Quantization of magnetoelectric fields

Transcription

1 Quantization of agnetoelectric fields E. O. Kaenetskii Microwave Magnetic Laboratory, Departent of Electrical and Coputer Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel January 22, 2018 Abstract The effect of quantu coherence involving acroscopic degree of freedo, and occurring in systes far larger than individual atos are one of the topical fields in odern physics. Because of aterial dispersion, a phenoenological approach to acroscopic quantu electrodynaics, where no canonical forulation is attepted, is used. The proble becoes ore coplicated when geoetrical fors of a aterial structure have to be taken into consideration. Magnetic-dipolar-ode (MDM) oscillations in a agnetically saturated quasi- 2D ferrite disk are acroscopically quantized states. In this ferriagnetic structure, long-range dipole-dipole correlation in positions of electron spins can be treated in ters of collective excitations of a syste as a whole. The near fields in the proxiity of a MDM ferrite disk have space and tie syetry breakings. Such MDM-originated fields called agnetoelectric (ME) fields carry both spin and orbital angular oentus. By virtue of unique topology, ME fields are different fro free-space electroagnetic (EM) fields. The ME fields are quantu fluctuations in vacuu. We call these quantized states ME photons. There are not virtual EM photons. We show that energy, spin and orbital angular oenta of MDM oscillations constitute the key physical quantities that characterize the ME-field configurations. We show that vacuu can induce a Casiir torque between a MDM ferrite disk, etal walls, and dielectric saples. PACS nuber(s): Jb, c, g I. INTRODUCTION Energy, linear oentu, and angular oentu are constants of otion, which constitute the key physical quantities that characterize the EM field configuration. Their conservation can be cast as a continuity equation relating to a density or a flux density, or a current, associated to the conserved quantity [1]. Recently, a new constant of otion and a current associated to this conserved quantity have been introduced in electrodynaics. For a circularly polarized EM plane wave, there are the EM-field chirality and the flux density of the EM-field chirality [2, 3]. Circularly polarized EM photons carry a spin angular oentu. EM photons can also carry an additional angular oentu, called an orbital angular oentu. Aziuthal dependence of bea phase results in a helical wavefront. The photons, carrying both spin and orbital angular oentus are twisted photons [4]. The twisted EM photons are propagatingwave, actual, photons. In the near-field phenoena, which are characterized by subwavelength effects and do not radiate though space with the sae range properties as do EM wave photons, the energy is carried by virtual EM photons. Virtual particles should also conserve energy and oentu. The question whether virtual EM photons can behave as twisting excitations is a subject of a strong interest. Recently, a new concept of the spin-orbit interactions in the evanescent-field region of optical EM waves has been proposed.

2 Theoretically, it was shown that an evanescent wave possesses a spin coponent, which is orthogonal to the wave vector. Furtherore, such a wave carries a oentu coponent, which is deterined by the circular polarization and is also orthogonal to the wave vector. The transverse oentu and spin push and twist a probe Mie particle in an evanescent field. This should allow the observation of ipossible properties of light, which was previously considered as virtual [5, 6]. The evanescent odes have a purely iaginary wave nuber and represent the atheatical analogy of the tunneling solutions of the Schrödinger equation. However, the quantization of evanescent waves is not a straightforward proble. Many fundaental optical properties rely on virtual photons to act as the ediator. A well-known exaple of the Casiir effect can be understood as involving the creation of short lived virtual photons fro the vacuu. Together with the calculation of the Casiir force between arbitrary aterials, in nuerous publications, the question of the angular-oentu coupling with the quantu vacuu fields is considered as a topical subject. Since photons also carry the angular oentu, the vacuu torque will appear between acroscopic bodies when their characteristic properties are anisotropic [7 13]. In particular, in Refs. [7, 8, 10, 13], it was discussed that vacuu can induce a torque between two axial birefringent dielectric plates. In this case, the fluctuating electroagnetic fields have boundary conditions that depend on the relative orientation of optical axes of the aterials. Hence, the zero-point energy arising fro these fields also has an angular dependence. This leads to a Casiir torque that tends to align two of the principal axes of the aterial in order to iniize the syste s energy. A torque occurs only if syetry between the right-handed and left-handed circularly polarized light is broken (when the edia are birefringent). The contributions of plasonic odes to the Casiir effect was analyzed in recent studies [14 17]. Plason polaritons are coupled states between electroagnetic radiation and electrostatic (ES) oscillations in a etal. Instantaneous Coulob interaction are suppleented with retardation effects associated with real electroagnetic fields. In a non-classical description, there is the effect of interaction between real and virtual photons. On the other hand, recently, it was shown that soe special-geoetric plasonic structures can be used to create so called superchiral fields [18, 19]. Could one consider these near fields as chiral virtual photons? Whether the contribution of such chiral virtual photons to the Casiir torque can be analyzed? To the best of our knowledge, no such effects of are discussed in literature. Magnetostatic (MS) oscillations in ferrite saples [20, 21] can be considered, to a certain extent, as a icrowave analog of plason oscillations in optics. The coupled states between electroagnetic radiation and MS oscillations ay be viewed as a icrowave MS-agnon polaritons, analogously to the ES-plason polaritons in optics. Such an analogy, however, is quite inaccurate, since dynaics of charge otion is copletely different in these cases. In contrast to ES-plason oscillations with linear electric currents, MS-agnon oscillations appear due to precessing agnetic dipoles [20 22]. The proble of MS [or, in other words, agnetic-dipolar-ode (MDM)] oscillations deonstrates fundaentally new properties when specific geoetrical fors of a aterial structure is taken into consideration. In a quasi-2d ferrite disk, one becoes evident with a fact that MDM oscillations are acroscopically (esoscopically) quantized states. The effects of quantu coherence involving acroscopic degrees of freedo and occurring in systes far larger than individual atos are one of the topical fields in odern physics [23]. There is evidence that acroscopic systes can under appropriate conditions be in quantu states, which are linear superpositions of states with different acroscopic properties. The Aharonov-Boh effect shows that the characteristically quantu effect is reflected in the behavior of acroscopic currents [24]. Much progress has been ade in deonstrating the acroscopic quantu behavior of superconductor systes, where particles for highly 2

3 correlated electron systes. The concept of coherent ixture of electrons and holes, underlying the quasiclassical approxiation based on the Bogoliubov-de Gennes (BdG) Hailtonian [25], well describes the topological superconductors. Macroscopic quantu coherence can also be observed in soe ferriagnetic structures. Long range dipole-dipole correlation in position of electron spins in a ferriagnetic saple with saturation agnetization can be treated in ters of collective excitations of the syste as a whole. If the saple is sufficiently sall so that the dephasing length of the agnetic dipole-dipole interaction exceeds the saple size, this interaction is non-local on the scale of L. This is a feature of a esoscopic ferrite saple, i. e., a saple with linear diensions ph saller than L ph but still uch larger than the exchange-interaction scales. In a case of a quasi-2d ferrite disk, the quantized fors of these collective atter oscillations MS (or MDM) agnons were found to be quasiparticles with both wave-like and particle-like behaviors, as expected for quantu excitations. The MDM oscillations in a quas-2d ferrite disk, analyzed as spectral solutions for the MS-potential scalar wave function, has evident quantu-like attributes. The oscillating state in a lossless ferrite-disk particle can be described in ters of a Hilbert space spanned by a coplete orthonoral set of eigenvectors of soe observable A with eigenvalues ai [26 28]. The discrete energy eigenstates of the MDM oscillations are well observed in icrowave experients [29 32]. Experientally, it was shown also that interaction of the MDM ferrite particles with its icrowave environent give ultiresonance Fano-type resonances [31, 32]. The observed interference between resonant and nonresonant processes is typical for quantu-dot structures [33]. In this case, the Fano regie eerges because of resonant tunneling between the dot and the channel. MDM oscillations in a quasi-2d ferrite disk can conserve energy and angular oentu. Because of these properties, MDMs strongly confine energy in subwavelength scales of icrowave radiation. In a vacuu subwavelength region abutting to a MDM ferrite disk, one observes the quantized state of power-flow vortices. Moreover, in such a vacuu subwavelength region, the tie-varying electric and agnetic fields can be not utually perpendicular. Such a specific near field so-called agnetoelectric (ME) field give evidence for spontaneous syetry breakings at the resonance states of MDM oscillations [34, 35]. The ME fields are quantu fluctuations in vacuu. We call the ME photons. The electric- and agnetic-field coponents of the ME photons have spin and orbital angular oenta. These twisted evanescent fields are neither virtual nor real (free-space propagating) EM photons in vacuu. The fact of the presence of the power flow circulation inside and in a subwavelength vacuu region outside a ferrite disk arises a question of the angular oentu balance. It is evident that at the MDM resonance, such a balance can be realized only when the opposite power flow circulations can be created due to etal (and also dielectric) parts of the icrowave structure, in which a ferrite disk is placed. It eans that at the MDM resonance in a ferrite disk, certain conductivity (or displaceent) electric currents should be induced in the etal (or dielectric) parts of the icrowave structure. The coupling between an electrically neutral MDM ferrite disk and an electrically neutral etal (or dielectric) objects in a icrowave subwavelength region, should involve the electroagnetic-induction or/and Coulob interactions through virtual photons ediation. In Ref. [36], it was shown that due to the topological action of the aziuthally unidirectional transport of energy in a MDMresonance ferrite saple there exists the opposite topological reaction on a etal screen placed near this saple. We call this effect topological Lenz s effect. The coupling energy depends on the angle between the directions of the agnetization vectors in a ferrite and electric-current vectors on a etal wall. A vacuu-induced Casiir torque [7 13] allows for torque transission between the ferrite disk and etal wall avoiding any direct contact between the. L ph ( rt, ) 3

4 In this paper, we analyze quantu confineent of MDM oscillations in a ferrite disk and coupling between MDM bound states and icrowave-field continuu. The paper is organized as follows. In section II, we consider MDM oscillations and ME fields based on a classical approach. We analyze the MS description in connection with the Faraday law and Sagnac effect in a MDM ferrite disk. We show that the near fields originated fro agnetization dynaics at MDM resonances the ME fields appear as the fields of axion electrodynaics. Sharp ultiresonance oscillations, observed experientally in icrowave structures with an ebedded quasi-2d ferrite disk give evidence for quantized states of the icrowave fields originated fro MDM oscillations. This is a subject of our detailed discussion in section III. In section IV, we consider MDM bound states in the waveguide continuu. The questions of PT syetry and PT-syetry breaking of MDM oscillations are the subject of section V. In section VI, we analyze the effects in non-heritian MDM structures with PT-syetry breaking. Section VII is a conclusion of the paper. II. MDM OSCILLATIONS AND MAGNETOELECTRIC FIELDS: CLASSICAL APPROACH A. Magnetostatic description and Faraday law In a frae of a classical description, MDM oscillations in sall ferrite saples are considered as an approxiation to Maxwell equations when a displaceent electric current is negligibly sall. The physical justification for such an approxiation arises fro the fact that in a sall (with sizes uch less than a free-space electroagnetic wavelength) saple of a agnetic aterial with strong teporal dispersion (due to the ferroagnetic resonance), one neglects a tie variation of electric energy in coparison with a tie variation of agnetic energy [20 22]. In this case, we have a syste of three differential equations for the electric and agnetic fields B 0, (1) H 0, (2) B E. (3) t In such a syste, there is evident electroagnetic duality breaking. The spectral solutions for MDM oscillations in a sall ferrite saple can be obtained based on the first two differential equations in the syste, Eqs. (1) and (2). With a foral use of equation H and a constitutive relation for a ferrite, one gets the Walker equation [37] inside and the Laplace equation 0 (4) 2 0 outside a ferrite saple. Here is a agnetostatic (MS) potential and (, H 0) is a tensor of ferrite pereability at the ferroagnetic-resonance frequency range. To obtain the MDM spectral solutions, the boundary conditions for the MS-potential scalar wave function ( rt, ) and its space derivatives should be iposed. In these spectral solutions, 4

5 we do not use the third equation in the syste the Faraday equation (3). Fro literature, one can see that for the MDM spectral proble forulated exceptionally for the MS-potential wave function, no use of the alternative electric fields is presued [20 22, 37 39]. Following a foral analysis [40, 41] it can be also shown that for MDM oscillations, the Faraday equation is incopatible with Eqs. (1) and (2). Really, fro Eq. (3), we obtain 2 E B. Excluding copletely the electric displaceent current, in a 2 t t ( rt, ) saple which does not possess any dielectric anisotropy, we have 2 B t 2 0 D 0 t. It follows that the agnetic field in sall resonant agnetic objects varies linearly with tie. This gives, however, arbitrary large fields at early and late ties which is excluded on physical grounds. An evident conclusion suggests itself at once: at the MDM resonances, the agnetic fields are constant quantities. Since such a conclusion contradicts the fact of teporally dispersive edia and any resonant conditions, one can state that the Faraday equation is incopatible with the MDM spectral solutions. In Ref. [42] it was erely stated that in a case of the MDM spectral proble, an electric field is copletely absent and thus B. Sagnac effect in a MDM ferrite disk E 0 It appears, however, that in а case of MDM oscillations in a quasi-2d ferrite disk, the Faraday equation plays an essential role. These oscillations are frequency steady states characterizing by power-flow aziuthal rotations. In a frae of reference co-oving with the orbital rotation, the fields are constant. In every steady state, the Faraday law appears as the Apere-law analog for agnetic currents as sources of the electric fields. The following consideration akes this question clearer. An analytical approach for MDM resonances in a quasi-2d ferrite disk is based on a forulation of a spectral proble for a acroscopic scalar wave function the MS-potential wave function [26, 28, 34]. In this approach, the description rests on two cornerstones: (i) Precession of all electrons in a agnetically ordered ferrite saple is deterines by function, and (ii) the phase of this wave function is well defined over the whole ferrite-disk syste, i.e., MDMs are acroscopic states aintaining the global phase coherence. For a ferrite agnetized along z axis, the pereability tensor has a for [20]:. ia 0 0 ia (5) In a quasi-2d ferrite disk with the disk axis oriented along z, the Walker-equation solution for the MS-potential wave function is written in a cylindrical coordinate syste as [26, 28, 34, 35]: C ( z) ( r, ), (6) where is a diensionless ebrane function, r and are in-plane coordinates, () z is a diensionless function of the MS-potential distribution along z axis, and C is a diensional aplitude coefficient. Being the energy-eigenstate oscillations, the MDMs in a ferrite disk are also characterized by topologically distinct structures of the fields. This becoes evident fro 5

