Optical generation of high-order harmonics is a prototypical example of nonlinear light–matter interactions in the high-field regime. Quantum optical effects have recently been demonstrated to have a significant influence on this phenomenon. These findings underline the importance of understanding the dynamics of the quantized electromagnetic field during high-order harmonic generation. In the following, we discuss the challenges that are related to the theoretical description of this process and summarize the results that were obtained using the high-field, multimode generalization of well-known quantum optical models that are based on the concept of the two-level atom.
Optical generation of high-order harmonics is a prototypical example of nonlinear light–matter interactions in the high-field regime. Quantum optical effects have recently been demonstrated to have a significant influence on this phenomenon. These findings underline the importance of understanding the dynamics of the quantized electromagnetic field during high-order harmonic generation. In the following, we discuss the challenges that are related to the theoretical description of this process and summarize the results that were obtained using the high-field, multimode generalization of well-known quantum optical models that are based on the concept of the two-level atom.
High-order harmonic generation is a highly nonlinear phenomenon in which ultraviolet photons are generated when an atom is subjected to a high-intensity infrared laser field. This process has been extensively studied, but most of the formulations employ a semiclassical approach in which the electron is treated quantum mechanically while the driving field is treated classically. Although some recent studies follow a quantum-optical approach, the existing descriptions do not offer much insight into the physical processes taking place, and are always limited to monochromatic driving fields. In this thesis, we construct a fully quantized framework for strong-field ionization and high-harmonic generation that allows for quantization of both the driving field and the ultraviolet emission. The presented formalism extends the Strong-Field Approximation theory to a quantum-optical framework, with which we can obtain an analytical expression for the electron and the driving field after the interaction.
The interaction of matter with a quantized electromagnetic mode is considered. Representing a strong exciting field, the mode is assumed to contain a large number of photons. As a result, the material response is highly nonlinear: the completely quantized description results in generation of high harmonics. In order to understand the essence of the physical processes that are involved, we consider a finite dimensional model for the material system. Using an appropriate description in phase space, this approach leads to a transparent picture showing that the interaction splits the initial, exciting coherent state into parts, and the rapid change of the populations of these parts (that are coherent states themselves) results in the generation of high-order harmonics as secondary radiation. The method we use is an application of the discrete lattice of coherent states that was introduced by von Neumann.
Abstract Traditionally, the theoretical description of the optical generation of high-order harmonics relies on the semiclassical approximation, i.e., on the assumption that the exciting radiation (driving) can be treated as a classical, time-dependent field. Recent results, however, show that the quantized nature of the electromagnetic radiation (”photon picture”) plays a significant, experimentally observable role in the process. Along this line, in this work we describe a general model, in which both the exciting field and the high harmonic modes are quantized, while the target material appears via parameters only. The time evolution of the quantized field modes clearly shows gradually increasing population of the harmonics at the expense of decreasing photon numbers in the excitation. The process is visualized in the phase-space, using Wigner functions.
Quantum-optical descriptions of strong-field processes have attracted significant attention in recent years. Typically, the theoretical modeling has been conducted in the Schrödinger picture, where results are only obtainable under certain approximations, while, in contrast, the Heisenberg picture has remained relatively unexplored. In this work, we develop an accurately controlled perturbative expansion of the time-evolution operator in the Heisenberg picture and derive beyond-semiclassical corrections to the emitter dynamics due to the coupling to the quantized electromagnetic field, capturing effects of the quantum fluctuations present in the latter. We focus on high-order harmonic generation (HHG), where the approach is accurate in parameter regimes of current interest and it gives closed-form expressions for key observables. This formulation not only simplifies numerical calculations compared to the Schrödinger-picture approach but also provides a clear correspondence between nonclassical features of the emitted light and the underlying induced dynamics of the generating medium including quantum fluctuations. Moreover, the Heisenberg framework naturally yields scaling relations with the number of independent emitters, enabling us to assess whether nonclassical behavior should persist under typical experimental conditions involving large emitter ensembles. Interestingly, we find that the degree of squeezing increases with the number of emitters, whereas the photon statistics approaches a classical Poissonian distribution in the many-emitter limit. We also find that the beyond-semiclassical emitter dynamics significantly enhances the degree of squeezing of the emitted light. Our work advances the theoretical understanding of quantum-optical HHG and introduces an accessible and well-controlled framework to describe realistic experiments.
The quantum optical description of high-order harmonic generation where both the electrons of the generating medium and the driving and generated light fields are described quantum mechanically has been of significant interest in the past years. The quantum optical formulation leads to equations of motion for the generated light field in which the quantum optical field couples to the time-dependent current of the electronic medium irrespectively of the specifics of the electronic system being an atom, molecule, or solid. These equations of motion are not solvable for any realistic system and accurate and verified approximations are hence needed. In this paper, we present a hierarchy of approximations for the equations of motion for the photonic state. At each level in this hierarchy, we compare it to the previous level justifying the validity using the Fermi-Hubbard model as an example of an electronic system with correlations. This model allows us to perform an accurate simulation of the electron motion of all the required states. We find that for the typical experimental situation of weak quantized-light-matter-coupling constant and at intensities well below the damage threshold, an explicit expression for the generated quantum light, referred to as the Markov-state approximation, captures the high-harmonic spectrum quantitatively and describes the single-mode quantum properties of the generated light as characterized by the Mandel-Q parameter and the degree of squeezing qualitatively.
Intense light-matter interaction largely relies on the use of high-power light sources, creating fields comparable to, or even stronger than, the field keeping the electrons bound in atoms. Under such conditions, the interaction induces highly nonlinear processes such as high harmonic generation, in which the low-frequency photons of a driving laser field are upconverted into higher-frequency photons. These processes have enabled numerous groundbreaking advances in atomic, molecular, and optical physics, and they form the foundation of attosecond science. Until recently, however, such processes were typically described using semi-classical approximations, since the quantum properties of the light field were not required to explain the observables. This has changed in the recent past. Ongoing theoretical and experimental advances show that fully quantized descriptions of intense light-matter interactions, which explicitly incorporate the quantum nature of the light field, open new avenues for both fundamental research and technological applications at the fully quantized level. These advances emerge from the convergence of quantum optics with strong-field physics and ultrafast science. Together, they have given rise to the field of quantum optics and quantum electrodynamics of strong-field processes.