Unlocking photodetection for quantum sensing with Bayesian likelihood-free methods and deep learning

TL;DR

The results pave the way for dynamical control of quantum sensors that leverage non-classical effects in photodetection that leverage non-classical effects in photodetection by comparing Bayesian likelihood-free methods with ones based on deep learning.

Abstract

To operate quantum sensors at their quantum limit in real time, it is crucial to identify efficient data inference tools for rapid parameter estimation. In photodetection, the key challenge is the fast interpretation of click-patterns that exhibit non-classical statistics -- the very features responsible for the quantum enhancement of precision. We achieve this goal by comparing Bayesian likelihood-free methods with ones based on deep learning (DL). While the former are more conceptually intuitive, the latter, once trained, provide significantly faster estimates with comparable precision and yield similar predictions of the associated errors, challenging a common misconception that DL lacks such capabilities. We first verify both approaches for an analytically tractable, yet multiparameter, scenario of a two-level system emitting uncorrelated photons. Our main result, however, is the application to a driven nonlinear optomechanical device emitting non-classical light with complex multiclick correlations; in this case, our methods are essential for fast inference and, hence, unlock the possibility of distinguishing different photon statistics in real time. Our results pave the way for dynamical control of quantum sensors that leverage non-classical effects in photodetection.

Figures (11)

• Figure 1: Quantum sensing scheme with photodetection. A quantum system is governed by a Hamiltonian $\hat{H}(\bm{\varphi}, \bm{\mu})$, where $\bm{\varphi}$ and $\bm{\mu}$ are vectors of estimated and control parameters, respectively. The system is driven by an external (classical, blue) field (e.g., strong light-beam) of strength $\Omega$ and dissipates through multiple (quantum, red) channels with rates $\gamma_i$. Photon emission constitutes a separate dissipation process, being further divided into detected and lost channels with rates $\kappa^{\mathrm{d}}_j$ and $\kappa^{\mathrm{l}}_j$, respectively. Upon registering $\ell$ photons, the most accurate estimator of parameters $\bm{\tilde{\varphi}}$ should be constructed based on the recorded photoclick pattern $D_\ell$---a vector of waiting times between consecutive clicks.
• Figure 2: Interpreting photodetection as a temporal point process (TPP). (a) Renewal TPP: Waiting times $X=\{X_1,X_2,\dots,X_\ell\}$ are independently and identically distributed (i.i.d.), sampled from a stationary waiting-time distribution $w(\tau)$; (b) History-dependent TPP: Waiting times are not i.i.d. and cannot be described by a single distribution. As illustrated, specific records may exhibit hierarchical temporal correlations---such as three-click patterns nested within larger five-click sequences. Any such multi-time correlations require a non-renewal framework to account for quantum effects like photon (anti)bunching Paul1982, which are essential for quantum-enhanced sensing tasks.
• Figure 3: Deep learning (DL) inference frameworks considered. The input of a neural network, $X=\{X_1,X_2,\dots,X_\ell\}$, is set to the sequence of inter-event waiting times for a given photoclick pattern $D_\ell$. The output $Y$ is adapted to a specific statistical task: (a) Within the regression framework, the network provides point estimates of the parameter vector $\tilde{\bm{\varphi}}$ by minimising the Mean Squared Error (MSE). (b) In the probabilistic regression framework, the network is assumed to yield a Gaussian posterior; thus, the Negative Log-Likelihood (NLL) is employed as the loss function, with the output providing both the parameter estimates and the corresponding errors covariance matrix, $\Sigma_{\tilde{\bm{\varphi}}}$. (c) In the classification framework, which uses Cross-Entropy (CE) as the loss function, the output directly provides a discrete grid-representation of the posterior distribution.
• Figure 4: Simultaneous estimation of the driving frequency $\Omega$ and the laser detuning $\Delta$ for a two-level atom. We compare overall RMSEs, $\sqrt{\textrm{MSE}[\tilde{\Omega}]+\textrm{MSE}[\tilde{\Delta}]}$, achieved by the DL and ABC methods considered with the true RMSE (red frame) computed analytically, as well as the errors predicted by the methods. In each case a 2D plot of error (RMSE) is shown as a function of estimated parameter values $\Delta,\Omega\in[0,2]$ (in units of $\kappa$). Top row (from left to right): RMSEs attained by the histogram-based regression network, Negative Log-Likelihood (NLL) histogram-based network, CNN classification network and by the ABC method. Bottom row: The true RMSE (in red) evaluated based on the waiting-time distribution followed by (from left to right) the error predicted by the NLL histogram-based network, CNN classification network and by the ABC method.
• Figure 5: Schematic of a generic optomechanical sensor. An external laser drive populates the optical cavity mode, which couples to a mechanical resonator (e.g., a vibrating mirror, a levitated nanoparticle, or a photonic crystal mode) via radiation pressure. This interaction imprints the mechanical dynamics onto the cavity field, which is then probed by monitoring the photons leaking out via photodetection. The resulting photoclick patterns provide the data for our likelihood-free inference methods.