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.
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.
Abstract 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. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained 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.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works, analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. However, most previous works have either consider low-intensity, or thermal sources. To broaden the applicability of this approach, it is important to extend these results to more general light sources. To this goal, we will present an analytical expression for the Quantum Fisher Information, determining the ultimate resolution limit, for the separation between two sources in an arbitrary Gaussian state. Applying this result to different quantum states, we can shine some light on some relevant questions. We can for example explore the role of partial coherence considering displaced and correlated thermal states, or investigate the importance of quantum correlations by considering squeezed light. In addition to the ultimate quantum limit, we will discuss a simple estimation technique, requiring access only to the mean value of a linear combination of demultiplexed intensity measurements, which is often sufficient to saturate these limits, and can easily be adapted to incorporate the most common noise sources. Finally, we will present our experimental setup that allows for the generation of the images of two sources with different photon statistics, as well as for spatial mode demultiplexing and we will discuss the first practical implementations if the above mentioned estimation techniques.
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.
In this thesis we make progress towards applications of quantum estimation theory to new physical systems. We first consider two commonly visited problems in quantum metrology: source optimisation and source localisation. For the first, we focus on estimating the distances, d, between neighbouring light sources along an array, which undergoes stretching deformations. We evaluate how changing the nature of the sources impacts the estimation precision of d by using the quantum Fisher information (QFI) as a figure of merit. By comparing this quantity for arrays of single photon emitters, coherent, thermal, and entangled sources, we find that the classical coherent and thermal states outperform the single photon emitters. This would be favourable since generating classical states is less resource-expensive to create. However, a quantum enhancement is observed when entanglement is employed. In agreement with separate work, the optimal state is that which entangles the eigenstates corresponding to the maximum and minimum difference eigenvalues of the generator. We demonstrate that not all entangled states can reproduce similar precision enhancements. This insight is reminiscent of previous studies, where entanglement was concluded as a necessary but insufficient resource for quantum metrology.
Next, we address the source localisation problem to detect any deformations applied to a grid of sources. Improving this detection depends on our ability to engineer grids that maximise the sensitivity of the QFI matrix. Hence, we derive the generators of local translations of unitary evolutions that describe any general grid deformation, and show that our result is a multi-parameter extension of other results in the literature. We obtain a general result for the quantum Fisher information matrix (QFIM) through these generators for any grid deformation and explore specific spatial maps, including composite stretches, shears, and rotations. Since the QFI matrix depends only on the properties of the probe state and the configuration of the emitters, we explore how we can modify both to enhance our estimation sensitivity to determine the applied grid deformation. Physically motivated, we find the best arrangement of sources that enhances the sensitivity of detection for a set number of sources.
Finally, we consider the optimal estimation of a complex squeezing operation in phase space. The use of squeezed light as a quantum resource is ubiquitous in quantum optics, and a complete characterisation of a complex squeezing operation is pivotal for fundamental reasons. This is a true multi-parameter quantum estimation problem of incompatible observables. Specifically, we find that the symmetric logarithmic derivates (SLDs) for amplitude and directional squeezing do not commute. This prohibits simultaneous optimal estimates of both parameters, even in the asymptotic limit. As a result, we focus on finding separable optimal estimates. The Cramer-Rao bound is determined to provide a theoretical benchmark on the bi-variate estimation precision for general single mode Gaussian probes. Using this and the SLDs, we present a practical experimental implementation that can realise the individual fundamental precision bounds.
The fundamental precision limit of an interferometer is crucial since it bounds the best possible sensitivity one could achieve using such a device. This thesis will focus on several different interferometers and try to give the ultimate precision bounds by carefully counting all the resources used in the interferometers. The thesis begins with the basics of the quantum state of light. The fundamentals of quantum metrology are also reviewed and discussed. More specifically, the terminology of classical and quantum Cram\'er-Rao bound and classical and quantum Fisher information are introduced. Chapter 3 discusses the conclusive precision bounds in two-mode interferometer such as Mach-Zehnder interferometer (MZI) and SU(1,1) interferometer. I revisit the quantum Fisher information approach of these two interferometers and show the discrepancy of phase sensitivity on the physically same setup. Then I establish fundamental precision estimation bounds for such device, due to the reason that many works of literature fail to accurately count the resources and knowledge of phase-to-be-estimated used in the interferometers. The analysis suggests that for a MZI, one can never do better than SNL in phase sensitivity, when an input to one of the two ports is the vacuum. If one does not allow the detector to use any external phase reference or power resource, then the precision is limited by the SNL. For a SU(1,1) interferometer, firstly, when one of the input states is restricted to be a vacuum state, I showed that by using either the phase-averaging method or the quantum Fisher information matrix method, different phase configurations of the SU(1,1) interferometer result in the same QFI. Secondly, I compared the results of the phase-averaging method and the quantum Fisher information matrix method, and then I argued that for an SU(1,1) interferometer, phase averaging or quantum Fisher information matrix method is generally required, and they are equivalent. Finally, I used the quantum Fisher information matrix method to calculate the precision limit for other common input states, such as two coherent state inputs or coherent state with squeezed vacuum inputs. In chapter 4, I will consider a passive multi-mode interferometer for multiparameter phase estimation. It was suggested that optical networks with relatively inexpensive overhead---single photon Fock states, passive optical elements, and single photon detection---can show significant improvements over classical strategies for single-parameter estimation, when the number of modes in the network is small. In this chapter, I analytically compute the quantum Cramer-Rao bound to show these
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.