Perturbation Equations Research Articles

Non-Markovian perturbation theories for phonon effects in strong-coupling cavity quantum electrodynamics

Phonon interactions are inevitable in cavity quantum electrodynamical systems based on solid-state emitters or fluorescent molecules, where vibrations of the lattice or chemical bonds couple to the electronic degrees of freedom. Due to the non-Markovian response of the vibrational environment, it remains a significant theoretical challenge to describe such effects in a computationally efficient manner. This is particularly pronounced when the emitter-cavity coupling is comparable to or larger than the typical phonon energy range, and polariton formation coincides with vibrational dressing of the optical transitions. In this Article, we consider four non-Markovian perturbative master equation approaches to describe such dynamics over a broad range of light-matter coupling strengths and compare them to numerically exact reference calculations using a tensor network. The master equations are derived using different basis transformations and a perturbative expansion in the new basis is subsequently introduced and analyzed. We find that two approaches are particularly successful and robust. The first of these is suggested and developed in this Article and is based on a vibrational dressing of the exciton-cavity polaritons. This enables the description of distinct phonon-polariton sidebands that appear when the polariton splitting exceeds the typical phonon frequency scale in the environment. The second approach is based on a variationally optimized polaronic vibrational dressing of the electronic state. Both of these approaches demonstrate good qualitative and quantitative agreement with reference calculations of the emission spectrum and are numerically robust, even at elevated temperatures, where the thermal phonon population is significant.

STABILITY OF A VISCOUS INCOMPRESSIBLE CONDUCTING LIQUID LAYER OF A CYLINDRICAL SHAPE IN AN INHOMOGENEOUS TEMPERATURE FIELD AND A MAGNETIC FIELD OF A VACUUM ARC CURRENT THROUGH IT

Convective mass transfer in a cylindrical viscous incompressible conductive fluid layer in an inhomogeneous temperature field and in the external magnetic field of the vacuum arc current through it is theoretically investigated in this work. For a horizontal layer of a viscous, incompressible, conducting liquid of a cylindrical shape, located in a temperature field inhomogeneous in height and in an external magnetic field of a vacuum arc current flowing through it, the original equations are written. These equations consist of linearized equations for small velocity perturbations, small deviations from the equilibrium values of temperature, pressure, and magnetic field strength. The considered boundary value problem is solved for the case of free boundaries. Comparison of the experimental data with theoretical calculations made it possible to determine the rotation velocity of the steel melt during vacuum arc melting.

Study on flow field and aerodynamic noise of electric vehicle rearview mirror

With the development of new energy and intelligent vehicles, aerodynamic noise problem of pure electric vehicles at high speed has become increasingly prominent. The characteristics of the flow field and aerodynamic noise of the rearview mirror region were investigated by large eddy simulation, acoustic perturbation equations and reduction order analysis. By comparing the pressure coefficients of the coarse, medium and dense grids with wind tunnel test results, the pressure distribution, and numerical accuracy of the medium grid on the body are clarified. It is shown from the flow field proper orthogonal decomposition of the mid-section that the sum of the energy of the first three modes accounts for more than 16%. Based on spectral proper orthogonal decomposition, the peak frequencies of the first-order mode are 19 and 97 Hz. As for the turbulent pressure of side window, the first mode accounts for approximately 11.3% of the total energy, and its peak appears at 39 and 117 Hz. While the first mode of sound pressure accounts for about 41.7%, and the energy peaks occur at 410 and 546 Hz. Compared with traditional vehicle, less total turbulent pressure level and total sound pressure level are found at current electric vehicle because of the limited interaction between the rearview mirror and A-pillar.

First saturation correction in high energy proton-nucleus collisions. Part II. Single inclusive semi-hard gluon production

Exploiting recently obtained analytic solutions of classical Yang-Mills equations for higher order perturbations in the field of the dilute object (proton), we derive the complete first saturation correction to the single inclusive semi-hard gluon production in high energy proton-nucleus collisions by applying the Lehmann-Symanzik-Zimmermann reduction formula. We thus finalize the program started by Balitsky (see ref. [1]) and independently by Chirilli, Kovchegov and Wertepny (see ref. [2]) albeit using a very different approach to carry out our calculations. We extracted the functional dependence of gluon spectrum on the color charge densities of the colliding objects; thus our results can be used to evaluate complete first saturation correction to the double/multiple inclusive gluon productions.

