Introduction

Manipulating photon transmission/absorption in a predesigned manner while keeping zero reflection (ZR) is important for applications in general wave optics. For practical applications, the transmitted photons are critical for signal transfer and filtering, and can reveal important light-matter interaction physics1 (such as in electromagnetically induced transparency2) or be encoded for information processing. The absorbed photons are crucial for signal conversion (e.g., photoelectric detection3,4,5,6,7) and energy harvesting8 (e.g., photothermal, photoelectric, photo-spin-voltaic9, or photomagnonic10 conversions). The reflected photons, however, are often harmful not only because the transmission/absorption efficiency can be sacrificed, but also because reflected photons can interfere with excitation and play detrimental effects to the performance of whole optical setup (e.g., the frequency stability of the excitation laser)11. Therefore, maintaining zero reflection, while simultaneously tailoring transmission/absorption is desirable for specific applications12,13,14.

To this aim, tremendous15,16,17,18,19,20 efforts have been made in several closely related fields. In metamaterials15,16,17 and optomechanics18,19, significant progresses such as perfect absorption, perfect transmission, and electromagnetically induced transparency/absorption (EIT/EIA)18,19,20,21,22 have been reported. Transformation optics has demonstrated cloaking and provided us a strategy for finding ZR condition with inhomogeneous, anisotropic materials. Optical systems with parity-time symmetry attract increasing interest partly because they can suppress reflection23. Horsley et al. introduced Kramers-Kronig complex potentials which can approach reflectionless propagation of photons24. The suppressed reflection in these works relates essentially to the rich mode interaction inside the system and their interference with incident beams. Although not explicitly mentioned, the non-Hermiticity in these works has played an important role. Meanwhile, the emergent non-Hermitian physics community has disclosed interesting mode interactions such as level repulsion (LR) and level attraction (LA). Several singular properties have also been reported, such as zero-damping states25,26, exceptional points27,28, skin effects29 and so on. These singular behaviors have been observed mainly in the transmission spectra of the reported system. For example, infinitely sharp transmission dips are observed in |S12(ω)| and |S21(ω)| spectra in the LA region of the non-Hermitian cavity magnonics systems25,26. So far, however, very few works30,31 have addressed the reflection singularity despite that ZR being critical for many applications.

Here, we propose a magnonic device with two low-damping yttrium iron garnet (YIG) spheres coupled to a same microwave transmission line (MTL). These two magnonic resonators couple only indirectly to each other being mediated by the traveling wave in MTL. By adjusting the microwave propagating delay phase (Φ) between the two resonators, we can control the hybridized states covering both level attraction/repulsion (LA/LR). We explicitly show that the ZR condition exists generally in the complex frequency plane and distinguishes from the resonant eigenstates of the effective non-Hermitian Hamiltonian. When ZR frequency becomes purely real, perfect ZR (PZR) becomes observable in our experiments as sharp reflection dips. This reflection singularity of PZR is attributed to the destructive interference between the direct and high-order reflected waves. It can be regulated to coincide with or deviate from the eigenmodes and accordingly the absorption/transmission of the coupled system can be managed. Our work expands the non-Hermitian physics in magnon-photon coupled systems and sheds light to the practical applications such as microwave circuits, photonic chips, and quantum information.

Results and discussion

Structure of the indirectly coupled two-magnon system

In our device (Fig. 1a) two 1-mm diameter yttrium iron garnet spheres (YIG 1 and YIG 2) are placed above a transmission line (MTL) with a distance of l = 25 mm. Microwave signals are loaded from either port 1 or port 2 to excite ferromagnetic resonance (FMR) in the two YIG spheres. An external magnetic field (H-field) is applied perpendicularly to the waveguide plane, which saturates the magnetizations of the two YIG spheres and determines the resonant frequencies of two magnon modes, ω1 and ω2. To fine tune their frequency detuning ΔH = ω1 − ω2, an additional small coil is placed under the position of YIG 1 so that its local magnetic field (δH ≪ H) can be precisely adjusted. As a result, two nonidentical resonators with excellent tunability of resonant frequencies (ω1 is determined by H + δH and ω2, by H) are configured. The very high accuracy (~0.1 MHz) in fine parameter tuning of ΔH enables the study of singular behaviors in spectrum of the non-Hermitian coupled systems32,33.

