Quantization Rule Research Articles

Quantization rule for transition suppression in dynamic two-level system

The transmission probability in a generic two-quasi-state system is addressed. An approximate expression for the transmission probability between the two quasi-states is derived for the case where the energy gap between the two states varies. Unlike the Landau-Zener formula, the temporal change can have a generic form and can vanish several times. In particular, when the energy gap vanishes twice, an analytical quantization rule for the suppression of transmission between two quasi-states is derived.

Revisiting Sommerfeld's atomic model using Euler–Lagrange dynamics

The purpose of this work is to present the atomic model proposed by Sommerfeld. We outline the classical calculation of the elliptical orbit and then apply the Wilson–Sommerfeld quantization rules to obtain expressions for the quantization of energy and orbit semi-axes. We then apply the relativistic theory to solve the problem of degenerate orbits. Thus, we see that the Sommerfeld atom is a very rich topic, involving the integration of different subjects ranging from Lagrangian mechanics to relativity, as well as plane geometry and differential equations.

‘IT FROM BIT’: How does information shape the structures in the Universe?

Based on a synthesis of three main ingredients: (i) the Shannon information in non-equilibrium systems, (ii) the semiclassical energy-time quantization rule and (iii) the quasi-static information-energy correspondence, a new general rule for the quantization of quasi-static information states supported by an environment away from equilibrium is introduced if the history of the environment is known as a function of time in terms of its thermodynamic potential for information T ( t ) Δ S ( t ) that is a free energy measuring the distance from equilibrium Δ S ( t ) , and T ( t ) is the mean temperature of the environment at time t . This all new quasi-static information-time quantization rule is applied to the expanding Universe using a phenomenological thermodynamic potential for information in the matter-dominated era in order to find the eigen-information of the persistent structures that are supported by the universe (or the local environments therein) at any given epoch, thus providing an information–theoretic foundation for formation of structures and rise of complexity with time that embodies the cosmic evolution as epitomized by the late Wheeler’s famous conjecture ‘it from bit’. This theoretical procedure must also open new avenues for further research into the quantum theory of information and complexity in non-equilibrium thermodynamics.

Quantum Spacetime: Proposed experiments would search for signs that space and time obey quantum rules.

Quantum Spacetime: Proposed experiments would search for signs that space and time obey quantum rules.

Affordable Highly Sensitive Fiber Optical Sensor Based on Coupled Fibers with Harmonically Varying Refractive Index

Optical fiber sensors consist of multiple Mach–Zehnder (MZ) interferometers and are common in the protection of different compounds. These sensors are very sensitive to any intrusion or threat. However, the spatial resolution is proportional to the number of MZ interferometers along the sensor. Consequently, a long sensor with a high resolution can be costly. In this paper, we suggest replacing the cascaded MZ interferometers with a couple of adjacent fibers, each of which have a harmonically varying refractive index. In this theoretical study, it is shown that two fibers with varying refractive indices demonstrate a sensitivity equivalent to that of multiple MZ interferometers. Furthermore, when the coupling coefficient between the fibers is weak, an analytical expression can be derived for the transmission between the fibers. This transmission reveals a quantization rule for which the light coupling between the two fibers vanishes.

Correction to: Analytical Solutions of the Schrödinger Equation with Generalized Hyperbolic Cotangent Potential for Arbitrary l States Via the New Quantization Rule

Correction to: Analytical Solutions of the Schrödinger Equation with Generalized Hyperbolic Cotangent Potential for Arbitrary l States Via the New Quantization Rule

Analytical Solutions of the Schrödinger Equation with Generalized Hyperbolic Cotangent Potential for Arbitrary l States Via the New Quantization Rule

Analytical Solutions of the Schrödinger Equation with Generalized Hyperbolic Cotangent Potential for Arbitrary l States Via the New Quantization Rule

Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to [Formula: see text] potentials.

Prediction of the non-SUSY AdS conjecture on the lightest neutrino mass revisited

We study the constraint of the nonsupersymmetric (non-SUSY) anti–de Sitter (AdS) conjecture on the three-dimensional vacua obtained from the compactification of the Standard Model coupled to Einstein gravity on a circle where the three-dimensional components of the four-dimensional metric are general functions of both noncompact and compact coordinates. We find from studying the wave function profile of the three-dimensional metric in the compactified dimension that the radius of the compactified dimension must be quantized. Consequently, the three-dimensional vacua are constrained by not only the non-SUSY AdS conjecture but also the quantization rule of the circle radius, leading to both upper and lower bounds for the mass of the lightest neutrino as 2≤mν/Λ4<3, where Λ4≃5.06×10−84 GeV2 is the observed cosmological constant. This means that the lightest neutrino should have a mass around 10−32 eV or it would be approximately massless. With this prediction, we reconstruct the light neutrino mass matrix that is fixed by the neutrino oscillation data and in terms of three new mixing angles and six new phases for both the normal ordering and inverted ordering. In the situation where the light neutrino mass matrix is Hermitian, we calculate its numerical value in the 3σ range. Published by the American Physical Society 2024

