WKB Approximation Research Articles

Quasinormal modes of the Einstein-Maxwell-aether black hole

We study the quasi-normal modes of the charged scalar perturbations in the background of the Einstein-Maxwell-aether black hole through three methods (WKB method, continued fraction method, generalized eigenvalue method). Then we propose the specific treatment for the generalized effective potential with $\omega$-dependence and the complete procedure of transforming calculation continued fraction method into finding the zero point of the corresponding complex function numerically. These methods are valid because the results from different methods are consistent. We also investigate the allowed region of the second kind aether black hole among the system parameters ($c_{13},c_{14},Q$). Finally we show the existence of quasi-resonances of massive perturbation for Einstein-Maxwell-aether black hole even with large aether parameter.

Influence of self-focused high power Cosh-Gaussian beam on second harmonic generation in cold quantum plasma

The objective of present work is to investigate influence of self-focused high power Cosh-Gaussian (ChG) beam on second harmonic generation (SHG) in cold quantum plasma (CQP). WKB and Paraxial approximations are utilized for acquiring 2nd order differential equation representing propagation characteristics of ChG beam with normalized propagation distance. The modification in dielectric function of plasma is observed on account of relativistic nonlinearity causing self-focusing of ChG beam. The density gradients are produced in transverse direction. These density gradients cause electron plasma wave (EPW) excitation at main wave frequency. Further, the nonlinear interaction between pump wave and excited EPW produces 2nd harmonics in plasma. We have performed numerical simulations for exploring impact of various laser & plasma parameters on beam waist of ChG beam and conversion efficiency of 2nd harmonics. We have also made the comparison of given work with classical relativistic plasma (CRP) regime.

Quasinormal modes of a Casimir-like traversable wormhole through the semi-analytical WKB approach

In this work, we implement the semi-analytical WKB method to explore the behaviour of a scalar field on a traversable wormhole space–time with a Casimir-like complexity reported in Eur. Phys. J.C 82, 420 (2022). We estimate the error in the computation of the quasi-normal frequencies of the scalar field at each order in the WKB and show that the order with the best accuracy is not unique. We compute the value of the quasinormal frequencies for the smallest estimated error. As an aside, we find that the imaginary part of the quasinormal modes frequencies approach to zero for the fundamental mode mimicking the so-called quasi-resonance observed for massive scalar fields in the Reissner–Nordström background.

Second harmonic generation of high power Cosh-Gaussian beam in cold collisionless plasma

The purpose of this study is to explore the second harmonic generation (SHG) of a high power Cosh-Gaussian beam in cold collisionless plasma. The ponderomotive force causes carrier redistribution from high field to low field region in presence of a Cosh-Gaussian beam thereby producing density gradients in the transverse direction. The density gradients so produced the results in electron plasma wave (EPW) generation at the frequency of the input beam. The EPW interacts with the input beam resulting in the production of 2nd harmonics. WKB and paraxial approximations are employed for obtaining the 2nd order differential equation describing the behavior of the beam’s spot size against normalized distance. The impact of well-established laser-plasma parameters on the behavior of the beam’s spot size and SHG yield are also analyzed. The focusing behavior of the beam and SHG yield is enhanced with an increase in the density of plasma, the radius of the beam and the decentred parameter, and with a decrease in the intensity of the beam. The results of the current problem are really helpful for complete information of laser-plasma interaction physics.

The classical limit of the Duffin-Kemmer-Petiau equation

The equivalent system of equations corresponding to the Duffin-Kemmer-Petiau (DKP) equation is derived and the WKB approximation of this system is found. It is proved that the Lorentz equation follows from the new DKP-Pardy system.

Image-force barrier lowering in top- and side-contacted two-dimensional materials

We compare the image-force barrier lowering (IFBL) and calculate the resulting contact resistance for four different metal–dielectric-two-dimensional (2D) material configurations. We analyze edge contacts in three different geometries (a homogeneous dielectric throughout, including the 2D layer; a homogeneous dielectric surrounding the 2D layer, both ungated and back gated) and also a top-contact assuming a homogeneous dielectric. The image potential energy of each configuration is determined and added to the Schottky energy barrier which is calculated assuming a textbook Schottky potential. For each configuration, the contact resistivity is calculated using the WKB approximation and the effective mass approximation using either SiO2 or HfO2 as the surrounding dielectric. We obtain the lowest contact resistance of 1 kΩμm by n-type doping an edge contacted transition metal-dichalcogenide (TMD) monolayer, sandwiched between SiO2 dielectric, with ∼1012 cm−2 donor atoms. When this optimal configuration is used, the contact resistance is lowered by a factor of 50 compared to the situation when the IFBL is not considered.