Fig. 1: Indirectly coupled two-magnon system.
figure 1

a Schematic diagram of the experimental setup. Two separated YIG spheres (l = 25 mm) coupled to a transmission line are placed under a vertical magnetic field (H-field), the local magnetic field (H + δH) around the YIG 1 is controllable independently by finely adjusting coil current. b Schematic diagram showing the two resonators are side-coupled to a transmission line and indirectly interact with each other mediated by traveling wave with the effective coupling strength of −iΓeiΦ. c Equivalent circuit of the coupled system. Circuit elements used to model the two YIG spheres are marked by the orange and blue box, respectively. d Schematic diagram showing the nearly full absorption (NFA) and nearly full transmission (NFT) under the perfect zero-reflection (PZR) condition.

Full size image

As shown in Fig. 1b, the excited resonators dissipate their energy through intrinsic damping rates (γ1,2), which are typically low and depend on the natural properties of YIG spheres including the surface roughness, impurities and defects, etc. Besides that, two resonators couple to the same dissipative channel (MTL)34,35 with the extrinsic damping rates of κ1,2, their cooperative effect would induce an effective indirect coupling −iΓeiΦ, where \(\varGamma=\sqrt{{\kappa }_{1}{\kappa }_{2}}\) and Φ = 2πl/λ. In contrast to earlier works, since l in our system is large enough to suppress the direct coupling between two resonators, the indirect coupling mediated by propagating photons dominates the system properties. Depending on the value of Φ, the indirect coupling strength can be either purely real (coherent coupling), purely imaginary (dissipative coupling), or mixed real and imaginary contributions. Experimentally, Φ can be adjusted by either λ or l. Here, we choose the non-mechanical method33 and change wavelength λ at a fixed distance l to avoid mechanical change of YIG spheres which may induce unwanted influence to the external damping rates κ1,2. When the H-field is swept over a large range, the resonant frequencies, and thereby, λ are modulated accordingly.

Hybridized eigenstates in the two-magnon non-Hermitian system

Under the rotating-wave approximation, the effective non-Hermitian Hamiltonian of the subsystem can be created by using Feshbach projection approach36 (see Supplementary Note 1 for details),

$${{{{{{\mathcal{H}}}}}}}_{{eff}}={{\hslash }}{\widetilde{\omega }}_{1}{\hat{a}}^{{{\dagger}} }\hat{a}+{{\hslash }}{\widetilde{\omega }}_{2}{\hat{b}}^{{{\dagger}} }\hat{b}-{{\hslash }}(i\varGamma {e}^{i\varPhi })({\hat{a}}^{{{\dagger}} }\hat{b}+{\hat{b}}^{{{\dagger}} }\hat{a}),$$
(1)

where \({\hat{a}}^{{{\dagger}} }\) (\(\hat{a}\)) and \({\hat{b}}^{{{\dagger}} }\) (\(\hat{b}\)) represent the creation (annihilation) operators of the two magnon modes, obeying the commute relations of \([\hat{a},{\hat{a}}^{{{\dagger}} }]=1\), \([\hat{b},{\hat{b}}^{{{\dagger}} }]=1\), and \([\hat{a},\hat{b}]=0\). \({\widetilde{\omega }}_{{{{{\mathrm{1,2}}}}}}={\omega }_{1,2}-i({\gamma }_{{{{{\mathrm{1,2}}}}}}+{\kappa }_{{{{{\mathrm{1,2}}}}}})\) are the complex frequencies of the uncoupled magnon modes, with the imaginary parts being related to the experimentally observed frequency linewidths. Similar non-Hermitian Hamiltonian has been used in various area including atomic system37, cavity quantum electrodynamics34, optomechanical systems35, and so on. The eigenvalues of Eq. (1) can be solved to be

$${\widetilde{\omega }}_{\pm }=\frac{1}{2}\left[{\widetilde {\omega }}_{1}+{\widetilde {\omega }}_{2} \pm \sqrt{{\left({\widetilde{\omega }}_{1}-{\widetilde{\omega }}_{2}\right)}^{2}+4{\left(i\varGamma {e}^{i\varPhi }\right)}^{2}}\right],$$
(2)

