In this thesis I examine how the statistical properties of radiation limit our ability to perform imaging and metrological procedures. In particular I focus on radiation in the far field zone of the source. The classical and quantum theories of parameter estimation are introduced and subsequently utilised throughout, along with the theory of optical coherence. Classical and quantum imaging protocols are examined with the aid of a resolution criterion and the criterion is shown to reproduce the results of previous works. This method is also extended to previously un-investigated situations and the effect of imperfect measurements is explored. Intensity correlation measurements are investigated in great detail and for the first time a rigorous comparison is made between higher-order intensity correlation measurements of the type introduced by Hanbury Brown and Twiss. The importance of considering covariances in intensity correlation data is demonstrated and I give a full, detailed account of how to include this in the formulation. I also show how the optimal arrangement for an intensity correlation measurement can be found, therefore allowing the best precision in parameter estimation to be achieved. A quantum mechanical description of blackbody radiation is used to examine the state arriving at a detector in the far field. By using the quantum Fisher information an interesting connection is found between the statistical independence of photons in the source plane and the acquisition of information in the far field.
In this thesis I examine how the statistical properties of radiation limit our ability to perform imaging and metrological procedures. In particular I focus on radiation in the far field zone of the source. The classical and quantum theories of parameter estimation are introduced and subsequently utilised throughout, along with the theory of optical coherence. Classical and quantum imaging protocols are examined with the aid of a resolution criterion and the criterion is shown to reproduce the results of previous works. This method is also extended to previously un-investigated situations and the effect of imperfect measurements is explored. Intensity correlation measurements are investigated in great detail and for the first time a rigorous comparison is made between higher-order intensity correlation measurements of the type introduced by Hanbury Brown and Twiss. The importance of considering covariances in intensity correlation data is demonstrated and I give a full, detailed account of how to include this in the formulation. I also show how the optimal arrangement for an intensity correlation measurement can be found, therefore allowing the best precision in parameter estimation to be achieved. A quantum mechanical description of blackbody radiation is used to examine the state arriving at a detector in the far field. By using the quantum Fisher information an interesting connection is found between the statistical independence of photons in the source plane and the acquisition of information in the far field.
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.
The research included in this thesis comes in two main bodies. In the first, the focus is on intensity interferometric schemes, and I attempt to identify the types of correlations dominant in their operation. This starts with the, now rather historical, Hanbury Brown and Twiss setup from the 1950s and progresses to more recent interests such as ghost imaging and a variant of `quantum illumination', which is a quantum-enhanced detection scheme. These schemes are considered in the continuous variable regime, with Gaussian states in particular. Intensity interferometry has been the cause of a number of disputes between quantum opticians over the past 60 years and I weigh in on the arguments using relatively recent techniques from quantum information theory. In the second half, the focus turns away from the optical imaging and detection schemes, and onto quantum estimation -- multiparameter quantum estimation to be precise. This is an intriguing area of study where one has to carefully juggle tradeoffs in choosing both the optimal measurement and optimal state for performing an estimation in two or more parameters. I lay out a framework for circumventing some of the difficulties involved in this and apply it to several physical examples, revealing some interesting and at times counterintuitive features of multiparameter estimation.
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.
We compare the performance of quantum imaging schemes based on spatially correlated photon pairs by formulating them as quantum multiparameter estimation problems, in which the object is characterized by transmission coefficients associated with different spatial modes. Our work focuses on standard quantum imaging techniques such as ghost imaging, two-photon imaging, and imaging with undetected photons. Specifically, we compute the quantum Fisher information matrices and show that they are saturated by Fisher information matrices corresponding to measurements in the object-mode basis, which generalize the common detection schemes employed in each imaging configuration. We find that ghost imaging and two-photon imaging generally provide higher precision for transmission estimation than imaging with undetected photons, but the latter is the only one that naturally does not couple transmission estimation across different spatial modes. These results identify which imaging protocols are best suited for specific tasks and provide practical guidelines for the design and optimization of quantum sensing technologies based on spatial correlations.
The foundational theory of quantum enhanced metrology for parameter estimation is of fundamental importance to the progression of science and technology as the scientific method is built upon empirical evidence, the acquisition of which is entirely reliant on measurement. Quantum mechanical properties can be exploited to yield measurement results to a greater precision (lesser uncertainty) than that which is permitted by classical methods. This has been mathematically demonstrated by the derivation of theoretical bounds which place a fundamental limit on the uncertainty of a measurement. Furthermore, quantum metrology is of immediate interest in the application of quantum technologies since measurement plays a central role. This thesis focuses on the role of quantum correlations and uncertainty relations which govern the precision bounds. We show how correlations can be distributed amongst limited resources in realistic scenarios, as permitted by current experimental capabilities, to achieve higher precision measurements than current approaches. This is extended to the setting of multiparameter estimation in which we demonstrate a more technologically feasible method of correlation distribution than those previously posited which perform as well as, or worse than, our scheme. Furthermore, a quantum metrology protocol is typically comprised of three stages: probe state preparation, sensing and then readout, where the time required for the first and last stages is usually neglected. We consider the more realistic sensing scenario of time being a limited resource which is divided amongst the three stages and demonstrate the most efficient use of this resource. Additionally, we take an information theoretic approach to quantum mechanical uncertainty relations and derive a one-parameter class of uncertainty relations which supplies more information about the quantum mechanical system of interest than conventional uncertainty relations. Finally, we demonstrate how we can use this class of uncertainty relations to reconstruct information of the state of the quantum mechanical system.
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--Twiss-type experiments. The improvement is achieved entirely by post-processing of the data. As a demonstrator, we simulate the far field intensity distribution of a circular aperture that emits thermal light and use maximum likelihood estimation to determine the radius of the aperture. We compare the achieved precision to the Cram\'er-Rao lower bound and find that the variance of measurements for the third- and fourth-order correlation functions are indeed closer to the Cram\'er-Rao bound than that of the second-order correlation function. The method presented here is general, and can be used for all kinds of incoherent emitters, geometries, and types of noise.