Magnetic-field controlled on-off switchable non-reciprocal negative refractive index in non-Hermitian photon-magnon hybrid systems

Abstract

Photon-magnon coupling, where electromagnetic waves interact with spin waves, and negative refraction, which bends the direction of electromagnetic waves unnaturally, constitute critical foundations and advancements in the realms of optics, spintronics, and quantum information technology. Here, we explore a magnetic-field-controlled, on-off switchable, non-reciprocal negative refractive index within a non-Hermitian photon-magnon hybrid system. By integrating an yttrium iron garnet film with an inverted split-ring resonator, we discover pronounced negative refractive index driven by the system’s non-Hermitian properties. This phenomenon exhibits unique non-reciprocal behavior dependent on the signal’s propagation direction. Our analytical model sheds light on the crucial interplay between coherent and dissipative coupling, significantly altering permittivity and permeability’s imaginary components, crucial for negative refractive index’s emergence. This work pioneers new avenues for employing negative refractive index in photon-magnon hybrid systems, signaling substantial advancements in quantum hybrid systems.

The authors demonstrate a magnetic-field-controlled, on-off switchable non-reciprocal negative refractive index in a non-Hermitian photon-magnon hybrid system, highlighting the crucial role of non-Hermitian dynamics in advancing photonics and magnonics in quantum hybrid technologies.

Introduction

In the rapidly advancing fields of optics, magnonics, and quantum information science, the interaction between photons and magnons within photon-magnon hybrid (PMH) systems, alongside the associated non-Hermitian dynamics, is at the forefront of innovative research^1–8. These systems reveal new complexities in optical and magnetic behaviors and their coupled interactions, attracting broad interest due to their potential applications in quantum information devices such as quantum networks⁹, transducers^10–12, memories¹³, entanglement^14,15, and sensors^16,17. A central focus of these investigations is non-Hermiticity, characterized by an imaginary component in the coupling constant^6,18–20. This leads to novel behaviors in non-reciprocal signal transmission, reflection, and absorption^7,8,21,22, challenging conventional Hermitian principles.

Over the past few decades, artificial metamaterials have emerged as a fascinating research area, particularly due to their unique ability to manipulate electromagnetic or acoustic waves in unprecedented ways^23–29. While metamaterials with negative refractive index (NRI) have been realized using various metastructures, such as continuous metal wires with periodic split-ring resonators^25,30, fishnet structures^31,32, and metal-dielectric-metal multilayers^33–35, the occurrence and mechanisms of NRI in PMH systems remain unexplored.

Here, we experimentally demonstrate a magnetic field-controlled, on-off switchable NRI in a non-Hermitian PMH system, uncovering a unique non-reciprocal NRI driven by the interplay of coherent and dissipative coupling. The non-Hermitian dynamics in the PMH system enhance anti-damping, thereby redefining dissipation mechanisms and establishing the optical conditions necessary for the emergence of NRI, as interpreted through an analytical circuit model that connects NRI with the non-Hermitian aspects of PMH systems. This study emphasizes the pivotal role of non-Hermitian dynamics in photonics and magnonics, particularly in enabling non-reciprocity and magnetic field-controlled on-off switchability, which are essential for the development of real devices in optical and magnonic quantum hybrid technologies.

