Nonzero Imaginary Part Research Articles

Nonreciprocal Pattern Formation of Conserved Fields

In recent years, nonreciprocally coupled systems have received growing attention. Previous work has shown that the interplay of nonreciprocal coupling and Goldstone modes can drive the emergence of temporal order such as traveling waves. We show that these phenomena are generically found in a broad class of pattern-forming systems, including mass-conserving reaction-diffusion systems and viscoelastic active gels. All these systems share a characteristic dispersion relation that acquires a nonzero imaginary part at the edge of the band of unstable modes and exhibit a regime of propagating structures (traveling wave bands or droplets). We show that models for these systems can be mapped to a common “normal form” that can be seen as a spatially extended generalization of the FitzHugh-Nagumo model, providing a unifying dynamical-systems perspective. We show that the minimal nonreciprocal Cahn-Hilliard equations exhibit a surprisingly rich set of behaviors, including interrupted coarsening of traveling waves without selection of a preferred wavelength and transversal undulations of wave fronts in two dimensions. We show that the emergence of traveling waves and their speed are precisely predicted from the dispersion relation at interfaces far away from the homogeneous steady state. Our work, thus, generalizes previously studied nonreciprocal phase transitions and shows that interfaces are the relevant collective excitations governing the rich dynamical patterns of conserved fields. Published by the American Physical Society 2024

Ghost Line Waves

Time-harmonic electromagnetic plane waves in anisotropic media can exhibit complex-valued wavevectors (with nonzero real and imaginary parts) even in the absence of material dissipation. These peculiar modes, usually referred to as “ghost waves”, hybridize the typical traits of conventional propagating and evanescent waves, displaying both phase accumulation and purely reactive exponential decay away from the direction of the power flow. Their existence has been predicted in several scenarios and has been recently observed experimentally in the form of surface phonon polaritons with complex-valued out-of-plane wavevectors propagating at the interface between air and a natural uniaxial crystal with slanted optical axis. Here, we demonstrate that ghost waves can arise also in lower-dimensional flat-optics scenarios, which are becoming increasingly relevant in the context of metasurfaces and in the field of polaritonics. Specifically, we show that planar junctions between isotropic and anisotropic metasurfaces can support “ghost line waves” that propagate unattenuated along the line interface, exhibiting phase oscillations combined with evanescent decay both in the plane of the metasurface (away from the interface) and out-of-plane in the surrounding medium. Our theoretical results, validated by finite-element numerical simulations, demonstrate a novel form of polaritonic waves with highly confined features, which may provide new opportunities for the control of light at the nanoscale and may find potential applications in a variety of scenarios, including integrated waveguides, nonlinear optics, optical sensing, and subdiffraction imaging.

Isolation effect analysis of periodic pile barriers for Rayleigh waves using complex band diagram

Isolating surface waves in soil by periodic underground barriers, such as group piles, has attracted growing interest in recent years. This paper presents and validates a numerical method to obtain the complex band diagram of a three-dimensional model with periodic underground surface wave barriers, from which both the propagative modes and the evanescent modes can be observed. The circular concrete pile was studied using this method. Results show that several evanescent modes with non-zero imaginary parts emerged, which can quantitively measure the attenuation of vibrations. The leaky surface waves (LSWs) are found by defining the normalized elastic strain energy along depth. They can be differentiated as their penetration depth is much greater than surface waves. A transmission model with semicircle boundary is established to calculate transmission spectra and investigate the direction of reflected vibration energy. The influence of pile length and number of rows on the isolation effect are studied. Moreover, both the evanescent modes and the existence of LSWs contribute to surface wave mitigation. The present study provides a tool to investigate the periodic underground barriers for surface waves and a new insight into the design of vibration barriers.

Controllable discrete Talbot self-imaging effect in Hermitian and non-Hermitian Floquet superlattices.

We investigate the discrete Talbot self-imaging effect in Floquet superlattices based on a mesh of directional couplers with periodically varying separation between waveguides, both theoretically and numerically. The modulated discreteness of the lattices sets strong constraints to ensure the Talbot effect generation. We show that discrete Talbot effect occurs only if the incident periods are N = 1, 2, and 4 in dispersive regimes of the Hermitian superlattices. In both dynamic localized and rectification regimes, self-imaging effect can occur for arbitrary input period N. For the rectification case, Talbot distance equals the input period. In the regime of dynamical localization, the Talbot distance remains unchanged irrespective of the pattern period. For non-Hermitian Floquet superlattices, due to the non-zero imaginary part of quasi-energy spectrum arising at the center of the Brillouin zone, where the mode degeneracy occurs, Talbot revival is not preserved when the input period is an even number, and exists only as N = 1 in the dispersive regime. The theoretical calculations and numerical simulations verify each other completely.