Quantum Limit for the Emittance of Dirac Particles Carrying Orbital Angular Momentum

In this article, we highlight that the interaction potential confining Dirac particles in a box must be invariant under the charge conjugation to avoid the Klein paradox, in which an infinite number of negative-energy particles are excited. Furthermore, we derive the quantization rules for a relativistic particle in a cylindrical box, which emulates the volume occupied by a beam of particles with a non-trivial aspect ratio. We apply our results to the evaluation of the quantum limit for emittance in particle accelerators. The developed theory allows the description of quantum beams carrying Orbital Angular Momentum (OAM). We demonstrate how the degeneracy pressure is such to increase the phase–space area of Dirac particles carrying OAM. The results dramatically differ from the classical evaluation of phase–space areas, that would foresee no increase in emittance for beams in a coherent state of OAM. We discuss the quantization of the phase–space cell’s area for single Dirac particles carrying OAM, and, finally, provide an interpretation of the beam entropy as the measure of how much the phase–space area occupied by the beam deviates from its quantum limit.

Geometric localization of waves on thin elastic structures.

We consider the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a singly curved shell as concrete examples. Previous studies on related problems have broadly focused on the localization of flexural waves on such structures. Here, using the semiclassical WKB approximation for multicomponent waves, we show that in addition to flexural waves, extensional and shear waves also form localized, bound states around points where the absolute curvature of the structure has a minimum. We also see excellent agreement between our numerical experiments and the semiclassical results, which hinges on the vanishing of two extra phases that arise in the semiclassical quantization rule. Our findings open up novel ways to fine-tune the acoustic and vibrational properties of thin elastic structures and raise the possibility of introducing new phenomena not easily captured by effective models of flexural waves alone.

An extension of localization operators

We review at first the role of localization operators as a meeting point of three different areas of research, namely: signal analysis, quantization and pseudo-differential operators. We extend then the correspondence between symbol and operator which characterizes localization operators to a more general situation, introducing the class of bilocalization operators. We show that this enlargement yields a quantization rule that is closed under composition. Some boundedness results are deduced both for localization and bilocalization operators. In particular for bilocalization operators we prove that square integrable symbols yield bounded operators on L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} and that the class of bilocalization operators with integrable symbols is a subalgebra of bounded operators on every fixed modulation space.

Efficient color image steganography based on new adapted chaotic dynamical system with discrete orthogonal moment transforms

In today's rapidly evolving digital landscape, safeguarding the secure transmission of sensitive information is paramount. Traditional steganography methods, while effective at data concealment, have grown vulnerable to brute force attacks in our age of powerful computing. This paper addresses the pressing need to bolster data transmission security. To address the challenges posed by limited key space and security concerns associated with 1-D chaotic maps in data transmission, we introduce a novel 1-D chaotic map, the Multi-Parameter Chebyshev Map (MCM). The MCM boasts four control parameters, each with a broad chaotic range and high chaos level, significantly enhancing security. Furthermore, we present an innovative steganography technique for color images that leverages the MCM in conjunction with discrete moment transforms. This method chaotically embeds the secret message within the moment coefficients of a cover color image using MCM and quantization rules. The expansive chaotic range and high chaos level of the MCM substantially reinforce the security of this approach. The considered moment transforms include Tchebichef, Krawtchouk, Hahn, Charlier, Meixner, Shmaliy, Racah, and Dual-Hahn. The performance of the proposed steganography method is thoroughly evaluated and compared with other state-of-the-art methods in terms of hiding capacity and image quality in both spatial and transform domains. The quantization of moment coefficients demonstrates its superior performance. This method holds crucial implications for practical applications requiring secure communication of color images, such as military and medical contexts. The combination of MCM and moment transforms in this method presents a promising avenue for future research in the field of information hiding.

Non-standard axion electrodynamics and the dual Witten effect

Standard axion electrodynamics has two closely related features. First, the coupling of a massless axion field to photons is quantized, in units proportional to the electric gauge coupling squared. Second, the equations of motion tell us that a time-dependent axion field in a background magnetic field sources an effective electric current, but a time-dependent axion field in a background electric field has no effect. These properties, which manifestly violate electric-magnetic duality, play a crucial role in experimental searches for axions. Recently, electric-magnetic duality has been used to motivate the possible existence of non-standard axion couplings, which can both violate the usual quantization rule and exchange the roles of electric and magnetic fields in axion electrodynamics. We show that these non-standard couplings can be derived from SL(2,ℤ) duality, but that they come at a substantial cost: in non-standard axion electrodynamics, all electrically charged particles become dyons when the axion traverses its field range, in a dual form of the standard Witten effect monodromy. This implies that there are dyons near the weak scale, leads to a large axion mass induced by Standard Model fermion loops, and dramatically alters Higgs physics. We conclude that non-standard axion electrodynamics, although interesting to consider in abstract quantum field theory, is not phenomenologically viable.