6 the boundary condition on a lateral surface of a ferrite disk of radius ebrane wave function as [26, 28, 34, 35]:, written for a a i 0. (7) r r r r r Evidently, in the solutions, one can distinguish the tie direction (given by the direction of the agnetization precession and correlated with a sign of ) and the aziuth rotation direction (given by a sign of a ). For a given sign of a paraeter a, there can be different MSpotential wave functions, and, corresponding to the positive and negative directions of the phase variations with respect to a given direction of aziuth coordinates, when 0 2. So a function is not a single-valued function. It changes a sign when angle is turned on 2. Inside a ferrite disk, the boundary-value-proble solution for Eq. (6) is written as ( ) ( ) r 1 i r,, z, t C, nj cos z sin z e e it. (8) Here is a wave nuber of a MS wave propagating in a ferrite along the z axis, is a positive integer aziuth nuber, and is the Bessel function of order for a real arguent. This equation shows that the odes in a ferrite disk are MS waves standing along the z axis and propagating along an aziuth coordinate in a certain (given by a direction of a noral bias agnetic field) aziuth direction. When the spectral proble for the MS-potential scalar wave function ( rt, ) is solved, distribution of agnetization in a ferrite disk is found as, where J is the susceptibility tensor of a ferrite [20, 21]. The agnetization has both the spin and orbital rotation. There is the spin-orbit interaction between these angular oenta. The electric field in any point inside or outside a ferrite disk is defined as [35, 43] 1 E( r ) 4 ( ) j r r r 3 r r dv, (9) V ( ) where j i0 is the density of a agnetic current. In Eq. (9), the frequency is a discrete quantity of the MDM-resonance frequency. The agnetic field inside a ferrite disk is easily defined fro the equation H. Based on the known agnetization inside a ferrite, one can find also the agnetic field distribution at any point outside a ferrite disk [1, 43]: rr r nrr r 1 H ( r ) dv ds V r r S r r. (10) 6

7 In Eqs. (9) and (10), V and S are a volue and a surface of a ferrite saple, respectively. Vector n is the outwardly directed noral to surface S. The aziuthally unidirectional wave propagation of MDMs are observed in echanically non-rotating quasi-2d ferrite disks. It is worth coparing these oscillations with Sagnac-effect resonances in icrocavities. The Sagnac effect is anifested when a saple body rotates echanically. In the case of the 2D disk dielectric resonator rotating echanically at angular velocity, the resonances are obtained by solving the stationary wave equation [44]: where r, ik n k r r r r c is an electric field coponent. The solutions for, (11) r, is given as i f r e, where is an integer. When the disk cavity is rotating, the wave function is the rotating wave J ( K r) e i, where K n k 2 k( c). The frequency difference between the counter-propagating waves is equal to 2( n). For a given direction of rotation, the clockwise (CW) and counterclockwise (CCW) waves inside a dielectric cavity experience different refraction index n. Their aziuthal nubers are. The wave equation for MDM oscillations in a quasi-2d ferrite disk, are siilar, to a certain extent, to the solutions describing Sagnac effect in echanically rotating optical icrocavities. When we copare Eq. (7) with Eq. (11) we see that in both cases, there are the ters with the first-order derivative of the wave function with respect to the aziuth coordinate. Siilar to the Sagnac effect in optical icrocavities, MDM oscillations in a ferrite disk are described by the Bessel-function aziuthally rotating MS-potential waves [26]. However, in our case of MDMs, the field rotation is due to topological chiral currents on a lateral surface of a echanically stable ferrite disk. These edge agnetic currents appear because of the spin-orbit interaction in agnetization dynaics. The above discussion and stateent that the Faraday equation is incopatible with the MDM spectral solutions we can clarify now for the case of a quasi-2d ferrite disk. It is known that in acroscopic electrodynaics, one can define three types of currents: density of the E electric displaceent current 0, the electric current density arising fro polarization, t and the electric current density arising fro agnetization [1, 22]. For a agnetic insulator [such as yttriu iron garnet (YIG)], we have the acroscopic Maxwell equation: p t E p B 00, (12) t t At the MDM resonances in YIG, orbitally rotating MS-potential waves ( rt, ) provide agnetization ( ) vectors with spin and orbital rotation and, as a result, the E and p vectors with spin and orbital rotation. We have the situation when the lines of the electric field E as well the lines of the polarization p are frozen in the lines of agnetization. It eans that there are no tie variations of vectors E and with respect to vector (and, certainly, with respect to space derivatives of vector ). In other words, all the tie variations of vectors E and p are synchronized with the tie variations of vector. p 7

8 In other words, the lines of the electric field E as well the lines of the polarization in a saple are frozen in the lines of agnetization. At this assuption, with taking into account the constitutive relation B 0 H, Eq. (12) is definitely reduced to Eq. (2). In such a case, the Faraday law (3) is not in contradiction with the MS equations (1) and (2). We can say that the entire syste of Eqs. (1) (3) is correct in the coordinate frae of orbitally driven field patterns. At the MDM resonance, the electric field originated fro a ferrite disk can cause electric polarization p in a dielectric saple situated outside a ferrite. In the reference frae corotating with the agnetization in a ferrite disk, this electric polarization in a dielectric saple is not tie varying. In the orbitally rotating field patterns, originated fro the agnetization dynaics in a ferrite disk, no displaceent electric current should be taken into consideration and, thus, the MS description is still valid in a dielectric saple. Fig. 1 illustrates the electricpolarization dynaics in a dielectric saple placed very closely to a ferrite disk. For an observer in a laboratory frae, the agnetic and electric fields outside a MDM ferrite disk are synchronically rotating fields. The entire field structure is called a ME field [35]. It is worth noting, however, that the basic condition p E p 0, (13) t t necessary for the MS description, can be violated when the axes of a ferrite disk and a dielectric disk, shown in Fig. 1, are shifted or, in general, a dielectric saple is not cylindrically syetric. In these cases the tie variations of vectors are not synchronized with the tie variations of vector and the electric polarization becoes tie varying with respect to the agnetization. As a result, the electric displaceent current is not negligibly sall and a coponent of a curl agnetic field appears. This, however, does not lead to restoration of a basic Maxwellian field structure. We concern this proble, ore in details, in the next section. C. Pseudoscalar helicity paraeters for ME fields In Ref. [36] it was shown that at MDM resonances, one has nonzero product * p. This product, characterizing the way in which the field lines of agnetization curl theselves, we call agnetization helicity paraeter. For agnetization defined as, with an assuption that MDMs are the fields rotating with an aziuth nuber obtain * 2 ( z) i e a i2 C ( z) a. (14) z r r r r Here and a are, respectively, diagonal and off-diagonal coponents of the agnetic susceptibility tensor [20, 21]. We can see that the agnetization helicity paraeter is purely an iaginary quantity:, we 8

9 * I. (15) This paraeter, appearing since the agnetization potential and curl ones [35], can be also represented as in a ferrite disk has two parts: the 1 Re e j 0 j * ( ) ( ), (16) ( ) ( e) where j i0 and j are, respectively, the agnetic and electric current densities in a ferrite ediu [1, 35]. The pseudoscalar paraeter gives evidence for the presence of two coupled and utually parallel currents the electric and agnetic ones in a localized region of a icrowave structure. The agnetization helicity paraeter can be considered as a certain source which defines the helicity properties of ME fields. An analysis of helicity properties of ME fields we should start with consideration of socalled optical chirality density. Recently, significant interest has been aroused by a rediscovered easure of helicity in optical radiation coonly tered optical chirality density based on the Lipkin's "zilch" for the fields [45]. The optical chirality density for propagating electroagnetic waves is defined as [2, 3]: 1. (17) 0 Copt E E B B 2 20 This is a tie-even, parity-odd pseudoscalar paraeter. Lipkin showed [45], that the chirality density is zero for a linearly polarized plane wave. However, for a circularly polarized wave, Eq. (17) gives a nonvanishing quantity. Moreover, for right- and left-circularly polarized waves one has opposite signs of paraeter. It is evident that for onochroatic electroagnetic waves we have C opt * * * I Copt I E E H H Re E H 0. (18) Following Ref. [46] we consider, forally, two separate ters in the left-hand side of Eq. (18). For EM fields, there are two pseudoscalar paraeters I 4 * ( E) 0 F E E I 4. (19) ( H ) 0 and F H H * To satisfy Eq. (18) we ay have two cases for onochroatic EM waves: ( E) F 0 and ( H ) F (20) 0 or F F. (21) ( E) ( H) Evidently, for Maxwellian fields, both Eqs. (20) and (21) are trivial. 9

10 For quasistatic ME fields, however, the situation appears quite different. Based on Eqs. (2) and (3), one obtains for the near fields outside a MDM ferrite saple [35]: * ( E) * F I E E Re E H (22) and F ( H ) 0. (23) ( ) We call the pseudoscalar paraeter F E ME the electric helicity of a ME field [35, 46]. This paraeter gives evidence for the presence of utually parallel coponents of the electric and agnetic fields in the ME-field structure. Both these, electric and agnetic, fields the fields in the right-hand side of Eq. (22) are potential fields. The outside electric field, however, has two coponents: the potential [defined by Eq. (9)] and the curl ones. The curl electric field is found fro the Faraday equation p H E 0 t, (24) where the agnetic field is described by Eq. (10). For a onochroatic field, the curl electric field is perpendicular to the potential agnetic field. This fields (the curl electric and potential agnetic) for power-flow vortices outside a MDM ferrite disk. Let us suppose now that in a cobined structure shown in Fig. 1, the axes of a ferrite disk and a dielectric disk are shifted or, in general, a dielectric saple is not cylindrically syetric. The electric polarization in this dielectric saple outside a ferrite is still induced by the electric field originated fro a ferrite disk at the MDM resonance. The electricpolarization dynaics in a dielectric is strongly deterined by the agnetization dynaics in a ferrite, however, the tie variations of vectors in are not synchronized with the tie variations of vector (which has the spin and orbital rotations in a ferrite). So, the condition (13) is not fulfilled. At the MDM resonances, the entire structure (a ferrite disk and a dielectric saple) is anifested as a strongly teporally dispersive syste. Assuing that a dielectric saple has sizes uch less than a free-space electroagnetic wavelength, we can neglect in a dielectric a tie variation of agnetic energy in coparison with a tie variation of electric energy. The situation appears to be dual with respect to the above considered quasi-agnetostatic approach in a ferrite saple: we can neglect the agnetic displaceent current and have a just only a potential electric field in a dielectric saple [35]. In such a quasi-electrostatic approach in a dielectric saple, the electric displaceent current is not negligibly sall and so the coponent of a curl agnetic field appears. In dielectric regions, where the condition (13) is not fulfilled, we have p D 0, (25) E 0, (26) D H, (27) t 10

11 where D 0E p. In these equations, a potential electric field E is defined by Eq. (9) and a curl agnetic field is accopanied also with a potential agnetic field described by Eq. (10). Siilar to the considered above a pseudoscalar paraeter of the electric helicity, we can introduce now pseudoscalar paraeter of the agnetic helicity,. Inside a dielectric, we have F ( H ) and * ( H ) * F I H H Re H E 0. (28) 4 4 F ( H ) F ( H ) F ( E) 0. (29) The paraeter also gives evidence for the presence of utually parallel coponents of the electric and agnetic fields. Siilar to the pseudoscalar paraeter [defined by Eq. (22)], in a case of the paraeter both the fields in the right-hand side of Eq. (28) are potential fields. It is worth noting, however, the electric and agnetic coponents of the ME fields (and, consequently, the entire structures of the ME fields), described by Eqs. (22), (23) fro one side and Eqs. (28), (29) fro the other side, are absolutely not the sae. There are two different solutions: the quasi-agnetostatic and quasi-electrostatic ones. Moreover, these two different solutions are related to two different space regions. Following Refs. [35, 46], we can introduce the noralized helicity paraeters. The noralized paraeter of the electric helicity is expressed as F ( E) E * I E cos E E, (30) while the noralized paraeter of the agnetic helicity is expressed as H * I H cos H H. (31) These noralized paraeters define angles between the electric- and agnetic-field coponents for two different structures of ME fields: the quasi-agnetostatic and quasielectrostatic ones. D. ME fields and axion electrodynaics The ME fields, being originated fro agnetization dynaics at MDM resonances, appear as the fields of axion electrodynaics [47]. In axion electrodynaics, the coupling between an axion field (which we, in general, denote ) and the electroagnetic field is expressed by an additional ter in the ordinary Maxwell Lagrangian 11