Results

Experimental observation of NRI

Figure 1a shows our measurement setup for the scattering parameters S_ij as functions of both the AC current frequency (f_AC) and the static magnetic field (μ₀H), as well as the geometry of our PMH sample. The sample is composed of a yttrium iron garnet (YIG) film and an inverted split-ring resonator (ISRR) patterned on the ground plane. Optical images of the sample’s front and back sides are displaced in Fig. 1b. The magnitude (first row) and phase (middle) of the experimentally measured S₂₁ (from port 1 to port 2) and S₁₂ (from port 2 to port 1) spectra are given in Fig. 1c. Owing to the sufficient-strength coupling between the YIG’s ferromagnetic resonance (FMR) mode and the ISRR’s photon mode, two split (higher- and lower-branch) hybrid modes appear in the |S₂₁| and |S₁₂| spectra, but with opposite asymmetry between |S₂₁| and |S₁₂|(see the varying contrasts denoted by the blue color around the coupling center in the field). These split hybrid modes represent anti-crossing dispersion at or near the common resonance frequency, as previously reported³⁶. Note that all the spectra shown in Fig. 1c were obtained by subtracting the background signal of the microstrip line from that of the entire structure to investigate the interest region only³⁷ (for details, see supplementary Note 3). In the magnitude and phase spectra, the intersecting solid black lines denote the uncoupled individual photon (${\tilde{\omega }}_{ISRR}$) and magnon (${\tilde{\omega }}_r$) modes versus μ₀H_DC, while the dashed lines correspond to the real component of the eigenvalues for their coupled modes, as expressed by:

$${\tilde{\omega }}_{\pm }=\frac{1}{2}\left\{\left({\tilde{\omega }}_{ISRR}+{\tilde{\omega }}_r\right)\pm \sqrt{{\left({\tilde{\omega }}_{ISRR}-{\tilde{\omega }}_r\right)}^2+{\left({\omega }_{ISRR}\kappa \right)}^2}\right\}$$ 1

Here, κ represents the coupling constant, characterized as a complex value^19,20,36. The resonance angular frequencies are defined as ${\tilde{\omega }}_{ISRR}={\omega }_{ISRR}-i\cdot \mathrm{\Delta }{\omega }_{ISRR}$ and ${\tilde{\omega }}_r={\omega }_r-i\cdot \mathrm{\Delta }{\omega }_r$, where ${\omega }_{ISRR}$ and ${\omega }_r$ correspond to the resonance frequencies, and $\mathrm{\Delta }{\omega }_{ISRR}$ and $\mathrm{\Delta }{\omega }_r$ to the linewidths of each mode (Supplementary Notes 1 and 2). The observed anti-crossing (or level repulsion) dispersion profiles predominantly arise from coherent coupling, as evidenced by the real component of κ, which is fitted as Re[κ] ~ 0.0296 using Eq. 1. On the other hand, the distinct asymmetries present in the profiles of the higher- and lower-branch modes, particularly when comparing |S₂₁| and |S₁₂| are attributed to the interaction between coherent and dissipative coupling, breaking time-reversal symmetry and thus leading to non-reciprocal signal transmission behaviors, as addressed in previous works^21,22,38. This dissipative coupling, mediated by traveling waves in the microstrip line³⁹, is characterized by the imaginary component of κ, with fitted values indicating Im[κ] at +0.0088i for |S₂₁| and −0.0088i for |S₁₂|, where the sign change is ascribed to a π phase difference in the traveling waves between the S₂₁ and S₁₂. Further discussion will be addressed in the ‘NRI mediated by non-Hermiticity’ section.

Fig. 1. Scattering parameter measurement and NRI estimation in an ISRR/YIG hybrid sample.

a Setup schematic for measuring S_ij scattering parameters versus AC current frequency (f_AC = ω/2π) and varying magnetic field strengths (μ₀H), applied at a 33° angle to the x axis. b Photo images of the ISRR/YIG sample, showcasing both front and backside views. c Experimental data presentation: Magnitude (dB scale, top row) and phase (radian, second row) for S₂₁ (left) and S₁₂ (right) transmission as functions of f_AC and μ₀H. Refractive index (third row) estimated from the scattering parameters, is highlighted in blue. The calculation of ${\epsilon }_{eff}^{\prime }\cdot {\mu }_{eff}^{″}+{\epsilon }_{eff}^{″}\cdot {\mu }_{eff}^{\prime }$ (last row) should be negative to indicate the occurrence of NRI. Solid lines trace the uncoupled photon and magnon modes (ω_ISRR/2π and ω_r/2π), with dashed lines indicating the split hybrid modes from photon-magnon interactions, as described by Eq. 1.

