Optimal sensing on an asymmetric exceptional surface

TL;DR

This work analyzes sensing of a cross-coupling perturbation $\epsilon$ in a two-mode microring resonator that exhibits an exceptional surface (ES) due to retro-reflection. Using a fully quantum input-output framework, it computes the quantum Fisher information $\mathcal{I_Q}(\epsilon)$ for coherent and NOON inputs, showing EP-related enhancements at $\epsilon=0$ with explicit bounds and optimal detection schemes that saturate the quantum Cramér-Rao bound. The results reveal that the ES can boost sensitivity (up to a factor of $4$ for coherent states and $\approx1.69$ for NOON states, with NOON states exhibiting Heisenberg scaling $\propto N^2$), but optimality is not restricted to the ES, and the generator $\mathbf{A}_{\epsilon}$ principally governs the achievable precision. The work also proposes practical saturation strategies (homodyne detection for coherent states and frequency-shifted photon counting for NOON states) and discusses how non-ES operating points can sometimes offer larger QFI, highlighting nuanced design considerations for non-Hermitian sensing platforms and potential explorations of $\mathcal{PT}$-symmetric schemes augmented by coherent feedback.

Abstract

We study the connection between exceptional points (EPs) and optimal parameter estimation, in a simple system consisting of two counter-propagating traveling wave modes in a microring resonator. The unknown parameter to be estimated is the strength of a perturbing cross-coupling between the two modes. Partially reflecting the output of one mode into the other creates a non-Hermitian Hamiltonian which exhibits a family of EPs, creating an exceptional surface (ES). We use a fully quantum treatment of field inputs and noise sources to obtain a quantitative bound on the estimation error by calculating the quantum Fisher information (QFI) in the output fields, whose inverse gives the Cramér-Rao lower bound on the mean-squared-error of any unbiased estimator. We determine the bounds for two input states, namely, a semiclassical coherent state and a highly nonclassical NOON state. We find that the QFI is enhanced in the presence of an EP for both of these input states and that both states can saturate the Cramér-Rao bound. We then identify idealized yet experimentally feasible measurements that achieve the minimum bound for these two input states. We also investigate how the QFI changes for parameter values that do not lie on the ES, finding that these can have a larger QFI, suggesting alternative routes to optimize the parameter estimation for this problem.

Figures (6)

• Figure 1: [color online] Set up for asymmetric coupling in a microring resonator (MRR). The resonator acts as a cavity that supports clockwise (CW) and counter-clockwise (CCW) whispering gallery modes with quantized field operators $\hat{a}_{cw}$ and $\hat{a}_{ccw}$, respectively. Left/right input photons (with field operator $\hat{a}_l$/$\hat{a}_r$) propagate in a waveguide and resonantly couple to the cavity modes with the rate $\gamma$. The presence of an external scatterer perturbs the resonator and induces a symmetric cross-coupling between the two whispering gallery modes, parameterized by the real value $\epsilon$. To the right of the resonator a beam splitter (BS) introduces the asymmetric coupling between the CW and CCW modes by reflecting CW output back as a CCW input. The BS is parameterized by reflection (transmission) coefficient $\rho$ ($\tau$), and phase angle $\phi$.
• Figure 2: [color online] Frequency-optimized quantum Fisher information (o-QFI) on the ES $\epsilon = 0$. This figure shows the o-QFI for a coherent state $\mathcal{I}_\beta$ (left column, panels (a) and (c)) and for a NOON state $\mathcal{I_N}$ (right column, panels (b) and (d)) as a function of the reflectivity $\rho$ and the reflection phase $\phi$. The ES is defined for all values of $\phi$ and $\rho > 0$. The first row, panels (a) and (b), shows results for $|\boldsymbol{\beta}|^2 = N = 2$, while the second row, panels (c) and (d), shows results for $|\boldsymbol{\beta}|^2 = N = 3$. For both input states, the peak QFI occurs at $\rho = 1$ and is periodic in $\phi$ where the coherent (NOON) state is maximal at even (odd) multiples of $\pi/4$, respectively. The maximal contrast is exactly a factor of 4 for a coherent state, a factor of $108/64 = 1.6875$ for a NOON state, and both are independent of the number of photons. The crossover between shot-noise limited and Heisenberg scaling occurs at $N = 3$, which shows a larger o-QFI for the NOON state.
• Figure 3: [color online] Panels (a) and (b) show contour plots of $A_{ll}(\omega, \epsilon)$ for complete reflection ($\rho = 1$), when $A_{ll}$ is the only non-zero matrix element of $\mathbf{A}_{\epsilon}$. In panel (a) the phase is fixed for the coherent state optimal value of $\phi = 0$, and in panel (b) the phase is fixed at the NOON state optimal value of $\phi = \pi/4$ (see Fig. \ref{['fig:QFI']}). Panels (c) and (d) show $A_{ll}(\omega)$ for constant values of $\epsilon$ with $\phi = 0$ in (c) and $\phi = \pi/4$ in (d). The coupling parameter values are for the ES $\epsilon = 0$ (blue lines) and the non-degenerate value of $\epsilon = -0.4 \gamma$ (orange lines). When $\phi = \pi/4$, $A_{ll}(\omega, \epsilon)$ becomes ill-defined at $\omega = 0$ and $\epsilon = -0.5\, \gamma$, showing sharply peaked oscillations in panel (d). At these values, $\tilde{ \mathbf{H} }_{\epsilon}$ has the eigenvalues $\{ -i\gamma, 0 \}$, and so the resolvent $\mathbf{R}_{\epsilon}(\omega)$ is undefined at $\omega = 0$.
• Figure 4: [color online] Balanced homodyne detection. The input coherent state $\boldsymbol{\beta}$, interacts with the system and is spatially separated from the input by a circulator. The $\epsilon$ dependent output $\boldsymbol{\beta}_\epsilon$, is then interfered on a symmetric 50:50 beam splitter with a large amplitude coherent state $\boldsymbol{\alpha}$. The difference in photon counts gives a mean signal $\mu = 2 \operatorname{Im} \langle \boldsymbol{\alpha}, \boldsymbol{\beta}_\epsilon \rangle$ and a variance $\sigma^2 = \|\boldsymbol{\alpha}\|^2 + \|\boldsymbol{\beta}\|^2$. When $\|\boldsymbol{\beta}\| \ll \|\boldsymbol{\alpha}\|$, the difference signal is a normal random variable of mean $\mu$ and variance $\sigma^2$.
• Figure 5: [color online] NOON state detection scheme. The reflected NOON state is spatially separated from the input by a circulator. The frequency content of the output is dispersed into two distinct spatial modes. The lower frequency $\omega_-$ is shifted to match $\omega_+$ via a high efficiency optical modulator. The common frequency beams interfere on a 50:50 beam splitter. The mixed outputs are measured at photon counters 1 and 2.