A Generalized Mode-Locking Theory for a Nyquist Laser With an Arbitrary Roll-off Factor PART I: Master Equations and Optical Filters in a Nyquist Laser

In this paper (PART I), we describe master equations and specific optical filters designed to generate a periodic Nyquist pulse train with an arbitrary roll-off factor α that can be emitted from a mode-locked Nyquist laser. In the first part, we derive a perturbative master equation for a Nyquist pulse laser with a “single” α ≠ 0 Nyquist pulse, where we obtain a new filter function F(ω) that determines filter shapes at both low and high frequency edges. A second-order differential equation that satisfies a periodic Nyquist pulse train with α ≠ 0 is derived and utilized for the direct derivation of a single Nyquist pulse solution in the time domain for a mode-locked Nyquist laser. Then, by employing the concept of Nyquist potential, we describe the differences between the spectral profiles and filter shapes of a Nyquist pulse when α = 0 and α ≠ 0. In the latter part, we describe a non-perturbative master equation that provides the solution to a “periodic” Nyquist pulse train with an arbitrary roll-off factor. We show first that the spectral profile of an α ≠ 0 periodic Nyquist pulse train, which consists of periodic δ functions, has a different envelope shape from that of a single α ≠ 0 Nyquist pulse. This is because a periodic α ≠ 0 pulse train consists of two independent periodic functions, which give rise to a different spectral envelope. Then, by Fourier transforming the master equation, we derive new filters consisting of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (ω) at a low frequency edge and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (ω) at a high frequency edge to allow us to generate arbitrary Nyquist pulses.

Strong coupling effects in quantum thermal transport with the reaction coordinate method

We present a semi-analytical approach for studying quantum thermal energy transport at the nanoscale. Our method, which is based on the reaction coordinate method, reveals the role of strong system-bath coupling effects in quantum energy transport. Considering as a case study the nonequilibrium spin-boson model, a collective coordinate is extracted from each thermal environment and added into the system to construct an enlarged system (ES). After performing additional Hamiltonian’s truncation and transformation, we obtain an effective two-level system with renormalized parameters, resulting from the strong system-bath coupling. The ES is weakly coupled to its environments, thus can be simulated using a perturbative Markovian quantum master equation approach. We compare the heat current characteristics of the effective two-state model to other techniques, and demonstrate that we properly capture strong system-bath signatures such as the turnover behavior of the heat current as a function of system-bath coupling strength. We further investigate the thermal diode effect and demonstrate that strong couplings moderately improve the rectification ratio relative to the weak coupling limit. The effective Hamiltonian method that we developed here offers fundamental insight into the strong coupling behavior, and is computationally economic. Applications of the method toward studying multi-level quantum thermal machines are anticipated.

A sharp interface immersed boundary method for flow-induced noise prediction using acoustic perturbation equations

In this paper, a hybrid computational aero/hydro-acoustic approach is proposed to deal with acoustic scattering and flow-induced noise problems based on the sharp interface immersed boundary method (IBM). For the flow field, the incompressible Navier–Stokes equations are solved by an in-house direct numerical simulation solver. The acoustic field is predicted by solving acoustic perturbation equations (APEs). Both flow and acoustic solid boundaries with complexity and mobility are dealt with by the sharp interface IBM. Benchmark acoustic problems with varied scatterers in two and three dimensions are presented to validate the accuracy of the acoustic codes and boundary treatments. Then, the feasibility and accuracy of the present hybrid approach are validated by considering the problem of flow past a circular cylinder at a Reynolds number of 200. Subsequently, the present method is used to predict the noise generated by flow around a four-cylinder array in two-dimensions with two arrangements (i.e., square array and diamond array), and the flow and acoustic physics are investigated in detail. The results show that the square array retains a monopole-like sound-radiation shape, while the directivity pattern of the diamond array produces a dipole-like shape. In both the square and diamond arrays, the propagation of acoustic waves is affected by the Doppler effect, and the latter array results in a larger alternation of the propagation angle compared with the single cylinder due to the influence of the geometric configuration. The intensity of the radiated acoustic pressure is much greater for the diamond array compared to the square one in most circumferential directions, and the acoustic intensity of both arrays is greater than that of the single cylinder. The spectrums of the far-field acoustic pressure indicate that the two arrays and the single cylinder have similar peak frequencies and profiles, with vortex shedding playing the predominant role in noise generation in all three configurations.

