674 research outputs found
Motion of vortex lines in nonlinear wave mechanics
We extend our previous analysis of the motion of vortex lines [I.
Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa, Phys. Rev. A 61, 032110
(2000)] from linear to a nonlinear Schroedinger equation with harmonic forces.
We also argue that under certain conditions the influence of the contact
nonlinearity on the motion of vortex lines is negligible. The present analysis
adds new weight to our previous conjecture that the topological features of
vortex dynamics are to a large extent universal.Comment: To appear in Phys. Rev. A, 4 page
Squeezing of electromagnetic field in a cavity by electrons in Trojan states
The notion of the Trojan state of a Rydberg electron, introduced by
I.Bialynicki-Birula, M.Kali\'nski, and J.H.Eberly (Phys. Rev. Lett. 73, 1777
(1994)) is extended to the case of the electromagnetic field quantized in
acavity. The shape of the electronic wave packet describing the Trojan state is
practically the same as in the previously studied externally driven system. The
fluctuations of the quantized electromagnetic field around its classical value
exhibit strong squeezing. The emergence of Trojan states in the cylindrically
symmetrical system is attributed to spontaneous symmetry braking.Comment: 9 pages, 8 figure
Roots of the affine Cremona group
Let k[x_1,...,x_n] be the polynomial algebra in n variables and let A^n=Spec
k[x_1,...,x_n]. In this note we show that the root vectors of the affine
Cremona group Aut(A^n) with respect to the diagonal torus are exactly the
locally nilpotent derivations x^a\times d/dx_i, where x^a is any monomial not
depending on x_i. This answers a question due to Popov.Comment: 4 page
Dynamical Casimir effect in oscillating media
We show that oscillations of a homogeneous medium with constant material
coefficients produce pairs of photons. Classical analysis of an oscillating
medium reveals regions of parametric resonance where the electromagnetic waves
are exponentially amplified. The quantum counterpart of parametric resonance is
an exponentially growing number of photons in the same parameter regions. This
process may be viewed as another manifestation of the dynamical Casimir effect.
However, in contrast to the standard dynamical Casimir effect, photon
production here takes place in the entire volume and is not due to time
dependence of the boundary conditions or material constants
Electromagnetic vortex lines riding atop null solutions of the Maxwell equations
New method of introducing vortex lines of the electromagnetic field is
outlined. The vortex lines arise when a complex Riemann-Silberstein vector
is multiplied by a complex scalar function
. Such a multiplication may lead to new solutions of the Maxwell
equations only when the electromagnetic field is null, i.e. when both
relativistic invariants vanish. In general, zeroes of the function give
rise to electromagnetic vortices. The description of these vortices benefits
from the ideas of Penrose, Robinson and Trautman developed in general
relativity.Comment: NATO Workshop on Singular Optics 2003 To appear in Journal of Optics
Collision entropy and optimal uncertainty
We propose an alternative measure of quantum uncertainty for pairs of
arbitrary observables in the 2-dimensional case, in terms of collision
entropies. We derive the optimal lower bound for this entropic uncertainty
relation, which results in an analytic function of the overlap of the
corresponding eigenbases. Besides, we obtain the minimum uncertainty states. We
compare our relation with other formulations of the uncertainty principle.Comment: The manuscript has been accepted for publication as a Regular Article
in Physical Review
Homogeneous components in the moduli space of sheaves and Virasoro characters
The moduli space of framed torsion free sheaves on the
projective plane with rank and second Chern class equal to has the
natural action of the -dimensional torus. In this paper, we look at the
fixed point set of different one-dimensional subtori in this torus. We prove
that in the homogeneous case the generating series of the numbers of the
irreducible components has a beautiful decomposition into an infinite product.
In the case of odd these infinite products coincide with certain Virasoro
characters. We also propose a conjecture in a general quasihomogeneous case.Comment: Published version, 19 page
Pinning and transport of cyclotron/Landau orbits by electromagnetic vortices
Electromagnetic waves with phase defects in the form of vortex lines combined
with a constant magnetic field are shown to pin down cyclotron orbits (Landau
orbits in the quantum mechanical setting) of charged particles at the location
of the vortex. This effect manifests itself in classical theory as a trapping
of trajectories and in quantum theory as a Gaussian shape of the localized wave
functions. Analytic solutions of the Lorentz equation in the classical case and
of the Schr\"odinger or Dirac equations in the quantum case are exhibited that
give precise criteria for the localization of the orbits. There is a range of
parameters where the localization is destroyed by the parametric resonance.
Pinning of orbits allows for their controlled positioning -- they can be
transported by the motion of the vortex lines.Comment: This version differs from the printed paper in having the full titles
of all referenced pape
Quantum lattice gases and their invariants
The one particle sector of the simplest one dimensional quantum lattice gas
automaton has been observed to simulate both the (relativistic) Dirac and
(nonrelativistic) Schroedinger equations, in different continuum limits. By
analyzing the discrete analogues of plane waves in this sector we find
conserved quantities corresponding to energy and momentum. We show that the
Klein paradox obtains so that in some regimes the model must be considered to
be relativistic and the negative energy modes interpreted as positive energy
modes of antiparticles. With a formally similar approach--the Bethe ansatz--we
find the evolution eigenfunctions in the two particle sector of the quantum
lattice gas automaton and conclude by discussing consequences of these
calculations and their extension to more particles, additional velocities, and
higher dimensions.Comment: 19 pages, plain TeX, 11 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages
Formulation of the uncertainty relations in terms of the Renyi entropies
Quantum mechanical uncertainty relations for position and momentum are
expressed in the form of inequalities involving the Renyi entropies. The proof
of these inequalities requires the use of the exact expression for the
(p,q)-norm of the Fourier transformation derived by Babenko and Beckner.
Analogous uncertainty relations are derived for angle and angular momentum and
also for a pair of complementary observables in N-level systems. All these
uncertainty relations become more attractive when expressed in terms of the
symmetrized Renyi entropies
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