Electromagnetic-field amplification in finite one-dimensional photonic crystals

The electromagnetic-field distribution in a finite one-dimensional photonic crystal is studied using the numerical solution of Maxwell’s equations by the transfer-matrix method. The dependence of the transmission coefficient T on the period d (or the wavelength λ) has the characteristic form with M–1 (M is the number of periods in the structure) maxima with T = 1 in the allowed band of an infinite crystal and zero values in the forbidden band. The field-modulus distribution E(x) in the structure for parameters that correspond to the transmission maxima closest to the boundaries of forbidden bands has maxima at the center of the structure; the value at the maximum considerably exceeds the incident-field strength. For the number of periods M ~ 50, more than an order of magnitude increase in the field amplification is observed. The numerical results are interpreted with an analytic theory constructed by representing the solution in the form of a linear combination of counterpropagating Floquet modes in a periodic structure.     

Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics

We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.     

Symmetries and Boundary Conditions with a Twist

Interest in finite-size systems has risen in the last decades, due to the focus on nanotechnological applications and because they are convenient for numerical treatment that can subsequently be extrapolated to infinite lattices. Independently of the envisioned application, special attention must be given to boundary condition, which may or may not preserve the symmetry of the infinite lattice. Here, we present a detailed study of the compatibility between boundary conditions and conservation laws. The conflict between open boundary conditions and momentum conservation is well understood, but we examine other symmetries, as well: we discuss gauge invariance, inversion, spin, and particle-hole symmetry and their compatibility with open, periodic, and twisted boundary conditions. In the interest of clarity, we develop the reasoning in the framework of the one-dimensional half-filled Hubbard model, whose Hamiltonian displays a variety of symmetries. Our discussion includes analytical and numerical results. Our analytical survey shows that, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian. The exception is twisted boundary condition with the special torsion Θ = πL/2, where L is the lattice size. Our numerical results for the ground-state energy at half-filling and the energy gap for L = 2–7 show how the breaking of symmetry affects the convergence to the L → ∞ limit. We compare the computed energies and gaps with the exact results for the infinite lattice drawn from the Bethe-Ansatz solution. The deviations are boundary-condition dependent. The special torsion yields more rapid convergence than open or periodic boundary conditions. For sizes as small as L = 7, the numerical results for twisted condition are very close to the L → ∞ limit. We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.     

Reflectance spectra and photonic band gaps of a one-dimensional photonic crystal based on a silicon-air periodic structure

The reflectance spectra of a one-dimensional photonic crystal based on a silicon-air periodic structure are calculated. A map of photonic band gaps is plotted, which makes it possible to deliberately choose the geometric parameters of the structure (the thickness of silicon partitions D _Si and the period A) for different ranges of the wavelength λ. To obtain structures with a photonic band gap in the range A/λ=0.15–0.5, the main region (as rule, corresponding to the lowest frequencies) can be used, and, taking into account the secondary photonic band gaps, the range A/λ can be extended to 1 and even more. In addition, it is found that, in the range D _Si/A=0.4–0.9, the secondary band gaps may be wider than the main ones (on the frequency scale). The influence of the filling factor D _Si/A on the formation of the edges of spectral bands is revealed.     

New Horizons in Multidimensional Diffusion: The Lorentz Gas and the Riemann Hypothesis

The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor \(\sqrt{t}\) are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by \(\sqrt {t\ln t}\), with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.     

Superlattices consisting of “lines” of adsorbed hydrogen atom pairs on graphene

The structures and electron properties of new superlattices formed on graphene by adsorbed hydrogen molecules are theoretically described. It has been shown that superlattices of the (n, 0) zigzag type with linearly arranged pairs of H atoms have band structures similar to the spectra of (n, 0) carbon nanotubes. At the same time, superlattices of the (n, n) type with a “staircase” of adsorbed pairs of H atoms are substantially metallic with a high density of electronic states at the Fermi level and this property distinguishes their spectra from the spectra of the corresponding (n, n) nanotubes. The features of the spectra have the Van Hove form, which is characteristic of each individual superlattice. The possibility of using such planar structures with nanometer thickness is discussed.     

