Exceptional Point Dynamics in Photonic Time Crystals for Enhanced Optical Sensing

TL;DR

This work transfers exceptional-point physics to the time domain by using a photonic time crystal with time-periodic index and balanced gain–loss to realize a non-Hermitian Floquet system. A minimal two-mode model yields a PT-symmetric dimer $H_{PT}(Δ,γ,κ)$ with an exact EP at $Δ=0$, $γ=±κ$, and eigenvalues $λ_{±}=±\sqrt{(Δ+iγ)^2+κ^2}$; encirclement of the EP yields mode exchange and a Berry phase $φ_B=π$, establishing robust topological signatures in the temporal domain. A non-Hermitian transmission model connects the EP to a concrete sensing protocol, enabling a CRB-based analysis that shows EP-enabled reductions in the temperature-estimation bound under identical resource constraints, with Monte Carlo validation confirming estimator optimality. The results demonstrate that temporal non-Hermiticity offers reconfigurable, broadband EP photonics for enhanced sensing on integrated platforms, with practical pathways in TFLN and SiN technologies.

Abstract

Exceptional points (EPs) in non-Hermitian photonics offer singular sensitivity enhancements but have thus far been realized almost exclusively in spatially engineered platforms with fixed geometries and limited tunability. Here we extend EP physics into the temporal domain by introducing balanced gain--loss modulation in a photonic time crystal (PTC). A time-periodic refractive-index modulation $n(t)=n_{0}+δn\cos(Ωt)$ generates an effective non-Hermitian Floquet Hamiltonian that supports coalescence of quasi-eigenmodes in frequency space, constituting a genuine \textit{temporal exceptional point}. Using a reduced two-mode model for the dominant frequency sidebands, we derive a non-Hermitian dimer Hamiltonian $H_{\mathrm{PT}}(Δ,γ,κ)$ that is strictly $\mathcal{PT}$-symmetric for $Δ=0$ and identify the exact EP condition. Numerical analysis reveals the associated Riemann-sheet topology, mode exchange and Berry-phase accumulation upon encirclement of the EP, and the characteristic $\sqrt{\varepsilon}$ perturbation response indicative of enhanced sensing. We further construct a non-Hermitian transmission model that is exact within the reduced two-mode description, compute the Cramér--Rao bound (CRB) for temperature estimation under an explicit noise model, and show that EP-enhanced sensitivity persists when compared to a linewidth-matched Hermitian reference under identical resource constraints. Monte Carlo simulations confirm that the CRB is saturable using spectral measurements. These results establish temporal non-Hermiticity as a new paradigm for dynamically reconfigurable, broadband, and geometry-independent exceptional-point photonics.

Figures (12)

• Figure 1: Conceptual comparison between spatial and temporal exceptional-point (EP) systems. (a) Spatial EP devices rely on engineered refractive-index profiles $n(x)$ and fixed coupling to realize non-Hermitian band structures in momentum space. (b) In photonic time crystals (PTCs), a uniform medium with time-periodic refractive-index modulation $n(t)=n_0+\delta n\cos(\Omega t)$ and dynamic gain--loss $\gamma(t)$ yields an effective non-Hermitian Floquet Hamiltonian in the frequency domain, enabling tunable temporal EPs.
• Figure 2: Exact eigenvalue landscape of the temporal non-Hermitian dimer. (a) and (b) Real and imaginary parts of $\lambda_{+}$ as functions of detuning $\Delta/g_0$ for several values of $\gamma/g_0$, illustrating the transition from the unbroken $\mathcal{PT}$ phase ($\gamma<\kappa$) through the EP ($\gamma=\kappa$) into the broken phase ($\gamma>\kappa$) at $\Delta=0$. (c) and (d) Two-dimensional maps of the mode splitting $|\lambda_+ - \lambda_-|/g_0$ and the magnitude of the imaginary part $|\mathrm{Im}\,\lambda_+|/g_0$ over the $(\Delta/g_0,\gamma/g_0)$ plane. The EP appears as a pinch point at $(\Delta,\gamma)=(0,\pm \kappa)$.
• Figure 3: Riemann topology of the exact eigenvalues. Real parts of the eigenvalue sheets for (a) $\lambda_{+}$ and (b) $\lambda_{-}$, as functions of $(\Delta/g_0,\gamma/g_0)$. The two sheets coalesce at the EP points along $\Delta=0$ and $\gamma=\pm\kappa$, forming a double-sheeted Riemann surface with a square-root branch point.
• Figure 4: Encirclement of the EP and Berry-phase accumulation. (a) Parametric loop in the $(\Delta/g_0,\gamma/g_0)$ plane encircling the EP. (b) Biorthogonal Berry phase $\phi_B(\theta)$ as a function of the loop angle $\theta$, obtained from the product of overlaps of left and right eigenvectors along the path according to Eq. \ref{['eq:Berry_discrete']}. A total phase of magnitude $\pi$ is accumulated after one encirclement, confirming the nontrivial EP topology. The calculation is purely parametric in $(\Delta,\gamma)$ and does not assume a specific dynamical encirclement protocol in time.
• Figure 5: Exact spectral response and estimator validation. (a) Transmission spectra $|T(\omega)|^2$ for several temperature shifts $\Delta T$, computed from the exact non-Hermitian scattering model in Eq. \ref{['eq:T_exact']}. (b) Eigenvalue splitting $R = |\lambda_+ - \lambda_-|/g_0$ versus $\Delta T$, showing excellent agreement between the true splitting (solid), the Cramér--Rao bound (CRB) prediction obtained from Eqs. \ref{['eq:Fisher']}--\ref{['eq:CRB_R']}, and Monte Carlo (MC) estimates from noisy spectra (error bars). (c1)--(c3) Representative residuals between noisy spectra and best-fit spectra at three values of $\Delta T$, $-3.0$ K, $0.0$ K, and $+3.0$ K. (d) Histogram of residuals at $\Delta T\approx0$, demonstrating Gaussian noise and negligible model mismatch.