The refractive index n of our hybrid sample, which exhibits significant PMC characteristics, was determined by analyzing the measured S_ij data, in accordance with Eq. 2^37,40–42:

$$n=n^{\prime }-i\cdot n^{″}=-\frac{1}{i{k}_0{l}_s}lnP$$ 2

where P is the time-independent wave function for microwaves propagating in the ISRR/YIG hybrid sample, l_s is the effective sample length, and k₀ the electromagnetic wave number in vacuum. P is given as $\frac{\overline{\mathrm{S}}+{\mathrm{S}}_{21}-\Gamma }{1-\left(\overline{\mathrm{S}}+{\mathrm{S}}_{21}\right)\Gamma }$ with Γ = $\frac{z-1}{z+1}$ the reflection coefficient at the interface between the ISRR/YIG hybrid structure and the microstrip line. z, the normalized impedance of the ISRR/YIG hybrid structure to 50 Ω, is given as z = $\sqrt{\frac{{\left(1+\overline{\mathrm{S}}\right)}^2-{\left({\mathrm{S}}_{21}\right)}^2}{{\left(1-\overline{\mathrm{S}}\right)}^2-{\left({\mathrm{S}}_{21}\right)}^2}}$). The relationships of ε_eff = n/z and μ_eff = n·z allow for the determination of the effective relative permittivity ε_eff and permeability μ_eff, with comprehensive results available in Fig. S5 (for further details, see Supplementary Notes 3).

In Fig. 1c, the third row illustrates the real components of the refractive index (n’) for the YIG/ISRR hybrid sample. A distinct area, where n’ manifests predominantly negative values, is identifiable in the anti-crossing region, indicated by a blue color along the higher-branch mode for S₂₁ and the lower-branch mode for S₁₂. These n’ values reach as low as −9.0 and −9.7 for S₂₁ and S₁₂, respectively, from an initial uncoupled state value of n’ = +0.2. The NRI spans a specific magnetic field region from μ₀H = 50.3 to 64.4 mT for S₂₁ and from μ₀H = 56.5 to 70.7 mT for S₁₂, extending over a wide frequency range of Δf_AC = 412 MHz and a field range of Δμ₀H = 19.9 mT for S₂₁, and Δf_AC = 498 MHz and Δμ₀H = 22.3 mT for S₁₂. Notably, the NRI characteristics resulting from the S₁₂ and S₂₁ measurements differ in their operating areas, demonstrating non-overlapping and functionally valuable non-reciprocal properties for opposite input-output signal directions.

To validate the optical prerequisites essential for NRI and elucidate the PMC-linked NRI mechanism, we revisited the NRI optical criteria. Contrary to the standard requirement for lossless materials, which mandates both ε‘_eff < 0 and μ‘_eff < 0 (double-negative condition)^23–26, our observations suggest an alternative mechanism for lossy materials. This mechanism adheres to the generalized optical condition: ${\epsilon }_{eff}^{\prime }\cdot {\mu }_{eff}^{″}+{\epsilon }_{eff}^{″}\cdot {\mu }_{eff}^{\prime }$ < 0, a prerequisite for backward phase velocity and hence indicative of NRI^43–47. The calculations, shown in the last row of Fig. 1c, based on both the real and imaginary components of ε_eff and μ_eff (Fig. S5), demonstrate negative values (indicated by blue color) coinciding with regions where n’ < 0, thereby validating this generalized condition. The positive ε“_eff and negative μ“_eff values observed in the NRI region, as shown in Fig. S5, confirm that PMC leads to sign changes in the electric loss (ε“_eff) and magnetic loss (μ“_eff) terms, through energy exchange between the YIG and ISRR resonators. This process aligns with the generalized optical condition necessary for achieving NRI.

