On a question regarding visibility of lattice points—III

AbstractFor a positive integer m, let ω(m) denote the number of distinct prime factors of m. Let h(n) be a function defined on the set of positive integers such that h(n)→∞ as n→∞ and let En(h)={d:disapositiveinteger,d⩽n,ω(d)⩾h(n)}. Writing Δn={(x,y):x,yareintegers,1⩽x,y⩽n}, in the present paper we show that one can give explicit description of a set Xn⊂Δn such that Δn is visible from Xn with at most 100|En(h)|2 exceptional points and for all sufficiently large n, one has|Xn|⩽800h(n)loglogh(n).As a corollary it follows that one can give explicit description of a set Yn⊂Δn such that for large n's, Δn is visible except for at most 100n2/(loglogn)2 exceptional points from Yn where Yn satisfies|Yn|=O((loglogn)(loglogloglogn))

Optical Signatures of Quantum Emitters in Suspended Hexagonal Boron Nitride

Hexagonal boron nitride (h-BN) is a tantalizing material for solid-state quantum engineering. Analogously to three-dimensional wide-bandgap semiconductors like diamond, h-BN hosts isolated defects exhibiting visible fluorescence, and the ability to position such quantum emitters within a two-dimensional material promises breakthrough advances in quantum sensing, photonics, and other quantum technologies. Critical to such applications, however, is an understanding of the physics underlying h-BN's quantum emission. We report the creation and characterization of visible single-photon sources in suspended, single-crystal, h-BN films. The emitters are bright and stable over timescales of several months in ambient conditions. With substrate interactions eliminated, we study the spectral, temporal, and spatial characteristics of the defects' optical emission, which offer several clues about their electronic and chemical structure. Analysis of the defects' spectra reveals similarities in vibronic coupling despite widely-varying fluorescence wavelengths, and a statistical analysis of their polarized emission patterns indicates a correlation between the optical dipole orientations of some defects and the primitive crystallographic axes of the single-crystal h-BN film. These measurements constrain possible defect models, and, moreover, suggest that several classes of emitters can exist simultaneously in free-standing h-BN, whether they be different defects, different charge states of the same defect, or the result of strong local perturbations

Percolation with long-range correlated disorder

Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as function of the Hurst exponent $H$ which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on $H$