21 research outputs found
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
Reflection from a free carrier front via an intraband indirect photonic transition
The reflection of light from moving boundaries is of interest both fundamentally and for applications in frequency conversion, but typically requires high pump power. By using a dispersion-engineered silicon photonic crystal waveguide, we are able to achieve a propagating free carrier front with only a moderate on-chip peak power of 6 W in a 6 ps-long pump pulse. We employ an intraband indirect photonic transition of a co-propagating probe, whereby the probe practically escapes from the front in the forward direction. This forward reflection has up to 35% efficiency and it is accompanied by a strong frequency upshift, which significantly exceeds that expected from the refractive index change and which is a function of group velocity, waveguide dispersion and pump power. Pump, probe and shifted probe all are around 1.5 ÎĽm wavelength which opens new possibilities for "on-chip" frequency manipulation and all-optical switching in optical telecommunications