where the real parts of \({\widetilde{\omega }}_{\pm }\) give the dispersions of two hybridized modes, and the imaginary parts \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{ \pm })\) correspond to their damping rates. The mode hybridization appears to be similar to previous works38,39,40, covering both level attraction (LA) and level repulsion (LR) regions as shown in Fig. 2a–d with four representative cases, Φ = nπ, (n + 1/4)π, (n + 1/2)π, (n + 3/4)π (n∈N), respectively. On the other hand, our work distinguishes from earlier works in the two aspects:

Fig. 2: The theoretical calculations of zero-reflection (ZR) condition and mode hybridization.
figure 2

a–d The frequency detunings of the real parts of ZR conditions, \(\Delta {{{{{\rm{Re}}}}}}({\widetilde{\omega }}_{{ZR}})={{{{{\rm{Re}}}}}}\left({\widetilde{\omega }}_{{ZR}}\right)-{\omega }_{2}\) (orange solid lines), and the frequency detunings of the real parts of eigenvalues, \(\Delta {{{{{\rm{Re}}}}}}\left({\widetilde{\omega }}_{\pm }\right)={{{{{\rm{Re}}}}}} \left({\widetilde{\omega }}_{\pm }\right)-{\omega }_{2}\) (blue dashed curves), as functions of ΔH in four Φ cases, Φ = nπ, (n + 1/4)π, (n + 1/2)π, (n + 3/4)π, n ∈ N, respectively. e–h Correspondingly, the imaginary parts of ZR conditions, \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{{ZR}})\) (orange solid lines) and the frequency detunings of the imaginary parts of eigenvalues, \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{\pm })\) (blue dashed curves), as functions of ΔH. In (e–h), \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{{ZR}})\) correspond to the left scale and \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{\pm })\) correspond to the right scale, the white and orange areas are divided by the horizontal line of \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})=0\). Circles in (b, f) and (d, h) represent the perfect zero-reflection (PZR) conditions. The top panels of (a–d) are the schematic diagrams at ΔH = 0 when Φ = π, 5/4π, 3/2π, 7/4π, respectively, assuming n = 1. The abbreviations LA and LR represent level attraction and level repulsion, respectively.

Full size image

First, traveling photon bath in this work always mediates the indirect coupling between two resonators and leads to both LA and LR; while in most of previous works, LR is often attributed to the direct coupling between resonators with large spatial wavefunction overlapping. The interaction mediated by a same photon bath here allows to reveal the essential difference between LA and LR: When Φ = nπ, two magnon modes are synchronized (first cartoon in Fig. 2) by traveling photons with either in-phase (n is even) or 180-degree out-of-phase (n is odd) spin precessions in the two YIG spheres. This results in the distinct feature of a dissipatively coupled system that the hybridized modes attract each other (blue dashed curves in Fig. 2a) while the corresponding imaginary parts repulse (blue dashed curves in Fig. 2e). In contrast, when Φ = (n + 1/2)π, two magnon modes have a phase difference of π/2 (third cartoon in Fig. 2), the hybridized modes show the distinct feature of a coherently coupled system where the real parts repulse each other (Fig. 2c) while the corresponding imaginary parts attract (Fig. 2g), in this situation, the coupled system behaves as if magnons keep hopping in-between two YIG spheres, although this long-range hopping of magnons cannot occur straightforwardly like conventional Rabi oscillation, but has to be mediated by traveling photons in MTL. For other Φ values, the synchronization and hopping coexist in the system thus lead to the mixed interaction of LA and LR, as shown in Fig. 2b, f and d, h, for Φ = (n + 1/4)π and (n + 3/4)π, respectively. Interestingly, the complex eigenvalues in Eq. (2) satisfy the unit round-way propagation of the trapped photons in-between two YIG spheres, implying the Fabry-Pérot (FP)-like resonant mechanism in our device (see Supplementary Note 1). In contrast to conventional FP cavity with simple dispersionless mirrors, the accumulated phase Φround = 2nπ has to take into account the reflection phase shift by both YIG 1(2), ϕ1(2) in addition to the propagating phase Φ. Namely, Φround ≡ ϕ1 + ϕ2 + 2Φ = 2nπ and this formula applies for single round-trip propagation of microwave photons but is found to be sufficient for reproducing our following experimental data (see Supplementary Note 1). Compared to previously reported Fabry-Pérot (FP) type interaction with identical resonant “mirrors” (ω1 = ω2, Φ ≠ 0)41,42, our system represents a more generalized coupling model with nonidentical resonant mirrors and arbitrary Φ which therefore unifies the FP-type and Friedrich-Wintgen (FW) type (ω1 ≠ ω2, Φ = 0) interactions43.

