Multiparameter Estimation Perspective on Non-Hermitian Singularity-Enhanced Sensing

Describing the evolution of quantum systems by means of non-Hermitian generators opens a new avenue to explore the dynamical properties naturally emerging in such a picture, e.g. operation at the so-called exceptional points, preservation of parity-time symmetry, or capitalizing on the singular behavior of the dynamics. In this Letter, we focus on the possibility of achieving unbounded sensitivity when using the system to sense linear perturbations away from a singular point. By combining multiparameter estimation theory of Gaussian quantum systems with the one of singular-matrix perturbations, we introduce the necessary tools to study the ultimate limits on the precision attained by such singularity-tuned sensors. We identify under what conditions and at what rate can the resulting sensitivity indeed diverge, in order to show that nuisance parameters should be generally included in the analysis, as their presence may alter the scaling of the error with the estimated parameter.

Figure 1

Non-Hermitian sensor model consisting of two coupled cavities bearing modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$ with effective loss and gain rates ${\gamma }_1=[({\eta }_1+\kappa )/2]$ and ${\gamma }_2=[({\eta }_2-\kappa )/2]$. The rates are controlled by “scattering channels” (modes), ${\widehat{B}}_1$ and ${\widehat{B}}_2$, coupled to respective cavities, whose dynamics is also affected by “probing channels”, ${\widehat{A}}_1$ and ${\widehat{A}}_2$, used to continuously monitor each cavity.

Figure 2

Space of parameters characterizing the dynamical generator $\mathbf{H}$ in Eq. (3), with $g$ and ${\gamma }_1$ (${\gamma }_2$) being the coupling constant and loss (gain) rates, respectively. (a) The green surface depicts values where the singularity condition $\det \mathbf{H}=0$ holds, and the triangular plane represents PT symmetry. Their intersection, where $g={\gamma }_1={\gamma }_2$, guarantees the EP condition (dashed blue line). (b) Cut at ${\gamma }_1=0.75$ is included to show the separation between stable and unstable regions of dynamics—with the lasing threshold (purple dashed line) following the singularity surface before becoming (at an EP, blue dot) equivalent to the PT condition for $g>{\gamma }_1$. In the multiparameter estimation setting, we consider nuisance parameters to preserve PT symmetry while either invalidating (black arrow) or maintaining (red arrows) the singularity condition.

Figure 3

Estimation errors attained by heterodyne detection (dashed lines), ${\delta }_{\mathrm{C}}{\theta }_0$ or ${\mathrm{\Delta }}_{\mathrm{C}}{\theta }_0$, vs quantum Cramér-Rao bounds applicable to any measurement (solid lines), ${\delta }_{\mathrm{Q}}{\theta }_0$ or ${\mathrm{\Delta }}_{\mathrm{Q}}{\theta }_0$. Within the ideal single-parameter scenario, both ${\delta }_{*}{\theta }_0$ (black) follow quadratic scaling. When ${\theta }_1$ constitutes a nuisance parameter whose variations maintain the singularity, the estimation errors ${\mathrm{\Delta }}_{*}{\theta }_0$ scale linearly with ${\theta }_0$, e.g. for ${\theta }_1=0$ (red) or ${\theta }_1=0.25$ (blue). In the above, all probe and scattering input modes are chosen to be in a thermal state with average photon-number $n_{\mathrm{A}}=n_{\mathrm{B}}=1$, while the intermode couplings are set to $\kappa =g=1$ and ${\eta }_1=1$, ${\eta }_2=3$.