The Rayleigh criterion has long served as a fundamental limit for the resolution of optical imaging. Recent advances in multiparameter quantum metrology have led to quantum superresolution that can break this limit and achieve nonvanishing precision in estimating the separation between a pair of closely located incoherent point sources. For two-dimensional optical systems, the quantum superresolution has been studied for the Cartesian components of separation between two incoherent point sources. However, the precision limit of estimating the full distance between two point sources remains unknown so far. In this paper, we investigate the estimation precision of the full distance between two incoherent point sources with arbitrary intensities in a two-dimensional imaging system. Through the multiparameter quantum estimation theory, we obtain the ultimate estimation precision for the distance and show that it remains nonzero when the distance approaches zero, which surpasses the Rayleigh criterion. We further show the dependence of the estimation precision on the relative orientation between the two point sources, which leads to a novel scheme that can enhance the precision by aligning the sources along proper directions if the point-spread functions are not circularly symmetric, and the enhancement is determined by the extent to which the point-spread functions deviate from circular symmetry. Finally, the results are illustrated by incoherent sources with Gaussian point-spread functions.
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 (3D) 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 ult…
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 alinear 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 …
Quantum superresolution and noise spectroscopy with quantum computing
Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）
Rayleigh's criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century. In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern farfield superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision. Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measuremen…
Using a semiclassical model of photodetection with Poissonian noise and insights from quantum metrology, we prove that linear optics and photon counting can optimally estimate the separation between two incoherent point sources without regard to Rayleigh's criterion. The model is applicable to weak thermal or fluorescent sources as well as lasers.
We construct optimal measurements, achieving the ultimate precision predicted by quantum theory, for the simultaneous estimation of centroid, separation, and relative intensities of two incoherent point sources using a linear optical system. We discuss the physical feasibility of the scheme, which could pave the way for future practical implementations of quantum inspired imaging.
Recent works identified resolution limits for the distance between incoherent point sources. However, it remains unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cramér-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, crosstalk and detector noise.
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 mid-point and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wavefunctions (PSWFs) on a centered line segment and its generalized version on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we subsequently calculate QFI with respect to the spatial parameters of the two sources. Zernik…
Analyses based on quantum metrology have shown that the ability to localize the positions of two incoherent point sources can be significantly enhanced through the use of mode sorting. Here we theoretically and experimentally investigate the effect of partial coherence on the sub-diffraction limit localization of two sources based on parity sorting. With the prior information of a negative and real-valued degree of coherence, higher Fisher information is obtained than that for the incoherent case. Our results pave the way to clarifying the role of coherence in quantum limited metrology.
The Rayleigh limit has so far applied to all microscopy techniques that rely on linear optical interaction and detection in the far field. Here we demonstrate that detecting the light emitted by an object in higher-order transverse electromagnetic modes (TEMs) can help achieving sub-Rayleigh precision for a variety of microscopy-related tasks. Using optical heterodyne detection in TEM01, we measure the position of coherently and incoherently emitting objects to within 0.0015 and 0.012 of the Rayleigh limit, respectively, and determine the distance between two incoherently emitting slits positi…
Sensitivity limits are usually determined using the Cramér-Rao bound. Recently this approach has been used to obtain the ultimate resolution limit for the estimation of the separation between two incoherent point sources. However, methods that saturate these resolution limits, usually require the full measurement statistics, which can be challenging to access. In this work, we introduce a simple superresolution protocol to estimate the separation between two thermal sources which relies only on the average value of a single accessible observable. We show how optimal observables for this techni…
Improving axial resolution is crucial for three-dimensional optical imaging systems. Here we present a scheme of axial superresolution for two incoherent point sources based on spatial mode demultiplexing. A radial mode sorter is used to losslessly decompose the optical fields into a radial mode basis set to extract the phase information associated with the axial positions of the point sources. We show theoretically and experimentally that, in the limit of a zero axial separation, our scheme allows for reaching the quantum Cramér-Rao lower bound and thus can be considered as one of the optimal…
Overcoming frequency resolution limits using a solid-state spin quantum sensor
Ulm University（乌尔姆大学）；Huazhong University of Science and Technology（华中科技大学）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；AWS Center for Quantum Computing（AWS量子计算中心）