Second, different from previous works using two different types of resonators, namely, metal cavity resonator and magnonic resonator25,26,38, the two same magnonic-type resonators in this work allows them more balanced in terms of the intrinsic and extrinsic damping rates (γ1,γ2,κ1,κ2). The balanced dampings, in particular κ1 ~ κ2, are important for establishing the balanced FP-like resonance since the reflection at the YIG 1(2) is proportional to the corresponding κ1(2) (see Supplementary Note 1). This balanced FP-like resonance allows the ZR condition can be observed near the coupling region, as will be seen in the next sections.

Zero-reflection (ZR) states in complex frequency plane of the non-Hermitian system

The reflection (S11 and S22) and transmission (S21 and S12) spectra can be derived from the effective Hamiltonian of Eq. (1) based on coupled-mode theory, or alternatively from the equivalent circuit model (Fig. 1c) using ABCD matrix44 method (see Supplementary Note 1 for details). These four S-parameters can be derived as

$${S}_{11}\left(\omega \right)=-2i\omega \frac{{\kappa }_{1}\left({X}_{2}+2i\omega {\kappa }_{2}\right)+{\kappa }_{2}{e}^{2i\varPhi }\left({X}_{1}-2i\omega {\kappa }_{1}\right)}{\left({X}_{1}+2i\omega {\kappa }_{1}\right)\left({X}_{2}+2i\omega {\kappa }_{2}\right)+4{\omega }^{2}{\kappa }_{1}{\kappa }_{2}{e}^{2i\varPhi }},$$
(3a)
$${S}_{22}\left(\omega \right)=-2i\omega \frac{{\kappa }_{1}{e}^{2i\varPhi }\left({X}_{2}-2i\omega {\kappa }_{2}\right)+{\kappa }_{2}\left({X}_{1}+2i\omega {\kappa }_{1}\right)}{\left({X}_{1}+2i\omega {\kappa }_{1}\right)\left({X}_{2}+2i\omega {\kappa }_{2}\right)+4{\omega }^{2}{\kappa }_{1}{\kappa }_{2}{e}^{2i\varPhi }},$$
(3b)
$${S}_{21}(\omega )={S}_{12}(\omega )=\frac{{X}_{1}{X}_{2}{e}^{i\varPhi }}{\left({X}_{1}+2i\omega {\kappa }_{1}\right)\left({X}_{2}+2i\omega {\kappa }_{2}\right)+4{\omega }^{2}{\kappa }_{1}{\kappa }_{2}{e}^{2i\varPhi }},$$
(4)

