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arXiv:2211.16395 · Physical Review A

Quantum limits on localizing point objects against a uniformly bright disk

Sudhakar Prasad

Published Mar 31, 2023• 3 citations
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Abstract

We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the two-dimensional location of a point source against a uniformly bright disk of known center and radius in the ideal photon-counting limit. We present both a perturbative calculation of the QFI in powers of the background-to-source brightness ratio and a numerically exact calculation of the QFI in the eigenbasis of the one-photon density operator. A related problem of the quantum limit on estimating the location of a small-area brightness hole in an otherwise uniformly bright disk, a problem of potential interest to the extrasolar planet detection community, is also treated perturbatively in powers of the ratio of the areas of the hole and the background disk. We then numerically evaluate the Cram\'er-Rao lower bound (CRB) for wavefront projections in three separate bases, those comprising Zernike, Fourier-Bessel, and localized point-source modes, for unbiased estimation of the two position coordinates of the point source and of the brightness hole center, respectively, for the two problems. By comparing these CRBs with the corresponding quantum-limited minimum error variances, given by inverting the QFI matrix, and with the CRBs associated with direct imaging, we assess the maximum efficiency of these wavefront projections in performing such estimations.

DOI
10.1103/physreva.107.032427

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We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the two-dimensional location of a point source against a uniformly bright disk of known center and radius in the ideal photon-counting limit. We present both a perturbative calculation of the QFI in powers of the background-to-source brightness ratio and a numerically exact calculation of the QFI in the eigenbasis of the one-photon density operator. A related problem of the quantum limit on estimating the location of a small-area brightness hole in an otherwise uniformly bright disk, a problem of potential interest to the extrasolar planet detection community, is also treated perturbatively in powers of the ratio of the areas of the hole and the background disk. We then numerically evaluate the Cram'er-Rao lower bound (CRB) for wavefront projections in three separate bases, those comprising Zernike, Fourier-Bessel, and localized point-source modes, for unbiased estimation of the two position coordinates of the point source and of the brightness hole center, respectively, for the two problems. By comparing these CRBs with the corresponding quantum-limited minimum error variances, given by inverting the QFI matrix, and with the CRBs associated with direct imaging, we assess the maximum efficiency of these wavefront projections in performing such estimations.
We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the length of a uniformly bright incoherent line source with a fixed midpoint and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wave functions (PSWFs) on a centered line segment and their generalized versions on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we calculate QFI with respect to the spatial parameters of the two sources. Zernike polynomials provide a good set into which to project the full source wavefront, and such classical wavefront projection data can realize quantum limited estimation error bound in each case. We subsequently generalize our approach to analyze sources of arbitrary brightness distributions and shapes using a certain class of Bessel Fourier functions that are closely related to the PSWFs. We illustrate the general approach by computing QFI for estimating the lengths of the principal axes of a uniformly bright, centered elliptical disk.
We investigate our ability to determine the mean position, or centroid, of a linear array of equally bright incoherent point sources of light whose continuum limit is the problem of estimating the center of a uniformly radiating object. We consider two receivers, an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) spatial-mode demultiplexing in the image plane, prior to shot-noise-limited photon detection. We compare the Fisher information (FI) for estimating the centroid achieved by these two receivers, which quantifies the information-accrual rate per photon, and compare those with the quantum Fisher information (QFI): the maximum attainable FI by any choice of measurement on the collected light allowed by physics. We find that focal-plane direct imaging is strictly suboptimal, although not by a large margin. We also find that not only is the HG mode sorter, which is the optimal measurement for estimating the separation between point sources (or the length of a line object), suboptimal, but it performs worse than direct imaging. We study the scaling behavior of the QFI and direct imaging's FI for a continuous uniformly bright object in terms of its length and find that both are inversely proportional to the object's length when it is sufficiently larger than the Rayleigh length. Finally, we propose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.
As a method to extract information from optical systems, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramér-Rao bound (QCRB). Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited optical field within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) in about one emitter’s position is independent on its true value. We calculate the QFIm of two unequal-brightness emitters’ relative positions and intensities; the results show that only when the relative intensity and centroids of two-point sources, including longitudinal and transverse directions, are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide use in applications from microscopy to astrometry.
Optically localizing a single quasimonochromatic source to subdiffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width www of the optical system's point-spread function (PSF) and the sought localization precision, ddd, i.e., as $\ensuremath{\alpha}{(w/d)}^{2}$. For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient $\ensuremath{\alpha}$ must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly $\ensuremath{\alpha}$ must increase with increasing width of a flat-top spectral profile of emission of a point source being localized in two dimensions by an imager with a clear circular aperture by calculating the quantum Fisher information (QFI), whose inverse yields the lowest possible unbiased-estimation variance of source-localization error. The novel use of prolate spheroidal wave functions as a basis for obtaining a solution of the eigenvalue problem of the single-photon density operator needed for the QFI calculation helps us develop the notion of an effective dimensionality of the continuous-state problem in terms of the associated space-bandwidth parameter. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.
The error in estimating the separation of a pair of incoherent sources from radiation emitted by them and subsequently captured by an imager is fundamentally bounded below by the inverse of the corresponding quantum Fisher information (QFI) matrix. We calculate the QFI for estimating the full three-dimensional pair separation vector, extending previous work on pair separation in one and two dimensions. We also show that the pair-separation QFI is, in fact, identical to source localization QFI, which underscores the fundamental importance of photon-state localization in determining the ultimate estimation-theoretic bound for both problems. We also propose general coherent-projection bases that can attain the QFI in two special cases. We present simulations of an approximate experimental realization of such quantum limited pair superresolution using the Zernike basis, confirming the achievability of the QFI bounds.
In recent years quantum metrology and quantum sensing have enabled the advancement of quantum technologies and research in fundamental physics. Through statistical analysis, in the context of parameter estimation, the amount of information about a parameter of interest encoded in a quantum state can be quantified. In this thesis we focus on the application of quantum estimation theory to imaging and spectroscopy with quantum light. We study the fundamental bound of the mean square error for an unbiased estimator in terms of the quantum Fisher information (QFI) for two problems, one in microscopy and one in spectroscopy. Firstly, we study the problem of localising multiple point sources below the diffraction limit. We show that localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. We obtain the QFI matrix elements analytically. In the regime of arbitrarily small separations we find it to be no more than rank two – implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits. Secondly, we study the estimation of the electric dipole moment (EDM) of a two-level atom through its interaction with quantum pulses of light in free space. We derive analytical expressions for the states of one-photon wavepackets and entangled photon pairs (EPP). We numerically calculate the QFI for different states of light, including coherent and squeezed pulses, and compare their performance in estimating the EDM. We find that the one-photon wavepacket and the EPP have similar performances, while the performance of the entangled photon pair is not improved by increasing the entanglement of the EPP. Our results indicate that using Fock states to estimate the EDM of a two-level atom is preferable to using entangled light.