NRI mediated by non-Hermiticity

The emergence of NRI in specific field regions is closely associated with the interplay of energy gain and loss in a non-Hermitian PMH system. Our analysis of the linewidths of the two distinct hybrid modes, derived from the |S₂₁| and |S₁₂| spectra (Fig. 2a), reveals that at particular field strengths, denoted as the lower field $H\_$ and higher field $H_{+}$, exceptionally narrow linewidths are observed exclusively in either the higher- or lower-branch modes: $H_{-}$ = 50.3 and $H_{+}$ = 64.4 mT for |S₂₁|, and $H_{-}$ = 56.5 and $H_{+}$ = 70.7 mT for |S₁₂|. Such narrow linewidths are indicative of zero damping, a distinct characteristic of non-Hermitian systems^{6,21,22,48,49}. Zero damping represents a transition state, where damping alternates between negative and positive values.

Fig. 2. Experimental confirmation of zero and negative damping in a non-Hermitian sample.

a Line profiles of |S₂₁| and |S₁₂| against frequency (ω/2π) at specific magnetic fields. Narrow linewidths marked in red indicate zero damping, while the blue-colored intermediate region denotes negative damping (anti-damping). b Analysis of the photon-magnon hybrid modes: The top panel shows the refractive index, the middle panel details real eigenvalues, and the bottom panel focuses on imaginary eigenvalues for ${\kappa }_c=0.0296+0.0088i$ (left) and ${{\kappa }^{*}}_c=0.0296-0.0088i$ (right), according to Eq. 1. Solid and dashed lines trace the eigenvalues of the higher- and lower-branch modes, respectively. Red dash lines denote zero linewidths, and the blue area represents regions of negative linewidth (indicative of negative damping). The vertical lines at field strengths, denoted as $H_{-}$ and $H_{+}$, highlight the points of zero damping and delineate the range within which NRI is observable.

To further verify zero and negative damping states, we plotted the real and imaginary components of the hybrid modes’ eigenvalues (Fig. 2b), employing Eq. (1) for both S₂₁ and S_12, with fitted values of $\kappa =0.0296+0.0088i$ and ${\kappa }^{*}=0.0296-0.0088i$, respectively. This approach, which involved extracting intrinsic values from each mode and fitting them to the experimental data (Fig. 2b), revealed that there exist two zero damping points at the marked $H_{-}$ and $H_{+}$ values in the linewidth (Δω) profiles for the higher-branch mode in S₂₁ and the lower-branch mode in S₁₂. Between these two fields, all Δω values are negative (indicated by the blue shaded area), denoting anti-damping zones, which coincide with the field regions, where experimentally observed NRI occurs, as shown in the top panel of Fig. 2b. These anti-damping regions, emerging from non-Hermitian dissipative coupling^18–22, coincide with transitions in the damping term from positive to negative. Correspondingly, the loss terms ε“_eff and μ“_eff alter their signs within this anti-damping region, with ε“_eff becoming negative and μ“_eff turning positive, as detailed in Fig. S5. Consequently, the generalized optical criterion for NRI, ${\epsilon }_{eff}^{\prime }\cdot {\mu }_{eff}^{″}+{\epsilon }_{eff}^{″}\cdot {\mu }_{eff}^{\prime }$< 0, is intrinsically associated to the anti-damping phenomenon in the non-Hermitian nature in our YIG/ISRR hybrid sample.

