47 research outputs found
On Di\'osi-Penrose criterion of gravity-induced quantum collapse
It is shown that the Di\'osi-Penrose criterion of gravity-induced quantum
collapse may be inconsistent with the discreteness of space-time, which is
generally considered as an indispensable element in a complete theory of
quantum gravity. Moreover, the analysis also suggests that the discreteness of
space-time may result in rapider collapse of the superposition of energy
eigenstates than required by the Di\'osi-Penrose criterion.Comment: 5 pages, no figure
Continuous Quantum Measurement and the Quantum to Classical Transition
While ultimately they are described by quantum mechanics, macroscopic
mechanical systems are nevertheless observed to follow the trajectories
predicted by classical mechanics. Hence, in the regime defining macroscopic
physics, the trajectories of the correct classical motion must emerge from
quantum mechanics, a process referred to as the quantum to classical
transition. Extending previous work [Bhattacharya, Habib, and Jacobs, Phys.
Rev. Lett. {\bf 85}, 4852 (2000)], here we elucidate this transition in some
detail, showing that once the measurement processes which affect all
macroscopic systems are taken into account, quantum mechanics indeed predicts
the emergence of classical motion. We derive inequalities that describe the
parameter regime in which classical motion is obtained, and provide numerical
examples. We also demonstrate two further important properties of the classical
limit. First, that multiple observers all agree on the motion of an object, and
second, that classical statistical inference may be used to correctly track the
classical motion.Comment: 12 pages, 4 figures, Revtex
Chaotic Diffusion on Periodic Orbits: The Perturbed Arnol'd Cat Map
Chaotic diffusion on periodic orbits (POs) is studied for the perturbed
Arnol'd cat map on a cylinder, in a range of perturbation parameters
corresponding to an extended structural-stability regime of the system on the
torus. The diffusion coefficient is calculated using the following PO formulas:
(a) The curvature expansion of the Ruelle zeta function. (b) The average of the
PO winding-number squared, , weighted by a stability factor. (c) The
uniform (nonweighted) average of . The results from formulas (a) and (b)
agree very well with those obtained by standard methods, for all the
perturbation parameters considered. Formula (c) gives reasonably accurate
results for sufficiently small parameters corresponding also to cases of a
considerably nonuniform hyperbolicity. This is due to {\em uniformity sum
rules} satisfied by the PO Lyapunov eigenvalues at {\em fixed} . These sum
rules follow from general arguments and are supported by much numerical
evidence.Comment: 6 Tables, 2 Figures (postscript); To appear in Physical Review
Bailout Embeddings, Targeting of KAM Orbits, and the Control of Hamiltonian Chaos
We present a novel technique, which we term bailout embedding, that can be
used to target orbits having particular properties out of all orbits in a flow
or map. We explicitly construct a bailout embedding for Hamiltonian systems so
as to target KAM orbits. We show how the bailout dynamics is able to lock onto
extremely small KAM islands in an ergodic sea.Comment: 3 figures, 9 subpanel
Interference in a Spherical Phase-Space and Asymptotic-Behavior of the Rotation Matrices
We extend the interference in the phase-space algorithm of Wheeler and Schleich [W. P. Schleich and J. A. Wheeler, Nature 326, 574 (1987)] to the case of a compact, spherical topology in order to discuss the large j limits of the angular momentum marginal probability distributions. These distributions are given in terms of the standard rotation matrices. It is shown that the asymptotic distributions are given very simply by areas of overlap in the classical spherical phase-space parametrized by the components of angular momentum. The results indicate the very general validity of the interference in phase-space concept for computing semiclassical limits in quantum mechanics
Quantum Chaos Versus Classical Chaos: Why is Quantum Chaos Weaker?
We discuss the questions: How to compare quantitatively classical chaos with
quantum chaos? Which one is stronger? What are the underlying physical reasons
Introduction to Quantum-Gravity Phenomenology
After a brief review of the first phase of development of Quantum-Gravity
Phenomenology, I argue that this research line is now ready to enter a more
advanced phase: while at first it was legitimate to resort to heuristic
order-of-magnitude estimates, which were sufficient to establish that
sensitivity to Planck-scale effects can be achieved, we should now rely on
detailed analyses of some reference test theories. I illustrate this point in
the specific example of studies of Planck-scale modifications of the
energy/momentum dispersion relation, for which I consider two test theories.
Both the photon-stability analyses and the Crab-nebula synchrotron-radiation
analyses, which had raised high hopes of ``beyond-Plankian'' experimental
bounds, turn out to be rather ineffective in constraining the two test
theories. Examples of analyses which can provide constraints of rather wide
applicability are the so-called ``time-of-flight analyses'', in the context of
observations of gamma-ray bursts, and the analyses of the cosmic-ray spectrum
near the GZK scale.Comment: 46 pages, LaTex. Based on lectures given at the 40th Karpacz Winter
School in Theoretical Physic
Level spacing statistics of classically integrable systems -Investigation along the line of the Berry-Robnik approach-
By extending the approach of Berry and Robnik, the limiting level spacing
distribution of a system consisting of infinitely many independent components
is investigated. The limiting level spacing distribution is characterized by a
single monotonically increasing function of the level spacing
. Three cases are distinguished: (i) Poissonian if ,
(ii) Poissonian for large , but possibly not for small if
, and (iii) sub-Poissonian if .
This implies that, even when energy-level distributions of individual
components are statistically independent, non-Poissonian level spacing
distributions are possible.Comment: 19 pages, 4 figures. Accepted for publication in Phys. Rev.
Quantum phase transition in the Frenkel-Kontorova chain: from pinned instanton glass to sliding phonon gas
We study analytically and numerically the one-dimensional quantum
Frenkel-Kontorova chain in the regime when the classical model is located in
the pinned phase characterized by the gaped phonon excitations and devil's
staircase. By extensive quantum Monte Carlo simulations we show that for the
effective Planck constant smaller than the critical value the
quantum chain is in the pinned instanton glass phase. In this phase the
elementary excitations have two branches: phonons, separated from zero energy
by a finite gap, and instantons which have an exponentially small excitation
energy. At the quantum phase transition takes place and for
the pinned instanton glass is transformed into the sliding
phonon gas with gapless phonon excitations. This transition is accompanied by
the divergence of the spatial correlation length and appearence of sliding
modes at .Comment: revtex 16 pages, 18 figure
Classical Evolution of Quantum Elliptic States
The hydrogen atom in weak external fields is a very accurate model for the
multiphoton excitation of ultrastable high angular momentum Rydberg states, a
process which classical mechanics describes with astonishing precision. In this
paper we show that the simplest treatment of the intramanifold dynamics of a
hydrogenic electron in external fields is based on the elliptic states of the
hydrogen atom, i.e., the coherent states of SO(4), which is the dynamical
symmetry group of the Kepler problem. Moreover, we also show that classical
perturbation theory yields the {\it exact} evolution in time of these quantum
states, and so we explain the surprising match between purely classical
perturbative calculations and experiments. Finally, as a first application, we
propose a fast method for the excitation of circular states; these are
ultrastable hydrogenic eigenstates which have maximum total angular momentum
and also maximum projection of the angular momentum along a fixed direction. %Comment: 8 Pages, 2 Figures. Accepted for publication in Phys. Rev.