Microscopic origin of level attraction for a coupled magnon-photon system in a microwave cavity

We discuss various microscopic mechanisms for level attraction in a hybridized magnon-photon system of a ferromagnet in a microwave cavity. The discussion is based upon the electromagnetic theory of continuous media where the effects of the internal magnetization dynamics of the ferromagnet are described using dynamical response functions. This approach is in agreement with quantized multi-oscillator models of coupled photon-magnon dynamics. We demonstrate that to provide the attractive interaction between the modes, the effective response functions should be diamagnetic. Magneto-optical coupling is found to be one mechanism for the effective diamagnetic response, which is proportional to photon number. A dual mechanism based on the Aharonov-Casher effect is also highlighted, which is instead dependent on magnon number.

Recently, for certain positions of a magnetic sample a mode attraction regime has been observed within a cylindrical [27] or a planar cavity [28]. Mode attraction is a general feature known already to be possible in optomechanical systems [29], and requires a mechanism related to non-Hermitian physics and exceptional points in dynamical systems [30,31]. In Ref. [27], the level attraction was interpreted as a manifestation of Lenz's law in a phenomenological electrical circuit model. The mechanism for level attraction between strongly interacting spin-photon excitations was proposed for a system with two driving terms with a phase offset [32], which was recently realized in a re-entrant cavity resonator using two separate drives [33].
The purpose of this paper is to discuss possible microscopic material mechanisms for mode attraction in ferromagnetic systems. Most of these mechanisms can be thought of in terms of an effective diamagnetic response of the system. To show this, we will use the approach based on the Maxwell's equations in dispersive media, where the internal magnetization dynamics is taken into account through response functions and constitutive relations [34].
Before going to specific details, we illustrate the essential ideas with a three coupled oscillator model. Such a model is obtained if we take a small magnetic specimen with the magnetization M (t) in a static magnetic field H 0 interacting with the microwave field h(t) of the cavity, as schematically shown in Fig. 1 (a). In the linear approximation for magnetization dynamics, the total interacting Hamiltonian can be written as (see Appendix A for details) where a † (a) is the creation (annihilation) operator for two degenerated cavity photon modes with right (R) and left (L) polarizations and the frequency ω c , b and b † describe the magnon oscillator with the frequency Ω m , g 0 denotes the magnon-photon coupling paramenter, and the last term is the external driving energy at the frequency ω, with the amplitude α in and coupling parameter κ ex .
The qualitative behavior of this model can be understood directly from the Hamiltonian (1). For Ω m ≈ ω c , the dominant contribution comes from the interaction term proportional to a † L b+ a L b † that describes a hybridization between the left polarized electromagnetic wave and the magnon mode precessing in the same direction, while a R remains decoupled [35]. More interesting behavior takes place if one can realize a situation with Ω m ≈ −ω c , where the term a R b + a † R b † is dominant, which is know as the 'two-mode squeezing' regime in optomechanics [1]. In this case, the level attraction regime is observed between the magnon and the right-polarized cavity mode in the region (Ω m + ω c ) 2 < 4g 2 [29]. The whole picture is summarized in Fig. 1 (b).
Another salient feature of the three oscillator response appears with varying the frequency of the drive around the point ω = −ω c when all three modes are degenerated, ω c = Ω m . This appears as a phase shift between the driving field α in and the driven cavity mode a R . If we introduce the dissipation in the magnon channel, Ω m − iκ m , and neglect cavity dissipation, phase shift between the driving term and the response field is estimated as which contains contributions for both electromagnetic waves of different polarizations coupled to the magnon mode. When all modes are degenerated, Ω m = −ω c , these two contributions provide the phase shifts of +π and −π around ω = −ω c that result in a characteristic 2π phase jump as shown in Fig. 1 (c). We believe that similar mechanism may be beyond the phase shift reported in Ref. [27]. We note, however, that within this illustration, 2π phase shift occurs at negative frequencies, whereas in the physical region ω ≈ ω c it equals to π. Below, we will consider the response functions that can bring 2π phase jump to positive ω. For our model in equation (1), the condition Ω m ≈ −ω c represents an unstable configuration where M and H 0 are antiparallel. Another possible scenario would require fully dissipative imaginary coupling constant g 0 . Either possibility explains why level attraction was not observed previously for cavity magnon-polaritons. In this paper, we discuss physical mechanisms that are not fully dissipative, but instead arise from effective 'diamagnetic' responses that may appear through other mechanisms. This can be somehow from the cavity itself, as proposed in Refs. [27,28], or through some mechanism in the sample response. We consider here possible mechanisms first via the electromagnetic pressure on the sample through electro-optics, and second via the magnon pressure on the cavity through the Aharonov-Bohm mechanism.
In the following we construct a classical electromagnetic theory for the measured responses in terms of cavity and sample impedances. This allows us to define a tensor magnetic response for cavity-magnon coupling through either magnetic or electro-optical mechanisms. We demonstrate that magneto-optical coupling, or more specifically, the inverse Faraday effect, provides one possibility for the effective diamagnetic response, which being quantized, leads to a linearized optomagnonic Hamiltonian [15,16]. Another possible scenario for the attractive regime comes from the Aharonov-Casher effect [36].
As we discuss at the end, this effect can be considered as dual to the magneto-optical coupling. In both scenarios, the the magnon-photon interaction is driven by the cavity electric field.