where \({X}_{{{{{\mathrm{1,2}}}}}}{=\omega }^{2}-{\omega }_{{{{{\mathrm{1,2}}}}}}^{2}+2i\omega {\gamma }_{{{{{\mathrm{1,2}}}}}}\). Under the rotating-wave approximation, \({\omega }^{2}-{\omega }_{1}^{2} \sim 2\omega (\omega -{\omega }_{1})\) and \({\omega }^{2}-{\omega }_{2}^{2} \sim 2\omega (\omega -{\omega }_{2})\) can be obtained. Note that, in the complex frequency plane, poles28 and zeros of the reflection (S11 and S22 in Eq. (3)) correspond to the resonant eigenvalues (\({\widetilde{\omega }}_{ \pm }\) in Eq. (2)) and the ZR states, respectively. The balanced damping rates together with the fine parameter tuning of ΔH = ω1 − ω2 allow us to observe both the eigenmodes (peaks in S11 and S22) and the ZR modes (dips in S11 and S22) in the same parameter space and range. In Fig. 2, the solutions of ZR, i.e., S11(ω) = 0, are plotted by orange solid lines in complex frequency plane (Supplementary Note 2), the real parts of \({{{{{\rm{Re}}}}}} ({\widetilde{\omega }}_{{ZR}})\) as functions of ΔH are shown in Fig. 2a–d, and the corresponding imaginary parts \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{ZR})\) are shown in Fig. 2e–h. \({{{{{\rm{Im}}}}}} ({\widetilde{\omega }}_{{ZR}})\) approaches two constant values but with opposite signs (one positive and one negative) at two conditions: in the case of pure LA with Φ = nπ, \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})=-i\frac{{\gamma }_{1}{\kappa }_{2}+{\gamma }_{2}{\kappa }_{1}}{{\kappa }_{1}+{\kappa }_{2}}=-{i} \, {{{{{\rm{MHz}}}}}}\) (Fig. 2e, left scale) and in the case of pure LR with Φ = (n + 1/2)π, \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})=i\frac{{\gamma }_{1}{\kappa }_{2}-{\gamma }_{2}{\kappa }_{1}-2{\kappa }_{1}{\kappa }_{2}}{{\kappa }_{1}-{\kappa }_{2}}=6{i} \, {{{{{\rm{MHz}}}}}}\) (Fig. 2g, left scale). For other Φ values, the lines of \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})\) are tilted (Fig. 2f, h), meanwhile, in Fig. 2b, d, the lines of \({{{{{\rm{Re}}}}}}({\widetilde{\omega }}_{{ZR}})\) intersect with \({{{{{\rm{Re}}}}}}\left({\widetilde{\omega }}_{\pm }\right)\), which imply the correlation between the eigenmodes and ZR modes can be tuned by Φ and ΔH.

Experimentally, we fix ω2/2π = 3.6 GHz, 4.24 GHz, 4.69 GHz, respectively, and sweep ω1 to obtain the reflection mappings as shown in Fig. 3. Reflection data at three different cases of ω2 are acquired and compared directly with the theoretical calculations: Φ = 1.07π, 1.25π, 1.37π, respectively. Here, the values of Φ are obtained by fitting the dispersion of reflection mappings in Fig. 3 according to Eq. (2) (also see Supplementary Note 3, fitted curves represented by dashed lines in FIG. S3). The correlation between the real parts of eigenmodes (\({\widetilde{\omega }}_{\pm }\)) and ZR modes (\({\widetilde{\omega }}_{{ZR}}\)) can be confirmed in the reflection spectra (both |S11| and |S22|), the experimental reflection peaks agree well with the calculated eigenvalues (blue dotted curves) for all the spectra (|S11| in Fig. 3a–c and |S22| in Fig. 3g–i), and the reflection dips match the calculated ZR lines (\({{{{{\rm{Re}}}}}}({\widetilde{\omega }}_{{ZR}})\) plotted by the white long dash lines). As a result, the final reflection spectra are always superimposed with both eigenmodes (peaks) and ZR modes (dips).

Fig. 3: Perfect zero-reflection (PZR) observed in experiment.
figure 3

a–c Mappings of |S11| as functions of ΔH and Δ2 in four cases: fixing ω2/2π = 3.6 GHz, 4.24 GHz, 4.69 GHz, respectively. d High-resolution zoom-in image of (a). g–i Mappings of |S22| as functions of ΔH and Δ2 in four ω2 cases. Blues dotted curves are the fitted hybridized modes, white long dashed lines are the calculated frequencies of the ZR states. The PZR conditions are marked by arrows. j High-resolution zoom-in image of (g). e and k are the group delay τg mappings corresponding to (d) and (j), the PZRs emerge as singularities, where the group delay abruptly switches between negative infinity and positive infinity. f and l are the PZR spectra (|S11| and |S22|) at the black arrows marked fields in (d) and (j), respectively.

