134 research outputs found
Uniform approximation for the overlap caustic of a quantum state with its translations
The semiclassical Wigner function for a Bohr-quantized energy eigenstate is
known to have a caustic along the corresponding classical closed phase space
curve in the case of a single degree of freedom. Its Fourier transform, the
semiclassical chord function, also has a caustic along the conjugate curve
defined as the locus of diameters, i.e. the maximal chords of the original
curve. If the latter is convex, so is its conjugate, resulting in a simple fold
caustic. The uniform approximation through this caustic, that is here derived,
describes the transition undergone by the overlap of the state with its
translation, from an oscillatory regime for small chords, to evanescent
overlaps, rising to a maximum near the caustic. The diameter-caustic for the
Wigner function is also treated.Comment: 14 pages, 9 figure
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
Universal quantum signature of mixed dynamics in antidot lattices
We investigate phase coherent ballistic transport through antidot lattices in
the generic case where the classical phase space has both regular and chaotic
components. It is shown that the conductivity fluctuations have a non-Gaussian
distribution, and that their moments have a power-law dependence on a
semiclassical parameter, with fractional exponents. These exponents are
obtained from bifurcating periodic orbits in the semiclassical approximation.
They are universal in situations where sufficiently long orbits contribute.Comment: 7 page
Decoherence of Semiclassical Wigner Functions
The Lindblad equation governs general markovian evolution of the density
operator in an open quantum system. An expression for the rate of change of the
Wigner function as a sum of integrals is one of the forms of the Weyl
representation for this equation. The semiclassical description of the Wigner
function in terms of chords, each with its classically defined amplitude and
phase, is thus inserted in the integrals, which leads to an explicit
differential equation for the Wigner function. All the Lindblad operators are
assumed to be represented by smooth phase space functions corresponding to
classical variables. In the case that these are real, representing hermitian
operators, the semiclassical Lindblad equation can be integrated. There results
a simple extension of the unitary evolution of the semiclassical Wigner
function, which does not affect the phase of each chord contribution, while
dampening its amplitude. This decreases exponentially, as governed by the time
integral of the square difference of the Lindblad functions along the classical
trajectories of both tips of each chord. The decay of the amplitudes is shown
to imply diffusion in energy for initial states that are nearly pure.
Projecting the Wigner function onto an orthogonal position or momentum basis,
the dampening of long chords emerges as the exponential decay of off-diagonal
elements of the density matrix.Comment: 23 pg, 2 fi
Duality between quantum and classical dynamics for integrable billiards
We establish a duality between the quantum wave vector spectrum and the
eigenmodes of the classical Liouvillian dynamics for integrable billiards.
Signatures of the classical eigenmodes appear as peaks in the correlation
function of the quantum wave vector spectrum. A semiclassical derivation and
numerical calculations are presented in support of the results. These classical
eigenmodes can be observed in physical experiments through the auto-correlation
of the transmission coefficient of waves in quantum billiards. Exact classical
trace formulas of the resolvent are derived for the rectangle, equilateral
triangle, and circle billiards. We also establish a correspondence between the
classical periodic orbit length spectrum and the quantum spectrum for
integrable polygonal billiards.Comment: 12 pages, 4 figure
Closed orbits and spatial density oscillations in the circular billiard
We present a case study for the semiclassical calculation of the oscillations
in the particle and kinetic-energy densities for the two-dimensional circular
billiard. For this system, we can give a complete classification of all closed
periodic and non-periodic orbits. We discuss their bifurcations under variation
of the starting point r and derive analytical expressions for their properties
such as actions, stability determinants, momentum mismatches and Morse indices.
We present semiclassical calculations of the spatial density oscillations using
a recently developed closed-orbit theory [Roccia J and Brack M 2008 Phys. Rev.
Lett. 100 200408], employing standard uniform approximations from perturbation
and bifurcation theory, and test the convergence of the closed-orbit sum.Comment: LaTeX, 42 pp., 17 figures (24 *.eps files, 1 *.tex file); final
version (v3) to be published in J. Phys.
Uniform approximations for pitchfork bifurcation sequences
In non-integrable Hamiltonian systems with mixed phase space and discrete
symmetries, sequences of pitchfork bifurcations of periodic orbits pave the way
from integrability to chaos. In extending the semiclassical trace formula for
the spectral density, we develop a uniform approximation for the combined
contribution of pitchfork bifurcation pairs. For a two-dimensional double-well
potential and the familiar H\'enon-Heiles potential, we obtain very good
agreement with exact quantum-mechanical calculations. We also consider the
integrable limit of the scenario which corresponds to the bifurcation of a
torus from an isolated periodic orbit. For the separable version of the
H\'enon-Heiles system we give an analytical uniform trace formula, which also
yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain
excellent agreement with the slightly coarse-grained quantum-mechanical density
of states.Comment: LaTeX, 31 pp., 18 figs. Version (v3): correction of several misprint
Bohr-Sommerfeld Quantization of Periodic Orbits
We show, that the canonical invariant part of corrections to the
Gutzwiller trace formula and the Gutzwiller-Voros spectral determinant can be
computed by the Bohr-Sommerfeld quantization rules, which usually apply for
integrable systems. We argue that the information content of the classical
action and stability can be used more effectively than in the usual treatment.
We demonstrate the improvement of precision on the example of the three disk
scattering system.Comment: revte
Accuracy of the Semi--Classical Approximation: the Pullen Edmonds Hamiltonian
A test on the numerical accuracy of the semiclassical approximation as a
function of the principal quantum number has been performed for the
Pullen--Edmonds model, a two--dimensional, non--integrable, scaling invariant
perturbation of the resonant harmonic oscillator. A perturbative interpretation
is obtained of the recently observed phenomenon of the accuracy decrease on the
approximation of individual energy levels at the increase of the principal
quantum number. Moreover, the accuracy provided by the semiclassical
approximation formula is on the average the same as that provided by quantum
perturbation theory.Comment: 12 pages, 5 figures (available upon request to the authors), LaTex,
DFPD/93/TH/47, to be published in Nuovo Cimento
Analytic approach to bifurcation cascades in a class of generalized H\'enon-Heiles potentials
We derive stability traces of bifurcating orbits in H\'enon-Heiles potentials
near their saddlesComment: LaTeX revtex4, 38 pages, 7 PostScript figures, 2 table
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