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Journal of Optics B Quantum and Semiclassical Optics

Quantum microscopy using photon correlations

Ashok Muthukrishnan, Marlan O. Scully, Muhammad Suhail Zubairy

Published May 29, 2004• 41 citations
Publisher

Abstract

Higher-order correlations of the radiation field improve resolution in stellar interferometers, as in the Hanbury-Brown–Twiss effect. It is also possible to improve microscopic resolution beyond the Rayleigh limit by using quantum light fields composed of entangled photons. Focusing on two photons, we distinguish two types of entanglement: frequency entanglement, where the photons in different paths are correlated in frequency, and path entanglement, where the correlation between paths is in photon number. Two paradigms of quantum microscopy are discussed: spectral microscopy, where path- and frequency-entangled photons produced in cascade decay of two atoms make possible sub-natural linewidth resolution of atomic levels, and spatial microscopy, where path-entangled photons emitted by an atomic array produce sub-wavelength diffraction resolution as compared to an equivalent classical grating. These scenarios require two-photon correlation or coincidence measurements. The connection between the two paradigms, and the two types of entanglement, highlights the link between the temporal and spatial aspects of quantum interferometry.

DOI
10.1088/1464-4266/6/6/017

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Higher-order correlations of the radiation field improve resolution in stellar interferometers, as in the Hanbury-Brown–Twiss effect. It is also possible to improve microscopic resolution beyond the Rayleigh limit by using quantum light fields composed of entangled photons. Focusing on two photons, we distinguish two types of entanglement: frequency entanglement, where the photons in different paths are correlated in frequency, and path entanglement, where the correlation between paths is in photon number. Two paradigms of quantum microscopy are discussed: spectral microscopy, where path- and frequency-entangled photons produced in cascade decay of two atoms make possible sub-natural linewidth resolution of atomic levels, and spatial microscopy, where path-entangled photons emitted by an atomic array produce sub-wavelength diffraction resolution as compared to an equivalent classical grating. These scenarios require two-photon correlation or coincidence measurements. The connection between the two paradigms, and the two types of entanglement, highlights the link between the temporal and spatial aspects of quantum interferometry.
Quantum correlation microscopy is an emerging technique for improving optical resolution. By taking advantage of the quantum statistics from single-photon fluorophores, more information about the emitters (including number and location) is obtained compared with classical microscopy. Although it is known that the resolution can be improved by increasing detector numbers, as well as using quantum correlation, the quantitative relationship between these two approaches is not immediately clear. Here we explore widefield quantum correlation microscopy using arrays of single-photon detectors. We explicitly compare the use of NNN detectors used in photon counting mode vs N/2N/2N/2 detectors used to measure quantum correlations. i.e., where there are N/2N/2N/2 Hanbury Brown and Twiss systems, using the same NNN detectors, on randomly generated two-emitter systems. We find regimes where N/2N/2N/2 Hanbury Brown and Twiss detectors provide improved localisation compared to NNN photon counting detectors, as a function of emitter position and number of photons sampled.
Correlation microscopy is an emerging technique for improving optical resolution. By taking advantage of the photon statistics from single-photon fluorophores, more information about the emitters (including number and location) is obtained compared with classical microscopy. Although it is known that the resolution can be improved by increasing detector numbers, as well as using photon correlations, the quantitative relationship between these two approaches is not immediately clear. Here we explore widefield photon correlation microscopy using arrays of single-photon detectors. We explicitly compare the use of N detectors used in photon counting mode vs N/2 detectors used to measure photon correlations. i.e., where there are N/2 Hanbury Brown and Twiss systems, using the same N detectors, on randomly generated two-emitter systems and triangular three-emitter systems. We find regimes where N/2 Hanbury Brown and Twiss detectors provide improved localisation compared to N photon counting detectors, as a function of emitter position and number of photons sampled.
In this thesis we discuss an extension of the historical Hanbury Brown and Twiss experiment and investigate spatial intensity correlations of higher-orders in the radiation field of statistically independent light sources. The light sources can be non-classical like single-photon emitters or classical, obeying thermal or coherent statistics. Although the light sources are assumed to emit the photons in an incoherent manner, we find spatial photon correlations in the far field which can be explained by the quantum mechanical concept of multi-photon interferences. We demonstrate that spatial correlation measurements between indistinguishable photons based on linear optical detection techniques can be used to overcome the resolution limit of classical optics (Abbe/Rayleigh limit) and also to manipulate the spatial radiation characteristic of the light sources. Both remarkable properties of higher-order intensity correlations have been experimentally verified using statistically independent pseudothermal light sources. Due to the fact that the reported correlation measurements require neither special quantum tailoring of light nor N-photon absorbing media, they are of particular interest for the field of quantum imaging and quantum information processing. The thesis is divided into three main parts: In the first part (Chapter 2 and 3) we introduce a quantum path formalism which allows to describe multi-photon interferences arising in both classical and nonclassical light fields. This quantum path approach explains the origin of the underlying interference mechanism of the higher-order spatial intensity correlation functions in a transparent and simple manner. Moreover, it enables us to compare the multi-photon interferences obtained by nonclassical single-photon emitters with those generated by classical light sources. In the course of this discussion it appears that Dirac's famous statement “Each photon interferes only with itself. Interference between two