We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of ...topological spaces with horizontal and vertical topologies is the fusion${\bf S4}\oplus {\bf S4}$. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers${\Bbb Q}\times {\Bbb Q}$with the appropriate topologies.
Perfect-crystal neutron interferometry which is analogous to Mach-Zehnder interferometry, uses Bragg diffraction to form interfering neutron paths. The measured phase shifts can be used to probe many ...types of interactions whether it be nuclear, electromagnetic, gravitational, or topological in nature. For a perfect-crystal interferometer to preserve coherence, the crystal must possess a high degree of dimensional tolerance as well as being relatively defect-free with minimal internal stresses. In the past, perfect-crystal neutron interferometers have been produced by a two step process. First, a resin diamond wheel would be used to remove excess material and shape the interferometer. Afterword, the crystal would be etched in order to remove surface defects and elevate strains. This process has had limitations in terms of repeatability and in maximizing the final contrast, or fringe visibility, of the interferometer. We have tested various fabrication and post-fabrication techniques on a single perfect-crystal neutron interferometer and measured the interferometer's performance at each step. Here we report a robust, repeatable non-etching fabrication process with high final contrast. For the interferometer used in this work, we achieved contrasts of greater than 90% several separate times and ultimately finished with an interferometer that has 92% contrast and a uniform phase distribution.
Neutron interferometry has played a distinctive role in fundamental science
and characterization of materials. Moir\'e neutron interferometers are
candidate next-generation instruments: they offer ...microscopy-like magnification
of the signal, enabling direct camera recording of interference patterns across
the full neutron wavelength spectrum. Here we demonstrate the extension of
phase-grating moir\'e interferometry to two-dimensional geometries. Our
fork-dislocation phase gratings reveal phase singularities in the moir\'e
pattern, and we explore orthogonal moir\'e patterns with two-dimensional
phase-gratings. Our measurements of phase topologies and gravitationally
induced phase shifts are in good agreement with theory. These techniques can be
implemented in existing neutron instruments to advance interferometric analyses
of emerging materials and precision measurements of fundamental constants.
Using the quantum information model of dynamical diffraction we consider a neutron cavity composed of two perfect crystal silicon blades capable of containing the neutron wavefunction. We show that ...the internal confinement of the neutrons through Bragg diffraction can be modelled by a quantum random walk. Good agreement is found between the simulation and the experimental implementation. Analysis of the standing neutron waves is presented in regards to the crystal geometry and parameters; and the conditions required for well-defined bounces are derived. The presented results enable new approaches to studying the setups utilizing neutron confinement, such as the experiments to measure neutron magnetic and electric dipole moments.
The incorporation of structured light techniques into vision science has enabled more selective probes of polarization related entoptic phenomena. Diverse sets of stimuli have become accessible in ...which the spatially dependant optical properties can be rapidly controlled and manipulated. For example, past studies with human perception of polarization have dealt with stimuli that appear to vary azimuthally. This is mainly due to the constraint that the typically available degree of freedom to manipulate the phase shift of light rotates the perceived pattern around a person's point of fixation. Here we create a structured light stimulus that is perceived to vary purely along the radial direction and test discrimination sensitivity to inwards and outwards radial motion. This is accomplished by preparing a radial state coupled to an orbital angular momentum state that matches the orientation of the dichroic elements in the macula. The presented methods offering a new dimension of exploration serve as a direct compliment to previous studies and may provide new insights into characterizing macular pigment density profiles and assessing the health of the macula.
Phase-grating moire interferometers (PGMIs) have emerged as promising candidates for the next generation of neutron interferometry, enabling the use of a polychromatic beam and manifesting ...interference patterns that can be directly imaged by existing neutron cameras. However, the modeling of the various PGMI configurations is limited to cumbersome numerical calculations and backward propagation models which often do not enable one to explore the setup parameters. Here we generalize the Fresnel scaling theorem to introduce a k-space model for PGMI setups illuminated by a cone beam, thus enabling an intuitive forward propagation model for a wide range of parameters. The interference manifested by a PGMI is shown to be a special case of the Talbot effect, and the optimal fringe visibility is shown to occur at the moire location of the Talbot distances. We derive analytical expressions for the contrast and the propagating intensity profiles in various conditions, and analyze the behaviour of the dark-field imaging signal when considering sample characterization. The model's predictions are compared to experimental measurements and good agreement is found between them. Lastly, we propose and experimentally verify a method to recover contrast at typically inaccessible PGMI autocorrelation lengths. The presented work provides a toolbox for analyzing and understanding existing PGMI setups and their future applications, for example extensions to two-dimensional PGMIs and characterization of samples with non-trivial structures.
The development of novel neutron optics devices that rely on perfect crystals and nano-scale features are ushering a new generation of neutron science experiments, from fundamental physics to ...material characterization of emerging quantum materials. However, the standard theory of dynamical diffraction (DD) that analyzes neutron propagation through perfect crystals does not consider complex geometries, deformations, and/or imperfections which are now becoming a relevant systematic effect in high precision interferometric experiments. In this work, we expand upon a quantum information (QI) model of DD that is based on propagating a particle through a lattice of unitary quantum gates. We show that the model output is mathematically equivalent to the spherical wave solution of the Takagi-Taupin equations when in the appropriate limit, and that the model can be extended to the Bragg as well as the Laue-Bragg geometry where it is consistent with experimental data. The presented results demonstrate the universality of the QI model and its potential for modeling scenarios that are beyond the scope of the standard theory of DD.
Methods of preparation and analysis of structured waves of light, electrons, and atoms have been advancing rapidly. Despite the proven power of neutrons for material characterization and studies of ...fundamental physics, neutron science has not been able to fully integrate such techniques due to small transverse coherence lengths, the relatively poor resolution of spatial detectors, and low fluence rates. Here, we demonstrate methods that are practical with the existing technologies, and show the experimental achievement of neutron helical wavefronts that carry well-defined orbital angular momentum (OAM) values. We discuss possible applications and extensions to spin-orbit correlations and material characterization techniques.
Neutrons have emerged as a unique probe at the forefront of modern material science, unrivaled in their penetrating abilities. A major challenge stems from the fact that neutron optical devices are ...limited to refractive indices on the order of \(n\approx 1 \pm 10^{-5}\). By exploiting advances in precision manufacturing, we have designed and constructed a micro-meter period triangular grating with a high aspect ratio of \(14.3\). The manufacturing quality is demonstrated with white-light interferometric data and microscope imaging. Neutron scattering experiment results are presented, showing agreement to refraction modelling. The capabilities of neutron Fresnel lenses based on this design are contrasted to existing neutron focusing techniques and the path separation of a prism-based neutron interferometer is estimated.
