Controlling the dynamics of cavity magnon polariton via microwave polarization

The strong interaction state formed by light and matter enables the exchange of physical properties between spatially separated systems under the modulation of an external field^1–3 and thus facilitates the coherent information transfer between different matter platforms. Among them, collective spin excitation, i.e., the magnon mode,^4–6 employs spins as information carriers, which can flexibly exchange energy with other quasiparticles through interfacial exchange, magnetostriction, dipole or exchange interactions, etc. This provides advantages in the fields of electronics, vibrations, and optics. Owing to this excellent compatibility, the coupled cavity photon–magnon systems^7–11 have attracted considerable attention from the research community. The control of energy flow in the coupled photon–magnon state is central to information transfer. For example, the photon–magnon unidirectional coupling, nonreciprocal transmission, and magnon accumulation effects have greatly facilitated the construction of coupled states for on-chip devices.^12–14 However, the regulation of microwave polarization on energy flow has still not been explored.

Cross cavity is a useful platform for manipulating coupled magnon–photon states.^15–18 Its orthogonal structure can support different photon states, including continuous standing waves or dipole and quadrupole cavity modes.¹⁶ Particularly, the adjustment of the phase delay between the orthogonal structures enables the control of microwave polarization from linear to left- and right-handed circular polarization at the cross center. Left- and right-handed circular polarization fields exhibit time-reversal asymmetry, whereas the linearly polarized ones exhibit time-reversal symmetry.^18,19 The breaking of the time reversal symmetry has also been discussed in atom-field coupling systems, which can be studied by considering the conservation of the system’s spin angular momentum.²⁰ The breaking of the time-reversal symmetry differentiates the coupling dynamics of the left- and right-handed circular polarized microwaves, thus leading to nonreciprocal transmission. Consequently, the coupling between the cross cavity and magnon mode can provide more degrees of freedom for controlling the nonreciprocal transmission, thereby showing potential for harnessing the information processing between light and matter.

In this work, we fabricate polarization controllable cavity magnon polaritons (CMPs) using the strong interaction between a cross cavity and a magnon mode. By regulating the cavity microwave polarization, the amplitudes of the left- and right-handed circularly polarized microwave fields can be controlled. We theoretically describe this coupled system using a classical harmonic oscillator model,²¹ and our calculations agree well with the experimental results (Part II). Cross-cavity-based CMPs possess three nondegenerate polariton modes in the transmission spectrum.¹⁵ We observe the chiral coupling phenomenon where the side modes (central mode) appear only when the right (left)-handed circularly polarized microwave fields occur, and the amplitudes of the side and central modes can be continuously tuned. Notably, we observe the nonreciprocal transmission behavior where the opposite transmission mapping differs from the transmission mapping. This study demonstrates the feasibility of regulating polarization in a coupled cavity–magnon system and presents a novel approach for realizing microwave nonreciprocity. Based on the polarization controlled CMPs, an on-chip device with one-way energy transfer, which has potential applications in switches, isolators, and logical gates, can be developed.

To manipulate the microwave polarization, as shown in Fig. 1(a), we design a two-dimensional cross cavity that has two identical orthogonal microstrip line resonators. Each resonator has a length and width of 20 and 1 mm, respectively, and is connected to the feed lines. The gaps between the resonators and feed lines are 2 mm. The cavity is fabricated on a 40 × 40 mm² and 1.55 mm-thick Rogers 5880 printed circuit board (PCB) (relative permittivity ε_r = 2.2). To suppress the radiative losses,²² we use an aluminum box as a shield on the cross cavity. We simulate the radio frequency (rf) magnetic field near the center of the cross cavity. As shown in Fig. 1(a), when inputting a microwave field at port A, the polarization of the microwave field toward to the x-axis is excited, which is termed the x-mode. Similarly, the linearly polarized microwave field with the polarization direction toward the y-axis can be excited when inputting a microwave field at port B, which is termed the y-mode.