12 E B, (32) where is a coupling constant. This coupling results in odified electrodynaics equations with the electric charge and current densities replaced by [47 49] ( E ) ( E) B, (33) t. (34) ( E ) ( E) j j B E The axion field transfors as a pseudoscalars under space reflection P and it is odd under tie reversal T. Iportantly, if is a space tie constant then its contribution to the classical equations of otion vanishes. The agnetization helicity V, defined by Eqs. (15), (16), is a pseudoscalar field. We reassign the agnetization helicity V as an axion field. In the absence of free charges and conduction currents we have two equations: and D B (35) E B 00 B E t t. (36) In a bulk agnetic insulator, the first ter in the right-hand side of Eq. (36) can be identified as the electric-polarization current, while the second ter as the agnetization current [49, 50]. The agnetization helicity, defined by Eqs. (15), (16), is a tie averaged quantity. So, in Eq. E (36) we have 0. Also, a displaceent current 0 0 is a negligibly sall quantity in t t our consideration. Taking into account Eq. (2), the constitutive relation B 0 H, and writing, where is a diensional coefficient, we represent Eqs. (35), (36) as ( E) B (37) and E. (38) 0 Eqs. (37), (38) clearly show that a MDM ferrite disk is a particle with the property of agnetoelectric polarizability. Based on these equations one can see how the distributions of electric charges and agnetization are related to the agnetization helicity factor and distributions of the electric and agnetic fields. For exaple, let us consider the ain (the 1 st radial) MDM in a ferrite disk [28]. For this ode, on a surface of a ferrite in the central region of the disk, we have for the fields are characterized by the following properties. The rotating 12

13 electric-field vectors have only in-plane coponents [35, 43]. Also, and in-plane vectors. has ainly the z coponent. Since vectors and -shifted in space and tie [36], we have fro Eq. (38): 90 zˆ E are rotating are utually, (39) where is unit vector along z axis. Eq. (39) shows that at the MDM resonance there is utual correlation of the distribution of vectors and E in a ferrite saple. It is evident that the inplane vectors and E are reciprocally perpendicular. At the MDM resonances, we observe special relationships between the agnetization and electric polarization, both inside a YIG disk and in dielectrics closely abutting to the ferrite. While, in classical electrodynaics, Maxwell equations well describe the dynaic relations between electric and agnetic fields, relations between electric polarization and agnetization appear as a highly nontrivial issue. One of the basic reason is that the electrons contribute to the electric polarization and agnetic oents in copletely different ways. If these two ways are specifically correlated and both the ways of contributions are long-range ordered, the ME coupling (the coupling between electric polarization and agnetization) could be enhanced. With such a ME coupling, the electric and agnetic properties should be understood and treated by a peculiar unified way. The ME-field helicity densities, defined by Eqs. (22), (30), fro the one side, and Eqs. (28), (31), fro the other side, transfor as pseudoscalars under space reflection P and are odd under tie reversal T. It eans that the ME fields are pseudoscalar axionlike fields. Whenever pseudoscalar axionlike fields, is introduced in the electroagnetic theory, the dual syetry is spontaneously and explicitly broken. This results in non-trivial coupling between pseudoscalar quasistatic ME fields and the EM fields in icrowave structures with an ebedded MDM ferrite disk. A special role in this coupling plays the effect of interaction of MDM oscillations with etal surfaces. ẑ E. MDM oscillations in ferrite-etal structures. The ME-EM field interaction in a icrowave waveguide In the probles, where one considers an interaction of MDM oscillations with a etal surface, the electric displaceent current is also neglected. On a etal surface, however, Eq. (2) is replaced by the Apere equation H j. (40) S S In the MS-wave probles, the current S j is considered as a surface electric current which just only gives discontinuity of the tangential coponent of the agnetic field on a etal [51 53]. Even so, it is evident, however, that to define the induced electric current in the Apere equation, the Faraday law should be used. Experiental results of agnetic induction probing of the fields near a ferrite saple [54, 55], show that the Faraday equation yields the electric field associated with the agnetic fields of MDMs. In a view of these experients, one can conclude that the near-field interaction of MDM resonances with external etal eleents cannot be analyzed without the Faraday law. Let us consider a flat etal screen placed in vacuu above a ferrite disk in parallel to the disk plane and at a distance uch less than the disk diaeter. The MDM ferrite disk is a particle with the property of ME polarizability. On a surface of a MDM ferrite disk, there are 13

14 both the regions of the orbitally driven noral agnetic and noral electric fields. Moreover, the axius (inius) of the electric and agnetic fields noral to the disk are situated at the sae places on the disk plane [36]. When a etal wall is placed closely to a ferrite-disk plane, this field structure is projected on the etal. Together with the surface electric charges induced on a etal wall by the electric field, there are also the Faraday-law eddy currents induced on a etal surface by a tie derivative of a noral coponent of the rotating MDM agnetic field. Evidently, the induced surface electric current should have two coponents. There are the linear currents arising fro the continuity equation for surface electric charges, for which, and the Faraday-law eddy currents, for j S S 0 whichs j S 0. As a result, we have the sources of the linear-current and eddy-current coponents situated at the sae place on a etal wall. Thus, the for of a surface electric current is not a closed line. It is a flat a spiral [36]. Unique properties of interaction of MDMs with a etal screen becoe ore evident when one analyzes the angular-oentu balance conditions for MDM oscillations in a ferrite disk in a view of the ME-EM field coupling in a icrowave waveguide. MDMs in a quasi-2d ferrite disk are icrowave energy-eigenstate oscillations with topologically distinct structures of rotating fields and unidirectional power-flow circulations. The active power flow of the field 1 * both inside and outside a ferrite disk P Re E H has the vortex topology. In the MDM 2 resonances, the orbital angular-oentu density is expressed as 1 Re 2 L * r E H, (41) where r is a radius vector fro the disk axis. For a given direction of a bias agnetic field, the power-flow circulations are the sae inside a ferrite and in the vacuu near-field regions above and below the disk. Scheatically, this is shown in Fig. 2. Depending on a direction of a bias agnetic field, we can distinguish the clockwise and counterclockwise topological-phase rotation of the fields. The direction of an orbital angularoentu of a ferrite disk 1 Re 2 * r E H dr (42) 0 is correlated with the direction of a bias agnetic field (along +z axis or z axis). When we consider a ferrite disk in vacuu environent, such a unidirectional power-flow circulation ight see to violate the law of conservation of an angular oentu in a echanically stationary syste. However, an angular oentu is seen to be conserved if topological properties of electroagnetic fields in the entire icrowave structure are taken into account. In Ref. [36] it was shown that due to the topological action of the aziuthally unidirectional transport of energy in a MDM-resonance ferrite saple there exists the opposite topological reaction (opposite aziuthally unidirectional transport of energy) on a etal screen placed near this saple. This effect is called topological Lenz s effect. In a icrowave structure with an ebedded ferrite disk, an orbital angular oentu, related to the power-flow circulation, ust be conserved in the process. Thus, if power-flow circulation is pushed in one direction in a ferrite disk, then the power-flow circulation on etal walls to be pushed in the other direction by the sae torque at the sae tie. Fig. 3 illustrates the orbital angular-oentu balance conditions at a given direction of a bias agnetic field. H 0 j S 14

15 In Ref. [36], an analysis of conservation of an angular oentu was ade in icrowave waveguiding structures with the etal walls situated very close to the ferrite-disk surfaces. A thin ferrite disk is placed inside a rectangular waveguide syetrically to its walls so that the disk axis is perpendicular to a wall, as it is shown in Fig. 4. Vacuu gaps between the etal and ferrite are uch less than a diaeter of a MDM ferrite disk and thus the entire icrowave structure (a ferrite disk and a waveguide) can be considered as a quasi-2d structure. In this structure, a role of a linear EM-wave oentu is negligibly sall. So, the orbital angularoentu balance does not depend, actually, on the direction of the electroagnetic wave propagation in a waveguide. With taking into account that MDMs in a quasi-2d ferrite disk are icrowave energy-eigenstate oscillations, we have the angular oentu quantization as well. The observed properties of interaction between a MDM ferrite disk and a etal screen rely on ME virtual photons to act as the ediator. There are two near-field eleents of this interaction: (a) the static electric force and (b) the electroagnetic induction. However, the fields in a ferrite disk rotate at icrowave frequencies and situation becoes ore coplicated when the vacuu-region scale is about the disk diaeter or ore and the finite speed of wave propagation in vacuu the retardation effects should be taken into consideration. This eans that for a brief period, the total angular oentu of the two topological charges (one in a ferrite, another on a etal) is not conserved, iplying that the difference should be accounted for by an angular oentu in the fields in the vacuu space in a waveguide. It eans that the agnetization dynaics have an ipact on the phenoena connected with fluctuation energy in vacuu. As the rotational syetry is broken in this case, the Casiir torque [56 60] arises because the Casiir energy now depends on the angle between the directions of the agnetization vectors in a ferrite and electric-current vectors on a etal wall. A vacuu-induced Casiir torque allows for torque transission between the ferrite disk and etal wall avoiding any direct contact between the. An experiental proof of the predicted above Casiir-torque effect in icrowaves is an iportant proble. However, even in optics, such an experient appears as an open question. While the Casiir force has been easured extensively, the Casiir torque has not been observed experientally though it was predicted over forty years ago. Soe ideas proposed recently to detect the Casiir torque with an optics (see, e. g. [61] and references therein) can be useful for the proposal of analogous experiental ethods in icrowaves. III. EVIDENCE FOR QUANTIZED STATES OF THE ME FIELDS ORIGINATED FROM MDM OSCILLATIONS The eigenvalues of MDM resonances are defined by discrete quantities of a diagonal coponent of the pereability tensor [26 28, 34]. For a ferrite saple, agnetized at saturation agnetization, the diagonal coponent of the tensor (5) is found as [20] M 0 MH i Hi, (43) where is the gyroagnetic ratio and H i is a DC internal agnetic field. In neglect of aterial anisotropy, the internal agnetic field is calculated as H H H, (44) i 0 d 15

16 H 0 H d where is a bias agnetic field and is a deagnetization field [20]. An analysis [26 28, 34] shows that real energy-eigenstate solutions for MS-potential wave functions in a quasi- 2D ferrite disk are obtained when, 0. This corresponds to the frequency range [20]: H, (45) where H 0 Hi H H. Here M M 0 M0. The agnetic-field range for negative quantities are defined as [20] H H, (46) i i M0 M0 where H i Sharp ultiresonance oscillations, observed experientally in icrowave structures with an ebedded quasi-2d ferrite disk [29 32], are related to agnetization dynaics in the saple. This dynaics have an ipact on the phenoena connected with the quantized energy fluctuation. For given sizes of a disk and a given quantity of saturation agnetization of ferrite aterial, there are two different echaniss of the MDM energy quantization: (i) by a M 0 signal frequency at a constant bias agnetic field and (ii) by a bias agnetic field at a constant signal frequency. Fig. 5 shows the FMR diagonal coponent of the pereability tensor versus frequency at a constant internal agnetic field. The FMR diagonal coponent of the pereability tensor versus an internal agnetic field at a constant frequency is shown in Fig. 6. For both these cases, the discrete quantities of, in the region where 0, are shown scheatically for the first four MDM resonances. Fig. 7 illustrates correlation between the two echaniss of the MDM energy quantization. For a certain frequency f, the energy quantization is observed at specific bias-field (1) quantities: H 0 (1), H 0 (1), H 0,. Evidently, one has to use a statistical description of the spectral response functions of the syste with respect to two external paraeters a bias agnetic field and a signal frequency and analyze the correlations between the spectral response functions at different values of these external paraeters. It eans that, in neglect of losses, there should exist a certain uncertainty liit stating that H 0 H 0 fh 0 uncertainty liit. (47) This uncertainty liit is a constant which depends on the disk size paraeters and the ferrite aterial property (such as saturation agnetization). Beyond the fraes of the uncertainty liit (47) one has continuu of energy. The fact that there are different echaniss of quantization allows to conclude that for MDM oscillations in a quasi-2d ferrite disk both discrete energy eigenstate and a continuu of energy can exist. In quantu echanics, the uncertainty principle says that the values of a pair of canonically conjugate observables cannot both be precisely deterined in any quantu state. In a foral haronic analysis in classical physics, the uncertainty principle can be sued up as follows: A nonzero function and its Fourier transfor cannot be sharply localized. This principle states also that there exist liitations in perforing easureents on a syste without disturbing it. Basically, H i H 0 16