Electronic band structure and x-ray spectra of boron nitride polytypes

The electronic band structures of boron nitride crystal modifications of the graphite (h-BN), wurtzite (w-BN), and sphalerite (c-BN) types are calculated using the local coherent potential method in the cluster muffin-tin approximation within the framework of the multiple scattering theory. The specific features of the electronic band structure of 2H, 4H, and 3C boron nitride polytypes are compared with those of experimental x-ray photoelectron, x-ray emission, and K x-ray absorption spectra of boron and nitrogen. The features of the experimental x-ray spectra of boron nitride in different crystal modifications are interpreted. It is demonstrated that the short-wavelength peak revealed in the total densities of states (TDOS) in the boron nitride polytypes under consideration can be assigned to the so-called outer collective band formed by 2p electrons of boron and nitrogen atoms. The inference is made that the decrease observed in the band gap when changing over from wurtzite and sphalerite to hexagonal boron nitride is associated with the change in the coordination number of the components, which, in turn, leads to a change in the energy location of the conduction band bottom in the crystal.     

Stability of a rotating pinch

The problem of pinch stabilisation is of interest in connection with plasma confinement and has been the subject matter of various investigations in recent years, (Kruskal andSchwarzschild, Taylor, Shafranov). The theoretical investigations in the past have been restricted to static equilibrium configurations. It is of some interest to explore whether an initial motion of plasma tends to make the pinch more, or less stable than when the fluid velocity is zero. Keeping this in mind let us investigate theaxisymmetric stability of an infinitely long cylindrical plasma column of radiusR rotating uniformly with angular velocityΩ about theZ-axis, the fluid being assumed to the incompressible, inviscid, and of infinite electrical conductivity. The plasma carries a uniform axial current of densityj ₀ and a uniform axial magnetic fieldH ₁. The field outside is taken to beH ₂ inZ-direction and a toroidal component which is continuous across the boundary.     

Electrocaloric response of a ferroelectric capacitor to a periodic electric field

The electrocaloric response of a ferroelectric capacitor to a periodic electric field has been analyzed in terms of the nonstationary heat conduction equation. A linear physical model is considered for an electrocaloric element in which one end (x = 0) is thermally insulated and a constant temperature T ₀ is maintained at the boundary x = l. The effect of a periodic electric field on the capacitor gives rise to temperature oscillations about a decreasing average value that reaches saturation. A relatively simple analytic expression is derived for the temperature distribution along the electrocaloric element and the heat flux density under stationary conditions. The calculations are carried out using the results obtained from measurements performed for a PMN-PT relaxor ferroelectric in the temperature range of the phase transition. A temperature gradient and a heat flux of ～150 W/cm² are observed in an electric field of 2.4 V/μm at a frequency of 10 Hz.     

Helical Quantum States in a Strongly Frustrated Two-Dimensional Magnet

Thermodynamic properties of the J₁–J₂–J₃ quantum Heisenberg model are investigated on a square lattice with spin S = 1/2. The calculation of spin–spin correlators, spin excitation spectra, susceptibility, and heat capacity within a spherically symmetric approach shows that the third exchange J₃ may qualitatively change the properties of the system. Along with standard short-range order (antiferromagnetic, ferromagnetic, and stripe) structures, various quantum helices arise. In particular, these structures may be isotropic with a local minimum of the spectrum along a circle in the Brillouin zone. The character of these states represents both ferromagnetic and antiferromagnetic “twisted” quantum spin ordering. Moreover, a range of parameters is determined in which heat capacity exhibits two-peak temperature behavior.     

Slow light effect with high group index and wideband by saddle-like mode in PC-CROW

Slow light with high group index and wideband is achieved in photonic crystal coupled-resonator optical waveguides (PC-CROWs). According to the eye-shaped scatterers and various microcavities, saddle-like curves between the normalized frequency f and wave number k can be obtained by adjusting the parameters of the scatterers, parameters of the coupling microcavities, and positions of the scatterers. Slow light with decent flat band and group index can then be achieved by optimizing the parameters. Simulations prove that the maximal value of the group index is > 10⁴, and the normalized delay bandwidth product within a new varying range of n _g > 10² or n _g > 10³ can be a new and effective criterion of evaluation for the slow light in PC-CROWs.     

