Abstract This tutorial outlines the theory of nonlinear spectroscopy with quantum light, and in particular with entangled photons. To this end, we briefly review molecular quantum electrodynamics, and discuss the approximations involved. Then we outline the perturbation theory underlying nonlinear spectroscopy. In contrast to the conventional semiclassical theory, our derivation starts from Glauber's photon counting formalism, and naturally includes the semiclassical theory as a special case. Finally, we review previous work, which we sort into work depending on the unusual features of quantum noise, and work relying upon quantum correlations in entangled photons. This work naturally draws from both quantum optics and chemical physics. Even though it is impossible to provide a comprehensive overview of both fields in one tutorial, this text aims to be self-contained. We refer to specialised reviews, where we cannot provide details. We do not attempt to provide an exhaustive review of all the literature, but rather focus on specific examples intended to elucidate the underlying physics, and merely cite the remaining publications.
Abstract This tutorial outlines the theory of nonlinear spectroscopy with quantum light, and in particular with entangled photons. To this end, we briefly review molecular quantum electrodynamics, and discuss the approximations involved. Then we outline the perturbation theory underlying nonlinear spectroscopy. In contrast to the conventional semiclassical theory, our derivation starts from Glauber's photon counting formalism, and naturally includes the semiclassical theory as a special case. Finally, we review previous work, which we sort into work depending on the unusual features of quantum noise, and work relying upon quantum correlations in entangled photons. This work naturally draws from both quantum optics and chemical physics. Even though it is impossible to provide a comprehensive overview of both fields in one tutorial, this text aims to be self-contained. We refer to specialised reviews, where we cannot provide details. We do not attempt to provide an exhaustive review of all the literature, but rather focus on specific examples intended to elucidate the underlying physics, and merely cite the remaining publications.
Two-photon absorption (TPA) and other nonlinear interactions of molecules with time-frequency-entangled photon pairs have been predicted to display a variety of fascinating effects. Therefore, their potential use in practical quantum-enhanced molecular spectroscopy requires close examination. This Tutorial presents a detailed theoretical study of one- and two-photon absorption by molecules, focusing on how to treat the quantum nature of light. We review some basic quantum optics theory and then we review the density-matrix (Liouville) derivation of molecular optical response, emphasizing how to incorporate quantum states of light into the treatment. For illustration, we treat in detail the TPA of photon pairs created by spontaneous parametric down conversion, with an emphasis on how quantum light TPA differs from that with classical light. In particular, we treat the question of how much enhancement of the TPA rate can be achieved using entangled states. This Tutorial includes a review of known theoretical methods and results as well as some extensions, especially the comparison of TPA processes that occur via far-off-resonant intermediate states only and those that involve off-resonant intermediate states by virtue of dephasing processes. A brief discussion of the main challenges facing experimental studies of entangled two-photon absorption is also given.
The application of quantum states of light such as entangled photons, for example, created by parametric down conversion, has experienced tremendous progress in the almost 40 years since their first experimental realization. Initially, they were employed in the investigation of the foundations of quantum physics, such as the violation of Bell's inequalities and studies of quantum entanglement. They later emerged as basic platforms for quantum communication protocols and, in the recent experiments on single-photon interactions, in photonic quantum computation. These applications aim at the controlled manipulation of the photonic degrees of freedom, and therefore rely on simple models of matter, where the analysis is simpler. Furthermore, quantum imaging with entangled light can achieve enhanced resolution, and quantum metrology can overcome the shot noise limit for classical light. This Account focuses on an entirely different emerging class of applications using quantum light as a powerful spectroscopic tool to reveal novel information about complex molecules. These applications utilize two appealing properties of quantum light: its distinct intensity fluctuations and its nonclassical bandwidth properties. These give rise to new and surprising behavior of nonlinear optical signals. Nonclassical intensity fluctuations can enhance nonlinear optical signals relative to linear absorption. For instance, the two-photon absorption of entangled photon pairs scales linearly (rather than quadratically) in the photon flux, just like a single photon absorption. This enables nonlinear quantum spectroscopy of photosensitive, for example, biological, samples at low light intensities. We will discuss how the two-photon absorption cross section becomes a function of the photonic quantum state, which can be manipulated by properties of the entangled photon pairs. In addition, the quantum correlations in entangled photon states further influence the nonlinear signals in a variety of ways. Apart from affecting the signal's scaling with intensity, they also constitute an entirely new approach to shaping and controlling excitation pathways in molecular aggregates in a way that cannot be achieved with shaped classical pulses. This is because between the two absorption events in entangled two-photon absorption, the light and material system are entangled. Classical constraints for the simultaneous time and frequency resolution can thus be circumvented, since the two are not Fourier conjugates. Here we review the simplest manifestation of quantum light spectroscopy, two-photon absorption spectroscopy with entangled photons. This will allow us to discuss exemplarily the impact of quantum properties of light on a nonlinear optical signal and explore the opportunities for future applications.
