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

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 $({\bm E} + i{\bm B})/\sqrt{2}$ is multiplied by a complex scalar function $\phi$. 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 $\phi$ 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

Heisenberg uncertainty relation for photons

The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. {\bf 108}, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of energy operator. The noncommutativity of the components of the center of energy operator results in the increase of the bound $3/2\,\hbar$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the limiting case of infinite momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics.Comment: Sequel to PRL, 108, 140401 (2012); Submitted to PR

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

Exponential beams of electromagnetic radiation

We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann-Silberstein vector and the Whittaker representation of the solutions of the Maxwell equations in terms of just one complex function. The connection between the Bessel beams and the exponential beams is made explicit by constructing the exponential beams as wave packets of Bessel beams.Comment: Dedicated to the memory of Edwin Powe

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

Vortex lines of the electromagnetic field

Relativistic definition of the phase of the electromagnetic field, involving two Lorentz invariants, based on the Riemann-Silberstein vector is adopted to extend our previous study [I. Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa, Phys. Rev. A 61, 032110 (2000)] of the motion of vortex lines embedded in the solutions of wave equations from Schroedinger wave mechanics to Maxwell theory. It is shown that time evolution of vortex lines has universal features; in Maxwell theory it is very similar to that in Schroedinger wave mechanics. Connection with some early work on geometrodynamics is established. Simple examples of solutions of Maxwell equations with embedded vortex lines are given. Vortex lines in Laguerre-Gaussian beams are treated in some detail.Comment: 11 pages, 6 figures, to be published in Phys. Rev.

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

Electromagnetic radiation by gravitating bodies

Gravitating bodies in motion, regardless of their constitution, always produce electromagnetic radiation in the form of photon pairs. This phenomenon is an analog of the radiation caused by the motion of dielectric (or magnetic) bodies. It is a member of a wide class of phenomena named dynamical Casimir effects, and it may be viewed as the squeezing of the electromagnetic vacuum. Production of photon pairs is a purely quantum-mechanical effect. Unfortunately, as we show, the emitted radiation is extremely weak as compared to radiation produced by other mechanisms.Comment: 6 page