Experimental Evidence-Based Sub-Rayleigh Source Discrimination

Abstract

We propose a Bayesian evidence-based inference framework based on relative belief ratios and apply it to discriminating between one and two incoherent optical point sources using spatial-mode demultiplexing (SPADE). Unlike the Helstrom measurement, SPADE require no collective detection and its optimal for asymptotically large samples. Our method avoids ad hoc statistical constructs and relies solely on the information contained in the data, with all assumptions entering only through the likelihood model and prior beliefs. Using experimental evidence, we demonstrate the superior resolving performance of SPADE over direct imaging from a new and extensible perspective; one that naturally generalizes to multiple sources and offers a practical robust approach to analyzing quantum-enhanced superresolution.

Figures (7)

• Figure 1: Schematic representation of the measurement setup. See the text for a detailed description.
• Figure 2: (a) Two individual sources (src 1 and src 2), located at positions not aligned with the demultiplexer, illuminate the scene either individually or simultaneously (combined, with brightness ratio $q=1/2$). The source centroids $x_c$ for all three cases are marked. (b) RB-based source discrimination for data from all three illumination cases. For the two-source hypothesis, all centroid positions $x_{c}$ within a sufficiently broad interval and all separations $d \neq 0$ are included, while brightness imbalance $q$ within a tolerance $q_{0}$ around 0 or 1 is attributed to the corresponding single-source hypotheses, while the mutually exclusive interval is assigned to the two-source hypothesis. With priors of $1/2$ for the two-source hypothesis and $1/4$ for each single-source hypothesis, exactly one hypothesis per panel yields $\mathrm{RB} >1$, with posterior probability near unity. The Rayleigh limit for these sources is approximately $326.57~\mathrm{\mu m}$, indicating that the results exhibit superresolution. The results shown remain unchanged if the 10000 data samples are partitioned into sets of 50 and RB analysis is performed on each set independently, indicating extremely small variance in the results of the analysis. The results, however, depend on the choice of nuisance parameter space, which is arbitrary.
• Figure 3: Sensitivity of RB in discriminating 'one-source’ and 'two-sources’ hypotheses. Variances of the mode-sorted intensities from the experimental data with $x_{c} = -1.06~\mathrm{\mu m}$ and $q = 0.501$, together with models generated from the experimental calibration data, were used to simulate 1000 experiments of two sources separated by distance $d$. For each simulated experiment, the hypothesis with the maximum RB was selected. The 'two-sources' success probability is plotted against the source separation using simulated experiments (blue) for (a) known fixed $x_c$ and $q$, or (b) averaging over their unknown values. The expectation from theory is shown in red for the simpler problem in (a). In both cases, superresolution is achieved with $d_c \ll 326.57~\mathrm{\mu m}$ and error probability of $\epsilon < 10^{-3}$.
• Figure 4: Parameter estimation using data-driven evidence from RB for equally bright $(q=0.5)$ sources and varying either (a) the source separation $d$, (b) the centroid position $x_\mathrm{c}$ in a $4~\mathrm{\mu m}$ interval with 5000 hypotheses using uniform priors, or (c,d) both in a $1~\mathrm{\mu m} \times 2~\mathrm{\mu m}$ region with $100 \times 100$ hypotheses around the parameter values determined for the experimental setup (true), $d = 19.63~\mathrm{\mu m}$ and $x_\mathrm{c} = -1.06~\mathrm{\mu m}$ using uniform priors and Gaussian priors with variances of $0.5~\mathrm{\mu m}^2$ and $0.125~\mathrm{\mu m}^2$ centered at the true values and randomly elsewhere respectively where the dashed contours represent their standard deviation. All priors are normalized over the depicted space. All 10000 intensity readings were used in (c), whereas 10 were randomly selected in (d) to simulate a small sampling regime.
• Figure 5: Sensitivity of RB in discriminating 'one-source’ and 'two-sources’ hypotheses. The parameters are identical to those in Fig. 3 of the main text. The simulated data and the theoretical model use calibration for only one source and assume the sources to be truly indistinguishable. As they match, we also see good agreement in the success probability of identifying the 'two-sources' hypothesis, and the critical distance criterion can be used.