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Physical Review Research

Resolving mutually-coherent point sources of light with arbitrary statistics

Ilya Karuseichyk, Giacomo Sorelli, Mattia Walschaers, Nicolas Treps, Manuel Gessner

Published Oct 7, 2022• 14 citations
Publisher

Abstract

We analyze the problem of resolving two mutually coherent point sources with arbitrary quantum statistics, mutual phase, and relative and absolute intensity. We use a sensitivity measure based on the method of moments and compare direct imaging with spatial-mode demultiplexing (SPADE), analytically proving advantage of the latter. We show that the moment-based sensitivity of SPADE saturates the quantum Fisher information for all known cases, even for non-Gaussian states of the sources.

DOI
10.1103/physrevresearch.4.043010

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We analyze the problem of resolving two mutually coherent point sources with arbitrary quantum statistics, mutual phase, and relative and absolute intensity. We use a sensitivity measure based on the method of moments and compare direct imaging with spatial-mode demultiplexing (SPADE), analytically proving advantage of the latter. We show that the moment-based sensitivity of SPADE saturates the quantum Fisher information for all known cases, even for non-Gaussian states of the sources.
We analyze the problem of resolving two point-sources in the case of mutually coherent sources with arbitrary quantum statistics, mutual phase, relative and absolute intensity. We use a sensitivity measure based on the method of moments and compare direct imaging with spatial mode demultiplexing, analytically proving the advantage of the latter. We show that the sensitivity of spatial mode demultiplexing saturates the quantum Fisher information, for all known cases, even for non-Gaussian states of the sources.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works, analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. However, most previous works have either consider low-intensity, or thermal sources. To broaden the applicability of this approach, it is important to extend these results to more general light sources. To this goal, we will present an analytical expression for the Quantum Fisher Information, determining the ultimate resolution limit, for the separation between two sources in an arbitrary Gaussian state. Applying this result to different quantum states, we can shine some light on some relevant questions. We can for example explore the role of partial coherence considering displaced and correlated thermal states, or investigate the importance of quantum correlations by considering squeezed light. In addition to the ultimate quantum limit, we will discuss a simple estimation technique, requiring access only to the mean value of a linear combination of demultiplexed intensity measurements, which is often sufficient to saturate these limits, and can easily be adapted to incorporate the most common noise sources. Finally, we will present our experimental setup that allows for the generation of the images of two sources with different photon statistics, as well as for spatial mode demultiplexing and we will discuss the first practical implementations if the above mentioned estimation techniques.
Spatial-mode demultiplexing (SPADE) has recently been adopted to measure the separation in the transverse plane between two incoherent point-like sources with sub-wavelength separation. It has been argued that this approach may yield extraordinary performances in the photon-counting regime. Here, we explore SPADE as a tool for precision measurements in the regime of bright, incoherent sources. First we analyze the general problem of estimating the second moments of the source's intensity distribution, for an extended incoherent source of any shape. Our theory predicts a substantial improvement in signal-to-noise ratio (SNR) of SPADE over direct imaging in the sub-wavelength regime. Second, we present an experimental application of SPADE to the case of two point-like, bright sources. We demonstrate the use of this setup for the estimation of the transverse separation and for the estimation of the relative intensity, confirming the expected improvement in SNR.
I present a semiclassical analysis of a spatial-mode demultiplexing (SPADE) measurement scheme for far-field incoherent optical imaging under the effects of diffraction and photon shot noise. Building on previous results that assume two point sources or the Gaussian point-spread function, I generalize SPADE for a larger class of point-spread functions and evaluate its errors in estimating the moments of an arbitrary subdiffraction object. Compared with the limits to direct imaging set by the Cram\'er-Rao bounds, the results show that SPADE can offer far superior accuracy in estimating second- and higher-order moments.
I propose a spatial-mode demultiplexing (SPADE) measurement scheme for the far-field imaging of spatially incoherent optical sources. For any object too small to be resolved by direct imaging under the diffraction limit, I show that SPADE can estimate its second or higher moments much more precisely than direct imaging can fundamentally do in the presence of photon shot noise. I also prove that SPADE can approach the optimal precision allowed by quantum mechanics in estimating the location and scale parameters of a subdiffraction object. Realizable with far-field linear optics and photon counting, SPADE is expected to find applications in both fluorescence microscopy and astronomy.
Recent works employing tools from quantum optics and quantum metrology proposed a new passive imaging technique that allows to resolve details far below the diffraction limit. This technique is based on replacing standard spatially-resolved intensity measurements, e.g. at each pixel of a camera, with spatial-mode demultiplexing (SpaDe) measurements that allow to acquire information in a more efficient way. In this contribution, we want to provide an intuitive explanation of why such a SpaDe approach is so effective, and illustrate how we used these idea to discriminate one point source from two, and to estimate the separation between two incoherent sources.