“Negative” gap in the spectrum of localized states of (In<Subscript>2</Subscript>O<Subscript>3</Subscript>)<Subscript>0.9</Subscript>(SrO)<Subscript>0.1</Subscript>

The reflection R(?ω), transmission t(?ω), absorption α(?ω), and refraction n(?ω) spectra of polycrystalline In₂O₃–SrO samples with low optical transparency, which contain In₂O₃ and In₂SrO₄ crystallites with In₄SrO_{6 + δ} interlayers, are examined. In the region of small ?ω values, the reflection coefficient decreases as the resistance of samples saturated with oxygen increases. Spectral dependences n(?ω) and α(?ω) are calculated using the classical electrodynamics relations. The results are compared to the data based on the t(?ω) spectra. The calculated absorption spectra are interpreted within the model with an overlap of tails of the density of states in the valence band and in the conduction band. A “negative” gap E _gn in the density of states with a width from–0.12 to–0.47 eV is formed in highly disordered samples in this model. It is demonstrated that the high density of defects and the band of deep acceptor states of strontium in the major matrix In₂O₃ phase are crucial to tailing of the absorption edge and its shift toward lower energies. The direct gap E _gd = 1.3 eV corresponding to the In₂SrO₄ phase is determined. The energy band diagram and the contribution of tunneling, which reduces the threshold energy for interband optical transitions, are discussed.     

Oscillations and Waves in a Nonlinear System with the 1/<Emphasis Type="Italic">f</Emphasis> Spectrum

Numerical methods are used to study a spatially distributed system of two nonlinear stochastic equations that simulate interacting phase transitions. Conditions for self-oscillations and waves are determined. The 1/f and 1/k spectra of extreme fluctuations are formed when waves emerge and move under the action of white noise. The distribution of the extreme fluctuations corresponds to the maximum entropy, which is proven by the stability of the 1/f and 1/k spectra. The formation and motion of waves under external periodic perturbation are accompanied by spatiotemporal chaotic resonance in which the domain of periodic pulsations is extended under the action of white noise.     

String order and degenerate entanglement spectrum of S = 1 / 2 and S = 1 Heisenberg bond-alternating chains

Generalized string orders and entanglement spectrum of S = 1/2 and S = 1 Heisenberg bond-alternating chains have been investigated by the infinite time-evolving block decimation (iTEBD) method. Generalized string order parameters with appropriate θ are capable of distinguishing all the topological phases. Central charges c ? 1 and critical exponents β ?1/12 indicate all the topological QPTs belong to the Gaussian universality class. Interestingly, odd- and even-fold degeneracies of the entanglement spectrum are observed. Even-fold (doubly) degenerate entanglement spectra and the typical two-fold degenerate lowest-lying level are found to exist in both the spin-1/2 dimer and the S = 1 Haldane phases. However, odd-fold degenerate entanglement spectra with three-fold degenerate lowest-lying level are observed in both the S = 1 dimer and the S = 2 Haldane phase. The degeneracy of the lowest-lying entanglement spectrum level, which can be understood by entanglement spectra in the dimer limit (J ₁ = 0), is adopted to estimate the lowest boundary of the bipartite entanglement. The entanglement spectrum and the generalized string orders are valuable for uncovering the underlying features of these symmetry-protect topological (SPT) states. Similar entanglement spectrum shows that the S = 1 (S = 2) Haldane phase is essentially the same as the S = 1/2 (S = 1) dimer phase.     

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ? _x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.     

