Criticality-enhanced global frequency sensing with a monitored Kerr parametric oscillator via extended Kalman filter

Abstract

We analyze a global sensing scenario in which the frequency of a monitored Kerr parametric oscillator is estimated assuming limited prior information. The frequency is estimated in real-time by continuously monitoring the oscillator quadrature through homodyne detection and processing the resulting photocurrent with an extended Kalman filter (EKF). Due to the sensor nonlinearity, individual EKF trajectories do not always converge to the true unknown frequency in the long-time limit. However, we show that the statistical distribution of the frequency estimates does exhibit a sharp peak around the true value in the same limit. Leveraging this key statistical property, we develop a global sensing protocol assisted by adaptive control of the sensor parameters to harness critical enhancement. We present numerical evidence that this criticality-enhanced frequency estimation remains robust under low detection efficiency.

Figures (5)

• Figure 1: Four typical trajectories of the EKF estimate $\tilde{\omega}_\text{ekf}(t)$ for ideal detection efficiency $\eta=1$ and $\varphi=0.1072$. The true frequency $\omega_\text{true}=\kappa$ is shown as the dashed line, and the prior interval $I_\text{prior} = (\omega_l, \omega_h)$, where $\omega_l=0.7\kappa$ and $\omega_h=2.3\kappa$, is displayed as the shaded gray area. The time instants where the filter divergence condition was triggered are marked. The insets show the corresponding photocurrents $\mathbf{Y}_{t}$, recorded with a time step $\kappa dt=0.02$. The EKF is initialized as $\tilde{\mathbf{x}}(0) = (0,0,1,1,0,(\omega_l+\omega_h)/2)^\intercal$ and $\tilde{\mathbf{\Sigma}}(0) = \text{diag}(10^{-3},10^{-3},10^{-3},10^{-3},10^{-3},\kappa^2v)$, with $v=1$ used here. The driving amplitude is $\epsilon=\kappa$ and the threshold parameter in Eq. (\ref{['eq:threshold']}) is set to $F_\text{max}=10^5$.
• Figure 2: (a) Distributions $P(\tilde{\omega}_\text{ekf}(t))$ at time $t=100/\kappa$ for two homodyne phases $\varphi=0.1072$ (blue) and $\varphi=1$ (red), obtained from $N_\text{traj}=2000$ trajectories under ideal HD ($\eta=1$). The vertical dashed line indicates the true frequency $\omega_\text{true}=\kappa$, and the gray shaded region corresponds to the prior interval $I_\text{prior}=(0.7\kappa,2.3\kappa)$. The inset shows the percentage of the $2000$ EKF estimates for $\varphi=0.1072$ that are larger than a certain frequency $\omega_0$. (b) Long-time growth rate $k_F$ of the CFI $F(t;\varphi,\omega,\epsilon,\eta)$ plotted as a function of the homodyne phase $\varphi$. The blue dot and red square mark the phase values $\varphi=0.1072$ and $\varphi=1$ used in panel (a), respectively. The inset shows the time evolution of the CFI $F$ at these two phase values. All other simulation parameters are the same as those in Fig. \ref{['fig:singletrajs']}.
• Figure 3: (a1)-(c1) Time evolution of the distribution $P_t\equiv P(\tilde{\omega}_\text{ekf}(t))$ for the first three iterations ($i=0,1,2$) of the proposed protocol, obtained with $N_\text{traj}=200$ trajectories of $\tilde{\omega}_\text{ekf}(t)$ for a low detection efficiency $\eta=0.2$ and the prior interval $I_\text{prior} = (0.7\kappa, 2.3\kappa)$. The control parameters $(\epsilon_i, \varphi_i)$ are selected via the rules (\ref{['eq:selectrule']}), using the update function $\mathcal{E}(\epsilon',\epsilon") = (\epsilon' + \epsilon")/2$. The true frequency is $\omega_\text{true}=\kappa$ (white dashed lines). (a2)-(c2) Distributions at time $t^*=300/\kappa$, together with their numerical fits (black dashed lines) to the skew‑normal function in Eq. (\ref{['eq:skn']}). Vertical dashed lines mark $\omega_\text{true}=\kappa$ and gray shaded areas indicate the prior interval $I_\text{prior}$.
• Figure 4: (a) Mean $\mathbb{E}[\omega_\text{est}^{(i)}(t)]$ (symbols) and standard deviation $\text{Std}[\omega_\text{est}^{(i)}(t)]$ (error bars) and (b) the MSE (symbols) of the estimator $\omega_\text{est}^{(i)}$ obtained using the bootstrapping technique for both low ($\eta=0.2$, solid symbols, upper panel) and ideal ($\eta=1$, open symbols, lower panel) detection efficiencies. Here $N_\text{traj}=200$ for each run of the estimation. The control parameters $(\epsilon_i,\varphi_i)$ ($i\in\{0,1,2\}$) are identical to those used in Fig. \ref{['fig:adaptiveEKF']}; the other parameter sets $(\epsilon_i',\varphi_i')$ are determined in Appendix \ref{['sec:sm_ideal']}, with the following values: $(\epsilon_0',\varphi_0') = (0.760\kappa,2.996)$, $(\epsilon_1',\varphi_1') = (0.928\kappa,0.048)$, $(\epsilon_2',\varphi_2') = (1.021\kappa,0.139)$.
• Figure 5: (a1)-(c1) Time evolution of the distribution $P_t\equiv P(\tilde{\omega}_\text{ekf}(t))$ for the first three iterations ($i=0,1,2$) of the proposed protocol, obtained from $N_\text{traj}=500$ trajectories of $\tilde{\omega}_\text{ekf}(t)$. Here, the detection efficiency is $\eta=1$ and the prior interval is $I_\text{prior} = (0.7\kappa, 2.3\kappa)$. The control parameters $(\epsilon_i', \varphi_i')$ are determined according to the rules (\ref{['eq:selectrule']}), using the update function $\mathcal{E}(\epsilon',\epsilon") = (\epsilon' + \epsilon")/2$ and the initial driving amplitude $\epsilon_0' = \epsilon_c(\omega_l=0.7\kappa)-0.1\kappa$. (a2)-(c2) Corresponding distributions at time $t^*=300/\kappa$ with black dashed lines showing numerical fits to the skew-normal distribution in Eq. (\ref{['eq:skn']}). Other simulation parameters are the same as those in Fig. \ref{['fig:adaptiveEKF']}.