A recent paper [Z. Yu and S. Prasad, "Quantum limited superresolution of an incoherent source pair in three dimensions," arXiv:1805.09227v2 [quant-ph] (2018)] considered the problem of quantum limited estimation of the separation vector for a pair of incoherent point sources in all three dimensions. Here we extend our analysis to treat the problem of simultaneously estimating the location of the centroid and the separation of the source pair, which is equivalent to localizing both sources simultaneously. We first calculate the quantum Fisher information for simultaneous pair centroid-separatio…
We investigate the localization of two incoherent point sources with arbitrary angular and axial separations in the paraxial approximation. By using quantum metrology techniques, we show that a simultaneous estimation of the two separations is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the sub-wavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the po…
I point out the mathematical correspondence between an incoherent imaging model proposed by my group in the study of quantum-inspired superresolution [Tsang, Nair, and Lu, Physical Review X 6, 031033 (2016)] and a noise spectroscopy model also proposed by us [Tsang and Nair, Physical Review A 86, 042115 (2012); Ng et al., Physical Review A 93, 042121 (2016)]. Both can be regarded as random displacement models, where the probability measure for the random displacement depends on unknown parameters. The spatial-mode demultiplexing (SPADE) method proposed for imaging is analogous to the spectral …
This paper extends our previous quantum Fisher information (QFI) based analysis of the problem of separating a pair of equal-brightness incoherent point sources in three dimensions to the case of a pair of sources that are unequally bright. When the pair's geometric center is perfectly known in advance, QFI with respect to the estimation of the three separation coordinates remains independent of the degree of brightness asymmetry. For the experimentally more relevant case of perfect prior knowledge of the pair's brightness centroid, however, such QFI becomes dependent on the pair separation ve…
Passive optical superresolution at the quantum limit
University of Oxford（牛津大学）；RTX BBN Technologies（RTX BBN技术公司）；University of Maryland, College Park（马里兰大学帕克分校）；National University of Singapore（新加坡国立大学）；University of Nottingham（诺丁汉大学）；Sorbonne Université（索邦大学）
Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a one- or two…
Superoscillations, i.e., the phenomenon that a bandlimited function can temporary oscillate faster than its highest Fourier component, are being much discussed for their potential for `superresolution' beyond the diffraction limit. Here, we consider systems that are driven with a time dependence that is off-resonance for the system, in the Fourier sense. We show that superoscillating sources can temporarily induce resonance during the period when the source is behaving superoscillatory. This observation poses the question as to how the system `undoes' the `false resonance' after the full sourc…
We study a two-level system (atom, superconducting qubit or quantum dot) strongly coupled to the single photonic mode of a cavity, in the presence of incoherent pumping and including detuning and dephasing. This system displays a striking quantum to classical transition. On the grounds of several approximations that reproduce to various degrees exact results obtained numerically, we separate five regimes of operations, that we term "linear", "quantum", "lasing", "quenching" and "thermal". In the fully quantized picture, the lasing regime arises as a condensation of dressed states and manifests…
An important topic of interest in imaging is the construction of protocols that are not diffraction limited. This can be achieved in a variety of ways, including classical superresolution techniques or quantum entanglement-based protocols. Here, we consider superresolving imaging in the far field using higher-order intensity correlations. We show that third and fourth order correlations can improve upon the first and second order correlations that are traditionally used in classical optics and Hanbury Brown and Twiss type experiments. The improvement is achieved entirely by post-processing of …
Many quantum mechanical experiments can be viewed as multi-round interactive protocols between known quantum circuits and an unknown quantum process. Fully quantum "coherent" access to the unknown process is known to provide an advantage in many discrimination tasks compared to when only incoherent access is permitted, but it is unclear if this advantage persists when the process is noisy. Here, we show that a quantum advantage can be maintained when distinguishing between two noisy single qubit rotation channels. Numerical and analytical calculations reveal a distinct transition between optim…
Spatially resolving two incoherent point sources whose separation is well below the diffraction limit dictated by classical optics has recently been shown possible using techniques that decompose the incoming radiation into orthogonal transverse modes. Such a demultiplexing procedure, however, must be perfectly calibrated to the transverse profile of the incoming light as any misalignment of the modes effectively restores the diffraction limit for small source separations. We study by how much can one mitigate such an effect at the level of measurement which, after being imperfectly demultiple…
Recently, Chen \em et al\rm.\ [New J. Phys. {\bf 13} (2011) 083018] presented experimental results, accompanied by quantum-mechanical analysis, showing that the quantum interference behavior of Bell states could be simulated in a modified Mach-Zehnder interferometer whose inputs are pseudothermal light beams obtained by passing laser light through a rotating ground-glass diffuser. Their experiments and their theory presumed low-flux operation in which the simulated quantum interference is observed via photon-coincidence counting. We first show that the Chen \em et al\rm.\ photon-coincidence co…
