Super-resolution of partially coherent bosonic sources

TL;DR

The paper addresses quantum-limited imaging of two partially coherent point sources by deriving the ultimate bounds for estimating their separation $s$, relative intensity $q$, and coherence $\gamma$. It shows that separation can be super-resolved across most parameter regimes using SPADE at the intensity center, with anti-correlated sources offering the largest gains, and that in the sub-Rayleigh limit simple boson counting suffices to estimate $q$ and $\gamma_R$ while enabling joint estimation of $s$. The analysis reduces the problem to a Bloch-vector description in a two-level system, and it reveals frame-dependent aspects of the estimation problem, notably the importance of choosing the intensity-centroid frame for accurate qubit-model results. Indirect estimation via purity is found suboptimal for nonzero coherence and can be ill-defined when $\gamma_R<0$, highlighting the practical relevance of direct quantum-limit measurements for high-precision imaging in partially coherent regimes.

Abstract

We consider the problem of imaging two partially coherent sources and derive the ultimate quantum limits for estimating all relevant parameters, namely their separation, relative intensity, as well as their coherence factor. We show that the separation of the two sources can be super-resolved over the entire range of all other pertinent parameters (with the exception of fully coherent sources), with anti-correlated sources furnishing the largest possible gain in estimation precision, using a binary spatial mode demultiplexing measurement positioned at the center of intensity of the joint point spread function for the two sources. In the sub-Rayleigh limit, we show that both the relative intensity, as well as the real part of the coherence factor, can be optimally estimated by a simple boson counting measurement, making it possible to optimally estimate the separation, relative intensity and real coherence factor of the sources simultaneously. Within the same limit, we show that the imaging problem can be effectively reduced to one where all relevant parameters are encoded in the Bloch vector of a two-dimensional system. Using such a model we find that indirect estimation schemes, which attempt to extract estimates of the separation of the two sources by measuring the purity of the corresponding state of the two-level system, yield suboptimal estimation precision for all non-zero values of the coherence factor.

Figures (9)

• Figure 1: The diagonal entries of the lower bound of Eq. (\ref{['eq:QBMSE']}) for $\xi(\boldsymbol{\theta})$ versus the separation between the sources $s/\sigma$. The plots are done for $q=1/2$ and several real values of $\gamma$, as shown by the legend. The $y$-axis is in units of $(\delta/4\sigma^2)$ with $\delta=10^{-2}$.
• Figure 2: QFI matrix diagonal elements (of $\xi(\bm\theta)$) versus the relative intensity of the sources $q$, in the limit $s/\sigma\ll 1$. The plots are done relative to the geometric frame of reference (i.e. $\alpha=1/2$) for several real values of $\gamma$, as shown by the legend. The QFI is in units of $(\delta/4\sigma^2)$ with $\delta=10^{-2}$.
• Figure 3: QFI matrix diagonal element (of $\xi(\bm\theta)$) for the separation versus the relative intensity, in the limit $s\to0$. The plots are done relative to the intensity centroid frame of reference for several values of $\gamma$, as shown by the legend. The QFI is in units of $(\delta/4\sigma^2)$, with $\delta=10^{-2}$.
• Figure 4: Comparison between the true QFI for $s$ measured from the intensity centroid (solid blue line) and the one measured from the $\alpha=0.7$ frame (orange dashed line). For this particular example we consider $q=0.75$ the true value of the relative intensity, and $\hat{q}=q+\epsilon(q)=\alpha=0.7$ the estimate for which we set the reference frame. The true QFI for $s$ at $q=0.75$ is marked by the green square, whereas the perceived QFI from the $\alpha=\hat{q}=0.7$ reference frame is marked by the blue triangle. For $\gamma = 0$ or $q = 1/2$ the difference between the real and perceived QFIs is of order $\mathcal{O}(\epsilon^2(q))$ (left plot), otherwise their difference is of order $\mathcal{O}(\epsilon(q))$ (right plot).
• Figure 5: Qubit and extra contributions to the diagonal element corresponding to the separation for the lower bound of Eq. (\ref{['eq:QBMSE']}). As in Fig. \ref{['fig:QFI_function_s']}, the plots are done for $q=1/2$ and several real values of $\gamma$, as denoted by the legend. The value of the lower bound is expressed in units of $(\delta/4\sigma^2)$ with $\delta=10^{-2}$ and the separation in units of $\sigma$.