To establish CMPs, we employed an yttrium iron garnet (YIG) sphere as the magnon source. The YIG sphere is a ferrimagnetic insulator that has a large spin density (ρ_s = 4.22 × 10²⁷ m⁻³), low Gilbert damping rate, and high Curie temperature,^7,23 and thus, it possesses high cooperativity and relatively narrow linewidths.^24,25 By adding an external magnetic field, the ferromagnetic resonance (FMR) mode, which describes the uniform Larmor precession of the spin aligned in the direction of the external magnetic field, can be excited.²⁶ The intrinsic frequency of the FMR mode can be tuned using the external magnetic field. By fixing a 1 mm-diameter YIG sphere at the center of the cross cavity, the FMR mode can be coupled with both the x-mode and y-mode. When the FMR mode is close to the cavity frequency, the indirect coupling between the x-mode and y-mode is mediated by the FMR mode.¹⁵ 

Before establishing CMPs, the transmission spectra of the cross cavity need to be characterized. The S-parameters measured using a vector network analyzer (VNA) describe the input–output relation of the system in the frequency domain. Particularly, S₂₁ denotes the transmission rate, i.e., the amplitude ratio of output to input, while S₁₂ denotes the opposite transmission rate with switching of the input/output ports. Figure 1(b) displays the measured transmission spectrum, wherein the green rhombi and the blue circles represent the experiment data of |S₂₁|² (input port: port A and output port: port C) and |S₁₂|² (input port: port C and output port: port A), respectively, and the red curve represents the fitting results. Clearly, the |S₂₁|² spectral line is highly coherent with the |S₁₂|² spectral line. A nonreciprocal (reciprocal) system is defined as a system that exhibits different (same) transmission ratios when its input and output ports are switched.²⁷ Herein, nonreciprocity (reciprocal) denotes |S₂₁| equals (does not equal) |S₁₂|. Therefore, our system exhibits good reciprocity at zero external magnetic field.

To obtain the coupling dynamic information, we first fit the measured data using the Lorentzian line shape, as the red curve shown in Fig. 1(b). Subsequently, we obtain the intrinsic frequency of the cavity ω_c/2π = 5.437 GHz and the cavity damping rate β = 0.001 458. Then, we measure the transmission spectrum of the FMR mode at far detune of the intrinsic frequencies of the cavity and magnon. Consequently, we obtain the Gilbert damping rate of FMR α = 0.000 2038 and obtain the linearly dependent relation between the intrinsic frequency of magnons ω_m and the external magnetic field H as follows: $ωm=γH+Ha$⁠. Here, the gyromagnetic ratio γ/2π = 29.39μ₀ GHz/T, where μ₀ is the vacuum permeability and the anisotropic field μ₀H_a = 8.7 mT. Using this relation, the calculation and experiment can be correlated.

Figure 1(c) displays the schematic of the apparatus setup. We split the microwave field produced by the VNA using a power splitter and input into the cavity–magnon system via ports A and B; then, we connect a phase shifter between the power splitter and port B to tune the phase difference θ between the inputs x-mode and y-mode. Thereafter, we connect port C to the VNA to read the transmission signal S₂₁ and the opposite transmission signal S₁₂ by switching the input and output.

The x-mode and y-mode simultaneously couple with the FMR mode. However, when we transform the linear polarization representation to the circular polarization representation, the old basis, x-mode and y-mode, is transferred to the new basis, left-mode (left-handed circularly polarized microwave) and right-mode (right-handed circularly polarized microwave), respectively. Due to the chiral coupling property where the coupling will only occur between magnons and a microwave field with the same chirality, the FMR mode will only couple with the right-mode.¹⁹ This property enables a new degree of freedom to modulate the system transmission, microwave polarization, which necessitates the control of the phase shift between the x-mode and y-mode.

We denote the x-mode and y-mode as a_x and a_y, respectively. By adding a phase shift θ between the x-mode and y-mode by tuning the phase shifter, the input hybrid mode a_in containing a mix of x-mode and y-mode can be represented as

The left-mode and right-mode are denoted as a_L and a_R, respectively. Using the Jones calculus,²⁸ a_L and a_R can be represented as

Using these relations, the representation of the hybrid modes can be transferred to the circular polarization as

Thus, the amplitude of the left-mode and right-mode, A_L and A_R, respectively, has the following relation:

Using a classic harmonic oscillator model, the dynamics of the coupling system and its dispersion behavior can be studied. In this case, as shown in Fig. 1(d), the two orthogonal cavity modes, x-mode and y-mode, act as two oscillators with the same frequency ω_c and an intrinsic damping rate β. Connected by springs with a stiffness factor κ, which is measured as 0.126, the two oscillators coupled with another oscillator, representing the FMR mode, and have a Gilbert damping rate α. The output signal read out from port C is connected to the x-mode oscillator through a spring with a stiffness factor κ_out. Since κ_out ≪ κ, the impact of the output on the coupling system is small and can be neglected. Let r_x, r_y, and r_m be the displacements of the oscillators of the x-mode, y-mode, and FMR mode from its equilibrium, respectively. Then, the dynamics of the oscillator model can be represented as

In addition, f is the amplitude of the forces. Here, −i denotes that the two inputs are orthogonal. Set r_x = A_xe^−iωt, r_y = A_ye^−iωt, and r_m = A_me^−iωt, where A_x, A_y, and A_m are the amplitudes of the x-mode, y-mode, and FMR mode, respectively. Equation (6) can be represented in the matrix form as ΩA = F, where the amplitude vector A = (A_x, A_m, A_y)^T, external force vector F = (f, 0, −i × e^−iθ × f)^T, and the dynamic matrix Ω, which describes the coupling dynamics among the three oscillators, can be represented as

The amplitude of the output A_out is proportional to A_x.²¹ Subsequently, we solve for the amplitude vector A as A = Ω⁻¹F, and $Aout$ can be derived as follows:

To characterize the coupling system, depicting its transmission mapping is necessary. The transmission mapping characterizes the dependence of the transmission rate (|S₂₁| or |S₂₁|) on the frequencies of the input microwave field and the FMR mode. Due to the linear dependence of ω_m on the external magnetic field H, we depict the transmission mapping by sweeping the external magnetic field H using the spectra (|S₂₁| or |S₂₁|) in both the experiment and calculation.

Since our system contains three harmonic oscillators, it should have three nondegenerate eigenmodes corresponding to three dispersion branches in the transmission mapping [Figs. 2(d), 2(h), and 2(f), 2(j)]: one central mode and two side modes.¹⁵ The amplitude of the central and side branches can be regulated by tuning the phase shifter. We track the amplitude of the central branch by tuning the phase shift under the zero-detune condition (ω = ω_c = ω_m). As shown in Fig. 2(a), the amplitude of the central mode reaches its maximum and minimum (|S₂₁| = 0) at phase shifts of 90° and 270°, respectively. The blue rhombi represent the experiment data, and the red line represents the calculation data. Using Eq. (5), |A_R| and |A_L| can be plotted at different phase shifts, as shown in Fig. 2(b). We select four phase shift angles (θ = 90°, 180°, 270°, and 360°, corresponding to the left-handed circular, linear, right-handed circular, and linear polarized microwave, respectively) and then calculate and measure the transmission mappings of the system at these four angles, as shown in Figs. 2(c)–2(j), respectively. The first row of transmission mapping [Figs. 2(c)–2(f)] denotes the calculation results, and the second [Figs. 2(g)–2(j)] denotes the measured results. As indicated by the red arrows, each column corresponds to a selected angle. At θ = 90°, the excited cavity mode is a pure left-mode, which decouples with the FMR mode. Thus, the input microwave can only excite the central cavity mode, as shown in Figs. 2(c) and 2(g). In contrast, at the 270° phase shift, the excited cavity mode is a pure right-mode. Thus, the cavity and magnon effectively couple and yield the CMP. Then, the two level-repulsion-like side branches of the CMP are excited, as shown in Figs. 2(e) and 2(i). At other phase shifts of 180° and 360°, the excited microwave field is linearly polarized and comprises both left-mode and right-mode. Therefore, the central mode and the two side modes simultaneously exist, as shown in Figs. 2(d), 2(h), and 2(f), 2(j). The experimental results are consistent with the calculation results, demonstrating the feasibility to fully regulate the microwave polarization of a coupling dynamics system.

The chiral cavity–magnon coupling causes the time reversal symmetry to break, which signifies the nonreciprocal transmission of the cavity–magnon system. The cavity–magnon system, as discussed in Sec. II A, is reciprocal at zero external magnetic field, but the nonreciprocity appears when the coupling between the magnon mode and cross cavity is established.