Full size image

Singularity of perfect zero-reflection (PZR)

For Φ ≠ nπ/2, the tilted ZR line (orange line in Fig. 2f or h) intersects with the real frequency axis and a critical value of \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})=0\) can be obtained (marked by circle in Fig. 2). This critical condition is termed as perfect zero-reflection (PZR) in order to be distinguished from the general ZR state with nonzero imaginary part, \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}}) \ne 0\). In Figs. 2e and 2g, the horizontal orange lines show \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})\) become nonzero constants in pure LA (Fig. 2e) and pure LR (Fig. 2g) regions, \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}}) \ne 0\) indicates there is no PZR exist. We notice that \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}}) < 0\) in pure LA region and \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}}) > 0\) in pure LR region, while in the general hybrid region (Figs. 2f and 2h) with Φ ≠ nπ/2, there exists the critical point where the sign of \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})\) changes abruptly leading to \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})=0\), ensuring the general existence of PZR. The PZR condition can be derived analytically to be

$${\omega }_{{PZR}}^{\pm }=\frac{1}{2}\frac{\sin \left(2\varPhi \right)({\omega }_{1}^{2}-{\omega }_{2}^{2})}{\left({\kappa }_{2}-{\kappa }_{1}\right)\left(\cos 2\varPhi -1\right)\pm \left[\left({\gamma }_{1}+{\gamma }_{2}\right)\cos \left(2\varPhi \right)+\frac{{\kappa }_{1}{\gamma }_{2}}{{\kappa }_{2}}+\frac{{\kappa }_{2}{\gamma }_{1}}{{\kappa }_{1}}\right]},$$
(5)

the superscript “±” represents the case that microwave signals are loaded from port 1 (|S11(ω)| = 0) and from port 2 (|S22(ω)| = 0), respectively. It’s worth noting that, in Eq. (5), ω1 is uniquely determined by ω2 due to the requirement of \({{{{{\rm{Im}}}}}}({\widetilde{\omega }}_{{ZR}})=0\) (see Supplementary Note 2), thus the PZR appears at a certain frequency point when the external magnetic field fixed. The underlying physical picture of the PZR condition is the destructive interference (Fig. 1d) between direct reflection by the first YIG sphere and the multiple reflection involving the second YIG sphere leads to the vanishing reflection amplitude (Supplementary Note 1). Since the strictly destructive interference occurs only at a single frequency, the linewidth of the PZR spectra becomes infinitely small with an effectively ultra-high Q-factor, for instance, ultra-sharp dip about 67 dB is observed with the measuring accuracy of ~0.1 MHz (Fig. 3f or Fig. 3l). Earlier work by Mukhopadhyay et al.30 proposed similar physics behind PZR by coupling a number of identical and lossless resonators to a common waveguide. Here, Eq. (5) provides a more general formalism for nonidentical and lossy resonators with traveling-photon-mediated indirect coupling.

Experimentally the general existence of the PZR condition has been verified. As can be seen from Fig. 3, yellow sharp dips appear in the reflection spectra. Figure 3a–c are the experimental mappings of |S11| as a function of ΔH = ω1 − ω2 and Δ2 = ω − ω2 for the cases of Φ = 1.07π, 1.25π, 1.37π, respectively, in which sharp reflection dips of the PZRs are clearly visible at the fields marked by arrows, \({\Delta }_{H}^{+}/2\pi=9.9,0.9,-2.2\,{{{{{\rm{MHz}}}}}}\). These values agree consistently with the theoretical prediction by Eq. (5). Similarly, in the experimental mappings of |S22| (Fig. 3g–i), sharp reflection dips show up again despite that the quantitative \({\Delta }_{H}^{-}\) values of the PZRs are different due to the asymmetric parameters of the two YIG spheres, i.e., κ1 ≠ κ2. In Fig. 3g–i, the PZR occurs at the field \({\Delta }_{H}^{-}/2\pi=-11.0,-4.1,-4.8\,{{{{{\rm{MHz}}}}}}\), respectively, which is also in good accord with Eq. (5). Figure 3d (3j) is the enlarged view of the PZR region in Fig. 3a (3g). Figure 3e (3k) is the group delay τg mapping corresponding to Fig. 3d (3j), τg = −∂P11(22)/∂ω is defined as the negative derivative of the reflection phase P11(22). Beside the ultra-sharp dip (~67 dB with ~0.1 MHz measuring accuracy) observed in the reflection spectrum (Fig. 3f (3l)), an additional intriguing feature of the PZR is the group delay τg, which is defined as the negative derivative of the reflection phase P11(22), τg = −∂P11(22)/∂ω. As revealed by Fig. 3e (3k), the PZR emerges in the position where τg switches abruptly between negative infinity and positive infinity, τg → ±∞, so that the group velocity vg = 1/τg becomes zero. This suggests that the PZR observed in our system is indeed a singularity and may be utilized for slowing microwave photons27.