Effect of PMC strength on NRI and operating area

To explore further how the occurrence of NRI varies with the PMC strength, we experimentally varied the coupling strength. This manipulation was achieved by inserting scotch tapes with different thicknesses (d), ranging from d = 0 to 3.0 mm, between the YIG and the microstrip line. The experimental data for |S₂₁|, |S₁₂|, and the calculated values of the corresponding n’ for d = 0, 0.2, 0.8, and 3.0 mm are depicted in Fig. 3. As d increases, resulting in a reduction in coupling strength κ/κ_c (where κ_c represents the coupling constant at d = 0), the region exhibiting negative n’ values diminishes, visually represented by shrinking blue-colored areas. Notably, at κ/κ_c = 0.5 (corresponding to d = 0.8 mm), NRI eventually disappears. These specific coupling strength values were determined by fitting to the higher- and lower-branch modes, as indicated by the dotted lines in Fig. 3. At d = 3 mm, the individual photon and magnon modes (dotted lines) are clearly separated, intersecting at a common resonant frequency. However, the zoomed insets in the left column of Fig. 3 still exhibit anti-crossing in the dispersion spectra, indicating persistent coupling with a non-zero value of κ/κ_c = 0.1. These experimental observations underscore the importance of significant PMC strength for enabling NRI in the hybrid system and establish a threshold coupling strength necessary for NRI manifestation.

Fig. 3. Coupling strength influence on S₂₁₍₁₂₎ and n’ measurements.

|S₂₁| spectra (first row) and |S₁₂| spectra (third row), along with the derived refractive indices n'₂₁ (second row) and n'₁₂ (fourth row), plotted in a range of 2.8 GHz <f_AC < 4.2 GHz and 40 mT <μ₀H < 80 mT. These measurements correspond to varied coupling strengths, κ/κ_c = 0.1, 0.5, 0.8, and 1.0, based on fitting to the higher and lower hybrid modes. The parameter d indicates scotch tape thickness, affecting the coupling constant κ with κ_c (at d = 0). Insets show the coupled mode for d = 3 mm in ranges of 3.3 GHz <f_AC < 3.5 GHz and 55 mT <μ₀H < 65 mT.

Analytical circuit model

To enhance our understanding of non-Hermitian PMC-associated NRI, quantitatively reproduce the experimentally observed NRI in the PMC region, and identify key controllable parameters, we developed an analytical circuit model based on transmission line theory (Fig. 4a)^30,50,51. This model incorporates the interactive dynamics between the YIG and ISRR resonators, as well as the microstrip line responsible for pumping and detecting AC transmission signals. The YIG’s excitation and its coupling with the AC current in the microstrip line (governed by Ampere’s law), are quantified by a coupling constant parameter M₀. The microstrip line’s effective inductance and capacitance, isolated from the ISRR through an insulating substrate, are denoted by L₀ and C₀, respectively. The ISRR’s effective inductance, capacitance, and resistance are expressed as L_ISRR, C_ISRR, and R_ISRR. Additionally, the YIG’s effective inductance is expressed as $L_{YIG}=L_0\left(\frac{{\omega }_m\left({\omega }_H+{\omega }_m+i\omega \alpha \right)}{{\omega }_r^2-{\omega }^2+i\omega \alpha \left(2{\omega }_H+{\omega }_m\right)}\right)$, as derived from the Landau-Lifshitz-Gilbert (LLG) equation. Here, ${\omega }_H$ = γ₀μ₀H is the angular frequency of AC magnetic field, ω_m = γ₀μ₀M_s is the effective characteristic frequency of magnetization, with ${\omega }_r^2={\omega }_H\left({\omega }_H+{\omega }_m\right)$ the FMR frequency, and α the FMR damping constant^52–54. The inductance of YIG, when coupled with microstrip line, is modified to $L_{YIG,ext}$

Fig. 4. Circuit model representation and numerical analysis of the ISRR/YIG sample.

a A circuit model based on the transmission line theory for coupled ISRR/YIG hybrid, illustrating the theoretical framework for the subsequent analyses. b Numerical calculations for S₂₁ (top) and S₁₂ (bottom) spectra, along with the derived real part of the refractive index n’ for coupling constants M_c = 0.0093–0.0028i and M^*_c = 0.0093 + 0.0028i. Dashed lines indicate the split hybrid modes (ω_±/2π) derived from Eq. 1. c Circuit model analysis of varying imaginary parts of the coupling constant (Im[M_c]) on |S_ij| and n’, for specific values, while maintaining the real part constant (M_c = 0.0093), displaying the dependence of spectral and refractive properties on the imaginary coupling constant, where (top) Im[M_c] = −0.0024, −0.002, 0, and (bottom) Im[M_c] = 0.0024, 0.002, 0. d Numerical calculation of the peak value of n’ (red) and the field region ΔH_AD indicating anti-damping (blue) as a function of κ/κ_c (solid lines) and κ*/κ*_c (dotted lines), comparing with experimental measurements (solid square and circles, respectively), to validate the model against observed NRI phenomena.