New benchmark scenarios of electroweak baryogenesis in aligned two Higgs double models

We discuss electroweak baryogenesis in aligned two Higgs doublet models. It is known that in this model the severe constraint from the experimental results for the electron electric dipole moment can be avoided by destructive interference among CP-violating effects in the Higgs sector. In our previous work, we showed that the observed baryon number in the Universe can be explained without contradicting current available data in a specific scenario in the same model. We here first discuss details of the evaluation of baryon number based on the WKB method taking into account all order of the wall velocity. We then investigate parameter spaces which are allowed under the current available data from collider, flavor and electric dipole moment experiments simultaneously. We find several benchmark scenarios which can explain baryon asymmetry of the Universe. We also discuss how we can test these benchmark scenarios at future collider experiments, various flavor experiments and gravitational wave observations.

Quantum tunneling from accelerating three-dimensional black hole

Hawking radiation as a quantum tunneling phenomena from accelerating BTZ black holes is presented in this work. We have calculated the Dirac particle's Hawking radiation from the horizon of the accelerating BTZ black hole. WKB approximation is applied to Dirac equation in the background of three dimensional black holes. This procedure gives us the tunneling probability, which we have used to calculate the Hawking temperature of the background three-dimensional black hole. Our study is consistent with the previous studies in the absence of the acceleration parameter. We have also studied the quantum corrections to the Hawking temperature of accelerating BTZ black hole.

Study of resonances in exotic neutron-rich 14Be by the use of isospectral potential

In this work, Hyperspherical Harmonic Expansion (HHE) formalism aided by supersymmetric quantum mechanics is used to study bound and resonant states of 14Be in the three-body (12Be+ n + n) cluster model, and the analysis of the resonant states close to the binding threshold utilizes supersymmetric quantum mechanics (SSQM). GPT nucleon-nucleon potential together with SBB core-nucleon potential is chosen for the solution of three-body Schrödinger equation to get the lowest bound state energy and wave function. In the next stage, energy and wave function of the bound state is used to derive an isospectral potential that exhibit deep well following a strong barrier facilitating confinement (or trapping) of particle inside it at some positive energy (E>0). The trapping probability when plotted against energy shows a prominent peak at the energy of resonance. WKB approximation is used to determine the width of resonance width. Calculated resonance energy and width of resonance are compared with those measured experimentally for the promising neutron halo candidate, the 14Be.

On the numerical evaluation of the prolate spheroidal wave functions of order zero

We describe a method for the numerical evaluation of the angular prolate spheroidal wave functions of the first kind of order zero. It is based on the observation that underlies the WKB method, namely that many second order differential equations admit solutions whose logarithms can be represented much more efficiently than the solutions themselves. However, rather than exploiting this fact to construct asymptotic expansions of the prolate spheroidal wave functions, our algorithm operates by numerically solving the Riccati equation satisfied by their logarithms. Its running time grows much more slowly with bandlimit and characteristic exponent than standard algorithms. We illustrate this and other properties of our algorithm with numerical experiments.

Asymptotic approximations for the phase space Schrödinger equation

We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the anisotropic Gaussian approximation, related to the nearby orbit approximation, which is identified with the phase space representation of the Littlejohn semi-classical propagator. By means of the semi-classical propagator, we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the complex WKB theory developed by Maslov, on the basis of a canonical system in double phase space, analogous to the Berezin–Shubin–Marinov Hamilton–Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in .

