3,636 research outputs found
Differential Inequalities and Univalent Functions
Let be the class of analytic functions in the unit disk \ID
with the normalization , and satisfying the condition \left
|z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)}
\right)^{2}-1\right |\leq 1, \quad z\in \ID. Functions in are
known to be univalent in \ID. In this paper, it is shown that the harmonic
mean of two functions in are closed, that is, it belongs again
to . This result also holds for other related classes of
normalized univalent functions. A number of new examples of functions in
are shown to be starlike in \ID. However we conjecture that
functions in are not necessarily starlike, as apparently
supported by other examples.Comment: 10 pages; To appear in Lobachevskii Journal of Mathematic
Should particle trajectories comply with the transverse momentum distribution?
The momentum distributions associated with both the wave function of a
particle behind a grating and the corresponding Bohmian trajectories are
investigated and compared. Near the grating, it is observed that the former
does not depend on the distance from the grating, while the latter changes with
this distance. However, as one moves further apart from the grating, in the far
field, both distributions become identical.Comment: 10 pages, 7 figure
Generalized Arago-Fresnel laws: The EME-flow-line description
We study experimentally and theoretically the influence of light polarization
on the interference patterns behind a diffracting grating. Different states of
polarization and configurations are been considered. The experiments are
analyzed in terms of electromagnetic energy (EME) flow lines, which can be
eventually identified with the paths followed by photons. This gives rise to a
novel trajectory interpretation of the Arago-Fresnel laws for polarized light,
which we compare with interpretations based on the concept of "which-way" (or
"which-slit") information.Comment: 14 pages, 6 figure
Hankel determinant for a class of analytic functions
Let be analutic in the unit disk and normalized so that
. In this paper we give sharp bound of Hankel
determinant of the second order for the class of analytic unctions satisfying
for and
Description of classical and quantum interference in view of the concept of flow line
Bohmian mechanics, a hydrodynamic formulation of quantum mechanics, relies on
the concept of trajectory, which evolves in time in compliance with dynamical
information conveyed by the wave function. Here this appealing idea is
considered to analyze both classical and quantum interference, thus providing
an alternative and more intuitive framework to understand the time-evolution of
waves, either in terms of the flow of energy (for mechanical waves, sound
waves, electromagnetic waves, for instance) or, analogously, the flow of
probability (quantum waves), respectively. Furthermore, this procedure also
supplies a more robust explanation of interference phenomena, which currently
is only based on the superposition principle. That is, while this principle
only describes how different waves combine and what effects these combinations
may lead to, flow lines provide a more precise explanation on how the energy or
probability propagate in space before, during and after the combination of such
waves, without dealing with them separately (i.e., the combination or
superposition is taken as a whole). In this sense, concepts such as
constructive and destructive interference, typically associated with the
superposition principle, physically correspond to more or less dense swarms of
(energy or probability) flow lines, respectively. A direct consequence of this
description is that, when considering the distribution of electromagnetic
energy flow lines behind two slits, each one covered by a differently oriented
polarizer, it is naturally found that external observers' information on the
slit crossed by single photons (understood as energy parcels) is totally
irrelevant for the existence of interference fringes, in striking contrast with
what is commonly stated and taught.Comment: 15 pages, 3 figure
Coherence loss and revivals in atomic interferometry: A quantum-recoil analysis
The coherence effects induced by external photons coupled to matter waves
inside a Mach-Zehnder three-grating interferometer are analyzed. Alternatively
to atom-photon entanglement scenarios, the model considered here only relies on
the atomic wave function and the momentum shift induced in it by the photon
scattering events. A functional dependence is thus found between the
observables, namely the fringe visibility and the phase shift, and the
transversal momentum transfer distribution. A good quantitative agreement is
found when comparing the results obtained from our model with the experimental
data.Comment: 18 pages, 4 figure
Understanding interference experiments with polarized light through photon trajectories
Bohmian mechanics allows to visualize and understand the quantum-mechanical
behavior of massive particles in terms of trajectories. As shown by
Bialynicki-Birula, Electromagnetism also admits a hydrodynamical formulation
when the existence of a wave function for photons (properly defined) is
assumed. This formulation thus provides an alternative interpretation of
optical phenomena in terms of photon trajectories, whose flow yields a
pictorial view of the evolution of the electromagnetic energy density in
