The Role of Exceptional Points and Transmission Peak Degeneracies in Non-Hermitian Sensing

TL;DR

This work addresses the limitations of exceptional-points (EPs) for sensing by introducing transmission peak degeneracies (TPDs) as robust, square-root-splitting alternatives that preserve an eigenbasis and mitigate noise amplification. The authors develop a unified semiclassical framework linking EPs and TPDs in a two-dimensional parameter space, derive analytic figures of merit, and map the EP–TPD landscape using a tunable cavity–magnon platform with a synthetic gauge field. They experimentally validate six EP–TPD configurations, derive robust TPD operating points (notably φ=0, κ_c=2) that suppress nuisance drift, and examine transmission-extrema degeneracies (TEDs) to show that third-order degeneracy provides a practical advantage for sensing under perturbations. The results yield practical design principles for TPD-based sensors, quantify noise performance via the Petermann factor and thermal-noise efficiency, and offer a versatile platform for exploring non-Hermitian dynamics, topology, and robust sensing strategies in hybrid magnon–photon systems. Overall, the paper establishes a comprehensive theory and experimental framework for TPD-based non-Hermitian sensing with principled guidance for real-world implementation.

Abstract

Transmission peak degeneracies (TPDs) have emerged as a promising alternative to exceptional points (EPs) for non-Hermitian sensing, providing square-root frequency splitting without the eigenbasis collapse and associated noise amplification that limit EP sensors. However, existing treatments of TPDs remain fragmented, lacking a unified theoretical framework, systematic figures of merit, or design principles for practical implementation. Here, we develop a comprehensive theory of two-dimensional TPDs that clarifies their relationship to EPs, maps their locations in parameter space, and provides analytic figures of merit for sensor design. We validate our theory using a tunable cavity-magnonics platform with in situ control of mode frequency, dissipation, and complex coupling via an effective synthetic gauge field. Our platform enables systematic exploration of six representative EP-TPD configurations spanning PT-symmetric, anti-PT-symmetric and anyonic-PT-symmetric regimes. Crucially, we show that TPDs, unlike EPs, retain square-root splitting even under nuisance parameter drift through generalized transmission extrema degeneracies (TEDs). We further identify specific robust TPD configurations that minimize the impact of nuisance drift. These findings establish a unified theoretical and experimental framework for TPD-based non-Hermitian sensing.

Figures (9)