Electromagnetic theory and diamagnetic response
A theory sufficiently complex to allow for different placements within a cavity in an actual experiment can be obtained using classical electrodynamics. This is most naturally formed in terms of impedances, and provides a direct connection to the scattering matrix element S 21 (ω) typically measured in experiment. Frequency shifts and phase can be obtained from the complex scattering elements. These are obtained using standard classical electromagnetic theory in terms of energies W e and W m , defined over the sample volume V S as where the complex electric permittivity,ε = 1+α, and magnetic permeability,μ = 1+χ are different from those for the free space due to the presence of the sample. The cavity electric and magnetic fields satisfy Maxwell's equations in free space and are E 0 and H 0 . The fields perturbed by the sample are E and H. Together the energies provide the sample contribution to the measured impedance Z S = W e − W m [37]. The perturbative impedance Z S can then be simply added to a cavity impedance Z c with parameters determined by unloaded measurements. The required properties needed to construct Z S from the response functions for the loaded cavity can be understood in terms of the electromagnetic theory of continuous media. In this approach, we ignore at the beginning the finite sample size and describe the field inside the cavity by the Maxwell's equations By expanding the field, E = E 0 + δE and H = H 0 + δH, where δE and δH are related to the sample, we rewrite the equations above in the form where the free space cavity field can be considered as driving terms, which provide the response determined by the material functions of the medium. Let us consider a situation where the plane electromagnetic wave is propagating in a homogeneous dispersive birefringent medium along the gyrotropic axis taken as z direction. In this case, we take magnetic susceptibility tensor in the form andε = 1 for simplicity. Transforming (5a) and (5b) to the ω-domain, and replacing ∂ z by iω c /c, where ω c is some characteristic frequency of the cavity, we obtain For a dissipative system, which parameters can be extended to the complex plane, this equation has the points, which satisfy the relation χ 2 (ω) − χ 1 (ω) = 2iω c /ω, where the matrix on the left-hand side does not have a diagonal form, which is a manifestation of exceptional points in non-Hermitian systems [31].
Calculating the circular components of the response magnetic field from equation (7), we obtain which is directly related to the impedance of the loaded cavity via equations (3a, 3b). If we take the components of the susceptibility for the ferromagnetic resonance [35], , we obtain a result consistent with equations (A.7)-(A.9) which shows level repulsion behavior around ω ≈ Ω m . To obtain level attraction instead of repulsion, we formally need a 'diamagnetic' response here, which corresponds to γM s < 0.

Inverse Faraday effect mechanism
From the physical point of view, the effective resonant diamagnetic response may be realized if we consider nonlinear effects in light-matter interactions. One particular example is the inverse Faraday effect, where circular polarized components of the electric field create an effective magnetic field is the electric field amplitude in the complex representation, E = (E + E * )/2. The effective magnetic field is able to drive the dynamics of the magnetization towards the resonance, which can be taken into account in Maxwell's equations through the modulation of the electric permittivity. For illustration, we consider the experimental setup shown in Fig. 2 (a), where the cavity electromagnetic wave is excited along the x-direction perpendicular to the saturation axis of the magnetization, which is taken as z-axis. In this case the displacement field generated by the inverse Faraday effect is given by δD i (t) = iε 0 f ǫ ijx m x (t)E j (t), where ǫ ijk denotes the Levi-Civita tensor. If we consider only linear terms in the fluctuating parts of the fields, and transform to Fourier space, the effective response that takes into account resonant magnetization behavior can be estimated as where D(ω) = [D(ω) + D * (ω)]/2, E 0 and ω L are the amplitude and the frequency of the driving field, and we neglected the terms with E(ω ± 2ω L ). The details can be found in Appendix B. Equation (10) shows that in the linear approximation the inverse Faraday effect provides an effective dielectric resonant permittivity proportional to the intensity of the driving field and consistent with the macroscopic Lenz's effect.
If we apply the formalism of equations (5a, 5b) together with the effective permittivity in (10), excluding the magnetic field, we obtain the following equations for the fluctuating cavity electric field where we have introduced effective coupling constants g i = ε 1/2 0 f E 0i /4 (i = y, z), and where∆ = ω − ω L with ω being the frequency of the probe field. The equation above shows that the magneto-optical coupling to a dynamic magnetization, in the linear approximation, is equivalent to an effective 'diamagnetic' response. This becomes apparent if consider the equation for δE z where the magneto-optical contribution competes with the ordinary ferromagnetic resonance [∼ χ yy (ω)] from the y-component of the fluctuating magnetic field.