Probing the reheating phase through primordial magnetic field and CMB

Inflationary magnetogenesis has long been assumed to be the most promising mechanism for producing large-scale magnetic fields in our universe. However, generically, such models are plagued with either backreaction or strong coupling problems within the standard framework. This paper has shown that the reheating phase can play a crucial role in alleviating those problems along with CMB. Assuming the electrical conductivity to be negligible during the entire period of reheating, the classic Faraday electromagnetic induction changes the magnetic field's dynamics drastically. Our detailed analysis reveals that this physical phenomenon not only converts a large class of magnetogenesis model observationally viable without any theoretical problem but also can uniquely fix the perturbative average inflaton equation of state, $\omega_{\phi} = (p+2)/(p+2)$ during reheating given a specific value of the large scale magnetic field. This observation hints the inflaton to assume the potential of form $V(\phi) \sim \phi^p$ near its minimum with $p \gtrsim 3.5$ if one considers the limit of the present-day strength of the large scale magnetic field to be $\mathcal{P}_{B0}^{\frac{1}{2}} \gtrsim 10^{-18}$ G. Our analysis opens up a new avenue towards constraining the inflationary and magnetogenesis model together via reheating.

Second-order perturbations of Kerr black holes: Formalism and reconstruction of the first-order metric

Motivated by gravitational wave observations of binary black hole mergers, we present a procedure to compute the leading-order nonlinear gravitational wave interactions around a Kerr black hole. We describe the formalism used to derive the equations for second-order perturbations. We develop a procedure that allows us to reconstruct the first-order metric perturbation solely from knowledge of the solution to the first-order Teukolsky equation, without the need of Hertz potentials. Finally, we illustrate this metric reconstruction procedure in the asymptotic limit for the first-order quasinormal modes of Kerr. In a companion paper [J. L. Ripley et al., Phys. Rev. D 103, 104018 (2021)] we present a numerical implementation of these ideas.

Nonlinear feedback algorithm for tearing-mode magnetic control in tokamak

Results of calculations and analysis of the tearing-mode evolution under the effect of resonant magnetic perturbation produced by a nonlinear feedback system in rotating tokamak plasma are presented. The TEAR-code used for calculations is based on the visco-resistive magnetohydrodynamic approximation that gives coupled diffusion-type equations for the magnetic flux perturbation and for the plasma rotation velocities in toroidal and poloidal directions. The code is supplemented by a computational unit simulating the nonlinear algorithm of the feedback system. The feedback gain and phase shift are automatically regulated in real time. The gain depends on the tearing-mode amplitude. At high feedback input signals, the gain does not exceed some finite value to reduce technical requirements for the feedback actuator and to mitigate the phase instability. In the process of the tearing-mode suppression, the gain gradually increases to provide the mode control under reduction of feedback input signal and rise of the intrinsic tearing-mode stability index. Besides that, the real-time control of the feedback phase shift is provided to additionally mitigate the phase instability. According to simulation, the used nonlinear characteristics of the feedback algorithm improve the efficiency of the tearing-mode suppression.

Hybrid Overset-LES Simulations of Noise Reduction Concepts of Loaded Airfoils

The noise reduction potential of a loaded airfoil is investigated by means of a novel Overset-LES simulation method. This aerodynamic noise prediction tool solves the compressible Navier-Stokes equations, supplemented with a sub-filter-stress model, in perturbation form over a background flow. The such obtained noise sources are subsequently propagated with wave propagation equations to the far-field (e.g., with the Acoustic Perturbation Equations). This hybrid prediction method is applied to a noise reduction investigation of a loaded airfoil. For this purpose, two geometries are considered, i.e., a reference geometry and a modified geometry with a long-chord slat. The effect of the long-chord slat on the turbulent sound sources is investigated by mainly considering turbulence statistics in the slat cove and shear-layer reattachment location. Moreover, the acoustic far-field propagation revealed the influence of a long-chord slat on the directivity. The slat noise reduction primarily results from the combined effect of the less intense shear-layer instabilities and the larger distance between reattachment location and upper trailing-edge, rather than a shielding effect by the long-chord slat. The predicted noise reduction matches well with values reported in literature.