• Figure 1: (a) Schematic of our architecture. The coupling amplifier and phase shifter ($G, \phi$) determine the coupling rate ($J$) and phase ($\phi$). Auxiliary controls ($g_c, g_y, \psi_c, \psi_y, B_z$) modify the mode parameters $f_c, f_y, \kappa_c, \kappa_y$. (b)-(g) Transmission spectra as a function of current through an external electromagnet, which tunes the YIG frequency $f_y$ through $B_z$ and thereby controls the detuning between the modes. Data illustrate level repulsion ($\phi = 0$) (b),(c),(f),(g) and level attraction ($\phi = \pi$) (d),(e) for varying coupling strengths $J$ and average dissipation rates $\bar{\kappa}$.
• Figure 2: Parameter landscape for $\phi = 0$ (a),(b), $\phi = \pi$ (c),(d), and $\phi = \pi/2$ (e),(f) overlaid atop the Petermann factor (clipped to 90th percentile). Each panel corresponds to the same $\phi$ and $\tilde{\kappa}_c$ used in the experimental panel in Fig. \ref{['fig:eps']}. Red crosses (circles) are EPs (TPDs) experimentally implemented in Fig. \ref{['fig:eps']}; "other" EPs and TPDs are not considered here. Gray diamonds represent TPDs not associated with an EP, referred to as rogue TPDs. Cyan contours are $\mathrm{Disc} = 0$ (Eq. \ref{['eq:disc_algebraic']}), separating the parameter space into single peak (outside cyan contours horizontally) and two peak (between cyan contours horizontally) regions. Magenta contours are $\tilde{q} = 0$ (Eq. \ref{['eq:q']}), used to determine the independent variable in Fig. \ref{['fig:eps']}. Lime contours are $\operatorname{Re}(\tilde{\Delta}_\lambda) = (\tilde{\kappa}_c - \tilde{\Delta}_\kappa)$, and separate the parameter space into stable (Inferno colorscale) and unstable (grayscale, linear model inapplicable) regimes. (g), an alternative representation of EPs and TPDs, representing a cross-section along the $\tilde{\Delta}_f$ axis for the $\tilde{\kappa}_c, \phi$ configuration in (a). The imaginary eigenvalues ($\operatorname{Im}(\tilde{\lambda})$) split at the EP (solid red vertical), while the transmission peak frequencies ($\tilde{\nu}$) split at the TPD (solid cyan vertical).
• Figure 3: Experimentally fit transmission peak locations $\tilde{\nu}_\pm$ overlaid atop theory (solid lines) and $|\operatorname{Im}(\tilde{\lambda}_\pm)|$ (dashed lines). The coupling phase ($\phi$) determines the relevant sweep parameter. (a),(b) $\phi = 0$ (blue) sweeps $\tilde{\Delta}_\kappa$, (c),(d) $\phi =\pi$ (purple) sweeps $\tilde{\Delta}_f$, and (e),(f) $\phi = \pi/2$ (green) sweeps along a hyperbolic path ($\tilde{\Delta}_\kappa \tilde{\Delta}_f = 2$) projected onto the $\tilde{\Delta}_\kappa$ axis. In all cases, the TPD location moves with $\tilde{\kappa}_c$, while the EP is static, with the left (right) column surveying smaller (larger) $\tilde{\kappa}_c$ values that place the TPD and EP closer to (further from) each other.
• Figure 4: (a)–(c) Experimental data from Fig. \ref{['fig:eps']} recast as peak splitting $\tilde{\Delta}_\nu$, versus the sweep parameter for $\phi = 0$ (a), $\phi = \pi$ (b), and $\phi = \pi/2$ (c). Markers distinguish small $\tilde{\kappa}_c$ (circles, data from left column from Fig. \ref{['fig:eps']}) and large $\tilde{\kappa}_c$ (triangles, data from right column from Fig. \ref{['fig:eps']}) configurations. (d) Semiclassical sensing figures of merit as functions of $\tilde{\kappa}_c$ for $\phi = 0$ (blue), $\phi = \pi$ (purple) and $\phi = \pi/2$ (green). (d.i) target signal splitting strength $\tilde{a}_{\text{sqrt}}^\text{TPD}$, (d.ii) nuisance scaling $\tilde{b}_{\text{cbrt}}^\text{TPD}$, (d.iii) thermal noise efficiency at the TPD, (d.iv) Petermann factor at the TPD, (d.v) distance from the TPD to the nearest instability transition, and (d.vi) distance to the nearest EP. Markers indicate the experimental configurations from (a)–(c).
• Figure 5: Monte Carlo simulation of nuisance propagation for $\phi = 0$ at three values of $\tilde{\kappa}_c$. (a) Local geometry of the $\mathrm{Disc} = 0$ contour (cyan dashed) near the TPD (red circle). The orange band depicts the standard deviation of a Gaussian-distributed nuisance fluctuation with $\sigma(\tilde{\delta}(\tilde{\Delta}_f)) = 10^{-3}$. In the non-robust configurations (a.i),(a.iii), the $\mathrm{Disc} = 0$ contour forms a cusp catastrophe that intersects the fluctuation band. (a.ii) At the robust TPD ($\tilde{\kappa}_c = 2$), the contour factorizes and the cusp vanishes. (b) Transduction of $\tilde{\delta}(\tilde{\Delta}_f)$ into transmission peak locations $\tilde{\nu}_\pm$. Solid lines show the mean peak frequencies, while shaded regions indicate $\pm 1\sigma(\tilde{\nu}_\pm)$ propagated from the nuisance fluctuations ($N = 10^4$ samples). Near the TPD, the cusp geometry (b.i),(b.iii) broadens $\sigma(\tilde{\nu}_\pm)$, whereas the robust configuration (b.ii) effectively decouples the peak splitting from $\tilde{\delta}(\tilde{\Delta}_f)$.