Unidirectional exceptional point of reflectionless states in a magnonic mirror array

Abstract

Exceptional points (EPs) in non-Hermitian systems are singularities where both eigenvalues and eigenvectors coalesce. In scattering systems, EPs correspond to the merging of scattering states, leading to reflectionless (RL) behavior. A reflectionless exceptional point (RL EP) arises when two RL states further coalesce, yielding an anomalous quartic spectral response. While RL EPs have been explored in bidirectional systems, their unidirectional realization remains elusive. Here, we experimentally demonstrate a unidirectional RL EP by engineering collective states in an anti-Bragg magnonic mirror array. Inversion symmetry is broken using a giant spin ensemble that couples to a waveguide at three spatially separated points, enabling unidirectional reflectionless. At the RL EP, the reflection spectrum flattens and broadens significantly beyond the Lorentzian profile. The observed spectral valleys also expose dark-state behaviors that are typically inaccessible through conventional measurements. Our results provide a route toward controlling collective coherence in open systems and developing broadband unidirectional devices.

Figures (12)

• Figure 1: Design of the asymmetric magnon mode array with giant spin ensemble. (A) Transmission mapping of $M_1$ as a function of the bias magnetic field, showing periodic modulation in dip depth due to phase-dependent coupling. (B) Extracted radiative damping rates of the giant spin ensemble (GSE) (dots) as a function of frequency, fitted using Eq. (\ref{['GSE']}) (solid curve). (C) Schematic of a GSE with three spatially separated coupling points, enabling collective constructive interference that significantly enhances the radiative coupling between magnon mode $M_1$ and the waveguide. (D) Experimental setup: YIG spheres $M_1$ (purple) and $M_2$ (pink) are mounted on the waveguide (orange). The frequency of $M_2$ is tuned via a local field $B_1$ generated by a coil beneath it. Sphere $M_3$ (cyan) is attached to a cantilever connected to a rotation stage. All spheres are magnetized by a global field $B_0$. (E) Side view showing the adjustable position of $M_3$ along the $x$ and $z$ axes. (F) Tunability of $M_3$’s resonance frequency via anisotropy field control, plotted versus rotation angle $\theta_{\rm H}$. (G) Extracted radiative damping rate $\kappa_3$ of $M_3$ versus its vertical distance $d$ from the waveguide.
• Figure 2: Unidirectional exceptional point of reflectionless states. (A) Schematic of the magnon mode array (MMA), where three magnon modes are coupled to a waveguide with external (intrinsic) dissipation rates $\kappa_{i}$ ($\beta$). Adjacent magnon modes are spaced by $\lambda_{\rm{m}}/4$. (B and C) Imaginary (B) and real (C) parts of the RL eigenfrequencies for rightward and leftward incidence, plotted as functions of the magnon radiative decay rates $\kappa_{\rm{1}}$ and $\kappa_{\rm{3}}$. Yellow dashed lines indicate the EP conditions. Only the imaginary parts of the RL eigenfrequencies for rightward incidence intersects the zero-imaginary-frequency (purple) plane, marking the unidirectional RL EP. Here, $\kappa_2/\beta = 0.93$. (D) Real and imaginary parts of the RL eigenfrequencies versus $\kappa_{\rm{3}}$, extracted from side views of panels B and C. (E) Reflection spectra for both incidence directions under conventional RL (dashed) and RL EP (solid) conditions. Orange and blue curves correspond to leftward and rightward incidence, respectively.
• Figure 3: Characterization of relative phases between magnon modes via reflection measurement. (A) Individually measured transmission spectra of the three magnon modes (dots) at the GSE resonance frequency, with theoretical fits (black curves). (B and C) Schematic illustrations of the relative phase configurations: (B) between $M_{\rm{1}}$ and $M_{\rm{3}}$, showing dissipative coupling with a $\pi$ phase difference; (C) between $M_{\rm{1}}$ and $M_{\rm{2}}$, showing coherent coupling with a $\pi/2$ phase difference. (D and F) Reflection spectra measured from port 1 (D) and port 2 (F) when only $M_{\rm{1}}$ and $M_{\rm{3}}$ are present and dissipatively coupled. (E and G) Reflection spectra measured from port 1 (E) and port 2 (G) when only $M_{\rm{1}}$ and $M_{\rm{2}}$ are present and coherently coupled. All measurements were performed with only the corresponding pair of magnon modes present in the system.
• Figure 4: Unidirectional detection in the effective magnonic mirror cavity system. (A) Schematic illustration of the effective magnonic mirror cavity system, where a probing magnon mode couples to a dark cavity mode formed by two dissipatively coupled magnonic mirrors. (B) Parameter space showing two coupling regimes, separated by the ratio between the coupling strength $J_{\rm D}$ and the linewidths of the cavity and magnon modes. The white line tracks the system's trajectory as $\kappa_3$ is tuned. Stars and circles mark the cases in panels E and F. (C and D) Fitted real (C) and imaginary (D) parts of the RL-state eigenfrequencies $\omega_{\rm{RL}}^{l,\pm}$ versus $\kappa_{3}/\beta$. (E and F) Reflection mappings of $|S_{11}|^2$ measured by gradually rotating sphere $M_3$ in two different coupling regimes: $\kappa_3/2\pi = 0.19$ MHz (E), and $0.55$ MHz (F). (G and I) Reflection spectra measured at resonance. Cyan and purple dots represent $|S_{11}|^2$ and $|S_{22}|^2$ extracted from E and F, respectively. Black curves show theoretical fits. (H and J) The $|S_{11}|^2$ reflection spectra in (G) and (I) plotted on a linear scale.
• Figure S1: Schematic of the GSE–waveguide system. Multiple YIG spheres are coupled to the waveguide at arbitrary positions. The waveguide has two ports labeled Port 1 and Port 2.