46 research outputs found
Correspondence in Quasiperiodic and Chaotic Maps: Quantization via the von Neumann Equation
A generalized approach to the quantization of a large class of maps on a
torus, i.e. quantization via the von Neumann Equation, is described and a
number of issues related to the quantization of model systems are discussed.
The approach yields well behaved mixed quantum states for tori for which the
corresponding Schrodinger equation has no solutions, as well as an extended
spectrum for tori where the Schrodinger equation can be solved.
Quantum-classical correspondence is demonstrated for the class of mappings
considered, with the Wigner-Weyl density going to the correct
classical limit. An application to the cat map yields, in a direct manner,
nonchaotic quantum dynamics, plus the exact chaotic classical propagator in the
correspondence limit.Comment: 36 pages, RevTex preprint forma
Irreversible Quantum Baker Map
We propose a generalization of the model of classical baker map on the torus,
in which the images of two parts of the phase space do overlap. This
transformation is irreversible and cannot be quantized by means of a unitary
Floquet operator. A corresponding quantum system is constructed as a completely
positive map acting in the space of density matrices. We investigate spectral
properties of this super-operator and their link with the increase of the
entropy of initially pure states.Comment: 4 pages, 3 figures include
QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS
We present a unified framework for the quantization of a family of discrete
dynamical systems of varying degrees of "chaoticity". The systems to be
quantized are piecewise affine maps on the two-torus, viewed as phase space,
and include the automorphisms, translations and skew translations. We then
treat some discontinuous transformations such as the Baker map and the
sawtooth-like maps. Our approach extends some ideas from geometric quantization
and it is both conceptually and calculationally simple.Comment: no. 28 pages in AMSTE
Quantum Multibaker Maps: Extreme Quantum Regime
We introduce a family of models for quantum mechanical, one-dimensional
random walks, called quantum multibaker maps (QMB). These are Weyl
quantizations of the classical multibaker models previously considered by
Gaspard, Tasaki and others. Depending on the properties of the phases
parametrizing the quantization, we consider only two classes of the QMB maps:
uniform and random. Uniform QMB maps are characterized by phases which are the
same in every unit cell of the multibaker chain. Random QMB maps have phases
that vary randomly from unit cell to unit cell. The eigenstates in the former
case are extended while in the latter they are localized. In the uniform case
and for large , analytic solutions can be obtained for the time
dependent quantum states for periodic chains and for open chains with absorbing
boundary conditions. Steady state solutions and the properties of the
relaxation to a steady state for a uniform QMB chain in contact with
``particle'' reservoirs can also be described analytically. The analytical
results are consistent with, and confirmed by, results obtained from numerical
methods. We report here results for the deep quantum regime (large ) of
the uniform QMB, as well as some results for the random QMB. We leave the
moderate and small results as well as further consideration of the
other versions of the QMB for further publications.Comment: 17 pages, referee's and editor's comments addresse
Periodic orbit spectrum in terms of Ruelle--Pollicott resonances
Fully chaotic Hamiltonian systems possess an infinite number of classical
solutions which are periodic, e.g. a trajectory ``p'' returns to its initial
conditions after some fixed time tau_p. Our aim is to investigate the spectrum
tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for
the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the
eigenvalues of the classical evolution operator. The density is naturally
decomposed into a smooth part plus an interferent sum over oscillatory terms.
The frequencies of the oscillatory terms are given by the imaginary part of the
complex eigenvalues (Ruelle--Pollicott resonances). For large periods,
corrections to the well--known exponential growth of the smooth part of the
density are obtained. An alternative formula for rho(tau) in terms of the zeros
and poles of the Ruelle zeta function is also discussed. The results are
illustrated with the geodesic motion in billiards of constant negative
curvature. Connections with the statistical properties of the corresponding
quantum eigenvalues, random matrix theory and discrete maps are also
considered. In particular, a random matrix conjecture is proposed for the
eigenvalues of the classical evolution operator of chaotic billiards
Anomalous particle-number fluctuations in a three-dimensional interacting Bose-Einstein condensate
The particle-number fluctuations originated from collective excitations are
investigated for a three-dimensional, repulsively interacting Bose-Einstein
condensate (BEC) confined in a harmonic trap. The contribution due to the
quantum depletion of the condensate is calculated and the explicit expression
of the coefficient in the formulas denoting the particle-number fluctuations is
given. The results show that the particle-number fluctuations of the condensate
follow the law and the fluctuations vanish when
temperature approaches to the BEC critical temperature.Comment: RevTex, 4 page
Casimir energy in multiply connected static hyperbolic Universes
We generalize a previously obtained result, for the case of a few other
static hyperbolic universes with manifolds of nontrivial topology as spatial
sections.Comment: accepted for publicatio
Scalar wave propagation in topological black hole backgrounds
We consider the evolution of a scalar field coupled to curvature in
topological black hole spacetimes. We solve numerically the scalar wave
equation with different curvature-coupling constant and show that a rich
spectrum of wave propagation is revealed when is introduced. Relations
between quasinormal modes and the size of different topological black holes
have also been investigated.Comment: 26 pages, 18 figure
Gravitational Faraday rotation in a weak gravitational field
We examine the rotation of the plane of polarization for linearly polarized
light rays by the weak gravitational field of an isolated physical system.
Based on the rotation of inertial frames, we review the general integral
expression for the net rotation. We apply this formula, analogue to the usual
electromagnetic Faraday effect, to some interesting astrophysical systems:
uniformly shifting mass monopoles and a spinning external shell.Comment: 10 pages; accepted for publication in Phys. Rev.
Distribution of Eigenvalues for the Modular Group
The two-point correlation function of energy levels for free motion on the
modular domain, both with periodic and Dirichlet boundary conditions, are
explicitly computed using a generalization of the Hardy-Littlewood method. It
is shown that ion the limit of small separations they show an uncorrelated
behaviour and agree with the Poisson distribution but they have prominent
number-theoretical oscillations at larger scale. The results agree well with
numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital,
but can be obtained upon request from the first Autho