Playing a prominent role in communications, quantum science and laser physics, quantum nonlinear optics is an increasingly important field. This book presents a self-contained treatment of field quantization and covers topics such as the canonical formalism for fields, phase-space representations and the encompassing problem of quantization of electrodynamics in linear and nonlinear media. Starting with a summary of classical nonlinear optics, it then explains in detail the calculation techniques for quantum nonlinear optical systems and their applications, quantum and classical noise sources in optical fibers and applications of nonlinear optics to quantum information science. Supplemented by end-of-chapter exercises and detailed examples of calculation techniques in different systems, this book is a valuable resource for graduate students and researchers in nonlinear optics, condensed matter physics, quantum information and atomic physics. A solid foundation in quantum mechanics and classical electrodynamics is assumed, but no prior knowledge of nonlinear optics is required.
We develop an input-output formulation of quantum light spectroscopy (QLS) by combining the input-output theory, commonly used in quantum optics, with the perturbative expansion method for nonlinear spectroscopy, commonly used in chemical physics. The use of the input-output relation provides physical intuition for how matter systems alter the photon input field. As examples, we consider linear spectroscopy, fluorescence, intra-beam G(2), and Hong-Ou-Mandel correlation. For certain initial photon states, we show that signal expectation values can be evaluated non-perturbatively. For general initial photon states, the signals can be computed via a perturbative expansion of the input-output relation. The input-output relation reduces the work needed to perturbatively expand the optical signal, allowing for cleaner derivations and clearer physical interpretations.
Quantum entanglement, a hallmark of quantum mechanics, offers unprecedented opportunities to revolutionize spectroscopic techniques. This paper explores the theoretical underpinnings and experimental prospects of quantum entanglement-enhanced spectroscopy for the precise probing of coherent dynamics in complex chemical systems. Traditional spectroscopic methods often face limitations in resolution, sensitivity, and the ability to distinguish subtle quantum phenomena due to their reliance on classical light sources. By harnessing entangled photons, it is possible to overcome these limitations, enabling sub-shot-noise measurements, enhanced spectral resolution beyond the diffraction limit, and access to non-classical correlations within molecular systems. We delve into how entangled light can be generated and engineered to interact with molecular ensembles, facilitating the study of processes such as exciton transport, charge transfer, and vibrational energy redistribution with extraordinary fidelity. The methodology section outlines conceptual experimental setups, including entangled two-photon absorption spectroscopy and quantum Ramsey interferometry, discussing the benefits of these approaches over their classical counterparts. We present hypothetical results demonstrating how entanglement can resolve overlapping spectral features, extend coherence lifetimes, and provide unique insights into the quantum pathways governing chemical reactions. The discussion addresses the challenges associated with maintaining entanglement in noisy chemical environments and outlines strategies for mitigating decoherence. Ultimately, entanglement-enhanced spectroscopy represents a transformative paradigm for chemical physics, promising a deeper understanding of fundamental quantum processes that dictate reactivity and function in biological and material sciences.
For decades, ultrafast light-matter interactions have provided a powerful means to observe electrons in motion across a wide range of physical systems. This dissertation builds new theoretical frameworks to make sense of the emergent phenomena that arise when light drives matter into the strong-field regime, spanning atomic, molecular, and condensed-matter systems. The resulting dynamics leave their fingerprints on a variety of observables, such as photoionization spectra, high-harmonic generation, and nonlinear spectroscopic signals, each of which provides a different window onto the underlying electronic motion. The work is organized into two parts.The first part focuses on the electric-field observables of nonlinear optical signals. Because both the amplitude and phase of the emitted field are experimentally accessible through techniques such as spectral interferometry and electro-optic sampling, the full field itself serves as a rich observable for characterizing nonlinear responses in atomic, molecular, and condensed-matter systems. Leveraging this, the simulations presented here uncover novel and largely unexplored physical phenomena encoded in these field observables. The second part turns to optical spectroscopy with quantum light. Although nonlinear optical processes are conventionally modeled within a semiclassical framework, treating the matter quantum mechanically while the driving ultrafast fields remain classical, the resulting signals can nonetheless exhibit intrinsically non-classical properties. Building on this insight, a framework for interpreting experimental data is introduced and used to show that four-wave mixing driven by ultrafast femtosecond laser pulses generates bright squeezed light. Beyond squeezed light, strong field-matter interactions can also produce attosecond entangled photons, a distinct form of quantum light. Several strategies are proposed to enhance the efficiency of their generation, with the developed theory predicting rates of energy-time entangled biphoton production orders of magnitude beyond those achievable with conventional spontaneous parametric down-conversion (SPDC) sources. The studies presented here are initial steps toward investigating quantum properties of these systems through quantum phynonlinear spectroscopy, which is not achievable with classical light.