High-order coherence effects between two first-order incoherent sources with fully independent phases have been well studied in the literature, which shows interference fringes with respect to the position separations among different space points. Here we show that this is not the whole story, and find that the high-order coherence effects depend on the mode of the phase random walk of the first-order incoherent sources, which can be controlled artificially and represented geometrically by vectorial polygons. Interestingly, by scanning the detectors along the same direction with the position s…
We establish a set of resource conversion relationships between quantum coherence and multipartite entanglement in many-body systems, where the operational measures of resource cost and distillation are focused. Under the multipartite incoherent operation, the initial coherence of single-party system bounds the maximal amounts of corresponding operational entanglement in an arbitrary bipartite partition as well as the genuine multipartite entanglement in many-body systems. Furthermore, the converted multipartite entanglement can be transferred to its subsystems and restored to coherence of a s…
Multi-time advantage for certifying quantum coherence
Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；Nordita, Stockholm University and KTH Royal Institute of Technology（诺迪塔，斯德哥尔摩大学和皇家理工学院）；Institute of Theoretical Physics, University of Warsaw（华沙大学理论物理研究所）；School of Mathematics and Physics, The University of Queensland（昆士兰大学数学与物理学院）
We consider the problem of energy transport in a chain of coupled quantum systems with the goal of shedding light on how nonclassical resources can affect transport. We study the cases for which either coherent or incoherent energy hopping takes place in the chain. Here, incoherent energy hopping is referred to as the "classical" scenario in allusion to its fully diagonal dynamics in the basis formed by the eigenstates of the decoupled sites. We focus on the case of a linear chain of two-level sites and find a hopping rate threshold above which the coherent quantum case is more efficient than …
The ability to distill quantum coherence is pivotal for optimizing the performance of quantum technologies; however, such a task cannot always be accomplished with certainty. Here we develop a general framework of probabilistic distillation of quantum coherence in a one-shot setting, establishing fundamental limitations for different classes of free operations. We first provide a geometric interpretation for the maximal success probability, showing that under maximally incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO) the problem can be simplified into efficiently…
We study dynamics of an artificial two-level atom in an open 1D space by measuring evolution of its coherent and incoherent emission. States of the atom -- a superconducting flux qubit coupled to a transmission line -- are fully controlled by resonant excitation microwave pulses. The coherent emission -- a direct measure of superposition in the atom -- exhibits decaying oscillations shifted by $π/2$ from oscillations of the incoherent emission, which, in turn, is proportional to the atomic population. The emission dynamics provides information about states and properties of the atom. By measur…
We develop the Keldysh formalism for the polarization dynamics of an open spin system. We apply it to the swapping between two qubit states in a model describing an NMR cross-polarization experiment. The environment is a set of interacting spins. For fast fluctuations in the environment, the analytical solution shows effects missed by the secular approximation of the Quantum Master Equation for the density matrix: a frequency decrease depending on the system-environment escape rate and the quantum quadratic short time behavior. Considering full memory of the bath correlations yields a progress…
We present a quantum theory for a fully coupled hybrid optomechanical system where all mutual couplings between a two-level atom, a confined photon mode and a mechanical oscillator mode are considered. In such a configuration, new quantum interference effects and correlations arise due to the interplay and competition between the different physical interactions. We present an analytical diagonalization of the related fully coupled Hamiltonian, showing the nature and energy spectra of the tripartite dressed excitations. We determine the driven-dissipative dynamics of such hybrid systems and stu…
We present a theory that efficiently describes the quantum dynamics of an electronic excitation that is coupled to a continuous, highly structured phonon environment. Based on a stochastic approach to non-Markovian open quantum systems, we develop a dynamical framework that allows us to handle realistic systems where a fully quantum treatment is desired yet the usual approximation schemes fail. The capability of the method is demonstrated by calculating spectra and energy transfer dynamics of mesoscopic molecular aggregates, elucidating the transition from fully coherent to incoherent transfer…
Motivated by the importance of optical microscopes to science and engineering, scientists have pondered for centuries how to improve their resolution and the existence of fundamental resolution limits. In recent years, a new class of microscopes that overcome a long-held belief about the resolution have revolutionized biological imaging. Termed "superresolution" microscopy, these techniques work by accurately locating optical point sources from far field. To investigate the fundamental localization limits, here I derive quantum lower bounds on the error of locating point sources in free space,…
In the case of quantum systems interacting with multiple environments, the time-evolution of the reduced density matrix is described by the Liouvillian. For a variety of physical observables, the long-time limit or steady state solution is needed for the computation of desired physical observables. For inverse design or optimal control of such systems, the common approaches are based on brute-force search strategies. Here, we present a novel methodology, based on automatic differentiation, capable of differentiating the steady state solution with respect to any parameter of the Liouvillian. Ou…
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The modified Bloch-Redfield master equation allows an unambiguous distinction between the regimes of quantum coherent vs. incoherent energy transfer under inco…