different photons never occurs.” should better read “A multi-photon state of independent photons only interferes with itself. Interference between different multi-photon states never occurs.” This general statement summarizes the physics behind all multi-photon interference phenomena investigated in this thesis. In the second part of the thesis (Chapter 4) we investigate particular multi-photon interference measurements of independent light sources in the context of quantum imaging. We demonstrate that the Nth-order spatial intensity correlation functions display at specific ‘magic’ detector positions the ability to filter sub-wavelength information out of the light fields arising from N incoherently radiating light sources. These multi-photon interference patterns achieve a higher resolution for N>2 than the classical Abbe/Rayleigh limit for imaging the light source. In case of N single-photon emitters we theoretically show that the super-resolving interference patterns obtained for the ‘magic’ detector positions are identical to the ones generated by NOON states with N-1 photons. Quite unexpected, the same is true for N thermal light sources, except for a reduced visibility. Experimental results with up to eight independent thermal light sources confirm this approach to improve the spatial resolution in imaging. In the last part of this thesis (Chapter 5) we finally demonstrate that multi-photon detections can lead to a strong angular focussing of incoherent photons emitted again by either uncorrelated single-photon sources or statistically independent classical light sources. For this effect the sources can be far away from each other and do not interact. The mth-order intensity correlation measurement causes source correlations which produce a heralded peaked emission pattern for the initially uncorrelated light sources. In particular, we show that if m-1 photons have been detected in a particular direction, the conditional probability to detect the mth photon in the same direction can be as high as 100 %. Owing to the great flexibility of detector positions for the correlation measurement, a wide variety of additional detection schemes exists which produces similar directional photon emission patterns. This theoretically predicted measurement-induced focussing was experimentally confirmed with up to eight pseudothermal light sources for three particular detection schemes.
The classical optical diffraction limit can be overcome by exploiting the quantum properties of light in several theoretical studies; however, they mostly rely on an entangled light source. Recent experiments have demonstrated that quantum properties are preserved in many fluorophores, which makes it possible to add a new dimension of information for super-resolution fluorescence imaging. Here, we developed a statistical quantum coherence model for fluorescence emitters and proposed a new super-resolution method using fluorescence quantum coherence in fluorescence microscopy. In this study, by exploiting a single-photon avalanche detector (SPAD) array with a time-correlated single-photon-counting technique to perform spatial-temporal photon statistics of fluorescence coherence, the subdiffraction-limited spatial separation of emitters is obtained from the determined coherence. We numerically demonstrate an example of two-photon interference from two common fluorophores using an achievable experimental procedure. Our model provides a bridge between the macroscopic partial coherence theory and the microscopic dephasing and spectral diffusion mechanics of emitters. By fully taking advantage of the spatial-temporal fluctuations of the emitted photons as well as coherence, our quantum-enhanced imaging method has the significant potential to improve the resolution of fluorescence microscopy even when the detected signals are weak.
Imaging is the estimation of the spatial distribution of a physical object by measuring the optical radiation it emits, or by making use of an optical wave that interacts with the object, via reflection or transmission, for example, before being measured by a detector. The resolution of an imaging system is limited by the inability to localize the optical field at points of the object. Under otherwise ideal conditions, resolution is limited by diffraction. The sensitivity of an imaging system is limited by the uncertainty in the measurement. Under ideal conditions, this is determined by photon noise, which depends on the statistical fluctuations of the light. In conventional imaging systems, an extended detector, such as a CCD camera or an array detector, measures the spatial distribution of the optical intensity, which is proportional to the photon flux density. In interferometric systems, the spatial distribution of the optical field is inferred from measurements of the optical intensity. With the emergence of coherence theory, imaging systems based on measurements of the second-order coherence function at pairs of points in the detection plane were developed. An example is the imaging of an incoherent object based on the van Cittert-Zernike theorem. Imaging systems based on measurement of intensity correlation, or the photon coincidence rate, at pairs of points, were developed in the 1960s. A classic example of the photon-correlation imaging of an object emitting thermal light is stellar imaging using a Hanbury-Brown-Twiss intensity-correlation interferometer. More recently, two-photon light, which may be generated via spontaneous parametric downconversion in a second-order nonlinear optical crystal, has been used for imaging. This type of two-photon (or biphoton) imaging, which has come to be called quantum imaging, is also based on the measurement of photon coincidence by the use of photon-counting array detectors or by scanning two photon-counting detectors at pairs of points. To compare the resolution and sensitivity of imaging systems based on the aforementioned types of measurements, it is necessary to derive expressions for the measured quantities in terms of the object distribution. The point-spread functions based on such expressions can be used to assess the resolution. One measure of the sensitivity of the imaging process is the signal-to-noise ratio (SNR) of the measured variables. The statistical nature of the light source must be known in order to determine the SNR. The purpose of this chapter is to compare the resolution and sensitivity of photon-correlation imaging systems that make use of thermal light and two-photon light. We will henceforth refer to these two imaging modalities as classical and quantum photon-correlation imaging, or simply classical and quantum imaging, respectively. Clearly, light in other quantum states can also be used for imaging.