17 forulation of the ain stateent of the MDM-oscillation theory is ipossible without using a classical icrowave structure. If a MDM particle is under interaction with a classical electrodynaics object, the states of this classical object change. The character and value of these changes depend on the MDM quantized states and so can serve as its qualitative characteristics. The icrowave easureent reflects interaction between a icrowave structure and a MDM particle. It is worth noting that for different types of subwavelength particles, the uncertainty principle ay acquire different fors. An interesting variant of Heisenberg s uncertainty principle was shown recently in subwavelength optics [62]. Being applied to the optical field, this principle says that we can only easure the electric or the agnetic field with accuracy when the volue in which they are contained is significantly saller than the wavelength of light in all three spatial diensions. As volues saller than the wavelength are probed, easureents of optical energy becoe uncertain, highlighting the difficulty with perforing easureents in this regie. Fro this stateent, we can see, once again, that a MDM particle, distinguished by strong ME properties in subwavelength icrowaves, is beyond the regular EM-field description. The fact that agnetization dynaics in a quasi-2d ferrite disk have an ipact on the energy quantization of the fields in a icrowave cavity, was confired experientally in Ref. [32]. In this work, the MDM-originated quantized states of the cavity fields were investigated with variation of a bias agnetic field at a constant operating frequency, which is a resonant frequency of the cavity. The observed discrete variation of the cavity ipedances are related to discrete states of the cavity fields. Since the effect was obtained at a certain resonant frequency, the shown resonances are not the odes related to quantization of the photon wave vector in a cavity. It is evident that these resonances should be caused by the quantized variation of energy of a ferrite disk, appearing due to variation of energy of an external source a bias agnetic field. In Ref. [32], the observed effect of energy quantization of the fields in a icrowave structure in relation to quantization of agnetic energy in a ferrite disk is analyzed qualitatively as follows. At the regions of a bias agnetic field, designated in Fig. 8 as A, a, b, c, d,, we do not have MDM resonances. In these regions, a ferrite disk is seen by electroagnetic waves, as a very sall obstacle which, practically, does not perturb a icrowave cavity. In this case, the cavity (with an ebedded ferrite disk) has good ipedance atching with an external waveguiding structure and a icrowave energy accuulated in a cavity is at a certain axial level. At the MDM resonances (the states of a bias agnetic fields designated in Fig. 8 by nubers 1, 2, 3, ), the reflection coefficient sharply increases. The input ipedances are real, but the cavity is strongly isatched with an external waveguiding structure. It eans that at the MDM resonances, the cavity accepts saller energy fro an external icrowave source. In these states of a bias agnetic field, the icrowave energy accuulated in a cavity sharply decreases, copared to its axial level in the A, a, b, c, d, Since the only external paraeter, which varies in this experient, is a bias agnetic field, such a sharp ejection of the icrowave energy accuulated in a cavity to an external waveguiding structure should be related to eission of discrete portions of energy fro a ferrite disk. It eans that at the MDM resonances, we should observe strong and sharp reduction of agnetic energy of a ferrite saple. The question on a proper explanation of the experientally observed quantized states of the fields in a icrowave structure with an ebedded ferrite disk reains still open. In Refs. [29, 30, 63], it was stated that the ain reason of appearing the ultiresonance oscillations is nonhoogeneity of an internal DC agnetic field in a ferrite-disk saple. To answer the question whether non-hoogeneity of an internal agnetic field can really lead to quantization of agnetic energy of a ferrite disk, we analyze briefly the odel suggested in Ref. [30] and odified in Ref. [64]. 17

18 In a supposition that saturation agnetization is unifor everywhere inside the saple, the deagnetization agnetic field in a quasi-2d ferrite disk varies only in the radial direction and is defined as [30] where I r d M 0 is a diensionless scalar quantity and 0 r 0 H r M I r, (48). In Ref. [30], the ultiresonance absorption peaks are interpreted to be caused by MS waves propagating radially across the disk with the internal agnetic field dependent on a radial coordinate in the plane of a YIG fil and the ode nubers are deterined based on the well-known Bohr Soerfeld quantization rule. This odel is illustrated in Figs. 9 and 10. Fig. 9 shows scheatically the Bohr Soerfeld quantization rule for a certain MDM. A qualitative picture of the levels of an internal agnetic field for standing waves corresponding to soe MDMs is shown in Fig. 10. With increasing the ode nuber, the effective diaeter of the disk increases as well. The in-plane MS-potential-function distribution is supposed to be aziuthally nondependent. However, the Bohr Soerfeld quantization used in Ref. [30] does not clarify the proble of the observed quantization of agnetic energy of a ferrite-disk saple. This quantization should be observed as a Zeean splitting of the energy levels created by a DC internal agnetic field. In an assuption that saturation agnetization is unifor inside the saple and is the sae for every ode, one cannot suggest any echanis of Zeean splittings at the MDM resonances. When a ferrite disk is placed in a hoogeneous external (bias) agnetic field, agnetic energy of the entire saple is varied onotonically with variation of H 0 H 0, even if the MS-wave nubers are quantized and the effective diaeter is quantized as well. Magnetic energy of a saple in an external (bias) agnetic field is deterined by the deagnetization field. The deagnetization field is the agnetic field generated by the agnetization in a agnet and the deagnetization factor deterines how a agnetic saple responds to an external (bias) agnetic field. It is evident that for the MDM spectra, obtained at variation of a bias agnetic field and at a constant signal frequency, a discrete reduction of agnetic energy of a ferrite disk at the MDM resonance should occur because of quantization of the deagnetization field. Contrary to Ref, [30], will consider of quantization of the deagnetization field due to quantization of DC agnetization. Following the technique described in Ref. [64], we can put aside the question on a role of nonhoogeniety of an internal agnetic field in a ferrite disk. With averaging of the paraeter Ir on the region of the actual diaeter of the disk, 2, one can write [64] where H 0 H H H i average 0 d 2 ( n) eff average M 0 2 ( n) eff, (49) Hd 0 average M I. (50) Here 0 1 I average. The quantity of average I depends on the disk geoetry. With an approxiation of a hoogeneous (averaged on the disk-diaeter region) internal agnetic average 18

19 field, one can obtain the diagonal pereability-tensor coponent fro Eq. (43). The spectral proble is analyzed with -function distributions in a for of Eq. (6). The eigenvalues of MDM resonances are defined by discrete quantities of the diagonal pereability-tensor coponent. In this case, one can classify MDMs as the thickness, radial, and aziuthal odes. The spectral proble for these odes gives evidence for energy eigenstate oscillations [26, 28]. The technique used in Ref. [64] gives a good agreeent with experiental results of the resonance ode position in the spectra. For siplicity of a further analysis, we will suppose that Iaverage 1. Assuing that quantization of agnetic energy of a ferrite-disk saple is due to quantization of the DC agnetization, the deagnetization agnetic field for a certain MDM with nuber n (n = 1, 2, 3, ), is found as where M ( n) 0 eff 0 H M, (51) ( n) ( n) d 0 eff M is a quantized DC agnetization. We can write M M K, (52) ( n) ( n) 0 eff 0 ( n) where the ode coefficient is a diensionless quantity: 0K 1. The nuber corresponds to the ain MDM [28]. For ode n, the agnetic energy of a saple of volue V, placed in a agnetic field calculated as ( n ) ( n ) V n W H MdV H M V. (53) eff Discreteness of agnetic energy in a ferrite disk is due to discrete reduction: W H M dv H K M V. (54) ( n) ( n) ( n) eff 2 V ( ) The energy W n is the icrowave energy extracted fro the agnetic energy of a ferrite disk at the n-th MDM resonance. The effect is illustrated in Fig. 11. Based on this odel, we can explain qualitatively how the ultiresonance states in a icrowave cavity, experientally observed in Ref. [32], are related to quantized variation of energy of a ferrite disk, appearing due to an external source a bias agnetic field. When accepting this odel, we can say that we have a quantu effect of electroagnetically generated deagnetization of a saple. What is the physics of quantization of a DC agnetization in a ferrite disk? Unidirectionally rotating fields with the spin and orbital angular oenta, observed at MDM resonances, are related to precessing agnetic dipoles in a ferrite and to a double-valued-function agnetic current on a lateral surface of a ferrite disk [34, 35]. Circulation of the chiral surface agnetic current results in appearance of a DC gauge electric field [34] and thus appearance of DC electric charges on the ferrite-disk planes. When a dielectric-disk saple is situated in a vacuu near a ferrite disk, it is subjected with an induced electric gyrotropy and orbitally driven electric polarization [35]. Because of the electric field originated fro a ferrite disk, H 0, is 19

20 every separate electric dipole in a dielectric disk precesses around its own axis and for all the precessing dipoles, there is an orbital phase running (see Fig. 1). Due to the orbital angular oentu of MDM oscillations, a torque exerting on the electric polarization in a dielectric saple should be equal to a reaction torque exerting on the agnetization in a ferrite disk. Because of this reaction torque, the precessing agnetic oent density of the ferroagnet will be under additional echanical rotation at a certain frequency. At dielectric loadings, the agnetization otion in a ferrite disk is characterized by an effective agnetic field [35, 65]. This is a DC gauge (topological) agnetic field caused by precessing and orbitally rotating electric dipoles in a dielectric saple. As a result, an external effective agnetic field becoes less than a bias agnetic field. So, at a dielectric loading of a ferrite saple, the Laror H H 0 frequency should be lower than such a frequency for an unloaded ferrite disk. In other words, one can say that due to a loading by an external dielectric saple, the effective DC agnetic charges appear on the ferrite-disk planes. At the sae tie, it is worth noting that a ferrite disk is ade of a agnetic dielectric yttriu iron garnet (YIG) which has sufficiently high perittivity, r 15. Inside the YIG disk, we also have the torque exerting on the electric polarization due to the agnetization dynaics of MDM oscillations. Because of the effective agnetic charges on a ferrite-disk planes (caused now by the induced electric gyrotropy and orbitally driven electric polarization inside a ferrite), the deagnetizing agnetic field is reduced. It eans that the DC agnetization of a ferrite disk is reduced as well. We have the frequency ( n) ( n) 0 M 0, which is less than such a frequency M 0M 0 in an unbounded M eff eff agnetically saturated ferrite. In connection with the effect discussed above, it is relevant to refer here to soe recent studies in optics. In Ref. [66], light-induced agnetization using nanodots and chiral olecules was experientally studied. It was shown that a torque transferred through the chiral layer to a ferroagnetic layer, can create local perpendicular agnetization. The experient is based on the chiral-induced spin-selectivity effect described in Refs. [67 70]. An iportant conclusion arises also fro the above consideration. When at a constant frequency of a icrowave signal we vary a bias agnetic field, at MDM resonances we can observe a DC agnetoelectric effect. We have quantized DC electric and agnetic charges on the ferrite-disk planes. Experients in Ref. [32] shows that for the quantized states of icrowave energy in a cavity and agnetic energy in a ferrite disk, the cavity input ipedances on the coplexreflection-coefficient plane (the Sith chart) [71] are real nubers (see Fig. 12). We have a two-state syste. As the energy swept through an individual resonance, one observes evolution of the phase the phase lapses. Fro the Sith chart one can also see that the phase jup of is observed each tie a resonant condition is achieved. In a 2D paraetric space the ipedance space on a Sith chart there is clockwise or counter clockwise circulation for the MDM states. Direction of the circulation should be correlated with the direction of a bias agnetic field H 0. In Fig. 12 we, contingently, showed the clockwise circulation. At the first glance, the analyzed above processes in a structure shown in Fig. 8 can be described based on a siple schee shown in Fig. 13. At a given frequency, deterined by a RF source, a icrowave waveguide is presented by a characteristic ipedance Z 0. The waveguide is loaded by an ipedance Z L, which depends on an external paraeter a bias agnetic field H 0. Such a siple schee, however, leads us to an evident contradiction. With acceptance of the fact that the icrowave energy extracted fro the agnetic energy of a ferrite disk at the MDM resonance is a quantized quantity, we cannot assue, at the sae tie, that the frequency of a icrowave photon propagating in a waveguide is a constant quantity. 20

21 IV. MDM RESONANCES AND BOUND STATES IN THE MICROWAVE CONTINUUM When analyzing the scattering of EM waves by MDM disks in icrowave waveguides and energy quantization of the field in a icrowave cavity, it is relevant also to dwell on soe basic probles of agnon-photon interaction and bound states in the icrowave continuu. We are witnesses that long-standing research in coupling between electrodynaics and agnetization dynaics noticeably reappear in recent studies of agnon-photon interaction. In a series of works [72 76], it was shown that agnetostatic odes in a sall YIG sphere can coherently interact with photons in a icrowave cavity. In a sall ferroagnetic particle, the exchange interaction can cause a very large nuber of spins to lock together into one acrospin with a corresponding increase in oscillator strength. This results in strong enhanceent of spin-photon coupling relative to paraagnetic spin systes [72]. The total Hailtonian of the syste incorporates the agnetic H and electric E fields of the cavity and the M agnetization of the ferroagnetic particle [1] H 0 E 0 H M d r 2. (55) The spatially unifor ode of the agnetization dynaics is called the Kittel ode. The Kittel and cavity odes for agnon-polariton odes, i.e., hybridized odes between the collective spin excitation and the cavity excitation. In the theory, the interaction between the icrowave photon and agnon is described by the Hailtonian with a rotating-wave approxiation (RWA) [72, 73]. In a structure of a icrowave cavity with a YIG sphere inside, the avoided crossing in the icrowave reflection spectra (obtained with respect to the bias agnetic field) verifies the strong coupling between the icrowave photon and the agnon. The Zeean energy is defined by a coherent state of the acrospin/photon syste when a agnetic dipole is in its antiparallel orientation to the cavity agnetic field. The coherent energy exchange occurs back and forth between photon and a acrospin states. It is pointed out that because of non-unifority of the RF agnetic field in sufficiently big YIG spheres, high-order agnon odes the MDMs can be excited. For such MDMs, the avoided crossing in the reflection spectra, obtained with respect to the bias agnetic field, is observed as well [73 76]. Iportantly, for these odes, characterizing by non-unifor agnetization dynaics, the odel of coherent states of the acrospin/photon syste discussed above, is not applicable. The MDMs in YIG spheres were observed long ago [77]. There are so-called Walker odes [37]. While for a ferrite sphere one sees a few broad MDM absorption peaks, the MDMs in a quasi-2d ferrite disk are presented with the spectra of ultiresonance sharp peaks. There are very rich spectra of both types, Fano and Lortenzian, of the peaks [29 32]. The ferrite disk is an open hi-q resonator ebedded in a icrowave waveguide or icrowave cavity. In such a structure, sharp MDM resonances can appear as bound states in a icrowave-field continuu. Bound states in the continuu (BICs), also known as ebedded trapped odes, are localized solutions which correspond to discrete eigenvalues coexisting with extended odes of a continuous spectru. The BICs are solutions having an infinitely long lifetie. Recent developents show that in a large variety of electroagnetic structures there can be different echaniss that lead to BICs [78]. One of the ain reasons for appearance of the MDM BICs is a syetry isatch. Modes of different syetry classes (such as reflection or rotation) are copletely decoupled. MDM oscillations do not exhibit a rotational syetry, while a 21