In the circuit model given in Fig. 4a, the hybrid system comprising YIG, ISRR, and microstrip is represented by integrating the per-unit-length series impedance (z_se) and shunt admittance (y_sh), incorporating the circuit parameters previously outlined^55,56:

$$z_{se}\cdot l_s=i\omega L_0+i\omega M_0^2{L}_{YIG,ext}$$ 3

$$y_{sh}\cdot l_s={\left[\frac{1}{i\omega C_0}+{\left(\frac{1}{i\omega L_{ISRR}+i\omega M_c^2{L}_{YIG}}+i\omega C_{ISRR}+\frac{1}{R_{ISRR}}\right)}^{-1}\right]}^{-1}$$ 4

where ω is the angular frequency of AC currents flowing along the microstrip line, and l_s (=0.005 m) is the length of the sample. The parameters of n, ε_eff, μ_eff, and z are reformulated as^55–57,

$$n=\frac{\sqrt{z_{se}{y}_{sh}}}{i\cdot k_0}\left[\frac{\frac{\theta }{2}}{\sin \left(\frac{\theta }{2}\right)}\right]=\frac{\theta }{{l_s\cdot k}_0}$$ 5

$${\epsilon }_{eff}=\frac{y_{sh}\cdot G}{i\cdot \omega \cdot {\epsilon }_0}\left[\frac{\frac{\theta }{2}}{\sin \left(\frac{\theta }{2}\right)\cdot \cos \left(\frac{\theta }{2}\right)}\right]$$ 6

$${\mu }_{eff}=\frac{z_{se}}{i\cdot \omega \cdot {\mu }_0\cdot G}\left[\frac{\left(\frac{\theta }{2}\right)\cdot \cos \left(\frac{\theta }{2}\right)}{\sin \left(\frac{\theta }{2}\right)}\right]$$ 7

By inputting numerical values obtained from the scattering parameters measured directly for each of the ISRR and YIG, along with the fitted values of the mutual coupling constants, M₀ = 0.085 and M_c = 0.0093−0.0028i for S₂₁ (M^*_c = 0.0093 + 0.0028i for S₁₂), into Eqs. 3–7, we were able to numerically calculate the analytical model for |S₂₁| and |S₁₂|, as well as the corresponding n’ (Fig. 4b). Note that we employed the fitted mutual coupling constant M^*_c = 0.0093 + 0.0028i for S₁₂, reflecting a sign inversion in the imaginary component of M_c for S_21. The difference between the coupling constants ${\kappa }_c$ observed in the experimental data and M_c used in the analytical circuit model can be attributed to the circuit model’s incorporation of mutual inductance. In the experimental data, ${\kappa }_c$ is determined directly from S parameters’ measurements, capturing the actual coupling effects and interactions between the components without explicitly considering the mutual inductance between them. However, in the analytical circuit model, mutual inductance, which refers to the inductive interaction between circuit elements, is explicitly included. This inclusion results in variations in M_c depending on the inductance level of the utilized circuit components. Despite this variation, the absolute ratio of the real (coherent coupling) to imaginary (dissipative coupling) terms of ${\kappa }_c$ matches the corresponding ratio of M_c applied in the circuit model. Figure 4b reveals that the negative values of n’ and the operating area and its position, highlighted in blue color along the higher- and lower-branch modes for S₂₁ and S₁₂, accurately reproduce the experimentally observed NRI regions as shown in the third row of Fig. 1c and at the top of Fig. 2b. Additionally, the analytical circuit model enabled us to calculate the real and imaginary parts of ε_eff and μ_eff, along with the generalized optical condition for NRI. The results of these calculations mirror trends observed in the experimental data (see Fig. S8), indicating that the model effectively captures the essential physics underlying the hybrid system and its NRI phenomena.