configuration space. This trajectory-based theoretical framework is considered
here to study and analyze the outcome from Young-type diffraction experiments
within the context of the Arago-Fresnel laws. More specifically, photon
trajectories in the region behind the two slits are obtained in the case where
the slits are illuminated by a polarized monochromatic plane wave. Expressions
to determine electromagnetic energy flow lines and photon trajectories within
this scenario are provided, as well as a procedure to compute them in the
particular case of gratings totally transparent inside the slits and completely
absorbing outside them. As is shown, the electromagnetic energy flow lines
obtained allow to monitor at each point of space the behavior of the
electromagnetic energy flow and, therefore, to evaluate the effects caused on
it by the presence (right behind each slit) of polarizers with the same or
different polarization axes. This leads to a trajectory-based picture of the
Arago-Fresnel laws for the interference of polarized light.Comment: 36 pages, 6 figure
Trajectory-based interpretation of Young's experiment, the Arago-Fresnel laws and the Poisson-Arago spot for photons and massive particles
We present a trajectory based interpretation for Young's experiment, the
Arago-Fresnel laws and the Poisson-Arago spot. This approach is based on the
equation of the trajectory associated with the quantum probability current
density in the case of massive particles, and the Poynting vector for the
electromagnetic field in the case of photons. Both the form and properties of
the evaluated photon trajectories are in good agreement with the averaged
trajectories of single photons observed recently in Young's experiment by
Steinberg's group at the University of Toronto. In the case of the
Arago-Fresnel laws for polarized light, the trajectory interpretation presented
here differs from those interpretations based on the concept of "which-way" (or
"which-slit") information and quantum erasure. More specifically, the
observer's information about the slit that photons went through is not relevant
to the existence of interference; what is relevant is the form of the
electromagnetic energy density and its evolution, which will model consequently
the distribution of trajectories and their topology. Finally, we also show that
the distributions of end points of a large number of evaluated photon
trajectories are in agreement with the distributions measured at the screen
behind a circular disc, clearly giving rise to the Poisson-Arago spot.Comment: 8 pages, 5 figure
On Wheeler's delayed-choice Gedankenexperiment and its laboratory realization
Here, we present an analysis and interpretation of the experiment performed
by Jacques et al. (2007 Science 315, 966), which represents a realization of
Wheeler's delayed-choice Gedankenexperiment. Our analysis is based on the
evolution of the photon state, since the photon enters into the Mach-Zehnder
interferometer with a removable beam-splitter until it exits. Given the same
incident photon state onto the output beam-splitter, BS_output, the photon's
state at the exit will be very different depending on whether BS_output is on
or off. Hence, the statistics of photon counts collected by the two detectors,
positioned along orthogonal directions at the exit of the interferometer, is
also going to be very different in either case. Therefore, it is not that the
choice of inserting (on) or removing (off) a beam-splitter leads to a delayed
influence on the photon behavior before arriving at the beam-splitter, but that
such a choice influences the photon state at and after BS_output, i.e., after
it has exited from the Mach-Zehnder interferometer. The random on/off choice at
BS_output has no delayed effect on the photon to behave as a wave or a
corpuscle at the entrance and inside the interferometer, but influences the
subsequent evolution of the photon state incident onto BS_output.Comment: 7 pages, 4 figure
On the influence of resonance photon scattering on atom interference
Here, the influence of resonance photon-atom scattering on the atom
interference pattern at the exit of a three-grating Mach-Zehnder interferometer
is studied. It is assumed that the scattering process does not destroy the
atomic wave function describing the state of the atom before the scattering
process takes place, but only induces a certain shift and change of its phase.
We find that the visibility of the interference strongly depends on the
statistical distribution of transferred momenta to the atom during the
photon-atom scattering event. This also explains the experimentally observed
(Chapman et al 1995 Phys. Rev. Lett. 75 2783) dependence of the visibility on
the ratio d_p/\lambda_i = y'_{12} (2\pi/kd\lambda_i), where y'_{12} is distance
between the place where the scattering event occurs and the first grating, k is
the wave number of the atomic center-of-mass motion, is the grating
constant and \lambda_i is the photon wavelength. Furthermore, it is remarkable
that photon-atom scattering events happen experimentally within the Fresnel
region, i.e. the near field region, associated with the first grating, which
should be taken into account when drawing conclusions about the relevance of
"which-way" information for the interference visibility.Comment: 9 pages, 1 figur
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