22 regular waveguide structure is rotationally syetric. With such a condition, the MDM bound states observed in icrowave structures can be classified as the syetry-protected BICs. The MDM bound states are ebedded in the icrowave continuu but not coupled to it. Certainly, if a bound state of one syetry class is ebedded in the continuous spectru of another syetry class their coupling is forbidden. Moreover, the ME fields, originated fro a MDM ferrite disk, and the EM fields in a icrowave structure are described by different types of equations. At any stable state, MDMs cannot radiate because there is no way to assign a farfield EM-wave polarization that is consistent with vortex ME fields near a MDM ferrite disk. In a short-range interaction, an iportant aspect concerns the topological nature of the MDM BICs. These topological properties can be understood through eigen power-flow vortices with corresponding topological charges [34 36]. Quantized topological charges cannot suddenly disappear. They are protected by special boundary conditions in a quasi-2d ferrite disk. The MDM BICs cannot be reoved unless MDM topological charges are cancelled with another structure carrying the opposite topological charges. Such opposite topological charges appear on etal walls of a icrowave waveguide. The probing of these BICs is due to topologicalphase properties of MDMs resulting in appearance of spiral electric currents induced on etal parts of a icrowave structure [36]. The coupling is possible via the continuu of decay channels at the condition that phases of the MDM bound states are strongly deterined phases. The region where a MDM ferrite disk is situated in a icrowave waveguide is a contact region. In this region, the conditions of tunneling and pairwise coalescence of waveguide coplex odes can be fulfilled. A odel of interaction of MDM resonances with the icrowave-waveguide continuu is shown scheatically in Fig. 14. In the syste, the whole wave function can be divided into an internal, localized MDM syste Q and an external part of environental icrowave states P [79 81]. The regions in a space between a ferrite disk and waveguide walls are contact regions. Surface electric currents on waveguide walls are edge states. For a whole spectru of waveguide odes, there should be a continuu of such edge states. A ferrite disk is considered as a defect interacting with these edge states. We suppose that the region where the defect is localized, is sall copared with the characteristic length scale of waveguide odes. Interaction of MDM resonances with the icrowave-waveguide continuu is realized by two ways (channels): (i) interaction of MDMs with etal walls and (ii) coupling of surface helical bound states (induced on the walls at MDM resonances [36]) with a continuu of waveguide edge states on etal walls. In a structure of a thin rectangular waveguide with an ebedded quasi-2d ferrite disk [36], the contact regions can be considered as two vacuu cylinders with the sae diaeter as a ferrite disk. This odel is inapplicable in a case of a thick icrowave waveguide [82, 83]. However, use of two dielectric cylinders with a high dielectric constant allows considering these cylinders as the contact regions in a thick icrowave waveguide. Such structures are shown in Ref. [83]. In the contact regions, above and below a ferrite disk, we have helical-ode tunneling. There is an evidence for a torsion structure of the fields in the contact regions [36, 83]. Rotations of the power-flow vortices along an axis of contact regions are at different directions. The odel is shown in Fig. 15. The appearance of BICs is directly related to the phenoenon of an avoided level crossing of neighbored resonance states. BICs can occur due to the direct and via-the-continuu interaction between quasistationary states and can be viewed as resonances with practically infinite lifeties. In a syste of one open channel and two discrete resonances, these resonances can be coupled via the continuu. There is a short-range interaction with a purely iaginary coupling ter. If two resonances pass each other as a function of a certain continuous paraeter, one of the resonance states can acquire zero resonance width. This resonance state becoes a BIC. Whether or not two resonances interfere is not directly related to whether or not they overlap. Such a echanis of cutting down of a discrete eigenstate fro 22

23 any connection with a continuu was described initially in Ref. [84] in a fraework of a twolevel odel. In electroagnetic systes, the phenoenon of avoided crossing of narrow resonance states under the influence of their coupling to the continuu is considered in any recent publications (see Ref. [78] and references therein). The BICs reain perfectly confined without any radiation. For an open resonance structure, we have non-heritian effective Hailtonian with coplex eigenvalues. The BIC can be found by the condition that at least one of the coplex eigenvalues becoes real. Fro a nuber of characteristic features inherent in the BICs, one should distinguish such a property as a Fano resonance collapse. For the MDM resonances the effect of Fano resonance collapse was clearly deonstrated in Ref. [85]. Thanks to the tenability of the ferrite-disk resonator by an external paraeter (the bias agnetic field, for exaple) the MDM oscillations can becoe close interacting odes. Fig. 16 shows how the physical paraeter a bias agnetic field tunes the shape of the MDM resonance. The structure used in Ref. [85] is a icrowave cavity with an ebedded thin-fil ferrite disk. It is seen that as we approach the top of the cavity resonance curve, the iniu of the icrowave transission approaches the axiu of the transission. At the top of the cavity resonance curve, these levels (iniu and axiu transission) are in contact. The Fano line shape is copletely daped and one observes a single Lorentzian peak. The scattering cross section corresponds to a pure dark ode. What are the neighbored resonance states in this MDM oscillation? There are the dipole (dark) and quadrupole (bright) resonances [86]. Such resonances are shown in Fig. 17. These resonances, being energy degenerated, appear due to orbital angular oentus originated fro edge agnetic currents on a lateral surface of a ferrite disk [34]. When the contact regions the space above and below a ferrite disk are dielectric cylinders with a high dielectric constant [83], the coupling of two resonance states is via the continuu of decay channels in a dielectric region. With variation of a perittivity of dielectric cylinders loading a ferrite disk, discrete eigenstates can be cut down fro any r connection with a continuu at a certain threshold paraeter r. This effect of the Fano resonance collapse in such a structure is shown in Fig. 18. One can see that at r 38, the Fano line shapes are copletely daped for the 1 st and 2 nd MDMs and single Lorentzian peak appear. The scattering cross section corresponds to pure bright odes. One of the features attributed to the BICs is also strong resonance field enhanceent. In the case of MDM oscillations in a icrowave-field continuu, such a strong field enhanceent is shown in nuerous nuerical studies [32, 35, 82, 86, 87]. For exaple, Fig. 19 shows passing the front of the electroagnetic wave, when the frequency is (a) far fro the frequency of the MDM resonance and (b) at the frequency of the MDM resonance in the disk. This is an evidence for strong resonance field enhanceent at the MDM resonance. Recently, the MDM BIC phenoena, found further developent in a novel technique based on the cobination of the icrowave perturbation ethod and the Fano resonance effects observed in icrowave structures with ebedded sall ferrite disks [85, 88]. When the frequency of the MDM resonance is not equal to the cavity resonance frequency, one gets Fano transission intensity. If the MDM resonance frequency is tuned to the cavity resonance frequency, by a bias agnetic field, one observes a Lorentzian line shape. The effect of Fano resonance collapse has no relations to the quality factor of a icrowave cavity. Use of an extreely narrow Lorentzian peak allows exact probing of the resonant frequency of a cavity loaded by a high lossy aterial saple. With variation of a bias agnetic field, one can see different frequencies of Lorentzian peaks for different kinds of aterial saples. This gives a picture of precise spectroscopic characterization of high absorption atter in icrowaves, including biological liquids. Iportantly, there is no influence of the dissipation effects in the 23

24 icrowave cavity on the quality of the MDM resonances. The poles in the transission aplitudes are connected with the bound states and their lifeties. V. PT-SYMMETRY OF MDMS IN A FERRITE DISK A MDM ferrite disk is an open resonant syste. In the description of such a syste, a non- Heritian effective Hailtonian can be derived fro the Heritian Hailtonian including the environent. Because of strong spin orbit interaction in the MDM agnetization dynaics in ferrite disks, the fields of these oscillations have helical structures. The right-left asyetry of MDM rotating fields is related to helical resonances: The phase variations for resonant functions are both in aziuth and in axial z directions. This shows that proper solutions of a spectral proble for MDMs should be obtained in a helical-coordinate syste. The helices are topologically nontrivial structures, and the phase relationships for waves propagating in such structures could be very special. Unlike the Cartesian- or cylindrical-coordinate systes, the helical-coordinate syste is not orthogonal, and separating the right-handed and left-handed solutions is aditted. In a helical coordinate syste, the solution for MS-potential wave function for MDM oscillations in an open ferrite-disk resonator have four coponents. In a single-colun atrix, this coponents are presented as (1) (2) (3) (4), (56) where the coponents of the atrix are distinguished as forward right-hand-helix wave, backward right-hand-helix wave, forward left-hand-helix wave, and backward left-hand-helix wave [89]. For helical odes, there are no properties of parity (P) and tie-reversal (T) invariance the PT invariance. The PT-syetry breaking does not guarantee real-eigenvalue spectra. It was shown, however, that by virtue of the quasi-2d of the proble for a thin-fil ferrite disk, one can reduce solutions fro helical to cylindrical coordinates with a proper separation of variables. As shown [89], for the double-helix resonance, one can introduce the notion of an effective ebrane function. This allows reducing the proble of parity transforation to a one-diensional reflection in space and describe the boundary-value-proble solution for the MS-potential wave function by Eq. (8). In such a case, the integrable solutions for MDMs in a cylindrical-coordinate syste can be considered as PT invariant. At MDM resonances, we have BICs with two degenerated real-eigenvalue resonances and an iaginary-ter coupling. The Fano-resonance collapse, observed at variation of a quantity of a bias agnetic field, corresponds to the BIC state. BICs are considered as resonances with zero leakage and zero linewidth (in other words, a resonance with an infinite quality factor). In our case of MDM oscillations, BICs appear when a icrowave structure is PT syetrical. A PT-syetric Hailtonian is in general not Heritian, but if the corresponding eigenstates are also PT-syetric, then the eigenvalues are real and eigenstates ay be coplete [90, 91]. In PT-syetry MDM structures, the tie direction is defined by the direction of the spin and orbital rotations of the fields in a ferrite disk [35, 43]. Depending on a direction of a bias agnetic field, we can distinguish the clockwise and counterclockwise topological-phase rotation of the fields. The direction of an orbital angular-oentu expressed by Eq. (42) is correlated with the direction of a bias agnetic field H 0 (along +z axis or z axis), deterines 24

25 the tie direction in our topological clock. Both directions of rotation are equivalent and the biorthogonality relations for MDMs can be used. Iportantly, the MDM resonance in a PT syetrical icrowave structure can reveal the properties analogous to the effect of spectral singularities of coplex scattering potentials in PT-syetric gain/loss structures [90]. The wave incident on a MDM ferrite disk induces outgoing (transitted and reflected) waves of considerably enhanced aplitude. The disk then uses a part of the energy of the agnetized ferrite saple to produce and eit a ore intensive electroagnetic wave. This spectral singularity-related resonance effect relies on the existence of a localized region with a PTsyetric coplex scattering potential. Because this is a characteristic property of resonance states, spectral singularities correspond to the resonance states having a real energy. In Refs. [34, 35], we analyzed two spectral odels for the MDM oscillations in a quasi-2d ferrite disk. There are odels based on the so-called Ĝ and differential operators the G- and L-ode spectral solutions, respectively. For G odes one has the Heritian Hailtonian for MS-potential wave functions. These odes are related to the discrete energy states of MDMs. The G-ode orthogonality condition is expressed as [34] * p q p q ˆL E E ds 0, (57) S where the sign * eans coplex conjugation, is the G-ode MS-potential ebrane function, E is noralized energy of the ode, and S is an area of the disk plane. In a case of the L odes, we have a coplex Hailtonian for MS-potential wave functions. For eigenfunctions associated with such a coplex Hailtonian, there is a nonzero Berry potential (eaning the presence of geoetric phases) [34, 35]. The ain difference between the G- and L-ode solutions becoes evident when one considers the boundary conditions on a lateral surface of a ferrite disk. In solving the energy-eigenstate spectral proble for the G-ode states, the boundary conditions on a lateral surface of a ferrite disk of radius is expressed as r r r r 0. (58) This boundary condition, however, anifests itself in contradictions with the electroagnetic boundary condition for a radial coponent of agnetic flux density B on a lateral surface of a ferrite-disk resonator. Such a boundary condition, used in solving the resonant spectral proble for the L-ode states, is written as Hr Hr ia H r r, where and H r are, respectively, the radial and aziuth coponents of a agnetic field on a border circle. In the MS description, for the L-ode ebrane function, this boundary condition on a lateral surface is described by Eq. (7). While the G odes are single-valued-function solutions, the L odes are double-valued-function solutions. It follows fro the fact that at a given direction of a bias agnetic field, the two (clockwise and counterclockwise) types of resonant solutions for L-ode states ay exist. When a ferrite disk is placed in a rectangular waveguide, so that the disk plane is in parallel to the wide waveguide walls and the disk position is syetrical to these walls, MDM oscillations are PT syetric. It follows fro the property of PT-invariance of the L-ode ebrane function. For a ferrite-disk axis directed along a coordinate axis y, on a lateral boundary of a ferrite disk we have the following condition [86] * ( z) ( z) ( z). (59) r r r H r 25