To establish a correlation between the non-reciprocal characteristics of the experimentally observed NRI, including its operating region and position, and the magnitude and sign of the imaginary component of M_c, we conducted further calculations of |S₂₁|, |S₁₂|, and the corresponding n’ values with reduced magnitudes of |Im[M_c]|, such as 0.0024, 0.002, and 0, while maintaining its positive real counterpart, Re[M_c] = 0.0093, for both M_c and M^*_c (Fig. 4c). As the real part of M_c remains unchanged across all calculations, the dispersion curves of the higher- and lower-branch modes including their opening gap largely remain unaffected by variations in the magnitude and sign of Im[M_c]. Nonetheless, the asymmetry in magnitude between the lower and higher modes intensifies with an increase in |Im[M_c]|. Most strikingly, the area exhibiting NRI shrinks as |Im[M_c]| decreases. With Im[M_c] = 0 (indicating only coherent coupling), NRI vanishes, even in the presence of substantial coupling strength. Furthermore, the sign of Im[M_c] specifically alters the NRI operational position, manifesting non-reciprocal behavior based on the direction of the input-output signal. Therefore, it becomes clear that both the sign and magnitude of the imaginary coupling constant component, indicative of the non-Hermitian nature, play a crucial role in the emergence of NRI, determining the extent of its operating area and its non-reciprocal behavior.

In Fig. 4d, we plotted the field range where NRI is found, along with the peak value of n’ as a function of κ/κ_c (κ^*/κ^*_c) within the range of 0 to 1.5 at an increment of 0.1. These calculations (solid and dashed lines) were conducted with M_c = 0.0093–0.0028i for S₂₁ (M^*_c = 0.0093 + 0.0028i for S₁₂). The analytical calculations are in quantitative agreement with the corresponding experimental data (denoted by symbols) for various $\kappa$ values with constant values of ${\kappa }_c=0.0296+0.0088i$ for S₂₁ and ${\kappa }_c^{*}=0.0296-0.0088i$ for S₁₂. Below κ/κ_c (κ^*/κ^*_c) = 0.5, the field range for NRI does not exist. However, as κ/κ_c (κ^*/κ^*_c) exceeds 0.6, n’ suddenly transitions to large negative values ranging from −8.7 to −10.4 for 0.7 <κ/κ_c (κ^*/κ^*_c) <1.5. As mentioned earlier, the field range is closely related to the anti-damping region, ΔH_AD, identified between two different fields exhibiting zero damping. For κ/κ_c (κ^*/κ^*_c) >0.6, two zero damping field points emerge, and the anti-damping range expands with increasing κ/κ_c. These analytical findings confirm that the NRI behaviors observed in the YIG/ISRR hybrid sample are significantly influenced by the non-Hermitian nature of our system. This influence is associated with dissipative coupling and its strength, as represented by the imaginary value of the coupling constant. The interplay between coherent and dissipative coupling is essential to the manifestation of characteristic non-reciprocal NRI behaviors.

Discussion

Our study represents a significant leap forward in PMC technologies, demonstrating remarkable non-reciprocal NRI and on-off switching capabilities across a tunable field and frequency ranges through non-Hermitian dynamics. A key innovation of our approach lies in the precise control over the refractive index with the application of variable magnetic fields, enabling adjustments spanning from positive through zero to significantly negative values, as well as the on-off switchability of NRI phenomena. This breakthrough has the potential to revolutionize the design and functionality of optical and magnonic devices, offering unprecedented levels of manipulation and performance.

