Fundamental Limits of Non-Hermitian Sensing from Quantum Fisher Information

Abstract

Exceptional points (EPs) exhibit strongly enhanced spectral responses and are therefore promising candidates for sensing applications. Whether these non-Hermitian degeneracies provide a genuine advantage in the quantum regime has been the subject of ongoing debate. Here, we address this issue within a scattering-matrix formalism for sensing with coherent light, which allows the quantum Fisher information (QFI) to be evaluated directly from experimentally accessible scattering data without introducing additional noise channels beyond those inherent to the scattering process. We analyze both nondegenerate and degenerate scattering-matrix poles, including EPs of arbitrary order, and show that the QFI per incoming photon flux is governed by three key factors: the decay rate of the resonant mode, the strength of the spectral response associated with non-normality, and the adjustment between the scattering states and the information source. For spatially localized perturbations, this implies that the Fisher information is fully determined by the local density of states at the perturbation site. Within this framework, we demonstrate that EPs can enhance the QFI compared to isolated modes or diabolic points with identical decay rates, and that the QFI can be further increased by moving away from the EP toward parameter regimes where non- Hermitian linewidth splitting reduces the decay rate of one mode. We further show that sufficiently small additional internal losses do not alter this overall picture, thereby providing a unified and experimentally relevant perspective on the design of quantum-limited non-Hermitian sensors.

Figures (7)

• Figure 1: Sketch of two coupled optical microrings attached to a single-mode waveguide with coupling strength $\gamma$. $V$ is the intercavity coupling coefficient. The information source is located in the lower ring, i.e., the system is perturbed by shifting its frequency with respect to the upper ring, for instance by a micro-heater underneath the lower ring, see Refs. HHW17FAB25.
• Figure 2: The maximal QFI for the two-ring setup as function of the intercavity coupling strength $|V|$ on a semi-logarithmic scale. Both quantities are made dimensionless by scaling with the waveguide coupling strength $\gamma$. The blue curve results from the full calculation according to Eq. (\ref{['eq:ecr0']}), while the red curve shows the contribution from the long-lived mode [one summand in Eq. (\ref{['eq:LDOSRL']})] in the weak coupling regime. The orange curve is the upper bound of this contribution, as given by Eq. (\ref{['eq:FIDPlocalized']}). The green dashed curve shows the decay-modified QFI from Eq. (\ref{['eq:FImod']}). The EP is indicated by the dashed vertical line.
• Figure 3: (a) The phase $\varphi$ in radians of the outgoing state as function of the normalized perturbation parameter $\varepsilon$, which equals a detuning of the lowermost ring for the EP in the two-ring setup illustrated in Fig. \ref{['fig:examplecoupledring']} and in the single-ring setup with the isolated mode. (b) Maximal QFI in Eq. (\ref{['eq:ecrphase']}) made dimensionless by scaling with the waveguide coupling strength $\gamma$. Without loss of generality, the phase $\varphi$ is gauged such that it is zero at $\varepsilon = 0$. The frequency is chosen to be on resonance, $\omega = \omega_0$.
• Figure 4: Sketch of an optical microring coupled to a semi-infinite single-mode waveguide with a partial mirror having a reflection coefficient $\rho$; $\gamma$ is the waveguide coupling strength. A small target particle induces a backscattering between clockwise and counterclockwise propagating waves inside the microring.
• Figure 5: The reduced QFI as function of (a) the internal losses $\kappa$ normalized by the waveguide coupling strength $\gamma$ and (b) $\gamma$ normalized by $\kappa$. Note the different scaling applied to the respective QFI to render it dimensionless. The two-ring setup at the EP [Eq. (\ref{['eq:ecr1lossesredEP']})] is shown in blue, the single-ring setup with the isolated mode [Eq. (\ref{['eq:ecrsinglelossesred']})] is shown in red, and the two-ring setup with optimized intercavity coupling strength [Eq. (\ref{['eq:optlosses']})] is shown in yellow.