Adaptive, symmetry-informed Bayesian metrology for precise quantum technology measurements

Abstract

High precision measurements are essential to solve major scientific and technological challenges, from gravitational wave detection to healthcare diagnostics. Quantum sensing delivers greater precision, but an in-depth optimisation of measurement procedures has been overlooked. Here we present a systematic strategy for parameter estimation in the low-data limit that integrates experimental control parameters and natural symmetries. The method is guided by a Bayesian quantifier of precision gain, enabling adaptive optimisation tailored to the experiment. We provide general expressions for optimal estimators for any parameter. The strategy's power is demonstrated in a quantum technology experiment, in which ultracold caesium atoms are confined in a micromachined hole in an optical fibre. We find a five-fold reduction in the fractional variance of the estimated parameter, compared to the standard measurement procedure. Equivalently, our strategy achieves a target precision with a third of the data points previously required. Such enhanced device performance and accelerated data collection will be essential for applications in quantum computing, communication, metrology, and the wider quantum technology sector.

Figures (6)

• Figure 1: (a) General adaptive, symmetry-informed, Bayesian estimation strategy. Information gained about system parameter $\theta$ from a detected signal $n_k$ determines a new optimal control parameter $y_k$ that maximises information gain in the next measurement shot, $k+1$. (b) Application to an ensemble of cold Cs atoms confined in an optical dipole trap (red cones) within a microscopic hole (left inset) that intersects the core of an optical fibre. Atom number is probed via photon-absorption detection of light travelling along the fibre. The light's frequency $\nu_k$ is the control parameter to be optimised. The final estimate of atom number is sequentially improved as photon-counts are recorded. The right inset shows an absorption image of the atoms passing through the optical fibre.
• Figure 2: Atom number estimates $\tilde{N}_k$ resulting from measurements using the standard (blue and orange) and Bayesian estimation strategies (red, purple and green) described in the text. Each estimate employs $k=30$ measurement shots. Error bars are given by the standard error for each $30$-shot measurement in the on-resonance and detuned cases and by the variance of Eq. \ref{['eq:marginal-posterior']} otherwise. The lower insets show the progression of the estimate with increasing $k$, for a single measurement run, in each case. The shading denotes the aforementioned errors. The small differences in the mean value obtained with each method are consistent with normal experimental drifts in loading efficiency. The dashed lines indicate final estimates in the insets and overall average across all methods in the main figure. The inset in the top right corner gives a visual Gaussian representation, $P(\tilde{N}_k)$, of the mean and the standard deviation of the data presented in the main body of the figure for each method.
• Figure 3: Noise-to-signal ratio (NSR) versus shot number $k$ for the on-resonance (blue inverted triangles), detuned (orange triangles), on-resonance Bayesian (red squares), a priori Bayesian (purple rhombuses), and adaptive Bayesian (green circles) strategies. The data points are obtained empirically from the distribution of $m$ estimates after each shot number $k$. The dashed line indicates the NSR for on-resonance method with $30$ shots (downward facing arrow), which is achieved by our adaptive protocol with only $9$ shots (upward facing arrow).
• Figure S1: (a) Example of how the optimal measurement frequency $\nu$ changes when the adaptive Bayesian feedback is applied. Three consecutive experimental runs are shown, where the sample has been prepared with an atom number of $N \sim 250$. (b-e) Examples of the gain function $\mathcal{G}_{\nu_k}$ used to find the optimal frequency [Eq. (8) in the main text]. Examples are shown for: (b) the a priori case, $k=1$; (c) high current atom number estimate, $N\sim340$ and $k=4$; (d) low current atom number estimate, $N\sim200$ and $k=4$; (e) at the end of a run, $N\sim250$ and $k=30$. Orange dots show the calculated values, while the blue line is a cubic interpolation.
• Figure S2: Example of an experimental run consisting of 30 measurements, with atoms present in the hole (red circles) and without (blue circles), using resonant light.