantum Systems, The University of Texas
at Austin C. JungCentro de Ciencias Fisicas,
UNAM, Mexico
Wavepacket Dynamics Send wavepacket from left
hand waveguide to right hand waveguide. Measure
the amplitude of the wavepacket at a point
(x,y) in the asymptotic region in the right
waveguide.
Chaotic Scattering and Tangles
Introduction Mixed phase space systems have shown
many interesting phenomena. An example is
scattering echoes. Classical chaotic scattering
is governed by a chaotic scattering layer
(homoclinic tangle) is built with stable and
unstable manifolds. Incoming trajectories
following the chaotic scattering layer rotate the
stable island. Depending initial conditions some
of them exit to asymptotic region at a unstable
fixed point (red arrow) and other resume to
rotate. Asymptotic outgoing flux oscillates in
time with the rotation frequency (Echoes).

• Pulse is observed
• is consistent
  with classical value, 7.50.5.
• Exponential decay in inset means we are unable
  to see hierarchy structure of the chaotic
  scattering layer

Husimi Plots
Stable and unstable manifolds of fixed points
(a) The state which lies on the large KAM island
at E11.1565858 (b) The state which lies on
period-6 island chain at E12.334 (c) The state
which lies on the chaotic scattering layer at
E12.502 High probability region covers
tangles near two outer fixed points
the mean return time of the Poincare
surface of section
Schematic view of a chaotic scattering described
by a ternary horseshoe 2
Conclusions The broad resonances in the
conductance fluctuations determine the period of
the scattering echoes. These resonances are due
to states in the chaotic scattering layer which
have high probability on heteroclinic
tangles. Three distinct families of resonances
are observed in the conductance
fluctuations. Each family is associated to
states lying on different region of the classical
Poincare surface of section (the central KAM
island, the island chain, and the chaotic
scattering layer) The transmission probability
for the wavepacket is time-periodic and the
period agrees with theoretical predictions
obtained from the underlying classical phase
space. The exponential decay shows the
hierarchy states do not play a significant role
in the quantum dynamics at the energies
considered.
with parameters given in previous figures
Waveguide and a ripple cavity Electron waveguide
has been widely studied for its conduction
properties of coherent electrons in two
dimensional space. A ripple cavity is useful to
study chaotic scattering because it has several
types of phase space depending on its geometrical
parameter. Both can be realized in semiconductor
heterostructures (ex. GaAs/AlGaAs). The
boundaries are hard wall at y0 and, Poincare
surface of section using Birkoff coordinate
at y0.
Scattering Echoes
Conductance and Resonances Using R-matrix Theory
Landauers formula to calculate conductance of
a waveguide with a ripple cavity.
Conductance Fluctuation

• Three resonance families are found
• O and represent resonance positions by the
  state lies on KAM island and island chain,
  respectively.
• All Fano type resonances
• . Broad resonances correspond to the state lies
  on a chaotic scattering layer. (partial transport
  layer)

References 1 Hoshik Lee, C. Jung and L. E.
Reichl, Phys. Rev. B. 73, 195315 (2006) 2 C.
Jung, C. Mejia-Monasterio, O. Merlo, and T.H.
Seligman, New J. Phys. 6, 48 (2004)

3 1.916 7.80
4 1.499 7.47
5 1.237 7.51
6 1.051 7.26
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http//order.ph.utexas.edu
Broad resonances looks to be periodic in energy
domain ? signature of scattering echoes !! (time
? energy)
The waveguide with a ripple cavity for
a0.0462,d0.1576, and Ld-a. Leads are attached
to the ripple cavity at x0 and x1. is the
reflection angle at the bottom. The classical
Poincare surface of section using Birkhoff
coordinates at the bottom boundary y0.