Non-Hermitian Sensing via a Divergent Quantum Metric

Abstract

The quantum metric, a geometric measure of state-space distance, has recently attracted growing attention for capturing anomalous state responses to parameter variations. Especially in non-Hermitian systems, the quantum metric has been observed to diverge when the eigenstates coalesce, a phenomenon identified as a remarkable resource for sensing. Here, by exploiting this divergence, we establish a non-Hermitian sensing scheme that leverages enhanced transient dynamics to provide a geometric gain for amplifying external field signals. We confirm the critical enhancement in the Fisher information using a trapped-ion 171Yb+ platform and demonstrate superior noise robustness over conventional eigenvalue-splitting--based non-Hermitian schemes by evaluating the minimum detectable signal. Moreover, this scheme can be naturally combined with non-Hermitian topological dynamics, revealing a unique unidirectional sensing response, which indicates its potential for directional signal discrimination. Our work establishes a new paradigm for sensing in open quantum systems through critical quantum geometry and opens a route toward robust topological quantum sensing.

Figures (3)

• Figure 1: Schematics of non-Hermitian sensing via a divergent quantum metric. (a) The energy levels of the non-Hermitian Hamiltonian $H_{\mathcal{PT}}(t)$. The shift of energy levels $\Delta$ induced by the external field being measured determines the variation of the phase $\dot{\Phi}$. (b) Conceptual figure illustrating quantum geodesic deviation driven by a divergent quantum metric. The state space is shown as a manifold, with blue and red surfaces denoting gain and loss regions governed by the imaginary part of the intraband Berry connection. The circle $A$ marks the initial state. Upper (lower) panel: far from (near) EPs, the quantum-metric variation is suppressed (enhanced). Solid and hollow black (purple) lines indicate two state trajectories evolved under slightly different $\dot{\Phi}$, terminating at $B$ and $B'$ ($C$ and $C'$) after dynamic sensing. The separation between the final states visualizes quantum geodesic deviation, where the divergent quantum metric amplifies small differences in $\dot{\Phi}$. (c) Schematics of non-Hermitian topological tunneling. Red (blue) branches show the real part of eigenenergies $E$ with positive (negative) values. $J$ is time-modulated to scan toward and then away from the EP line, as traced by the paths in the lower $J$–$\Phi$ plane. From the initial state $A$, dashed (solid) arrows denote process under small (large) $\dot{\Phi}$. The final states $C$ and $C'$ exhibit different populations in both branches ($C_{-}$, $C'_{+}$, $C'_{-}$), indicated by circle sizes. Near EPs, metric enhancement amplifies a minute signal difference into a pronounced separation of the final states after tunneling. (d) Modulation function $\beta(t)$ during dynamic sensing. The system starts at the "tail" far from the EP line (orange dashed line, $\beta=1$), scans to the "pin" near the EP line (red line, $\beta_p=0.75$), and then returns to the "tail".
• Figure 2: Experimental results of the time-dependent response of dynamic non-Hermitian sensing. (a) Non-Hermitian: using the modulation parameters in Fig. \ref{['one']}(a), for various detuning $\Delta$, the time evolution of $P_z$ represented by the color map. (b) Hermitian: using the same modulation, while setting the dissipation to zero, the system becomes Hermitian, $P_z$ is also measured. (c) Comparison of the final-state $P_z(T=200\mu s)$ population with the different signal $\Delta$ being measured. The orange points represent the non-Hermitian sensing in (a), while the blue points correspond to the Hermitian case in (b). The non-Hermitian regime shows higher signal sensitivity.
• Figure 3: Experimental tests of criticality enhancement, unidirectionality, and robustness. (a) The state is initialized in $\ket{\phi_2}$ with $\dot{\Phi}(t) > 0$. The solid red curves represent the simulation of the gain of the CFI, and the points with error bars represent measured ones from the experimental data. For each point, the CFI obtained from the tunneling data across the detuning range $\Delta/2\pi \in (-5.9, 5.9)$ kHz. Insets: state evolution over time for $\beta_p=0.1$ and $0.8$. Axes correspond to coupling strength $J$ (X), phase $\Phi$ (Y), and energy magnitude (Z). The red (blue) surface depicts positive (negative) real eigenvalues in the $\mathcal{PT}$-symmetric regime, associated with eigenstate $\ket{\phi_1}$ ($\ket{\phi_2}$). Black circles indicate initial states, lines show evolution trajectories, and arrows mark direction. (b) The state is initialized in $\ket{\phi_2}$ with $\dot{\Phi}(t) < 0$. (c) Minimum resolvable signal $\delta\Delta_{\min}$ under different noise strengths $\lambda$. Blue circles and pink squares represent experimental results for $\beta_p = 0.2$ and $0.6$, respectively; green triangles show EV-based simulation results. Solid lines indicate corresponding linear fits. Each noise strength was tested with 15 measurements ($N_r = 15$).