Second harmonic generation of zeroth-order Bessel-Gaussian laser beam in collisionless plasma

The objective of present work is to investigate second harmonic generation (SHG) of zeroth-order Bessel-Gaussian (B-G) laser beam in collisionless plasma. The ponderomotive force produces density gradients in plasma by expelling the plasma electrons from high field portion to low field portion thereby modifying plasma’s refractive index. This modification causes self-focusing of main beam. The density gradient further causes excitation of electron plasma wave (EPW) at the main beam frequency. The excited EPW couples with input beam thereby generating second harmonics. The well known WKB approximation and moment theory are employed for obtaining non-linear differential equation for laser beam’s spot size and expression for SHG yield. Further, numerical solutions of these equations are obtained by well known RK-4 method. It has been concluded from the present analysis that transverse component of wave parameter μ, normalized laser intensity and plasma density plays a significant role in affecting laser beam self-focusing and its SHG yield. Results from the present study are important for the laser induced plasma.

Quasinormal modes and the correspondence with shadow in black holes with a deficit solid angle and quintessence-like matter

In this paper, we investigate the photon sphere, shadow radius and quasinormal modes of a 4-dimensional black hole with a deficit solid angle and quintessence-like matter. We find that the radii of the photon sphere and shadow decrease with the decreases of the deficit solid angle and density of quintessence-like matter. The quasinormal modes are gotten by the sixth order WKB approximation method and shadow radius, respectively. The values of the real part and imaginary parts of the quasinormal modes increase with the decrease of the values of the deficit solid angle and density of quintessence-like matter when the multipole number is fixed. The quasinormal modes gotten by these two methods are in good agreement, especially when the multipole number is large. It shows the correspondence between the quasinormal modes in the eikonal limit and shadow.

Landau level collapse in graphene in the presence of in-plane radial electric and perpendicular magnetic fields

It is known that in two-dimensional relativistic Dirac systems placed in orthogonal uniform magnetic and electric fields, the Landau levels collapse as the applied in-plane electric field reaches a critical value $\pm E_c$. We study this phenomenon for a distinct field configuration with in-plane constant radial electric field. The Dirac equation for this configuration does not allow analytical solutions in terms of known special functions. The results are obtained by using both the WKB approximation and the exact diagonalization and shooting methods. It is shown that the collapse occurs for positive values of the total angular momentum quantum number, the hole (electron)-like Landau levels collapse as the electric field reaches the value $ +(-) E_c/2$. The investigation of the Landau level collapse in the case of gapped graphene shows a number of distinctive features in comparison with the gapless case.

The ringing of quantum corrected Schwarzschild black hole with GUP

Schwarzschild black holes with quantum corrections are studied under scalar field perturbations and electromagnetic field perturbations to analyze the effect of the correction term on the potential function and quasinormal mode (QNM). In classical general relativity, spacetime is continuous and there is no existence of the so-called minimal length. The introduction of the correction items of the generalized uncertainty principle, the parameter β, can change the singularity structure of the black hole gauge and may lead to discretization in time and space. We apply the sixth-order WKB method to approximate the QNM of Schwarzschild black holes with quantum corrections and perform numerical analysis to derive the results of the method. Also, we find that the effective potential and QNM in scalar fields are larger than those in electromagnetic fields.

Cross-focusing of two chirped Gaussian laser beams and THz wave generation in plasmas of multi-ion species under a weakly relativistic and ponderomotive regime

The generation of terahertz (THz) waves via the beating of two high-intensity chirped Gaussian lasers in a multi-ion-species plasma is numerically studied by taking into account the weak relativistic and ponderomotive regime of interaction. The coupled differential equations for beamwidth parameters are extracted by introducing the dielectric function of such plasma and using WKB and paraxial ray approximations. The amplitude of THz radiation at beat frequency resulting from the nonlinear current density induced by the beat ponderomotive force of the cross-focusing of beams was obtained. The impacts of the chirp frequency parameter, initial laser intensity and initial ionic species density (specifically, the presence of singly and doubly charged ions) in the plasma on THz generation were discussed. Our numerical results reveal that THz radiation generation strongly depends on the chirp frequency parameter. A specific range of chirp frequencies exists for self-focusing as well as THz generation with a ‘turning point’, where the THz emission reaches its maximum value. The results show that the strength of self-focusing and consequently the generated THz radiation are reduced by increasing the density of doubly charged ionic species in the plasma due to the suppression of the nonlinear effects.

