Recent theoretical and experimental work has shown that the quantum Fisher information associated with estimating the separation between two optical point sources remains finite at small separations, effectively opening new routes to super-resolution imaging of simultaneously emitting sources. Most studies to date, however, implicitly invoke the scalar approximation, which is not appropriate in the context of high-numerical-aperture microscopy. Utilizing parameter estimation theory, here we consider the estimation of separation between two closely spaced dipole emitters, a commonly employed model for single-molecule optical beacons. We consider two limiting cases: one in which the orientations of the emitters are fixed and equal, and another in which both dipoles freely sample all of orientation space over the course of the measurement. We quantify precision limits using quantum and classical variants of the Fisher information and Cramér-Rao bound. In all cases, the vectorial nature of the emission complicates the analyses, but with appropriate filtering of the collected light in the azimuthal-radial polarization basis, a previously proposed scheme to saturate the quantum Fisher information via image inversion interferometry can be salvaged.
Recent theoretical and experimental work has shown that the quantum Fisher information associated with estimating the separation between two optical point sources remains finite at small separations, effectively opening new routes to super-resolution imaging of simultaneously emitting sources. Most studies to date, however, implicitly invoke the scalar approximation, which is not appropriate in the context of high-numerical-aperture microscopy. Utilizing parameter estimation theory, here we consider the estimation of separation between two closely spaced dipole emitters, a commonly employed model for single-molecule optical beacons. We consider two limiting cases: one in which the orientations of the emitters are fixed and equal, and another in which both dipoles freely sample all of orientation space over the course of the measurement. We quantify precision limits using quantum and classical variants of the Fisher information and Cramér-Rao bound. In all cases, the vectorial nature of the emission complicates the analyses, but with appropriate filtering of the collected light in the azimuthal-radial polarization basis, a previously proposed scheme to saturate the quantum Fisher information via image inversion interferometry can be salvaged.
According to Rayleigh's criterion, two incoherent emitters with a separation below the diffraction limit are not resolvable with a conventional fluorescence microscope. One method of superresolution microscopy (SRM) circumvents the diffraction-limited resolution by precisely estimating the position of spatiotemporally independent emitters. However, these methods of SRM techniques are not optimal for estimating the separation of two simultaneously excited emitters. Recently, a number of detection methods based on modal imaging have been developed to achieve the quantum Cramér-Rao lower bound (QCRB) to estimate the separations between two nearby emitters. The QCRB determines the minimum achievable precision for all possible detection methods. Current modal imaging techniques assume a scalar field generated from a point source, such as a distant source from an optical fiber or a pinhole. However, for fluorescently labeled samples, point emitters are single fluorophores that are modeled as dipole emitters and, in practice, are often freely rotating. Dipole radiation must be described by vectorial theory, and the assumption of a scalar field no longer holds. Here, we present a method to numerically calculate the QCRB for measuring the separation of two dipole emitters, incorporating the vectorial theory. Furthermore, we propose a near-quantum optimal detection scheme based on one of the modal imaging techniques, superlocalization by image inversion interferometry (SLIVER), for estimating the separation of two freely rotating dipoles. In the proposed method, we introduce a vortex wave plate before the SLIVER detection to separate the radial and azimuthal components of the dipole radiation. With numerical simulations, we demonstrated that our method achieves nondivergent precision at any separation between two dipole emitters. We investigated several practical effects relevant to experimental measurements in superresolution microscopy, including numerical aperture, detection bandwidth, number of estimation parameters, background, and misalignment on separation estimation. Our proposed measurement provides a near-quantum-limited detection scheme for measuring the separation of two freely rotating dipole emitters, such as fluorescently tagged molecules, which are commonly used in superresolution microscopy.
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.
Precisely measuring molecular orientation is key to understanding how molecules organize and interact in soft matter, but the maximum theoretical limit of measurement precision has yet to be quantified. We use quantum estimation theory and Fisher information (QFI) to derive a fundamental bound on the precision of estimating the orientations of rotationally fixed molecules. While direct imaging of the microscope pupil achieves the quantum bound, it is not compatible with wide-field imaging, so we propose an interferometric imaging system that also achieves QFI-limited measurement precision. Extending our analysis to rotationally diffusing molecules, we derive conditions that enable a subset of second-order dipole orientation moments to be measured with quantum-limited precision. Interestingly, we find that no existing techniques can measure all second moments simultaneously with QFI-limited precision; there exists a fundamental trade-off between precisely measuring the mean orientation of a molecule versus its wobble. This theoretical analysis provides crucial insight for optimizing the design of orientation-sensitive imaging systems.
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.
We report the experimental super-resolution of pairs of point-like fluorescent sources using a modified image inversion interferometer microscope. The technique is inspired by recent developments in the application of quantum parameter estimation theory to semiclassical imaging problems. We find that the image inversion technique requires special polarization filtering to account for the dipolar nature of the emission. Using an azimuthal polarizer, we obtain improvements in the Fisher information of source separation by over an order of magnitude relative to direct imaging. Unlike established super-resolution fluorescence techniques, the method does not require sequential photoswitching/blinking of the fluorophores, and thus could facilitate significant speed-ups for certain biological imaging/tracking tasks. The authors propose and experimentally demonstrate a new route to passive super-resolution fluorescence microscopy inspired by quantum parameter estimation theory, which does not require sequential photoswitching of the fluorophores.
Telescope systems with distributed apertures are a well-established approach for boosting resolution in astronomical imaging. However, theoretical limits on quantitative imaging precision, and the fundamentally best possible mode-sorting, beam-combining, and detection schemes to use with such arrays, remain largely unexplored. Using mathematical tools of the quantum and classical Crámer-Rao bounds, we perform analyses showing the fundamental origins of the enhancement provided by distributed imaging systems, over and above a single monolithic telescope, and consider the precision with which one can estimate any desired parameter embedded in a scene's incoherent radiation with a multi-aperture imaging system. We show how quantum-optimal measurements can be realized via beam-combination strategies of two classes: (1) , where light from different apertures is directed to a common focal plane, e.g., of a segmented-aperture telescope; and (2) , where light collected at each aperture, e.g., telescope sites of a long-baseline array, is routed to an optical interferometer. As an example, we show an explicit calculation of the quantum Fisher information (QFI) for estimating the angular separation between two-point emitters using two identical apertures separated by a baseline distance. We show that this QFI splits instructively into additive contributions from the single apertures and from the baseline. We quantify the relative benefits of intratelescope (e.g., spatial-mode) optical processing and intertelescope beam combination. We show how both receiver designs can be used to capture both sources of information and discuss how similar methods could be extended to more general imaging tasks. We discuss translating QFI-attaining measurements to explicit receiver designs, and the use of preshared entanglement to achieve the QFI when it is impractical to colocate and combine light collected by the apertures. Published by the American Physical Society 2024