Magneto-optical coupling mechanism
A possible origin for the effective diamagnetic contribution in (10) and (11) can be understood from a quantized optomagnonic Hamiltonian [15,16]. For this purpose, we consider a spin S = Sδ(r − r 0 ) coupled to the electric field through the magneto-optical coupling where we take δε ij = if ǫ ijk S k in the form, which gives the Faraday effect. We take the z-axis as a quantization axis for the spin S, and apply the electric field along the x-direction, as shown in Fig. 2 (a), so that the magneto-optical coupling becomes where g 0 = f ω c V −1 (S/8) 1/2 sin 2 (πnx 0 /L x ) is the coupling constant (x 0 is for the position of the sample, and n is the cavity mode index), and the electric field is taken in the form of cavity standing waves E( For a ferromagnetic specimen, the effective S can be estimated as 10 10 , which gives g 0 ≈ 10 5 Hz [15]. The interaction energy in (13) is the same as in optomechanical applications. In the rotating wave approximation (RWA), the total Hamiltonian can be written as (14) where we have transformed to the linearly polarized basis, a L = (a 1 − ia 2 )/ √ 2 and a R = (a 1 + ia 2 )/ √ 2. The last term in this expression is the external driving, which becomes time independent under the RWA [1]. Compared to the magneto-dipole interaction in (1), the optomechanical coupling in RWA leads to the detunning parameter ∆ = ω L − ω c that may have an arbitrary sign depending on the ration of the driving frequency, ω L , to ω c .
The interaction part of the Hamiltonian (14) can be linearized as follwos: where n 1 and n 2 denote the average numbers of correspondent cavity photons. The linearized Hamiltonian is written as where we have introduced renormalized coupling parameters g 1 = g 0 √ n 2 and g 2 = g 0 √ n 1 . In a situation where the driving field is coupled only to the cavity photon of one polarization, we would normally expect g 2 ≫ g 1 . The Heisenberg equations of motion for the Hamiltonian (16) can be written in the matrix form: which has a pair of trivial eigenvalues ω 0 = ±∆, and two pairs of hybridized eigenmodes In accordance with [29], if ∆ > 0, ω (±) demonstrate level attraction in the region where (∆ 2 − Ω 2 m ) 2 < 16(g 2 1 + g 2 2 )∆Ω m , which corresponds to the frequency of the cavity modes in the co-rotating frame being negative. This region is bounded by the exceptional points, where the eigenmodes coalesce, and the matrix on the right-hand side of (17) is characterized by the Jordan form:  Remarkably, the energy-level picture for the dispersion relations (18a, 18b) shown in Fig. 2 (b) becomes similar to that in Fig. 1 (b) for the three-oscillator model with inverted horizontal axis. Solutions of equations of motion (17) for the cavity modes have the following form , where we used real α for simplicity. If g 1 ∼ √ n 2 is nonzero, driving of a 1 excites also a 2 .
In this situation, if we take into account the dissipation of the cavity mode, ∆ → ∆+iκ c , the response in (20) has 2π-phase shift with respect to the drive due to the degeneracy of the cavity modes with different polarizations.