26 The biorthogonality relation for the two L odes can be written as * p q p q, ( z) ( z) dl r r r r l = p ( z) q ( z) dl, r r l (60) where is a contour surrounding a cylindrical ferrite core. This relation ay presue the absence of coplex eigenvalues for L odes. The phase acquired by the MDM fields orbitally rotating in a ferrite disk is, where k is an integer [86]. The relation (60) has different eanings for even and odd quantities k. For even quantities k, the edge waves show reciprocal phase behavior for propagation in both aziuthal directions. Contrarily, for odd quantities k, the edge waves propagate only in one direction of the aziuth coordinate. In a l 2 k general for, the inner product (60) can be written as,, 1 k p r q r p q, where is the Kronecker delta. While a PT-syetric Hailtonian is, in general, not Heritian, in the proble under consideration we can introduce a certain operator, that the action of together with the PT transforation will give the heriticity condition and real-quantity energy eigenstates [90]. The operator is found fro an assuption that it acts only on the boundary conditions of the ˆ a i, ˆ L-ode spectral proble. This special differential operator has a for r where is the spinning-coordinate gradient. It eans that, for a given direction of a bias ˆ field, operator acts only for a one-directional aziuth variation. The eigenfunctions of operator are double-valued border functions [34]. This operator allows perforing the transforation fro the natural boundary conditions of the L odes, expressed by Eq. (7), to the essential boundary conditions of G odes, which take the for of Eq. (58) [34, 92, 93]. Using operator, we construct a new inner product structure for the boundary functions, ˆ ˆ ˆ ˆ ˆ pq,, n ( z) n ( z) dl, r r r (61) r l As a result, one has the energy eigenstate spectru of MDM oscillations with topological phases accuulated by the double-valued border functions [34, 86]. The topological effects becoe apparent through the integral fluxes of the pseudoelectric fields. There are positive and negative fluxes corresponding to the clockwise and counterclockwise edge-function chiral rotations. For an observer in a laboratory frae, we have two oppositely directed anapole oents e a. This anapole oent is deterined by the ter a i r r in Eq. (7). For a e e given direction of a bias agnetic field, we have two cases: a H0 0 and a H0 0. In addition to the above consideration, other evident aspects prove the PT syetry in a geoetrically syetric structure with a MDM ferrite disk. It follows fro the PT syetry of a scalar paraeter characterizing properties of a ME field. At the MDM resonance, energy ( E) density of the ME-field structure is defined by the helicity factor F [94]. There is the energy density of a quantized state of the near field originated fro agnetization dynaics in a 26

27 quasi-2d ferrite disk. In a geoetrically syetric structure, the distribution of a helicity factor, expressed by Eq. (22), is syetric with respect to in-plane x, y coordinates and antisyetric with respect to z coordinate [35, 43]. The helicity factor is a pseudoscalar. It changes a sign under inversions (also known as parity transforations). On the other hand, the factor changes its sign when one changes the direction of a bias agnetic field, that is, the direction of tie (see Fig. 20). In a view of PT-syetry of MDM oscillations (which is proven by an analyses of the MS-wave operator [86] and the tie-space syetry properties of the helicity factor ), the presence of an axial-vector orbital angular-oentu of a ferrite disk presues also existence a certain polar vector directed parallel/antiparallel to the vector. Such a polar vector exists. This is an anapole oent originated fro the - flux resonances due to surface agnetic currents on a lateral surface of a ferrite disk [34, 95]. There can be different types of the icrowave structures where PT-syetry of MDMs in a ferrite disk is broken. The effect of the PT-syetry breaking can be observed when, for exaple, a ferrite disk is placed in a rectangular waveguide non-syetrically with respect to its walls. In such a case, the distribution of a helicity factor becoes assyetric with respect to z coordinate [35]. This is due to the role of boundary conditions on a etal surface. Inside a etal a helicity factor is zero. Fig. 21 shows scheatically four cases of the helicity-factor distributions in a waveguide structure when two external paraeters a disk position on z axis and a direction of a bias agnetic field change. At such a nonsyetric disk position, the PT syetry of MDM oscillations is broken. Other cases are related to icrowave waveguiding structures with geoetric nonsyetry or the structures with inserted special chiral objects. As an iportant criterion for PT-syetry breaking, there is the syetry breaking of the helicity-factor distribution. Nonreciprocity effects, observed in the scattering atrix paraeters of a icrowave structure at MDM resonances, are also due to PT-syetry breaking. Nuerous studies of these properties for MDM structures, both nuerical and experiental, are shown in Refs. [35, 65, 96, 97]. To a certain extent, the above questions on the PT syetry breaking are akin to recent research in optics. For exaple, in publications [98 100], related to nonreciprocity in optics, it was shown that this effect takes place due to the PT syetry breaking. F ( E) F ( E) F ( E) VI. PREDICTION OF EXCEPTIONAL POINTS FOR A STRUCTURE OF MDMs IN THE CONTINUUM MDM oscillations exhibit eigenstate chirality which is related to the Berry phase. The phase of the MDM wave function is not restored after one encircling around a lateral surface of the disk, but is changed by. Only the second encircling restores the MS wave function including its phase [34]. There is the degeneracy of Heritian operators (G odes), at which the eigenvalues coalesce while the eigenvectors reain different. The L-ode analysis [34, 89] shows that the eigenvector basis is skewed and the eigenvalues theselves are coplex. Depending on a direction of a bias agnetic field, only one MDM state (clockwise or counterclockwise) doinates. In an analysis of MDM resonant structures, there are not only paraeters controlling internal properties of the separated L-ode ferrite disk. There are also paraeters by eans of which the coupling strength between the MDM disk and the environent can be controlled. When we consider our icrowave structures with an ebedded MDM ferrite disk and inserted chiral objects, we also should consider the chirality of the eigenstates and the external-paraeter chirality. In Ref. [96], the observations support the odel which assigns a special role of the MDM chiral edge states in the unidirectional ME field ulti-resonant tunneling. In Ref. [97], we have experiental observation of icrowave F a ( E) e 27

28 chirality discriination in liquid saples due to MDM eigenstate chirality. There is an evident correspondence between topological edge odes of MDM oscillations and external chirality of inserted objects. For open resonant structures, an iportant aspect concerns a behavior of non-heritian systes at the spontaneous breaking PT syetry, arked by the exceptional point (EP). EPs are exhibited as a distinct class of spectral degeneracies, which are branch points in a 2D paraeter space. The EP is the degeneracy intrinsic to non-heritian Hailtonians at which two eigenvalues and corresponding eigenvectors coalesce. The wave function at an EP is a specific superposition of two configurations. The phase relation between the configurations is equivalent to a chirality [ ]. The PT-syetry breaking is related to chirality at the external-paraeter loop. The sign of the chirality is defined via the direction of tie. In the experient in Ref. [102, 103], the positive direction of tie is given by the decay of the eigenstates. In our case, the tie direction is given by the direction of a bias field. This is an external paraeter of a type that can be controlled. In the prediction of exceptional points for a structure of MDMs in the continuu, we will use two tunable external paraeters: (i) a bias agnetic field H0 and (ii) the disk shift d in a waveguide along z axis, when the disk plane is parallel to waveguide walls. Evidently, if a MDM ferrite disk is placed in a waveguide syetrically to its wall (z = 0), the direction of z axis is arbitrary and either (positive or negative) tie directions are the sae. When the disk is shifted along z axis, we can distinguish a definite direction of tie. We chose the positive direction of tie by an orientation of a bias agnetic (1) (2) field along z axis. Now, let a bias agnetic field is varied in a range H H H. A MDM ferrite disk be shifted along z axis and its positions are defined as d1d d2. The two external paraeters copose a certain interaction paraeter H d. We suppose that, siilar to the structure shown in Fig. 8, the fields of a icrowave continuu are the fields of a icrowave cavity. Also, siilar to the resonances shown in Figs. 15 and 16, we have two odes. Following the picture in Fig. 16, the odes are related to (a) the single-rotating-agnetic-dipole (SRMD) resonance; (b) double-rotating-agnetic-dipole (DRMD) resonance. By variation of the quantity of a bias agnetic field, we have an adjustable frequency difference f1 f2 for these two odes. When a ferrite disk is shifted, with variation of quantity d, one observes variation of the helicity distribution. For the above resonances, we have variation of the field aplitudes and variation of the phases between the electric and agnetic fields [35, 43]. In the case of f1 = f2, there should be the possibility to adjust the phase difference between the odes (by variation of the disk shift along z axis). One of the ain evidence for existence of the EP is exhibition of chirality when one encircles this point in the space of [101, 102]. When the EP is encircled, the eigenvectors pick up geoetric phases. One of the evidence for this is the presence of the - flux resonances. There ( EP ) ( EP ) ( EP ) H d, where the two odes coalesce. should exist a critical value 0, The EP can be found by looking at the behavior of the real and iaginary parts of the eigenvalues. Such a basic analysis is beyond the fraes of the present paper. It is a purpose of our future research. However, a qualitative analysis allows to predict encircling exceptional points along two paths on a paraetric space created by a bias agnetic field and a disk shift. We predict four types of contours on such a 2D paraetric space (see Fig. 22). When a ferrite disk is shifted along z axis at the positions defined as d1d d2, two types of contours (1) (2) exist. For the range of positive agnetic field H0 H0 H0, we have an exceptional point CW EP 1 inside the clockwise contour: A A B B A. For the range of negative agnetic (1) (2) CCW field H H H, an exceptional point EP 2 is inside the counterclockwise , 28

29 contour: C C D D C. Fro Fig. 21, one can see that a siultaneous change of a sign of the bias agnetic field H0 and a sign of the disk shift d in a waveguide gives syetrically the sae pictures of the helicity distribution. It eans that there exist two other contours on our 2D paraetric space. When a ferrite disk is shifted along axis at the positions (1) (2) d1 d d2 and the range of positive agnetic field is H0 H0 H0, we have an CCW exceptional point EP 3 inside the counterclockwise contour:. At the (1) (2) positions d1 d d2 and the range of negative agnetic field H0 H0 H0, an CW exceptional point EP 4 is inside the counterclockwise contour: G G H H G. It is CW CW CCW CCW evident that EP 1 is the sae as EP 4 and EP 2 is the sae as EP 3. We can see that in a case of a waveguide with a non-syetrically ebedded ferrite disk, the ME near-field distribution becoes not PT syetrical. However, for the entire icrowave structure we have PT syetry together with chiral syetry of contours on a 2D paraetric space. To verify the above prediction of the EPs in a icrowave waveguide with a nonsyetrically ebedded ferrite disk, we will do the following qualitative analysis of the proble. Energy density of the ME near-field structure is defined by the helicity-factor density [94]. We have the regions with the positive and negative ME-field densities, which are characterized by the positive and negative helicity-factor densities, and, respectively. We define the positive helicity (the positive ME energy) as an integral of the MEfield density over the entire near-field vacuu region with the helicity-factor density : F ( E)( ) F ( E) ( ) z E E F F E F ( E)( ) F ( E)( ) F d. (62) ( ) ( E)( ) ( ) Siilarly, the negative helicity (the negative ME energy) is defined as an integral of the MEfield density over the entire near-field vacuu region with the helicity-factor density F ( E)( ) : ( ) F d. (63) ( ) ( E)( ) ( ) ( ) ( ) In a case of PT syetrical ME near-field distribution,. So, the total helicity (the total ME energy) of the entire near-field vacuu region surrounding a ferrite disk ( ) ( ) is equal to zero [94]: ( ) ( ) ( E)( ) ( E)( ) F F d 0. (64) If, however, the PT syetry is broken (in a case, for exaple, when a ferrite disk is shifted in ( ) ( ) a waveguide along z axis), we have. It eans that we ay have predoinance of the positive or negative ME energy. In particular, in Fig. 21 (a) we have predoinance of the negative ME energy, while in Fig. 21 (b), there is predoinance of the positive ME energy. On a etal wall, the helicity-factor density (and so, the ME-energy density) is zero. Where does the excess of ME energy go? The ME fields are quantized state of the near fields originated fro the agnetization dynaics in a quasi-2d ferrite disk. There are pseudoscalars, which appear as the fields of axion electrodynaics. The coupling between an axion field and the electroagnetic field 29