Transmission and absorption under the PZR condition

For arbitrary value of Φ, once the PZR condition of Eq. (5) is satisfied, the nearly full absorption (NFA) and nearly full transmission (NFT) can be realized by trading-off the absorption and transmission (A1(2) + |S21(12)|2 = 1) and experimentally, by cooperatively adjusting Φ and ΔH. On the premise of satisfying Eq. (5), the Eq. (4) can be written as

$${{S}_{11}({\omega }_{{PZR}}^{+})=0,\, S}_{21}({\omega }_{{PZR}}^{+})=\frac{{e}^{i\varPhi }}{1+\frac{2i{\omega }_{{PZR}}^{+}{\kappa }_{2}(1-{e}^{2i\varPhi })}{{({\omega }_{{PZR}}^{+})}^{2}-{\omega }_{2}^{2}+2i\omega {\gamma }_{2}}},$$
(6a)
$${{S}_{22}({\omega }_{{PZR}}^{-})=0,\, S}_{12}({\omega }_{{PZR}}^{-})=\frac{{e}^{i\varPhi }}{1+\frac{2i{\omega }_{{PZR}}^{-}{\kappa }_{1}(1-{e}^{2i\varPhi })} {{({\omega }_{{PZR}}^{-})}^{2}-{\omega }_{1}^{2}+2i\omega {\gamma }_{1}}}.$$
(6b)

We have numerically analyzed the transmission and absorption by Eq. (6), as shown in Fig. 4a, b, the transmittivity t1(2) = |S21(12)| 2 (blue curves) and absorptivity A1(2) = 1 − |S21(12)|2 (red curves) show complementary oscillation tendency with Φ. (Gray lines represents r1(2) = |S11(22)|2 = 0 due to the PZR). The color bar shows the hybridized states at different Φ values, at the purely coherent (LR) and purely dissipative (LA) coupling regions (Φ = nπ/2, n∈N), the PZR does not exist, and hence the full transmission cannot be reached strictly but can be approached infinitely when Φ → nπ/2. On the other hand, in the region of LR and LA coexistence, the nearly full absorption (NFA) can be approached when Φ gets close to (but still different from) nπ/2 + π/4. All experimental datapoints depicted by symbols in Fig. 4a, b show excellent agreement with the numerical results (detail measurement data see Supplementary Note 3, Fig. S5). Since in the two-port configuration45, the PZR (S11(22) = 0) and zero-transmission (S21(12) = 0) cannot be satisfied simultaneously and therefore the full absorption cannot be realized strictly. In Fig. 4a, the achievable maximum absorption under the PZR is ~96% when microwave signals are loaded from port 1, and in Fig. 4b, when microwave signals are loaded from port 2, the achievable maximum absorption is ~80%. This implies to realize the NFA, κ2 > κ1 is necessary if microwave signals are loaded from port 1 and vice versa. By increasing the ratio of κ2/κ1 (see Supplementary Note 4 for detail), the maximum absorption of A1 can be approached to unity (while A2 is sacrificed). In the limit of κ2/κ1 → ∞, the two-port configuration evolves into one-port configuration where the full absorption condition exists. Figure 4c displays a three-dimensional plot of the absorption/transmission when sweeping ω1 and Φ, while keeping ω2/2π = 4.24 GHz. The nearly full absorption appears in the small detuning region (ω1 ≅ ω2) as colored by yellow, and the nearly full transmission takes place in the large detuning region. As shown schematically in Fig. 1d, in the nearly full absorption (NFA) region (ω1 ≅ ω2), FP-like mechanism for photons trapped between the two resonators assists the magnon exchange between two YIG spheres, and eventually, most of the microwave energy is dissipated through the spin processions in the two YIG spheres despite the intrinsic losses of two magnon modes are rather small (~1 MHz). While in the nearly full transmission (NFT) region, the large frequency detuning (ω1 ≠ ω2) makes two magnon modes are nearly uncoupled with each other at the PZR frequency, so that little microwave energy can be absorbed by YIG spheres since ωPZR ≠ ω1 and ωPZR ≠ ω2. From Fig. 4c, in the negative detuning region (ω1 < ω2), ωPZR increases as ω1 increases with an approximate slope of κ2/(κ2 − κ1). In the positive detuning region (ω1 > ω2), the slope changes to κ2/(κ1 + κ2), this result provide us a guidance to manipulate the microwave propagating and absorption for any target frequency. To sum up, by adjusting H-field (affects propagating delay phase Φ) and δH-field through coil current (affects frequency detuning ΔH = ω1 − ω2), we demonstrate the microwave behavior at the PZR frequency is controllable and realize the waveguide magnonic device allowing the continuous control between nearly full transmission (infinitely close to 100%) and nearly full absorption (~96%).

