Download Uzy Smilansky

Stationary scattering on non-linear networks (qm-graphs)
Sven Gnutzmann (Nottingham), Stas Derevyanko (Aston)
and
Uzy Smilansky
Warsaw May, 2011
Abstract:
Transmission through a complex network of nonlinear one-dimensional leads is discussed
by extending the stationary scattering theory on quantum graphs to the nonlinear regime.
The resonances which dominate linear scattering are shown to be extremely sensitive to the
nonlinearity and display multi-stability and hysteresis. This work provides a framework for
the study of light propagation in complex optical networks, and for studying universal
properties of Bose-Einstein Condensate (BEC) in connected chaotic traps.
PHYS. REV. A 83, 033831 (2011)
Motivations:
1. Develop a paradigm for studying the effects of non linearity in physical systems.
2. Applications for fiber lasers consisting of inter connected network of optical fibers.
Introduction : Scattering on linear quantum graphs
Example:
Amplification of the interior wave-function at resonance
k
Nonlinear Schroedinger equation on metric graphs – stationary solutions.
x
In this way the NLSE can be written as Euler-Lagrange equations
for a particle in two dimensions with a central potential and polar coordinates
Classical particle in an axially symmetric potential with two constants of motion
H and l
Elliptic integrals: manageable but not “user-friendly”
Explicit solution on the line
attractive
repulsive
NLSE on graphs
Graph: V vertices, B internal bonds, N external leads
Linear wave equation on the external leads
Nonlinear wave equation on the internal bonds
From the classical picture on the bonds: Conservation of H and of
Boundary (Matching) condition at the vertex :
1. Continuity:
2. Flux conservation:
Using this formalism one can easily prove that scattering from the graph conserves flux:
However
and only in the linear limit:
The nonlinear scattering operator is a norm preserving mapping
Scattering from a star graph
Back to a more complex graph
Numerical simulations for the graph
Conclusions:
1. Nonlinear effects are magnified near resonances, the narrower the resonance
the stronger the magnification.
2. Graphs are cavities which support narrow resonances and therefore are more
susceptible to nonlinear effects.
3. This may have important effect on e.g., fiber-optic lasers.
Open problems:
0. Linear theory: statistics of the amplification factor.
1. The non-linear scattering operator may have interesting properties which are not yet
investigated.
2. Connection to time dependent theory.
3. Physical basis of the boundary conditions.
4. The applicability of the NLSE for the description of optical fibers.
5. Inclusion of dissipation, optically active materials…
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