The precise localization of particles yields structural and dynamical information in applications ranging from fundamental physics to molecular biology. Here, we discuss how the concept of (Quantum) Fisher information can guide the development of localization techniques. Starting from dipole scattering, we discuss imaging based on elastic and inelastic (e.g. fluorescence) scattering and present experimental designs that increase the information obtained per detected photon.
The precise localization of particles yields structural and dynamical information in applications ranging from fundamental physics to molecular biology. Here, we discuss how the concept of (Quantum) Fisher information can guide the development of localization techniques. Starting from dipole scattering, we discuss imaging based on elastic and inelastic (e.g. fluorescence) scattering and present experimental designs that increase the information obtained per detected photon.
Precisely measuring the three-dimensional position and orientation of individual fluorophores is challenging due to the substantial photon shot noise in single-molecule experiments. Facing this limited photon budget, numerous techniques have been developed to encode 2D and 3D position and 2D and 3D orientation information into fluorescence images. In this work, we adapt classical and quantum estimation theory and propose a mathematical framework to derive the best possible precision for measuring the position and orientation of dipole-like emitters for any fixed imaging system. We find that it is impossible to design an instrument that achieves the maximum sensitivity limit for measuring all possible rotational motions. Further, our vectorial dipole imaging model shows that the best quantum-limited localization precision is 4%-8% worse than that suggested by a scalar monopole model. Overall, we conclude that no single instrument can be optimized for maximum precision across all possible 2D and 3D localization and orientation measurement tasks.
Much of our progress in understanding microscale biology has been powered by advances in microscopy. For instance, super-resolution microscopes allow the observation of biological structures at near-atomic-scale resolution, while multi-photon microscopes allow imaging deep into tissue. However, biological structures and dynamics still often remain out of reach of existing microscopes, with further advances in signal-to-noise, resolution and speed needed to access them. In many cases, the performance of microscopes is now limited by quantum effects – such as noise due to the quantisation of light into photons or, for multi-photon microscopes, the low cross-section of multi-photon scattering. These limitations can be overcome by exploiting features of quantum mechanics such as entanglement. Quantum effects can also provide new ways to enhance the performance of microscopes, such as new super-resolution techniques and new techniques to image at difficult to reach wavelengths. This review provides an overview of these various ways in which quantum techniques can improve microscopy, including recent experimental progress. It seeks to provide a realistic picture of what is possible, and what the constraints and opportunities are.
Technological advancements in the creation, manipulation and detection of quantum states of light have motivated the application of such states to overcome classical limits in sensing and imaging. In particular, there has been a surge of recent interest in super-resolution imaging based on principles of quantum optics. However, the application of such schemes for practical imaging of biological samples is demanding in terms of signal-to-noise ratio, speed of acquisition and robustness with respect to sample labeling. Here, we re-introduce the concept of quantum image scanning microscopy (Q-ISM), a super-resolution method that enhances the classical image scanning microscopy (ISM) method by measuring photon correlations. Q-ISM was already utilized to achieve super-resolved images of a biological sample labeled with fluorescent nanoscrystals whose contrast is based entirely on a quantum optical phenomenon, photon antibunching. We present here an experimental demonstration of the method and discuss with further details its prospects for application in life science microscopy.
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.
Multiphoton fluorescence microscopy is now a well-established technique, currently attracting much interest across all fields of biophysics - especially with regard to enhanced focal resolution. The fundamental mechanism behind the technique, identified and understood through the application of quantum theory, reveals new optical polarization features that can be exploited to increase the information content of images from biological samples. In another development, based on a newly discovered, fundamentally related mechanism, it emerges the passage of off-resonant probe laser pulses may characteristically modify the intensity of single-photon fluorescence, and its associated optical polarization behavior. Here, the probe essentially confers optical nonlinearity on the decay transition, affording a means of optical control over the fluorescent emission. Compared to a catalogue of other laser-based techniques widely used in the life sciences, most suffer limitations reflecting the exploitation of specifically lifetime-associated features; the new optical control mechanism promises to be more generally applicable for the determination of kinetic data. Again, there is a prospect of improving spatial resolution, non-intrusively. It is anticipated that tight directionality can be imposed on single-photon fluorescence emission, expediting the development of new imaging applications. In addition, varying the optical frequency of the probe beam can add another dimension to the experimental parameter space. This affords a means of differentiating between molecular species with strongly overlapping fluorescence spectra, on the basis of their differential nonlinear optical properties. Such techniques significantly extend the scope and the precision of spatial and temporal information accessible from fluorescence studies.
The importance of the Fisher information metrics and its quantum generalisations is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich range of possible quantum Fisher informations, only a handful are typically used and studied. This review aims at collecting a number of results scattered in the literature and provide a cohesive treatment to people who begin the study of Fisher information and to those who are already working on it to have a more organic understanding of the topic. Moreover, we complement the review with new results about the relation between Fisher information and physical evolutions. Extending previous works, we show that dynamical properties such as (complete) positivity, Markovianity, detailed balance, retrodictive power of evolution maps can be characterised in terms of their relation with respect to the Fisher information metrics. These results show a fact that was partially overseen in the literature, namely the inherently dynamical nature of Fisher information.