The compatibility of our system with current CMOS circuit technology, along with its simplicity, significantly enhances its practicality for easy incorporation into a wide array of electronic and photonic devices. Additionally, the planar configuration of our PMH system marks a departure from traditional, complex designs, playing a crucial role in miniaturizing quantum hybrid structures. This approach enables the realization of materials with NRI in the beyond-terahertz-range, under effective magnetic field control. However, achieving comprehensive control across such a broad spectrum poses substantial challenges, necessitating ongoing exploration to overcome material-specific limitations and expand operational frequency ranges. Moreover, mitigating losses inherent in devices utilizing NRI materials is crucial for optimizing overall device efficiency.

Beyond conventional NRI measurements, future research should prioritize direct optical demonstrations of negative refraction across a variety of frequency ranges. This can be facilitated through arrays of PMH units or blocks, leveraging circuit models to adjust parameters such as damping loss, coupling constants, and resonance frequencies flexibly. By analytically varying these parameters, one can design and optimize model systems, advancing the fabrication and validation of PMH unit arrays and specific unit structures. This approach not only enhances our understanding of related mechanisms but also holds promise for transformative applications in magnonics and photonics, such as invisible cloaking devices⁵⁸, perfect lenses⁵⁹, and sensitive signal detection and processing^30,60 in optical research areas.

This work represents a significant step forward in the development of photonics and magnonics, setting a foundation for the development of ultra-compact, high-frequency devices. Additionally, it paves the way for further advancements in quantum hybrid systems, promising a future abundant in technological innovations and superior device functionalities.

Methods

Sample fabrication

The hybrid sample used in this study is depicted in Fig. 1b. A YIG film (green color) was placed on top of the 50 Ω characteristic impedance microstrip line on the front side of the sample, while an ISRR was fabricated on the ground plane just below the microstrip line on the backside. The dimensions of the ISRR, microstrip line, and YIG film used in this study are depicted in Fig. S3. The dimensions of the ISRR were determined through simulation using CST-STUDIO. The samples of the microstrip line and the ISRR on the ground plane were fabricated using a conventional photolithography process.

Scattering parameters measurement

The measurement setup used in this study, as illustrated in Fig. 1a, involved exciting and detecting the dynamic modes of the YIG film by applying microwave AC currents along the microstrip line and coupling with the electrodynamic photon mode of the ISRR. The input and output of the microstrip feeding line were connected to a two-port vector network analyzer (VNA, Agilent PNA series E8362C) through microwave connectors. The entire ISRR-YIG hybrid sample was subjected to a DC bias magnetic field at room temperature. S- parameters were measured experimentally as a function of the frequency (f_AC = ω/2π) of AC currents flowing along the microstrip line for different strengths of μ₀H applied at an angle ψ = 33° with respect to the x-axis (microstrip line) in the sample plane. The angle ψ was chosen because, at this angle, all excited spin-wave modes exhibit a zero-group velocity (v_g = 0). This configuration results in a singular FMR line without subsidiary lines from non-zero wave number modes that would occur at other field angles, thereby simplifying the NRI analysis and reducing complexity⁶¹. To isolate the S parameters of the ISRR/YIG hybrid in the region of interest, the background signal of the empty microstrip line was subtracted from the entire signal measured from the whole sample³⁷.

Author contributions

S.-K.K. B.K. conceptualized the idea and designed the project. J.K., B.-J.K., and H.J. prepared the samples and conducted the experiments, with B.K. providing essential support. J.K. developed the analytical circuit model under the guidance of B.K. and S.-K.K., in collaboration with B.-J.K. S.-K.K., led and supervised the project. S.-K.K., J.K., and B.K. wrote the manuscript, with all authors participating in providing feedback to refine and edit it.

Peer review

Peer review information

Nature Communications thanks Alessandro Pitanti, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data presented in this study are available within the article and its Supplementary Information files.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-53328-9.