Quantum Limits to Incoherent Imaging are Achieved by Linear Interferometry

We solve the general problem of determining, through imaging, the three-dimensional positions of $N$ weak incoherent pointlike emitters in an arbitrary spatial configuration. We show that a structured measurement strategy in which a passive linear interferometer feeds into an array of photodetectors is always optimal for this estimation problem, in the sense that it saturates the quantum Cramér-Rao bound. We provide a method for the explicit construction of the optimal interferometer. Further explicit results for the quantum Fisher information and the optimal interferometer design that attains it are obtained for the special case of one and two incoherent emitters in the paraxial regime. This work provides insights into the phenomenon of superresolution through incoherent imaging that has attracted much attention recently. Our results will find a wide range of applications over a broad spectrum of frequencies, from fluorescence microscopy to stellar interferometry.

Figure 1

A collection of $N_S$ incoherent pointlike emitters (left) and the apparatus used to measure them (right). The light emitted or scattered by the objects is collected at $N_C$ specific locations in the collection plane. The collected light is coherently processed in a general interferometer and measured using photodetection.

Figure 2

Schematic of two sources with a separation of $\mathrm{\Delta }x$ in the object plane, and a separation $\mathrm{\Delta }z$ in the axial direction. Two collectors at $u_1$ and $u_2$ direct light into a two-mode interferometer consisting of a phase shift $\alpha$ and a $50:50$ beam splitter, followed by two photon counters.

Figure 3

Schematic for estimating simultaneously $\mathrm{\Delta }x$ and $\mathrm{\Delta }z$ using four collectors which are evenly spaced, centered at position 0. The optimal linear optical transformation is a four-mode quantum Fourier transform.