30 results in odification of the electric charge and current densities in electrodynaics equations. Such odified sources of electroagnetic fields were observed on etal walls of a icrowave waveguide with an ebedded MDM ferrite disk [36, 82, 86]. It is worth noting here that our ME-field structures are fundaentally different fro the ME fields considered in Ref. [104]. In Ref. [104] it has been proposed, hypothetically, that local ME fields can be realized due to the interference of several plane waves. In such a case, however, the question, how one can create a pseudoscalar field fro interaction of regular electroagnetic waves, reains open. Absorption of ME energy takes place only at the PT-syetry breaking. When, in particular, the PT-syetry-breaking-behavior is realized by shifting a ferrite disk in a waveguide along z axis, the excess of ME energy (positive or negative) leads to appearance of topological charges and currents induced on the upper and lower walls of a waveguide. In Refs. [36, 82], it was shown nuerically that in a structure with an ebedded ferrite disk, surface electric currents on a waveguide wall are the right-handed and left-handed flat spirals. For the PT-syetric structure, we have chiral syetry of such currents on the upper and lower waveguide walls. In a case of the PT-syetry breaking, spontaneous chiral syetry breaking occurs and the net chirality of the surface currents can fall into either the left-handed or the right-handed regie. This results in absorption of ME energy for MDMs. Let in the MDM oscillation spectra, there be two Heritian-Hailtonian G odes with energies and, which are sufficiently close one to another. We assue that coupling between these odes is reciprocal and is characterized by a real coefficient. For an open structure with PT syetry breaking, the a non-heritian Hailtonian of the two-state syste is expressed as E 1 E 2 E1 i1 E. (65) i and 2 Real quantities characterize the absorption rates. For the two odes, the absorption is due to the radiation inside a waveguide continuu caused be the currents induced on the waveguide walls of the PT-syetry-breaking structure. The signs in Eq. (65) characterize different kinds of the ME-energy excess (positive or negative). When the signs of iaginary coponents in Eq. (65) are the sae, an analytical treatent of encircling of EPs can be siilar to that used in Ref. [103]. As we discussed above, due to the topological action of the aziuthally unidirectional transport of energy in a MDM-resonance ferrite saple there should exist the opposite topological reaction (opposite aziuthally unidirectional transport of energy) on a etal screen placed near this saple. Conservation of the orbital angular oentu, related to the powerflow circulation, has been proven in a PT-syetrical icrowave structure with an ebedded ferrite disk [36]. It was also discussed that a vacuu-induced Casiir torque allows for torque transission between the ferrite disk and etal wall avoiding any direct contact between the. For the PT-syetrical icrowave structure, the Casiir energy is the sae above and below a ferrite disk. When the PT is broken the Casiir energy above and below a ferrite disk is not the sae. The orbital angular oentu is not conserved now. All this eans that together with chirality of the MDM ferrite disk we should observe chirality of the icrowave continuu. At the MDM resonances, the eigenvector basis of continuu becoes skewed and the eigenvalues of continuu becoe coplex. This appears clearer fro the fact that in icrowave structures with MDM ferrite particles we can observe path-dependent interference [82]. In general, we have a case of non-integrable systes. Following Ref. [105], there is the 30

31 possibility to introduce two topological nubers: (i) related to the Berry-curvature chirality of the eigenstates and (ii) the exceptional-point chirality in the external-paraeter space. It is worth noting that when a MDM ferrite disk is placed in a rectangular waveguide syetrically with respect to its walls, PT syetry is valid when one can neglect asyetry in the ME-field structure due to the wave propagation direction in a waveguide. This is possible in a case of a quasi-2d icrowave structure with a thin waveguide (see Fig. 4), analyzed in Ref. [36]. In a icrowave structure with a thick waveguide, there is an evident asyetry in the distribution of a helicity factor with respect to x, y coordinates [35, 82, 83]. Such asyetry due to the wave propagation direction in a waveguide can lead to unique topological effects. In Ref. [82], it was shown that in a icrowave structure of two ferrite disks placed on a waveguide axis syetrically to the waveveguide walls, the coupling of two identical MDM resonances is nonreciprocal. VII. DISCUSSION: ON THE COMPLEX-WAVE INTERACTION BETWEEN EM AND ME PHOTONS In Ref. [106], an optical syste that cobines a transverse spin and an evanescent electroagnetic wave in sei-space geoetry, has been proposed. In a theoretical odel, it is assued that in such surface odes with strong spin-oentu locking the real and iaginary EM wave vectors are utually perpendicular. Nevertheless, it becoes clear that a slight deviation of the spin vector fro the noral to the surface-wave propagation direction, will lead to appearance of a leaky wave, which is treated, atheatically, as a guided coplex wave. The question of a coplex-wave behavior is very iportant in our case of a icrowave structure with an ebedded ferrite disk. In the near-field region, ME photons, originated fro a quasi-2d ferrite disk, are characterized by the fields orbitally rotating in a plane parallel to the disk plane and decaying along the disk axis. Such a field structure can be described by the MS-wave function with utually perpendicular a real wave vector (in a plane parallel to the disk plane) and an iaginary wave vector (directed along the disk axis). When analyzing in Ref. [36] the electric current induced on a etal wall in a waveguide with an ebedded MDM ferrite disk, we found the vortex structure of this current, we can also assue that the field near to this wall has utually perpendicular real (in a plane parallel to the etal wall) and iaginary (along a noral to the wall) wave vectors. However, when we take into account the waveguide thickness, that is when we consider a 3D icrowave structure, the real and iaginary wave vectors of the ME-field structure will have co-parallel coponents. It eans that in the near-field region, the ME field is viewed as a coplex-wave field. This coplexwave behavior is observed together with the spin-orbit coupling in the ME-field structure. It becoes clear that together with localized and quantized ME-field coplex waves, we have to presue the presence of the coplex-wave continuu in a icrowave waveguide. In the above discussions, regarding experiental results obtained in Ref. [32] (and shown in the present paper in Fig. 8), we stated that for the icrowave photon propagating in a waveguide at a constant frequency, the energy extracted fro the agnetic energy of a ferrite disk at the MDM resonances cannot be a quantized quantity. So, the question of which icrowave continuu we have when observing discrete states of MDM resonance in a icrowave cavity with a constant frequency, reains open. To answer this question, we should consider the spectru of so-called coplex odes in lossless icrowave waveguides. In a closed lossless waveguide, there can be a finite nuber of real positive eigenvalues and an infinite nuber of coplex conjugate pairs of eigenvalues [107, 108]. Following the paraeter circulation in Fig. 12, we can see that for a given frequency, the input ipedance of the cavity passes through coplex quantities for every separate MDM. Assuing the presence of 31

32 coplex waves in the waveguide spectru we should conclude that in Fig. 13 not only ipedance is a coplex quantity but also a characteristic ipedance of a lossless Z L Z 0 waveguide is a coplex quantity. Generally, we have to take into account the possibility of interaction between discrete-state coplex-wave MDMs and the icrowave coplex-wave continuu. Let us consider general relations for ode orthogonality in a lossless regular waveguide assuing the presence of coplex waves in the waveguide spectru. We write hoogeneous Maxwell equations in an operator for: MU 0, (66) where M is the Maxwell operator: M i0 i 0 (67) and U is a vector function of the fields: U E. (68) H In a regular waveguide, the solution of Eq. (66) for electroagnetic wave propagating along z axis is presented based on the ode expansion: ˆ U А U е z, (69) where i is a propagation constant of a noral ode and functions on the waveguide cross section: ˆ U x, y ˆ E ˆ H For hoogeneous Maxwell equations, we can write: x, y x, y ˆ U is a ebrane. (70) ˆ ˆ MU QU, (71) where M is differential-atrix operator siilar to the operator M but operating over cross section coordinate of a waveguide and 0 Q ez ez. (72) 0 32

33 e z Here is the unit vector directed along z axis. We juxtapose now the differential proble described by Eq. (71) with the conjugated proble. For the conjugated proble, we have: ˆ ˆ MV QV, (73) where i is a propagation constant of a noral ode function of the ebrane-function fields of the conjugated proble: and ˆ V is a vector ˆ V x, y ˆ E ˆ H x, y x, y. (74) A for of the conjugate operator M is defined fro integration by parts [92]: S ˆ ˆ * ˆ ˆ ˆ ˆ S * * M U V ds U M V ds U RV d, (75) where S is the waveguide cross-section area, has a for is the contour surrounding S. The operator R n 0 n R n 0, (76) where is the unit vector directed along the external noral to contour. For a lossless waveguide and hoogeneous boundary conditions, operator M is the sae as operator M. As a result, one has the orthonorality condition ˆ QU ˆ V ds 0. (77) * * S Eq. (68) is correct for a general case of coplex waves in a lossless waveguide. In this case, we have biothogonality for conjugate odes. The conjugate odes are and are orthogonal * when 0. Otherwise, these odes constitute a nor: ˆ ˆ * * * ˆ ˆ ˆ ˆ. (78) N QU V ds E H E H e ds z S S In the spectru of a lossless waveguide, there are four coplex odes: i, * i, i, and i, which exist in fours:, * * *,,. Positions of these coplex wave nubers are shown in Fig. 23. An interaction of odes with and or and cause appearance of active power flows; eanwhile an interaction of odes with and or and lead to appearance 33

34 of reactive power flows. The pairs of odes which realize the carrying over of energy are characterised by the sae direction of phase velocity and different sign of aplitude changing [107, 108]. In a particular case of a propagating-wave behavior, we have i. For propagating odes E ˆ ˆ, ˆ ˆ E H H, the electric and agnetic fields oscillate in phase. So, the nor (78) is a real quantity. On the other hand, in a particular case of an evanescent wave, the electric-field oscillations are 90 phase shifted in tie with respect to the agnetic-field oscillations. So, the nor defined by Eq. (78) is an iaginary quantity. At such an evanescentwave behavior no transission of energy occurs. However, one can have power transission (tunneling) at certain boundary conditions of a section of a below-cutoff waveguide. Evanescent waves are non-local waves. So, noralization is non-local as well. To get transission in a below cut-off section we have to take into account the load conditions. Let the incident field be the positive decaying ode and the reflected field be the negative decaying ode (see Fig. 24). Such reflected and incident fields can be related through a certain load-dependent reflection coefficient [109]. Fig. 24 shows the coplex-plane cobination of incident and reflected fields that will exist at the end of a below-cutoff waveguide section. The port 1 is considered as a source. The section is terinated in a certain load at port 2. The above expressions involve both noral odes, with fields decaying in positive and negative, ˆ ˆ ˆ ˆ respectively. We consider E Ei, E Er and H Hi, H Hr, where are the incident and reflected electric and agnetic fields, respectively. We also assue that both incident and reflected electric and agnetic fields are linearly polarized in space and that. When, due to a load in port 2, the electric- and agnetic-field oscillations of odes and E, E i r H, H becoe 90 phase shifted in tie, the total electric Ei Er and total agnetic Hi Hr fields are in phase and the nor (78) is a real quantity. So, an interaction of odes with and causes appearance of active power flows through a below-cutoff waveguide section. This effect is siilar to the tunneling effect in quantu echanics ˆ ˆ structures. The two orthogonal basis states U and V oscillate in tie if they are 1 2 ˆ ˆ superiposed according to relation 1 i 2 U iv. This coalescent state is a chiral state. As an interaction paraeter of a syste a below-cutoff waveguide section there is a boundary condition at port 2. The pairwise coalescence occurs only at a certain boundary condition at port 2 that is at a certain value of an interaction paraeter. Now, we coe back to the structure shown in Fig. 8. Following the above analysis, we suppose that the port 1 is the cavity input. The localized region in a waveguide where a MDM ferrite disk is situated, is the port 2 with a certain load. We also suppose that in a waveguide section between ports 1 and 2 the coplex waves can be observed. Fig. 25 shows a possible region of existence of such coplex waveguide odes on the k diagra, where k is a real wavenuber. The phases of the MDM bound states are strongly deterined phases. We can assue that for every MDM there are certain quantities of a bias agnetic field that provide 90 phase shift in tie for the waveguide fields at the region of a ferrite disk, that is the region of the port 2. This exactly corresponds to points 1, 2, 3, in Figs. 8 and 12. Discretization of the icrowave response is only due to this coplex-wave behavior of waveguide odes. The conditions for noralization of these coplex-wave odes is illustrated in Fig. 23. How can we view a general picture of scattering of these coplex waveguide odes by the resonant MDM particle? We suppose, initially, that a agnonic-resonance in a sall ferrite i r 34