Aharonov-Casher effect
Here, we discuss another mechanism of mode attraction for a coupled magnon-photon dynamics inside a ferromagnetic insulator based on the Aharonov-Casher effect [36]. This effect is related to a dualism between the electrodynamics of electric charges in a magnetic field and that of charge-neutral magnetic dipoles in an electric field. When applied to magnon dynamics this leads e. g. to the Landau quantization of the magnon states under applied electric field gradient [38]. Since the diamagnetism of a conventional electron gas follows from Landau quantization of electron motion, by analogy, we expect that a sort of 'diamagnetic' response should also exist for magnons in the electric filed, which, in turn, can provide us a mechanism for the level attraction, as explained below. For this purpose, we consider the following spin Hamiltonian for a small ferromagnetic specimen where J is the ferromagnetic exchange constant, K is the anisotropy parameter, and the summation is over the nearest neigbouring sites limited by the size of the specimen. The first term contains a field phase associated with the cavity electric field. The phase factor is given by [38] θ where the integration is taken along the path connecting ith and jth sites, and z plays the role of a spin quantization axis. If we suppose that the characteristic wavelength of the electric field is much larger than the sample size, then the interaction part in the Hamiltonian (21) can be written is the spin current in the sample (see Fig. 3). By quantizing the spin operators, S If the sample is populated into a state with n k 0 ≫ n −k 0 for some k 0 , where n k = b † k b k is the magnon number, and magnon states with different k are well separated, we can consider a simplified model where we added the driving term H , with the amplitude of the driving field β (in) k 0 . The Hamiltonian (23) can be considered as dual to the model with magneto-optical interactions in equations (13,14). It can be interpreted in terms of the pressure force created by the magnon flow with finite k 0 on the cavity photon oscillator, with the position operator proportional to a + a † .
Similar to optomagnonic Hamiltonian (14), equation (23) can be transformed to the time-independent frame by applying the unitary transformation to the magnon . In this representation, the magnon frequency is replaced by the magnon detunning parameter, Ω k 0 → −∆, where∆ = ω − Ω k 0 , with which we can reach the attraction regime in the region (∆ − ω c ) 2 < 4g 2 k 0 n k 0 . We note that, in contrast to optomagnonic case, the single magnon coupling constant is small compared to Ω k 0 . For realistic parameters, J = 100 K, ω c = 10 12 s −1 , and S = 1, we estimate g k 0 = 10 −4 s −1 , which means that the effect is difficult to realize experimentally. However, we believe that is can be relevant to ultrafast magnetization dynamics in metamaterials with S ≫ 1, and can provide an experimental evidence for the spin current electrodynamics proposed in [36]

Summary
We considered several mechanisms of mode attraction for a coupled magnon-photon system inside a microwave cavity. Using the Maxwell's equations of continuous media, we demonstrated how the level attraction can be described in terms of response functions of the medium. In particular, we derived the effective permittivity for the inverse Faraday effect, and showed that it can be interpreted as an effective diamagnetic effect in the equations of motion for the electromagnetic fields inside the cavity. This approach has been supported the quantum picture based on the optomagnonic Hamiltonian [15,16], which has reasonable parameter values for experimental demonstration of the effect.
Also, we discussed another mechanism, which is based on the electro-dipole effect of the magnon spin current [39]. In the context of level attraction, this mechanism can be considered as dual to optomagnonic approach because it requires driving of the magnon mode with finite wave vector. This effect is relativistically small compared to the magneto-optical mechanism.
where ω c and N c denote the frequency and the index of the cavity mode, V is the volume of the cavity, L z is the size of the cavity along the z-axis, and a λ (a † λ ) is the photon creation (annihilation) operator in the helicity basis e λ = (λ, i, 0)/ √ 2 with λ = 1 (−1) for the right (left) polarized wave. Using this quantization of the magnetic field in (A.1), and adding the electromagnetic energy of cavity photons, we obtain the Hamiltonian (1) with g 0 = (ω c M 2 s / µ 0 V ) 1/2 if the sample is at the position of maximum magnetic field.
The Heisenberg equations of motion for the Hamiltonian (1) with the driving have the following form: The stationary response of the system to the driving at frequency ω can be found from the algebraic system of equations for the field amplitudes which has the following solutions The where A = Ω 3 m −9Ω m ω 2 c +27g 2 0 ω c and B = Ω 2 m +3ω 2 c , and C = √ B 3 − A 2 . The frequencies are real in the region where B 3 ≥ A 2 . Otherwise, ω 1 and ω 2 become complex. In the limit g 0 = 0, we have ω 0 = −Ω m , ω 1 = −ω c , and ω 2 = ω c .

Appendix B. The effective permittivity for the inverse Faraday effect
In order to derive the effective response for the inverse Faraday effect, we consider the electric displacement field δD i (t) = iε 0 f ǫ ijx m x (t)E j (t). (B.1) In the Fourier space, this expression can be written as where the magnetization dynamics in the effective field is taken into account through the ferromagnetic susceptibility, m x (ω) = χ xx (ω)H eff x (ω), and the Fourier components of the fields are determined as follows F (ω) = ∞ −∞ dte iωt F (t). The Fourier component of the effective field is given by which gives after substitution into (B.2) We imply that the electric field is driven by the external field E 0 (t) = [E 0 exp(−iω L t) + E * 0 exp(iω L t)]/2, and expand (B.4) up to the linear order in the fluctuating field, E = E 0 + δE, which gives after some algebra Neglecting the last terms with ω ± 2ω L and the first term, which corresponds to the static component of the effective field, we obtain equation (10).