Experimental Observation of an Exceptional Surface in Synthetic Dimensions with Magnon Polaritons

Exceptional points (EPs) are singularities of energy levels in generalized eigenvalue systems. In this Letter, we demonstrate the surface of EPs on a magnon polariton platform composed of coupled magnons and microwave photons. Our experiments show that EPs form a three-dimensional exceptional surface (ES) when the system is tuned in a four-dimensional synthetic space. We demonstrate that there exists an exceptional saddle point (ESP) in the ES which originates from the unique couplings between magnons and microwave photons. Such an ESP exhibits unique anisotropic behaviors in both the real and imaginary parts of the eigenfrequencies. To the best of our knowledge, this is the first experimental observation of ES, opening up new opportunities for high-dimensional control of non-Hermitian systems.

Figure 1

(a) Picture of the microwave cavity and the simulated cavity magnetic fields for ${\mathrm{TE}}_{101}$ mode (fields inside the slot not shown). A slot is cut near the field maximum to host the YIG sphere. The bias magnetic field is applied in the $x\text{−}y$ plane with an angle $\theta$ to the $y$ direction. (b) Normalized cavity reflection spectra at different bias fields, showing the anticrossing. The background signals are removed during normalization for clarity. A small anticrossing induced by a high order magnon mode is visible at around 3500 Oe. Slot length: 1 mm. (c),(d) Riemann surfaces for the real (resonance frequency) and imaginary (resonance linewidth) parts of the eigenfrequency, respectively, reconstructed from measured reflection spectra and averaged over multiple measurements. EP is indicated by the circle, showing the bifurcation of the Riemann surface.

Figure 2

(a) Coupling strength as a function of $x$ and $y$. The black dot indicates the saddle point. Slot length: 5 mm. (b) Simulated microwave magnetic field distribution for the cavity mode around the slot. Dashed black line indicates the slot area. (c) Coupling strength as a function of $\theta$ at $x=0\mathrm{mm}$ and $y=0.05\mathrm{mm}$, where $\theta$ is the angle of the bias magnetic field relative to the $y$ direction. Circles and the solid line are the measurement and cosine fitting results, respectively.

Figure 3

(a)–(c) Experimentally obtained Riemann surfaces for the real (top) and imaginary (bottom) part of the eigenfrequencies at different $x$ positions of the YIG sphere: $x=0.66\mathrm{mm}$ ($g>g_c$), $x=0.13\mathrm{mm}$ ($g=g_c$), $x=0.00\mathrm{mm}$ ($g<g_c$). EP coalescence is observed at $x=0.13\mathrm{mm}$ when $g=g_c$. (d) Coupling strength $g$ as a function of $y$ positions at various $x$ locations. (e),(f) Cross-sectional views at the EP for the lower figure of (b). Dashed lines are plotted to guide the eye.

Figure 4

(a) Slices of ES in the 3D parameter space ($x$, $y$, $\theta$). The fourth dimension $H$ is hidden (fixed at $H=3327\mathrm{Oe}$). White dots: results extracted from experimental data; black lines: calculation from numerical fittings. Planes at given $\theta$s are guides to the eye. The yellow star represents the exceptional saddle point. (b)–(d) Cross sections of the Riemann surfaces at the ESP. Top figures: real part of the eigenfrequencies (resonance frequencies). Bottom figures: imaginary part of the eigenfrequencies (linewidth). Circles: experimental results; solid lines: numerical fittings; dashed lines: guides to the eye. Black dots represent EPs. The slight mismatch between experimental results and numerical fittings stems from the imperfect zero-detuning condition ($f_c-f_m\ne 0$) in the experiment.