WKB approximation with conformable operator

In this paper, the Wentzel–Kramers–Brillouin (WKB) method is extended to be applicable for conformable Hamiltonian systems, where the concept of conformable operator with fractional order [Formula: see text] is involved. The WKB approximation for the [Formula: see text]-wave function is derived for potentials which slowly vary in space. Some illustrative examples to demonstrate the method are presented. The quantities of the conformable form are found to be in exact agreement with the corresponding traditional quantities when [Formula: see text].

Magnetic Vortex Core String Gyrotropic Oscillations in Thick Cylindrical Dots

The nonuniform magnetic vortex gyrotropic oscillations along the cylindrical dot thickness were calculated. A generalized Thiele equation was used for describing the vortex core motion including magnetostatic and exchange forces. The magnetostatic interaction was accounted for in a local form. This allowed reducing the Thiele equation of motion to the Schrödinger differential equation and analytically determining the spin eigenmode spatial profiles and eigenfrequencies using the Liouville–Green method for the high-frequency modes. The mapping of the Schrödinger equation to the Mathieu equation was used for the low-frequency gyrotropic mode. The lowest-frequency gyrotropic mode transformed to the dot faces localized mode, increasing the dot thickness. The vortex gyrotropic modes are described for a wide range of the dot thicknesses according to the concept of the turning points in the magnetostatic potential. This approach allows treating the vortex localized modes (turning points) and nonlocalized modes within a unified picture.

Theoretical analysis of sound propagation and entropy generation across a distributed steady heat source

Acoustic and entropy waves interacting in a duct with a steady heat source and mean flow are analysed using an asymptotic expansion (AE) for low frequencies. The analytical AE solutions are obtained by taking advantage of flow invariants and applying a multi-step strategy. The proposed solutions provide first-order corrections to the compact model in the form of integrals of mean flow variables. An eigenvalue system is then built to predict the thermoacoustic modes of a duct containing a distributed heat source or sink. Predictions from the AE solutions agree well with the numerical results of the linearised Euler equations for both frequencies and growth rates, as long as the low-frequency condition is satisfied. The AE solutions are able to accurately reconstruct the acoustic and entropy waves and correct the significant errors in the predicted entropy wave associated with the compact model. The analysis illustrates that the thermoacoustic system needs to account for the entropy wave generated by the interaction of acoustic wave and the distributed steady heat source, especially when density- or entropy-dependent boundary conditions are prescribed at the duct ends. Furthermore, a combination of the AE method and the modified WKB approximation method is discussed for a cooling case. The AE solutions remedy the disadvantage of the WKB solution in the low and very low-frequency domain and facilitate full-frequency theoretical analyses of sound propagation and entropy generation in inhomogeneous duct flow fields.

Ultrahigh on/off-current ratio γ-graphyne-1 nanotube-based sub-10-nm TFET modeling and simulation

The use of γ-graphyne-1 nanotubes (GyNTs) in tunneling field effect transistors (TFETs) suppresses ambipolarity and enhances the subthreshold swing (\(\mathrm{SS}\)) of TFETs, due to the large energy band gap and high electron effective mass of GyNTs. In this research, the analysis of the structural, electronic and thermoelectric properties of the γ-graphyne-1 family under the deformation potential (DP) approach reveals that the electron–phonon mean free path (MFP) of an armchair GyNT (3AGyNT) and zigzag GyNT (2ZGyNT) are \(24\) and \(279\) nm, respectively. Therefore, ballistic transport of sub-10-nm 3AGyNT-TFETs and 2ZGyNT-TFETs in different channel lengths is investigated utilizing the non-equilibrium Green’s function (NEGF) formalism in the DFTB platform. An ultrahigh \(\mathrm{on}/\mathrm{off}\)-current ratio (\(\mathrm{OOCR}\)) value of 1.6 × 1010 at \(V_{DD} = 0\). \(2 \;{\text{V}}\) and very low point \({\text{SS}}\) of \(5 \;{\text{mV}}/{\text{dec}}\) were demonstrated by the 3AGyNT-TFET with a channel length of 9.6 nm. 2ZGyNT-TFETs show higher on-state current and \({\text{SS}}\) and lower \({\text{OOCR}}\) than those of 3AGyNT-TFETs. A linear relationship was found between channel length and logarithmic off-state current that is consistent with the WKB approximation. The obtained results along with the ultralow power consumption of the proposed GyNT-TFETs make them candidates to replace digital silicon MOSFETs in next-generation nanoelectronic devices.