Absorption of light by exchange-coupled ions in a 2D antiferromagnet

The optical absorption spectra of Rb₂Mn_xCd_1?xCl₄ crystals are experimentally studied in the vicinity of a magnon sideband of the exciton band at a manganese content x ranging from 1.0 to 0.4. Additional absorption bands are observed with an increase in the magnetic structural disorder upon replacement of manganese ions by cadmium ions. An analysis of the evolution of the additional absorption bands in a magnetic field during the spin-flop phase transition and the change in the intensity with variations in the manganese content x demonstrates that these bands are associated with the excitation of the exchange-coupled pairs of manganese ions located in different environments in a plane square lattice. The phase boundary between the antiferromagnetic and spin-flop phases is constructed using the results of optical measurements. The manganese content corresponding to the magnetic percolation point is evaluated.     

Waves in a superlattice with arbitrary interlayer boundary thickness

The transmittance D(ω), reflectance R(ω), and dispersion ω(k) are investigated for waves of various nature propagating through a one-dimensional superlattice (multilayer structure) with arbitrary thickness of the interlayer boundary. The dependences of the band gap widths δω_m and their positions in the wave spectrum of the superlattice on the interlayer boundary thickness d and the band number m are calculated. Calculations are performed in terms of the modified coupled-mode theory (MCMT) using the frequency dependence of R(ω), as well as in the framework of perturbation theory using the function ω (k), which made it possible to estimate the accuracy of the MCMT method; the MCMT method is found to have a high accuracy in calculating the band gap widths and a much lower accuracy in determining the gap positions. It is shown that the m dependence of δω _m for electromagnetic (or elastic) waves is different from that for spin waves. Furthermore, the widths of the band gaps with m=1 and 2 are practically independent of d, whereas the widths of all gaps for m>2 depend strongly on d. Experimental measurements of these dependences allow one to determine the superlattice interface thicknesses by using spectral methods.     

Electronic and optical properties of Fe<Subscript>2</Subscript>SiO<Subscript>4</Subscript> under pressure effect: ab initio study

We report first-principles studies the structural, electronic, and optical properties of the Fe₂SiO₄ fayalite in orthorhombic structure, including pressure dependence of structural parameters, band structures, density of states, and optical constants up to 30 GPa. The calculated results indicate that the linear compressibility along b axis is significantly higher than a and c axes, which is in agreement with earlier work. Meanwhile, the pressure dependence of the electronic band structure, density of states and partial density of states of Fe₂SiO₄ fayalite up to 30 GPa were presented. Moreover, the evolution of the dielectric function, absorption coefficient (α(ω)), reflectivity (R(ω)), and the real part of the refractive index (n(ω)) at high pressure are also presented.     

Limiting rate of secret-key generation in quantum cryptography in spacetime

The fundamental restrictions on the maximum admissible rate of secret-key commitment in quantum cryptography in real time are discussed. It is shown that the maximum rate in a quantum channel with limited transmission band is achieved in a cryptosystem on orthogonal states. The dimensionless rate (the number of bits per unit time frequency band through unit of the channel) is determined by the universal function C(λ₀(ΔkT))/ΔkT [where C(λ₀(ΔkT)) is the transmission capacity of a classical binary channel, Δk is the transmission band width, 1/T is the transmission frequency of quantum states, and λ₀ is the maximum eigenvalue of a certain integral equation].     

Anisotropy of the Complex Permittivity of the Kagome-Staircase Compounds Co<Subscript>3</Subscript>V<Subscript>2</Subscript>O<Subscript>8</Subscript> and Ni<Subscript>3</Subscript>V<Subscript>2</Subscript>O<Subscript>8</Subscript>: Experiment and Ab Initio Calculations

The anisotropy of the components of the complex permittivity of vanadate Co₃V₂O₈ and Co₃V₂O₈ single crystals in the paramagnetic phase are studied by optical ellipsometry in the spectral region 0.5–5.0 eV. Our experimental results support the weak anisotropy of the optical response detected earlier for axes a and c. The optical properties are also investigated along axis b. The properties of both compounds are compared. The optical spectra of both compounds along axis b are shifted toward low energies as compared to axes a and c. The maximum of the main interband absorption band of Co₃V₂O₈ is shifted toward low energies by 0.25–0.3 eV as compared to Co₃V₂O₈. The electronic structure parameters of both compounds are determined. Optical function spectra are analyzed using the results of ab initio band calculations.