35 particle is deprived of any orbital rotations (in other words, there are no vortices in the resonant odes). There is a resonant agnetic particle dual to plasonic-resonance subwavelength particle. In this case, the scattering of propagating EM waves is well described by Mie theory. The particle near field is a decaying field described by Laplace equation. Now, we take into account that the quasistatic fields of this agnonic-resonance particle are orbitally rotating fields. Because of such a behavior, we have strongly deterined localized phases. At the MDM resonances, this gives a proper scattering for evanescent EM waves. All this eans that at MDM resonances, we observe scattering of waveguide coplex-wave odes. In other words, we have an interaction of MDM discrete states with the coplex-wave icrowave continuu. Till now, we do not have direct experiental and/or nuerical proves of the proposed odel of the coplex-wave interaction between EM and ME photons. Nevertheless, soe unique topological-phase properties shown in Ref. [82] should deserve our special attention regarding this aspect. It was shown that in a icrowave structure of two sall ferrite disks placed on a waveguide axis syetrically to the waveguide walls [82], the frequency split of the interacting MDM resonances is extreely narrow and (what is especially unique) is quite independent on the distance between the disks. Since this distance varied fro the near-field to far-field regions of the waveguide ode, the coupling between MDM disks is not due to the propagating EM wave, but because of the presence of the evanescent part of the coplex EM wave. One of specific properties of evanescent and tunneling odes is that they are non-local. For EM waves, the MDM disks appear as localized-phase singularities and interaction between the disks is due to the icrowave coplex-wave continuu. VIII. CONCLUSION In sall ferroagnetic-resonance saples, acroscopic quantu coherence can be observed. Long range agnetic dipole-dipole correlation can be treated in ters of collective excitations of the syste. In a case of a quasi-2d ferrite disk, the quantized fors of these collective atter oscillations the MDM agnons were found to be quasiparticles with both wave-like and particle-like behaviors, as expected for quantu excitations. In an electroagnetically subwavelength ferrite saple one neglects a tie variation of electric energy in coparison with a tie variation of agnetic energy. In this case, the Faraday-law equation is incopatible with the spectral solutions for agnetic oscillations. It appears, however, that in а case of MDM oscillations in a quasi-2d ferrite disk, the Faraday equation plays an essential role. In a ferrite-disk saple, the agnetization has both the spin and orbital rotations. There is the spin-orbit interaction between these angular oenta. This results in unique properties when the lines of the electric field as well the lines of the polarization in a saple are frozen in the lines of agnetization. In such a case, the Faraday law is not in contradiction with the agnetostatic equations describing the MDM-oscillation spectra. The MDMs in a ferrite disk are characterized by the pseudoscalar agnetization helicity paraeter which gives evidence for the presence of two coupled and utually parallel currents the electric and agnetic ones in a localized region of a icrowave structure. The near fields of a MDM saple are so-called ME fields. The agnetization helicity paraeter can be considered as a certain source which defines the helicity properties of ME fields. The ME fields, being originated fro agnetization dynaics at MDM resonances, appear as the pseudoscalar axionlike fields. Whenever the pseudoscalar axionlike fields, is introduced in the electroagnetic theory, the dual syetry is spontaneously and explicitly broken. This results in non-trivial coupling between pseudoscalar quasistatic ME fields and the EM fields in icrowave structures with an ebedded MDM ferrite disk. 35

36 Unique properties of interaction of MDMs with a etal screen becoe ore evident when one analyzes the angular-oentu balance conditions for MDM oscillations in a ferrite disk in a view of the ME-EM field coupling in a icrowave waveguide. MDMs in a quasi-2d ferrite disk are icrowave energy-eigenstate oscillations with topologically distinct structures of rotating fields and unidirectional power-flow circulations. Due to the topological action of the aziuthally unidirectional transport of energy in a MDM-resonance ferrite saple there exists the opposite topological reaction (opposite aziuthally unidirectional transport of energy) on a etal screen placed near this saple. This effect is called topological Lenz s effect. In a icrowave structure with an ebedded ferrite disk, an orbital angular oentu, related to the power-flow circulation, ust be conserved in the process. Thus, if power-flow circulation is pushed in one direction in a ferrite disk, then the power-flow circulation on etal walls to be pushed in the other direction by the sae torque at the sae tie. A vacuuinduced Casiir torque allows for torque transission between the ferrite disk and etal wall avoiding any direct contact between the. The fact that agnetization dynaics in a quasi-2d ferrite disk have an ipact on the energy quantization of the fields in a icrowave cavity, was confired experientally. Sharp ultiresonance oscillations, observed experientally in icrowave structures with an ebedded quasi-2d ferrite disk, are related to agnetization dynaics in the saple. This dynaics have an ipact on the phenoena connected with the quantized energy fluctuation. Inside the YIG disk, we have the torque exerting on the electric polarization due to the agnetization dynaics. Because of the effective agnetic charges on a ferrite-disk planes, the deagnetizing agnetic field is reduced. It eans that the DC agnetization of a ferrite disk is reduced as well. At the MDM resonances, we observe quantization of the DC agnetization of a ferrite disk. It is worth noting also that at the MDM resonance, a quasi-2d disk is anifested as a ME particle with two DC oents directed along the disk axis: (i) a DC agnetic oent (due to saturation agnetization) and (ii) a DC electric oent the anapole oent. In the paper, we proved PT-syetry of MDMs in a ferrite disk and analyzed the conditions of the PT-syetry breaking. When analyzing the scattering of EM waves by MDM disks in icrowave waveguides and energy quantization of the field in a icrowave cavity, we dwelled on soe basic probles of agnon-photon interaction and bound states in the icrowave continuu. For a PT-syetrical structure, one of the features attributed to the BICs is a strong resonance field enhanceent. In the case of MDM oscillations in a icrowave-field continuu, such a strong field enhanceent was confired in nuerous nuerical studies. In icrowave-cavity structures with ebedded sall ferrite disks, the effect of Fano resonance collapse was shown at variation of an external paraeter a bias agnetic field. When a PT syetry of a icrowave structure with an ebedded MDM ferrite disk is broken, one can predict existence of certain branch points the exceptional points in a 2D paraeter space of the structure. To answer the question of which icrowave continuu we have when observing discrete states of MDM resonance in a icrowave cavity with a constant frequency, we take into consideration the spectru of so-called coplex odes in lossless icrowave waveguides. The near fields of a MDM-resonance particle are orbitally rotating fields. Because of such a behavior, we have strongly deterined localized phases. For incident EM waves, this particle appears as a localized phase singularity. As an iportant proble, we discussed the possibility of interaction between discrete-state coplex-wave MDMs and the icrowave coplex-wave continuu. Finally, we have to note that interdisciplinary studies of EM-field chirality and agnetis is a topical subject in optics. The agnetic dipole precession has neither left-handed nor righthanded quality, that is to say, no chirality [20, 22]. The synthesis of chiral and agnetic 36

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42 FIG. 1. The fields in a coposed ferrite-dielectric structure at MDM resonances. Due to agnetization dynaics, an electric field inside a ferrite disk has both spin and orbital angular oentus. Electric dipoles induced in a dielectric saple precess and accoplish an orbital rotation. A bias agnetic field is directed norally to the disk plane. For opposite directions of a bias agnetic field, one has opposite rotations, both orbital and spin, of the fields inside a ferrite and a dielectric. Fig. 2. An orbital angular oentu of a ferrite disk at a MDM resonance. For a given direction of a bias agnetic field, the power-flow circulations are the sae inside a ferrite and in the vacuu near-field regions above and below the ferrite disk. 42

43 Fig. 3. The ME-EM field interaction in a icrowave waveguide. Evidence for the angular oentu balance conditions at a given direction of a bias agnetic field. In a view along z axis, there are opposite power flow circulations in a ferrite disk and on a etal surface. Fig. 4. A quasi-2d odel of a thin rectangular waveguide with an ebedded thin-fil ferrite disk. An insert shows the disk position inside a waveguide. 43

44 Fig. 5. FMR diagonal coponent of the pereability tensor versus frequency at a constant bias agnetic field. The discrete quantities, shown for the first four MDM resonances, are in the region where 0 H. 44

45 Fig. 6. FMR diagonal coponent of the pereability tensor versus a DC internal agnetic field at a constant frequency. The discrete quantities, shown for the first four MDM resonances, are in the region where 0 H H. i i 0 Fig. 7. Correlation between two echaniss of the MDM energy quantization: Quantization by signal frequency and quantization by a bias agnetic field. 45

46 Fig. 8. Relationship between quantized states of icrowave energy in a cavity and agnetic energy in a ferrite disk. (a) A structure of a rectangular waveguide cavity with a norally agnetized ferrite-disk saple. (b) A typical ultiresonance spectru of odulus of the reflection coefficient. (c) Microwave energy accuulated in a cavity; are jups of electroagnetic energy at MDM resonances. ( n) w RF Fig. 9. A odel showing a radial distribution of the MS-potential wave function based on the Bohr Soerfeld quantization rule. The region of an internal agnetic field where such a standing wave takes place is defined as H i Hi 0. The internal field increases with increase of a bias agnetic field. H 0 H i 46

47 Fig. 10. A qualitative picture of the Bohr Soerfeld-quantization levels of an internal agnetic field for standing waves corresponding to the first three MDMs. With increasing H i the ode nuber, the effective diaeter 2 ( n) eff increases as well. 47

48 const Fig. 11. A odel illustrating discretization of agnetic energy in a ferrite disk at. The slope of the straight lines is deterined by DC agnetization. is saturation agnetization of a hoogeneous ferrite aterial. At MDM resonances in a ferrite disk, the slope of the straight lines is deterined by. shows the icrowave energy M ( n) 0 eff extracted fro a ferrite disk at the n-th MDM resonances. Discretization of the agnetic energy is shown for the first four MDMs. W ( n) M 0 Fig. 12. Scheatic representation of the cavity input ipedances at MDM resonances on the coplex-reflection-coefficient plane (the Sith chart). The nubers and letters correspond to nubers and letters in Fig. 8 (b). As the energy swept through an individual resonance, one observes evolution of the phase the phase lapses. The phase jup of is observed each tie a resonant condition is achieved. 48

49 Fig. 13. An equivalent schee describing a structure shown in Fig. 8. At a given frequency, deterined by a RF source, a icrowave waveguide is presented by a characteristic ipedance. The waveguide is loaded by an ipedance, which depends on an external paraeter Z 0 a bias agnetic field H 0. Z L Fig. 14. An interaction of a MDM ferrite disk with a icrowave waveguide. The structure is viewed as the P + Q space. It consists of a localized quantu syste (the MDM ferrite disk), denoted as the region Q, which is ebedded within an environent of scattering states (the icrowave waveguide), denoted as the regions P. The coupling between the regions Q and P is regulated by eans of the two contact regions in the waveguide space. Fig. 15. Directions of rotations of the power-flow vortices along the axis of a contact region. In the contact regions, above and below a ferrite disk, we have helical-ode tunneling. 49

50 Fig 16. Modification of the Fano-resonance shape. At variation of a bias agnetic field, the Fano line shape of a MDM resonance can be copletely daped. The scattering cross section of a single Lorentzian peak corresponds to a pure dark ode. Reference to [85]. (a) (b) Fig. 17. An exaple of the power-flow density distributions in a near-field vacuu region above a ferrite disk. A ferrite disk is placed in a rectangular waveguide syetrically to waveguide walls. (a) Single-rotating-agnetic-dipole (SRMD) resonance; (b) double-rotatingagnetic-dipole (DRMD) resonance. There is the dissiilar radiation rates of the SRMD and DRMD resonances. Radiation of the SRMD results in strong reflection while at the DRMD ode one has electroagnetic-field transparency and cloaking for a ferrite particle. Reference to [86]. (a) 50

51 (b) (c) (d) 51

52 Fig. 18. Fano resonance collapse observed in the icrowave transission. (a) The structure of a rectangular waveguide with a MDM ferrite disk and loading dielectric cylinders. (b) Transission characteristics at a dielectric paraeter, (c) at, (d) at. For 30 r 38 r 50 the 1 st and 2 nd MDMs, the Fano line shapes are copletely daped at r 38. In this case, single Lorentzian peaks appear. The scattering cross section corresponds to pure bright odes. Reference to [83]. r Fig. 19. Evidence for strong resonance field enhanceent at the MDM resonance. Passing the front of the electroagnetic wave, when the frequency is (a) far fro the frequency of the MDM resonance and (b) at the frequency of the MDM resonance in the disk. Reference to [82]. (a) (b) Fig. 20. The helicity factor distribution near a ferrite disk in a geoetrically syetrical ( E) ( E) ( E) structure. In a red region, F 0, in a blue region, F 0, in a green region, F 0. The factor is characterized by antisyetrical distribution with respect to z axis. Along with this, the helicity factor changes a sign at tie reversal. F ( E) F ( E) (a) (b) 52

53 (c) (d) Fig. 21. Four cases of the helicity-factor distributions when two external paraeters the disk position on z axis and the direction of a bias agnetic field change. When a MDM ferrite disk is placed in a geoetrically nonsyetric structure, the distribution of a helicity factor ( E) F becoes nonsyetric as well. Reflection with respect to the xy plane with siultaneous change of a direction of a bias agnetic field copletely restore the sae picture of a helicity factor [look at the distributions (a) and (d) and the distributions (b) and (c)]. At such a nonsyetric disk position, the PT syetry (with respect to the syetry plane of the disk) of MDM oscillations is broken and the biorthogonality condition is not satisfied. F ( E) Fig. 22. Four types of contours encircling exceptional points along the paths on a 2D paraetric space, a bias agnetic field and a disk shift. CW Exceptional point EP 1 is inside the clockwise contour: A A B B A. CCW Exceptional point EP 2 is inside the counterclockwise contour: C C D D C. CCW Exceptional point EP 3 is inside the counterclockwise contour: E E F F E. CW Exceptional point EP 4 is inside the clockwise contour: G G H H G. 53