Effect of small heat release and viscosity on thermal-diffusive instability

The linear stability of a thermal reaction front has been investigated based on the thermal-diffusive model proposed by Zel’dovich and Frank-Kamenetskii, which is called ZFK model. In the framework of ZFK model, heat-conduction and mass-diffusion equations are treated without the effect of hydrodynamic flow. Then, two types of instability appear: cellular and oscillatory instabilities. The cellular instability has only positive real part of growth rate, while the oscillatory instability is accompanied with non-zero imaginary part. In this study, the effect of heat release and viscosity on both instabilities is investigated asymptotically and numerically. This is achieved by coupling mass-conservation and Navier–Stokes equations with the ZFK model for small heat release. Then, the stable range of Lewis number, where the real part of growth rate is negative, is widened by non-zero values of heat release for any wavenumber. The increase of Prandtl number also brings the stabilization effect on the oscillatory instability. However, as for the cellular instability, the viscosity leads to the destabilization effect for small wavenumbers, opposed to its stabilization effect for moderate values of wavenumber. Under the limit of small wavenumber, the property of viscosity becomes clear in view of cut-off wavenumber, which makes the real part of growth rate zero.

Inverse Faraday Effect for Superconducting Condensates.

The Cooper pairs in superconducting condensates are shown to acquire a temperature-dependent dc magnetic moment under the effect of the circularly polarized electromagnetic radiation. The mechanisms of this inverse Faraday effect are investigated within the simplest version of the phenomenological dynamic theory for superfluids, namely, the time-dependent Ginzburg-Landau (GL) model. The light-induced magnetic moment is shown to be strongly affected by the nondissipative oscillatory contribution to the superconducting order parameter dynamics, which appears due to the nonzero imaginary part of the GL relaxation time. The relevance of the latter quantity to the Hall effect in the superconducting state allows us to establish the connection between the direct and inverse Faraday phenomena.

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials

This paper studies second order elliptic equations in both divergence and non-divergence forms with measurable complex valued principle coefficients and measurable complex valued potentials. The PDE operators can be considered as generalized Schrodinger operators. Under some sufficient conditions, we prove existence, uniqueness, and regularity estimates in Sobolev spaces for solutions to the equations. We particularly show that the non-zero imaginary parts of the potentials are the main mechanisms that control the solutions. Our results can be considered as limiting absorption principle for Schrodinger operators with measurable coefficients and they could be useful in applications. The approach is based on the perturbation technique that freezes the potentials. The results of the paper not only generalize known results but also provide a key ingredient for the study of \begin{document}$ L^p $\end{document} -diffusion phenomena for dissipative wave equations.

Coherent control of dissipative dynamics in a periodically driven lattice array

We find a different mechanism for suppression of decay in an open one-dimensional lattice system, which originates from a dark Floquet state, a sink state to which the system is asymptotically driven, whose overall probability is determined only by the parameters of the periodic driving field. The zero-quasienergy of dark Floquet state has been shown to be not a real zero, but a vanishingly small negative imaginary number which will cause undesirable physical effect in long-time evolution of quantum states, which is extremely different from the conservative counterpart. Another important finding is that the value of the system's effective decay, determined by the size of the non-zero imaginary part of the dark-Floquet-state-related quasienergy, depends not on how many localized lossy sites there are but on which one of the lossy sites is nearest to the driven site. Thus, for specially designed local dissipation, by controlling the driving parameters, it is possible for us to drive the system to a dark Floquet state with a much lower level of overall probability loss as compared to the undriven case and with good stability over enough longer evolution time. These results are applicable to the multisite lattice system with an odd number of sites and may be significant for long-time control of decay in a vast family of multistate physical systems with localized dissipation.

Searches for the Σ-nucleus Potential

The determination of the Σ-nucleus potential is the main orientation of the late Σ-hypernuclear physics. For this purpose, the last (Κ-,π+) light Σ-hypernuclear spectra were studied together with the recent (π-, Κ+)28Si_Σ spectrum in the context of the Green function method with a square well optical potential and a delta function spin-orbit interaction using plane waves. It was found that these spectra with a broad enhancement mainly at large energies are well described by a repulsive potential with nonzero imaginary and spin-orbit parts.

