In this talk we describe an effective method of electromagnetic field quantization based on the characteristic modes. The diagonal form of the scattering matrix in the characteristic mode basis is the key to significant advantages of the proposed approach. It allows the effective calculation of functions of the scattering matrix, in particular high-order correlation functions, generalized Wigner-Smith matrix and quantum Wigner-Smith operator. Furthermore, it opens the way for effective simulations in different fields of quantum metrology such as the time spent by a wave packet in the process of scattering, forces and torques in optomechanics, Fisher information, and Cramer-Rao bounds in quantum sensing.
In this talk we describe an effective method of electromagnetic field quantization based on the characteristic modes. The diagonal form of the scattering matrix in the characteristic mode basis is the key to significant advantages of the proposed approach. It allows the effective calculation of functions of the scattering matrix, in particular high-order correlation functions, generalized Wigner-Smith matrix and quantum Wigner-Smith operator. Furthermore, it opens the way for effective simulations in different fields of quantum metrology such as the time spent by a wave packet in the process of scattering, forces and torques in optomechanics, Fisher information, and Cramer-Rao bounds in quantum sensing.
The interaction of quantum light with matter is of great importance to a wide range of scientific disciplines, ranging from optomechanics to high-precision measurements. A central issue we discuss here, is how to make optimal use of both the spatial and the quantum degrees of freedom of light for characterizing and manipulating arbitrary observable parameters in a linear scattering system into which suitably engineered light fields are injected. Here, we discuss a comprehensive framework based on a quantum operator that can be assembled solely from the scattering matrix of a system and its dependence on the corresponding local parameter, making this operator experimentally measurable from the far field using only classical light. From this, the effect of quantum light in the near field, i.e., in the vicinity of the target object, can be inferred. Based on this framework, it is straightforward to formulate optimal protocols on how to jointly design both the spatial shape and the quantum characteristics of light for micromanipulation as well as for parameter estimation in arbitrarily complex media. Also, the forces of the quantum vacuum naturally emerge from this formalism. The aim of our tutorial is to bring different perspectives into alignment and thereby build a bridge between the different communities of wave control, quantum optics, micromanipulation, quantum metrology, and vacuum physics.
Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
Recently, distributed sensing has been extensively studied using squeezed states. While this is an excellent development, it is desirable to investigate the use of other quantum probes, such as entangled states of light. In this paper, we focus on distributed sensing, i.e., estimating multiple unknown phases at different spatial nodes using multiphoton polarization-entangled Greenberger-Horne-Zeilinger states distributed across different nodes. We utilize tools of quantum metrology and calculate the quantum Fisher information matrix (QFIM). However, the QFIM turns out to be singular, hindering the determination of quantum Cramér-Rao bounds for the parameters of interest. Recent experiments have contended with a weaker form of the Cramér-Rao bound, which does not require the inversion of the QFIM. It is desirable to understand how relevant these weaker bounds are and how closely they approach the exact Cramér-Rao bounds. We thus analyze the reason for this singularity and, by removing a redundant phase, obtain a nonsingular QFIM, allowing us to derive exact quantum Cramér-Rao bounds. Using the nonsingular QFIM, we show that the arithmetic average of the distributed phases is Heisenberg limited. We demonstrate that quantum metrological bounds can be saturated by projective measurements, enabling us to determine the Fisher information matrix, which is also singular. We then show how this singularity can be resolved.
We propose a numerical technique for modeling quantum multimodal light scattering by a perfectly conducting body. Using the special quantization technique, we develop the quantum adaptation of the characteristic mode approach widely used in classical electrodynamics. The method is universal with respect to the body's configuration, as well as its dimensions relative to the wavelength. Using this method and calculating the first- and second-order field correlation functions, we demonstrate how scattering affects quantum-statistical features of the field. As an example, we consider the scattering of two single-photon incident Gaussian beams on a cylinder with a circular cross section. We show that the scattering is accompanied by two-photon interference and demonstrates the Hong-Ou-Mandel effect. It is shown that the scattered two-photon field and its correlations manifest directive propagation, which is controllable by various means (angles of incidence, configuration of the body, relations between its dimensions and the wavelength). We expect that this method will be useful for designing quantum-optical devices.
Fisher information plays a central role in statistics and quantum metrology, providing the basis for the celebrated Cramér-Rao bound. In this work, we introduce a new information measure based on higher-order Fisher information and show that it naturally leads to a generalized uncertainty relation for parameter estimation, which can be regarded as an extension of the Cramér-Rao bound. As an application, we analyze the case of quantum phase estimation with a single qubit and compare our theoretical bounds with the well-known established hierarchical bounds. Finally, we experimentally validate the proposed framework using a photonic platform.
The emergence and development of quantum optics as a discipline in the latter half of the twentieth century has established photons as one of the most promising candidates for future technologies in computing, metrology, communication, and imaging. In imaging and sensing, a series of recent theoretical works and proof-of-concept experiments have shown that by examining the mode structure of photons, one can build optical systems with resolutions below the Rayleigh–Abbe limit. Building upon these developments, in one of our works, we investigate the use of higher-order Hermite–Gauss (HG) modes for sensing optically induced transverse displacements. We show that projective measurements onto neighboring modes are optimal and sufficient in the small-displacement regime, and that the sensitivity—quantified by the Fisher information—scales linearly with the order of the mode used. We generalize the calculations to arbitrary displacements and varying degrees of coherence between the displaced modes, and perform a proof-of-concept experiment with higher-order HG modes, obtaining an order-of-magnitude enhancement in sensitivity compared to the fundamental mode. This approach enables enhanced displacement sensitivity with minimal measurements and highlights the potential of structured light for ultrasensitive displacement sensing, particularly for birefringence measurements with broadband or low-coherence light sources. We also consider displacement sensing in the light field originating from a biphoton source, which possesses rich high-dimensional correlations in its biphoton mode structure. We show that the Fisher information is proportional to the degree of entanglement, or equivalently, its Schmidt rank. In addition to sensing, photons structured in their transverse degrees of freedom also provide a convenient toolkit for quantum computing and communication due to the high dimensionality of their Hilbert space. In one of the works presented, we introduce a technique for nonlocal transfer of high-dimensional computational results using spatial correlations in photon pairs. We expect these findings to be useful in future photonic quantum networks, where computational resources are distributed and users can access the results of complex computations via a centrally located quantum processor. We also experimentally implement the protocol in down-converted photon pairs and demonstrate the transfer of unitaries in one-dimensional and two-dimensional transverse momentum spaces. In another work, we address the noise inherent in quantum computing circuits and simulate the interaction between a qubit and its environment by manipulating transverse momentum modes. This study, performed by fabricating three liquid-crystal metasurfaces, provides a practical solution for quantum state purification, demonstrating a versatile approach for simulating open-qubit dynamics. Combining the transverse spatial structure with polarization yields the so-called vector modes of photons. Structuring two-qubit interference, or the Hong–Ou–Mandel (HOM) effect, offers significant potential to extend the applications of such modes beyond the classical regime. We demonstrate the simultaneous generation of all four Bell states in polarization by exploiting HOM interference of vector modes. These results represent a significant step toward manipulating HOM interference within structured photons, offering promising applications in high-dimensional quantum information processing, quantum communication, and sensing. On the algorithmic side of photonic information processing, we propose a quantum pattern recognition algorithm to identify images obtained from quantum imaging techniques with correlated photons—such as “interaction-free” or “ghost” imaging. Interfacing these imaging techniques with quantum pattern recognition promises a significantly improved image acquisition and identification system, with potential applications in medical diagnostics and biological imaging.