Multiparameter quantum metrology of incoherent point sources: Towards realistic superresolution

Abstract

We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost important in realistic scenarios, such as observational astronomy.The time-honored Rayleigh criterion [1] specifies the minimum separation between two incoherent optical sources using a linear imaging system. As a matter of fact, it is the size of the point spread function [2] that determines the resolution: two points closer than the PSF width will be difficult to resolve due to the substantial overlap of their images.Thus far, this Rayleigh criterion has been considered as a fundamental limit. Resolution can only be improved either by reducing the wavelength or by building higher numericalaperture optics, thereby making the PSF narrower. Nonetheless, outstanding methods have been developed lately that can break the Rayleigh limit under special circumstances [3][4][5][6][7][8][9][10][11][12]. Though promising, these techniques are involved and require careful control of the source, which is not always possible, especially in astronomical applications.Despite being very intuitive, the common derivation of the Rayleigh limit is heuristic and it is deeply rooted in classical optical technology [13]. Recently, inspired by ideas of quantum information, Tsang and coworkers [14][15][16][17] have revisited this problem using the Fisher information and the associated Cramér-Rao lower bound (CRLB) to quantify how well the separation between two point sources can be estimated. When only the intensity at the image is measured (the basis of all the conventional techniques), the Fisher information falls to zero as the separation between the sources decreases and the CRLB diverges accordingly; this is known as the Rayleigh curse [14]. However, when the Fisher information of the complete field is calculated, it stays constant and so does the CRLB, revealing that the Rayleigh limit is not essential to the problem.These remarkable predictions prompted a series of experimental implementations [18][19][20] and further generalizations [21][22][23][24][25], including the related question of source localization [26][27][28]. All this previous work has focused on the estimation of the separation, taking for granted a highly symmetric configuration with identical sources. In this Letter, we approach the issue in a more realistic scenario, where both sources may have unequal intensities. This involves the si...

Realistic sub-Rayleigh imaging with phase-sensitive measurements

“…First, we see in figure 5(a) that the error when estimating δ diverges at low separations when using either IPC or adapted SPLICE. This was to be expected since it is now known that the quantum CRB for estimating δ between unequal intensity emitters diverges at low δ [20,21]. However, we also see that the unadapted SPLICE measurement, which assumes equal intensity emitters, does surprisingly well in comparison.…”

“…Until recently these works have assumed the intensities of the two point sources to be equal. It has been shown [20,21] that when this assumption is relaxed-the two sources have unknown, disparate intensitiesthat the quantum Cramér-Rao bound (CRB) for an unbiased estimate of the separation diverges as separation decreases below the Rayleigh limit 5 . However, this divergence happens quadratically slower than the CRB for IPC, suggesting that measurement schemes exist which still give an advantage over directly imaging the sources.…”

Resolving starlight: a quantum perspective

“…Reháček and co-workers studied the quantum limits and the optimal measurements for two point sources with unequal brightnesses in Refs. [59,60]. They found that, while significant enhancements over direct imaging remain possible, the performance gets worse for unequal sources.…”

Quantum Fisher information with coherence

“…As shown in [4], however, the model used in [2] is not robust with respect to the inclusion of other parameters. When the intensities of the bright spots are considered as estimated parameters, together with the separation and the centroid, the QFI remains constant only if the two intensities are equal, but it drops to zero for unequal intensities.…”