Measurement of strange baryon–antibaryon interactions with femtoscopic correlations

Two-particle correlation functions were measured for pp‾, pΛ‾, p‾Λ, and ΛΛ‾ pairs in Pb–Pb collisions at sNN=2.76 TeV and sNN=5.02 TeV recorded by the ALICE detector. From a simultaneous fit to all obtained correlation functions, real and imaginary components of the scattering lengths, as well as the effective ranges, were extracted for combined pΛ‾ and p‾Λ pairs and, for the first time, for ΛΛ‾ pairs. Effective averaged scattering parameters for heavier baryon–antibaryon pairs, not measured directly, are also provided. The results reveal similarly strong interaction between measured baryon–antibaryon pairs, suggesting that they all annihilate in the same manner at the same pair relative momentum k⁎. Moreover, the reported significant non-zero imaginary part and negative real part of the scattering length provide motivation for future baryon–antibaryon bound state searches.

Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts

Weak values are quantities accessed through quantum experiments involving weak measurements and postselection. It has been shown that ``anomalous'' weak values (those lying beyond the eigenvalue range of the corresponding operator) defy classical explanation in the sense of requiring contextuality [M. F. Pusey, Phys. Rev. Lett. 113, 200401 (2014)]. Here we elaborate on and extend that result in several directions. First, the original theorem requires certain perfect correlations that can never be realized in any actual experiment. Hence, we provide theorems that allow for a noise-robust experimental verification of contextuality from anomalous weak values, and compare with a recent experiment. Second, the original theorem connects the anomaly to contextuality only in the presence of a whole set of extra operational constraints. Here we clarify the debate surrounding anomalous weak values by showing that these conditions are tight: if any one of them is dropped, the anomaly can be reproduced classically. Third, whereas the original result required the real part of the weak value to be anomalous, we also give a version for any weak value with nonzero imaginary part. Finally, we show that similar results hold if the weak measurement is performed through qubit pointers, rather than the traditional continuous system. In summary, we provide inequalities for witnessing nonclassicality using experimentally realistic measurements of any anomalous weak value, and clarify what ingredients of the quantum experiment must be missing in any classical model that can reproduce the anomaly.

Path of Momentum Integral in the Skorniakov-Ter-Martirosian Equation**Supported by the National Key R&D Program of China under Grant No. 2018YFA0306502, the Natural Science Foundation of China under Grant Nos. 11434011, 11674393, 11604300, and 11835011, and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China under Grant No. 16XNLQ03

The Skorniakov-Ter-Martirosian (STM) integral equation is widely used for the quantum three-body problems of low-energy particles (e.g., ultracold atom gases). With this equation these three-body problems can be efficiently solved in the momentum space. In this approach the boundary condition for the case that all the three particles are gathered together is described by the upper limit of the momentum integral, i.e., the momentum cutoff. On the other hand, in realistic systems, the three-body recombination (TBR) process can occur when all these three particles are close to each other. In this process two particles form a deep dimer and the other particle can gain high kinetic energy and then escape from the low-energy system. In the presence of the TBR process, the momentum-cutoff in the STM equation would include a non-zero imaginary part. As a result, the momentum integral in the STM equation should be done in the complex-momentum plane. In this case the result of the integral depends on the choice of the integral path. Obviously, only one integral path can lead to the correct result. In this paper we consider how to correctly choose the integral path for the STM equation. We take the atom-dimer scattering problem in a specific ultracold atom gas as an example, and show the results given by different integral paths. Based on the result for this case we explore the reasonable integral paths for general case.

Interband and intraband transition, dynamical polarization and screening of the monolayer and bilayer silicene in low-energy tight-binding model

We investigate the interband and intraband transition of the monolayer and AB-stacked bilayer silicene in low-energy tight-binding model under the electric field, where we focus on the dynamical polarization function, screening due to the charged impurity, and the plasmon dispersion. We obtain the logarithmically divergen polarization function within the random-phase-approximation (RPA) whose logarithmic singularities corresponds to the discontinuities of the first derivative which is at the momentum ${\bf q}=2{\bf k}_{F}$ in static case and indicate the topological phase transition point from the gapless semimetal to the gapped band insulator. We also obtain the power-law-dependent Friedel oscillation which can be enhanced by increasing the Rashba-coupling, that can contribute to the screened potential of the charged impurity which scale as $\sim r^{-1/2}$ in the short distance from the impurity and scale as $\sim r^{-1/3}$ in the long distance from the impurity. In the single-particle excitation regime with the electron-hole continuum, the interband and intraband transition happen, and the plasmon dispersion, which we mainly focus on the optical plasmon (which $\sim\sqrt{{\bf q}}$ in long-wavelength limit) in this paper, start to damped into the electron-hole pairs due to the nonzero imaginary part of the polarization function.