In the opposite transmission case, the harmonic oscillator model is similar to the previous case, as shown in Fig. 1(d), but the input and output ports are switched. The microwave field is input through port C and output through ports A and B; then, the signal coherently mixed at the two output ports is read out. The phase shift of the phase shifter connected between ports A and B changes from θ to −θ due to the inversion of the transmission direction. The form of the dynamic matrix Ω is the same since the coupling dynamic relations of the three oscillators remain unchanged. However, the excitation force vector changes to F = (f, 0, 0)^T due to the switching of the input port. Since the output signals at ports A and B are coherently mixed, the amplitude of the output $Aoutop$ is proportional to $Ax−ieiθAy$ [similar to Eq. (7), −i denotes that the two output ports are orthogonal]. Using the same approach of solving A = Ω⁻¹F, $Aoutop$ can be derived. Consequently, the transmission rate of the opposite transmission case has a relation with that of the original system: $Aoutop(θ)=Aout(−θ)$⁠.

Experimentally, we observe the nonreciprocal transmission property of the coupling system by measuring the transmission mapping |S₂₁| and the opposite transmission mapping |S₁₂|. Figures 3(a) and 3(b) display the measured transmission mapping |S₂₁| and opposite transmission mapping |S₁₂| of the system at zero-detune (ω_c = ω_m) with continuous tuning of the phase shift. The nonreciprocal transmission phenomenon shows that |S₂₁| and |S₁₂| are not equivalent. Instead, |S₁₂| is equivalent to |S₂₁| with an inverted sign of the phase shift θ. To observe this nonreciprocal dispersion, we measure the transmission mappings of |S₂₁| and |S₁₂| at θ = 90° and 270°. The two left-side mappings shown in Figs. 3(c) and 3(d) are the transmission mappings of |S₂₁| corresponding to the mapping slides of θ = 270° and 90°, respectively, as indicated by the red dashed lines and the arrows. Furthermore, the two right-side mappings shown in Figs. 3(e) and 3(f) are the opposite transmission mappings of |S₁₂| corresponding to the mapping slides of θ = 270° and 90°, respectively. Notably, |S₁₂| at θ = 90° (θ = 270°) is consistent with |S₂₁| at θ = 270° (θ = 90°). These results agree with the theoretical conclusions. This study presents a new approach for realizing the nonreciprocal transmission and reveals its potential applications in constructing new types of microwave devices, such as switches, isolators, and logical gates, based on polarization controllable CMP dynamics.

In summary, we built a cross cavity wherein two orthogonal circularly polarized microwave fields can be arbitrarily regulated. Using the strong coupling between the cross cavity and YIG sphere, microwave polarization controllable CMPs were constructed. We analyzed the coupling system via both theoretical calculation using a harmonic oscillator model and experimental measurement. By continuously regulating the microwave polarization, we demonstrated the chiral coupling phenomena of CMPs. The amplitude of the central and side branches can be tuned and the side modes (central mode) will disappear in dispersion when the input signal is pure left-mode (right-mode). This phenomenon indicates that the coupling between the microwave and FMR mode with different levels of chirality will be effectively canceled. Interestingly, the chiral coupling causes the time reversal symmetry to break, yielding nonreciprocal transmission. The opposite transmission mapping is equivalent to the transmission mapping with an inverted sign of the phase shift angle θ. This paper demonstrates the feasibility to fully regulate the microwave polarization of CMPs and reveals the potential to construct on-chip devices with one-way energy transfer based on the polarization controllable CMPs, which may have applications in novel switches, isolators, and logical gates.

This work was funded by the National Natural Science Foundation of China (Grant Nos. 11974369 and 12122413), the STCSM (Grant Nos. 21JC1406200 and 22JC1403300), the Youth Innovation Promotion Association CAS (Grant No. 2020247), the SITP Innovation Foundation (Grant No. CX-350) and the SITP Independent Research Project, and the Strategic Priority Research Program of CAS (Grant No. XDB43010200). We would like to thank J. W. Rao for the discussions.

The authors have no conflicts to disclose.

Lihua Zhong: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Resources (lead); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Chao Zhang: Conceptualization (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (supporting). B.M. Yao: Supervision (lead); Validation (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.