Fig. 4: Theoretical prediction and experimental verification: transition between the nearly full transmission (NFT) and nearly full absorption (NFA).
figure 4

a For microwaves are loaded from port 1, the calculated reflectivity (r1), transmittivity (t1) and absorptivity (A1) as functions of Φ under the PZR condition. b For microwaves are loaded from port 2, the calculated reflectivity (r2), transmittivity (t2), and absorptivity (A2) as functions of Φ under the PZR condition. The square (r1(2)), diamond (t1(2)), and triangle (A1(2)) symbols in (a) and (b) are the experimental results. Color bar shows the hybridized states for different values of Φ, the abbreviations LA and LR represent level attraction and level repulsion, respectively. Open circles plotted in (a, b) indicate the PZRs do not exist at these points. c The calculated three-dimensional plot of the absorption/transmission as a function of Φ, ω1, and ω. Here, γ1/2π = γ2/2π = 1 MHz, κ1/2π = 1.7 MHz, κ2/2π = 3.3 MHz are set based on the fitting results of experimental spectra.

Full size image

In conclusion, we report the non-Hermitian control of reflectionless photon propagation, from nearly full transmission to nearly full absorption regions in a magnon-photon coupled system. By freely adjusting the frequency detuning between the two magnonic resonators and the microwave propagating delay phase, the PZR condition can be always achieved with tailored transmission/absorption. The flexible tunability and the general existence of the reflection singularity in our non-Hermitian magnonic device make the system promising for actual applications, such as lossless signal conversion and avoiding signal disturbances. Our work reveals the rich non-Hermitian physics in magnon-photon coupled systems, and meanwhile suggests the broad prospect of non-Hermitian control in the fields of microwave circuits, photonic chips, wave optics, and future quantum information technologies.

Methods

Device description

The waveguide magnonic device consists of a transmission line and two YIG spheres. The transmission line with a width of 1.14 mm and a length of 50 mm is fabricated on a 0.762-mm-thick Rogers RO4350B substrate. The dimensions of the substrate are 20 × 50 mm with a copper thickness of 35 μm on both sides. In the measured frequency range (3–5 GHz), the insertion loss of the bare waveguide is less than 0.5 dB, and the reflection keeps below −20 dB. Two 1 mm diameter YIG spheres separated by 2.5 mm are glued to the end of a displacement cantilever and adjusted accurately close to the copper waveguide through a three-dimensional adjustable stage. The crystallographic directions of two YIG spheres are not identified but remain fixed in all experiments. The magnon mode in the YIG sphere follows the Kittel dispersion equation ωm = γeμ0(|H| + |H0|), where μ0 is the vacuum permeability. For the YIG 1, the gyromagnetic ratio and the anisotropy field are γe/2π = 30 GHz/T and μ0|H0| = 7.9 mT, respectively. For the YIG 2, γe/2π = 29 GHz/T and μ0|H0| = 8.4 mT. The static magnetic field is applied perpendicularly to the waveguide plane.

Measurement setup

The reflection and transmission spectra are measured using a vector network analyzer (VNA) with an input power of −5 dBm.