Half-Spectrum Suppression in Dynamic Resonant Tunneling

It is well known that in a process of Dynamic Resonant Tunneling, where the energy level of the quasi-bound state varies in time, the tunneling current can be drastically suppressed at specific energies. These energies obey a generic quantization rule (QR). However, these systems exhibit two types of current suppression. In the first type, the current vanishes completely, and in the second the current is suppressed but does not vanish. We investigate these two types of current suppression and their relations to the quantization rule.

The Maslov index for composite power-law potentials: The case of the first Exton potential

Using the exact solution of the one-dimensional Schrödinger equation, we demonstrate that the Maslov index involved in the Bohr-Sommerfeld semiclassical quantization rule for the first Exton potential depends on the intensity parameters of the potential. As a function of the intensity of the potential term proportional to x−3/2, it varies within a rather wide range, from −1 to 0, adopting −1/6 when this term and its conjugate term ∼x−1, are absent, which is the case when the Exton potential reduces to the inverse-square-root potential.

An Energy-Aware Nanoscale Design of Reversible Atomic Silicon Based on Miller Algorithm

Atomic silicon and reversible logic are domain field-coupled nanocomputing (FCN) techniques that have drawn significant attention for their lower power consumption, area, and design overhead. As atomic silicon and reversible logic reach dramatically reduced occupied area and power consumption, they can be a suitable alternative to CMOS technology. These technologies can significantly reduce the occupied area and energy consumption in all kinds of digital circuits, which are the two most challenging aspects of developing digital circuits. On the other hand, the Miller algorithm is a crucial synthesis for suggesting reversible circuits with extraordinary techniques in nanotechnology. It is an exceptionally effective and systematic method based on quantum rules for designing and proposing reversible circuits that can help suggest a reversible gate with low energy and a low occupied area. This study aims to construct novel nano-scale circuits with a focus on low-occupied area and minimal energy consumption as essential factors while designing digital circuits. In this paper, we propose a reversible gate with the well-known Miller algorithm and atomic silicon technology. Then it is used to develop a reversible full adder, 4-bit ripple carry adder, and 4:2 compressor. Finally, the proposed structures are simulated using the SiQAD tool.

Generic temporal quantization criterion in dynamic resonant tunneling for current suppression

It is well known that resonant tunneling can significantly increase the tunneling current when the incident particle's energy (Ω) is equal to the resonant energy (ΩR). However, recent studies have shown that complex current structures can emerge when the resonant energy varies in time. While these temporal variations typically increase the current, there is a specific condition under which the current is significantly suppressed. This condition, which has previously only been presented for specific cases, is characterized by the equation12∫t1t1|Ω−ΩR(t′)|dt′=π(n−14) for =1,2,... , where t1,2 are the solutions of ΩR(t1,2)=Ω.In this paper, we present for the first time a full derivation of this expression using the Quasi-Bound Super State method and demonstrate it for various perturbation scenarios with the following temporal profiles: Gaussian, hyperbolic secant, harmonic, and smooth rectangular.The derivation of this quantization rule has practical implications beyond the intellectual realm. The specific spectral structure of these systems has implications for nanotechnology (e.g., frequency-controlled transistors) and the understanding of biological and biochemical processes (e.g., odor detection).

Asymptotic theory of not completely integrable soliton equations.

We develop the theory of transformation of intensive initial nonlinear wave pulses to trains of solitons emerging at asymptotically large time of evolution. Our approach is based on the theory of dispersive shock waves in which the number of nonlinear oscillations in the shock becomes the number of solitons at the asymptotic state. We show that this number of oscillations, which is proportional to the classical action of particles associated with the small-amplitude edges of shocks, is preserved by the dispersionless flow. Then, the Poincaré-Cartan integral invariant is also constant, and therefore, it reduces to the quantization rule similar to the Bohr-Sommerfeld quantization rule for a linear spectral problem associated with completely integrable equations. This rule yields a set of "eigenvalues," which are related to the asymptotic solitons' velocities and their characteristics. It is implied that the soliton equations under consideration give modulationally stable solutions; therefore, these "eigenvalues" are real. Our analytical results agree very well with the results of numerical solutions of the generalized defocusing nonlinear Schrödinger equation.

Quantized MPSC for switched systems based on permissible type-switching mechanism

The investigation of the mixed time/event-triggered quantized model predictive security control (MPSC) for switched systems with DoS attacks is the emphasis of our study. To lessen the DoS attacks’ impact, a unique structure that combines a network loop and a local loop is adopted. The operation sequence of permissible controllers coordinated by the permissible type-switching mechanism (PTM) is further designed. To handle the effect of limited bandwidth and save network communication resources, a time-varying uniform quantization rule and a mixed time/event-triggered scheme are designed. Meanwhile, a buffer is introduced in the observer, and the stored predicated observation data in which will be used to update the observer state with different weighting coefficients when DoS attacks occur. By introducing a new coefficient into the performance index, the upper bound of which is obtained. Then the model predictive control (MPC) design is transformed into an optimization problem minimizing such constant upper bound. On this basis, the sufficient conditions are deduced for the exponential convergence of closed-loop systems. Ultimately, two systems are adopted to testify the effectiveness of the proposed method, and the optimal weighting coefficients which guarantees the fast system convergence speed are obtained for the specific system.