46 research outputs found
Fractal Weyl Law for Open Chaotic Maps
This contribution summarizes our work with M.Zworski on open quantum open
chaoticmaps (math-ph/0505034). For a simple chaotic scattering system (the open
quantum baker's map), we compute the "long-living resonances" in the
semiclassical r\'{e}gime, and show that they satisfy a fractal Weyl law. We can
prove this fractal law in the case of a modified model.Comment: Contribution to the Proceedings of the conference QMath9,
Mathematical Physics of Quantum Mechanics, September 12th-16th 2004, Giens,
Franc
Quantum Transport on KAM Tori
Although quantum tunneling between phase space tori occurs, it is suppressed
in the semiclassical limit for the Schr\"{o}dinger equation
of a particle in \bR^d under the influence of a smooth periodic potential.
In particular this implies that the distribution of quantum group velocities
near energy converges to the distribution of the classical asymptotic
velocities near , up to a term of the order \cO(1/\sqrt{E}).Comment: 21 page
Resonant cyclotron acceleration of particles by a time periodic singular flux tube
We study the dynamics of a classical nonrelativistic charged particle moving
on a punctured plane under the influence of a homogeneous magnetic field and
driven by a periodically time-dependent singular flux tube through the hole. We
observe an effect of resonance of the flux and cyclotron frequencies. The
particle is accelerated to arbitrarily high energies even by a flux of small
field strength which is not necessarily encircled by the cyclotron orbit; the
cyclotron orbits blow up and the particle oscillates between the hole and
infinity. We support this observation by an analytic study of an approximation
for small amplitudes of the flux which is obtained with the aid of averaging
methods. This way we derive asymptotic formulas that are afterwards shown to
represent a good description of the accelerated motion even for fluxes which
are not necessarily small. More precisely, we argue that the leading asymptotic
terms may be regarded as approximate solutions of the original system in the
asymptotic domain as the time tends to infinity
Stability of the electron cyclotron resonance
We consider the magnetic AC Stark effect for the quantum dynamics of a single
particle in the plane under the influence of an oscillating homogeneous
electric and a constant perpendicular magnetic field. We prove that the
electron cyclotron resonance is insensitive to impurity potentials.Comment: version to appear in Comm. Math. Phy
Spectral Stability of Unitary Network Models
We review various unitary network models used in quantum computing, spectral
analysis or condensed matter physics and establish relationships between them.
We show that symmetric one dimensional quantum walks are universal, as are CMV
matrices. We prove spectral stability and propagation properties for general
asymptotically uniform models by means of unitary Mourre theory
Energy-time uncertainty principle and lower bounds on sojourn time
One manifestation of quantum resonances is a large sojourn time, or
autocorrelation, for states which are initially localized. We elaborate on
Lavine's time-energy uncertainty principle and give an estimate on the sojourn
time. For the case of perturbed embedded eigenstates the bound is explicit and
involves Fermi's Golden Rule. It is valid for a very general class of systems.
We illustrate the theory by applications to resonances for time dependent
systems including the AC Stark effect as well as multistate systems.Comment: Version to appear in Annales Henri Poincar\'
Engineering stable quantum currents at bulk boundaries
We study transport properties of discrete quantum dynamical systems on the lattice, in particular Coined Quantum Walks and the Chalker-Coddington model. We prove existence of a non trivial charge transport and that the absolutely continuous spectrum covers the whole unit circle under mild assumptions. For Quantum Walks we exhibit explicit constructions of coins which imply existence of stable directed quantum currents along classical curves. The results are of topological nature and independent of the details of the model
A constant of quantum motion in two dimensions in crossed magnetic and electric fields
We consider the quantum dynamics of a single particle in the plane under the
influence of a constant perpendicular magnetic and a crossed electric potential
field. For a class of smooth and small potentials we construct a non-trivial
invariant of motion. Do to so we proof that the Hamiltonian is unitarily
equivalent to an effective Hamiltonian which commutes with the observable of
kinetic energy.Comment: 18 pages, 2 figures; the title was changed and several typos
corrected; to appear in J. Phys. A: Math. Theor. 43 (2010