Rogue waves and solitons of the coherently-coupled nonlinear Schrödinger equations with the positive coherent coupling

A set of the coherently-coupled nonlinear Schrödinger equations with the positive coherent coupling terms, which are related to the optical fiber communication, are studied through the binary Darboux transformation with the dimensional reduction. Formalisms of the solutions appear as the mixtures of the polynomial functions with exponential functions. When the spectral parameter is real, we obtain different kinds of the solutions, such as the soliton, degenerate-soliton, periodic, and soliton-like rational solutions. When the spectral parameter is complex with a non-zero imaginary part, we obtain the rogue waves and twisted rogue-wave pairs, and show that an eye-shaped rogue wave splits into a twisted rogue-wave pair.

Rational solutions in Grammian form for the [formula omitted]-dimensional generalized shallow water wave equation

In this paper, the (3+1)-dimensional generalized shallow water wave equation is investigated using the Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction. The explicit rational solutions for such an equation have been presented in the Grammian form. Based on the Grammian form solution for the equation, the one-rational, two-rational and three-order rational solutions are obtained. When complex parameters pi with nonzero real and imaginary parts are chosen, the lump soliton solutions which are localized in all directions for the (3+1)-dimensional generalized shallow water wave equation can be derived from the corresponding rational solutions. As the figures illustrate, the one-lump soliton solution with one peak and one trough propagates stably on the (x,y) plane. The two-lump solitons with different velocities interact with each other and separate with their original shapes and propagation directions. Different from the case of two-lump solitons, the propagation directions of the third-order lump solitons change after the interaction.

Realizable lists on a class of nonnegative matrices

A square matrix of order n with n≥2 is called permutative matrix when all its rows are permutations of the first row. In this paper recalling spectral results for partitioned into 2-by-2 symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given. In particular, we study NIEP and PNIEP when some complex elements in the lists under consideration have non-zero imaginary part. Realizability regions for nonnegative permutative matrices are obtained. A Guo's realizability-preserving perturbations result is obtained.

Singularities of the transmission coefficient and anomalous scattering by a dielectric slab

We prove the existence and describe the distribution on the complex plane of the singularities, resonant states (RSs), of the transmission coefficient in the problem of the plane wave scattering by a parallel-plate dielectric slab in free space. It is shown that the transmission coefficient has isolated poles all with nonzero imaginary parts that form countable sets in the complex plane of the refraction index or permittivity of the slab with the only accumulation point at infinity. The transmission coefficient never vanishes and anomalous scattering, when its modulus exceeds unity, occurs at arbitrarily small loss of the dielectric filling the layer. These results are extended to the cases of scattering by arbitrary multi-layer parallel-plane media. Connections are established between RSs, spectral singularities, eigenvalues of the associated Sturm–Liouville problems on the line, and zeros of the corresponding Jost function.

Speed-of-light pulses in the massless nonlinear Dirac equation with a potential.

We consider the massless nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction g^{2}/2(Ψ[over ¯]Ψ)^{2} in the presence of three external electromagnetic real potentials V(x), a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find different scenarios depending on initial conditions, namely, propagation of the initial pulse along one direction, splitting of the initial pulse into two pulses traveling in opposite directions, and focusing of two initial pulses followed by a splitting. For all considered cases, the final waves travel with the speed of light and are solutions of the massless linear Dirac equation. During these processes the charge and the energy are conserved, whereas the momentum is conserved when the solutions possess specific symmetries. For the case of the constant potential, we derive exact analytical solutions of the massless NLD equation that are also solutions of the massless linearized Dirac equation. Decay or growth of the initial pulse is also predicted from the evolution of the charge for the case of a non-zero imaginary part of the potential.

Graphene based 1D photonic crystals bands via the Fourier Modal Method

A theoretical approach for computing the photonic band structure of 1D graphene based photonic crystal (1DGPC) using the Fourier Modal method (FMM) is presented. It is based on the resolution of the Maxwell's equations and the Bloch theorem. In the model, the graphene sheet is considered as layer with atomic thickness characterized by a dielectric function, which is frequency dependent and has a non-zero imaginary part. This frequency dependency is given by the Drude dielectric function. Under these conditions, we show that within the framework of the FMM, it is possible to obtain a polynomial eigenvalue problem allowing the calculation of the band structure which reveals photonic band structures with photonic bandgaps. It is concluded that the existing photonic bandgaps are highly tunable by varying the graphene chemical potential. Furthermore, we explore the spatial field-structure of certain modes and show that the modes associated with the lower edges of the bandgaps can be considered as quasi-modes generated by a cavity formed by the graphene and the dielectric medium.