Analytical solutions of acoustic field in annular combustion chambers with non-uniform cross-sectional surface area and mean flow

Low-order acoustic network models, treating the complex combustor geometry as a network of simple geometry elements, are typically used to analyse circumferential combustion instabilities in annular combustion chambers. These elements are typically assumed to have uniform cross-sectional surface area and average radius so that the analytical solutions of the acoustic field within them can be directly obtained. However, this may lead to errors in the combustion instability analyses. The present work derives the analytical solutions of the acoustic field in annular combustion chambers with both varying cross-sectional surface area and average radius sustaining a mean flow. A wave equation for the pressure perturbation is firstly derived based on very few assumptions. Analytical solutions of the acoustic field are then derived based on a modified WKB approximation. These solutions are then validated by comparing them to results by numerically resolving the linearised Euler equations. Results show that accurate predictions can always be obtained for a smooth change of the cross-sectional surface area and small-to-moderate subsonic axial Mach numbers as long as the frequency is larger than a certain value.

Phenomenological gravitational phase transition: Reconciliation between the late and early Universe

In this work we investigate whether a certain phenomenological extension to the general relativity (GR), in the form of a gravitational phase transition (GPT) in the Universe, can reduce the external $Planck$ tensions with the local Hubble measurements and the distribution of matter, characterized by $\sigma_8$, as well as its internal inconsistencies in the lensing amplitude and the low--high $\ell$ parameter estimates. We introduce new degrees of freedom into the background and the two scalar perturbation equations in the Newtonian gauge, with simultaneous transitions from an early gravitational phase equivalent to GR toward a late phase. We model the transition as a "tanh" parametrized by the transition redshift $z_{\rm t}$ and width $\alpha$, with amplitudes $\Lambda(z)$ and $(\mu(z),\gamma(z))$ for the background and perturbations respectively. We verify the consistency of the datasets used in this work in the GPT framework and confirm that the individual tensions do not require conflicting transitions. We find that the joint datasets prefer a recent transition at $z_{\rm t}\approx 0.9$ in the background and perturbed Einstein equations, driven mainly by the local Hubble measurement. This transition relaxes all the tensions considered in this work.

Perturbations and quasinormal modes of black holes with time-dependent scalar hair in shift-symmetric scalar-tensor theories

We study odd parity perturbations of spherically symmetric black holes with time-dependent scalar hair in shift-symmetric higher-order scalar-tensor theories. The analysis is performed in a general way without assuming the degeneracy conditions. Nevertheless, we end up with second-order equations for a single master variable, similarly to cosmological tensor modes. We thus identify the general form of the quadratic Lagrangian for the odd parity perturbations, leading to a generalization of the Regge-Wheeler equation. We also investigate the structure of the effective metric for the master variable and refine the stability conditions. As an application of our generalized Regge-Wheeler equation, we compute the quasi-normal modes of a certain nontrivial black hole solution. Finally, our result is extended to include the matter energy-momentum tensor as a source term.

Spatiotemporal linear instability analysis of collective neutrino flavor conversion in four-dimensional spacetime

In an environment with high-density neutrinos formed in a core-collapse supernova (CCSN), the neutrinos exhibit nonlinear and complex oscillation behaviors due to their self-interactions. The onset of this nonlinear oscillation can be investigated by linearizing the evolution equation for small perturbations around the flavor eigenstates. While the condition under which the flavor eigenstates are unstable has been investigated in many studies, how the perturbations evolve in spacetime has yet to be elucidated. In this paper, we analytically and correctly derive the asymptotic behaviors of the linear perturbations in 4-dimensional spacetime in the linear regime for a 2-beam neutrino model using the recently proposed Lefschetz thimble formulation. The result suggests that the perturbations grow in the directions between the two neutrino beams. We also briefly discuss the possible effects of neutrino flavor conversion on the explosion mechanism of a CCSN. In particular, the result implies that the flavor instability in the preshock region may propagate into the postshock region, contrary to the previous study focusing on the group velocity in 1-dimensional space. How to treat the case of a more realistic continuous spectrum is also discussed.

Theoretical analysis on the static and dynamic performances of a squeeze film air journal bearing with three separate pads structure

Owing to their distinct non-contact and oil-free characteristics, squeeze film air bearings have been introduced to satisfy ultra-precision, low wear, and ultra-clean requirements. This paper proposes an analytical model of a three-pad squeeze film bearing to study its static and dynamic performance. The bearing force is calculated by integrating the pressure distribution over the bearing surface, which is governed by the Reynolds equation. The stable equilibrium position of the rotor is obtained by the Newton-Raphson method. A numerical method to acquire the bearing dynamic coefficients is originally proposed by considering the vibration of its pad. These dynamic coefficients are determined by solving the perturbation equations derived from the combination of the Reynolds equation and the modified film thickness and pressure. The predicted static and dynamic results show good agreement with experimental results. The parameter study shows that the variation in eccentricity with respect to the rotational speed can be controlled by reasonably adjusting the vibration amplitude or the nominal clearance of the bearing. In addition, the results indicate that the direct stiffness and damping coefficients are increased by decreasing the rotation speed or increasing the vibration amplitude of the bearing pad.

ПОСТУПАТЕЛЬНО-ВРАЩАТЕЛЬНОЕ ДВИЖЕНИЕ НЕСТАЦИОНАРНОГО ОСЕСИММЕТРИЧНОГО ТЕЛА

A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The masses of the central body and the satellite vary isotropically at different rates and do not occur reactive forces and additional rotational moments. The nonstationary axisymmetric body have an equatorial plane of symmetry. Thus, it has three mutually perpendicular planes of symmetry. The axes of its intrinsic coordinate system coincide with the principal axes of inertia and they are directed along the intersection lines of these three mutually perpendicular planes. This position remains unchangeable during the evolution. Equations of motion of the satellite in a relative system of coordinates are considered. The translational- rotational motion of the nonstationary axisymmetric body in the gravitational field of the nonstationary ball is studied by perturbation theory methods. The equations of secular perturbations reduces to the fourth order system with one first integral. This first integral is considered and three-dimensional graphs of this first integral are plotted using the Wolfram Mathematica system.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model.

Effects of Rastall parameter on perturbation of dark sectors of the Universe

In recent years, Rastall gravity is undergoing a considerable surge in popularity. This theory purports to be a modified gravity theory with a non-conserved energy–momentum tensor (EMT) and an unusual non-minimal coupling between matter and geometry. This work looks for the evolution of homogeneous spherical perturbations within the Universe in the context of Rastall gravity. Using the spherical Top-Hat collapse model, we seek for exact solutions in linear regime for density contrast of dark matter (DM) and dark energy (DE). We find that the Rastall parameter affects crucially the dynamics of density contrasts for DM and DE and the fate of spherical collapse is different in comparison to the case of general relativity (GR). Numerical solutions for perturbation equations in nonlinear regime reveal that DE perturbations could amplify the rate of growth of DM perturbations depending on the values of Rastall parameter.

1 + 3 covariant perturbations in power-law f(R) gravity

We applied the 1+3 covariant approach around the Friedmann–Lemaître–Robertson–Walker (FLRW) background, together with the equivalence between f(R) gravity and scalar-tensor theory to study cosmological perturbations. We defined the gradient variables in the 1 + 3 covariant approach which we used to derive a set of evolution equations. Harmonic decomposition was applied to partial differential equations to obtain ordinary differential equations used to analyse the behavior of the perturbation quantities. We focused on dust dominated area and the perturbation equations were applied to background solution of alpha R+beta R^{n} model, n being a positive constant. The transformation of the perturbation equations into redshift dependence was done. After numerical solutions, it was found that the evolution of energy-density perturbations in a dust-dominated universe for different values of n decays with increasing redshift.