Strong-field Physics Encounters Quantum Electro Dynamics

When an intense mid-infrared (MIR) pulse interacts with matter, it can trigger the emission of high-energy photons at harmonic multiples of the laser’s fundamental frequency, a process known as high-order harmonic generation (HHG). This phenomenon is typically described by the three-step model (TSM): first, an electron is tunnel-ionized and released into the continuum; next, it is accelerated by the strong electromagnetic field, gaining kinetic energy; finally, the electron recombines with its parent atom, releasing the acquired energy as harmonics of the driving field's central frequency. This phenomenon has recently attracted increasing attention for its ability to upconvert light that carries quantum properties, enabling frequency conversion of quantum light without altering its quantum state, a process known as quantum frequency conversion (QFC) [PhysRevA.110.063118; Nat Commun. 11 4598, (2020); Nat. Phys. 19, 1689–1696 (2023)]. Only recently has HHG been observed in liquids [Nat Commun 9, 3723 (2018)], bringing to a growing interest due to the spectroscopic capabilities this process offers, as well as its potentially higher conversion efficiency compared to its gas-phase counterpart, owing to the higher density of the liquid medium. We already demonstrated the generation of visible high-harmonic radiation from thermotropic liquid crystals (LCs) [APL Photonics 9, 060801 (2024)], a unique phase of matter that exhibits properties intermediate between liquids and solids, depending on temperature. Furthermore, the application of an external field enables control over molecular orientation, allowing access to the harmonic response as a function of the driving field's polarization direction. This characterization revealed the sensitivity of HHG to the degree of molecular order in LCs. Ferroelectric nematic liquid crystals, such as RM734, are of particular interest due to their spontaneous net dipole moment, which breaks inversion symmetry and enables the generation of both even and odd harmonics. Remarkably, this LC has also been shown to generate quantum light via spontaneous parametric down-conversion (SPDC) [Nature 631, 294–299 (2024)], paving the way for a unique platform for producing quantum light with tunable properties. Whether quantum light can be generated via quantum frequency conversion (QFC) in this system remains an open question. If achievable, it would enable the production of quantum light at higher photon energies and potentially with shorter pulse durations compared to SPDC. To investigate the quantum optical nature of HHG emitted from ferroelectric liquid crystals, this work presents high-harmonic generation from RM734, along with its characterization as a function of the relative orientation between the driving field and the molecular alignment. These measurements establish a foundation for future studies on the quantum properties of the emitted radiation and demonstrate the feasibility of generating harmonic fields without perturbing the system.

High-harmonic generation (HHG) in strong-field laser–matter interactions provides a platform to probe non-perturbative quantum dynamics and generate coherent extreme ultraviolet light. While its semiclassical interpretation is well-established, a complete quantum optical description of the field modes involved remains an active area of research, particularly in relation to squeezing, entanglement, and nonclassical light generation. In this work, we model the quantum optical signatures of HHG by treating the coupled dynamics of an atomic system and the quantized electromagnetic field. Starting from a soft-core potential calibrated to reproduce the ionization energy of Argon, we solve the time-dependent Schr¨odinger equation using a split-step Fourier method (SSFM) to capture the electron’s evolution under intense laser fields. For the initial stages, analytical approximations are employed using the forced harmonic oscillator framework, enabling the computation of the evolution of the Husimi Q distributions and identification of shifts from vacuum states. We further explore numerical simulations for both the atomic and field degrees of freedom, analyzing probability densities, phase-space distributions, and discretization effects. By extracting reduced density matrices for field modes, we compute Husimi Q distributions to characterize squeezing and non-classicality, corroborating recent predictions of entanglement among optical modes in HHG processes. Our methodology is extended from one-dimensional to higher-dimensional models to incorporate multiple harmonics, paving the way for a comprehensive description of the quantum state of light generated in HHG. These findings contribute to understanding the interplay between strong-field physics and quantum optics, offering insights for future experiments aiming to harness HHG for quantum information and coherent control applications.

We present the on-going development of a cavity-enhanced HHG source devoted to the study of high harmonic generation (HHG) in the photon counting regime. For each infrared driving pulse, we will target to produce statistically less than a single extreme ultraviolet (XUV) photon through HHG. The extremely low signal per driving pulse will be compensated by the high repetition rate of the source in the 100MHz range, thanks to the cavity-enhanced HHG technique. Several experimental schemes will be presented to study the quantum optical nature of the HHG process.

In this study, we investigate the generation of high-order harmonics in two-dimensional hexagonal materials. We demonstrate that a simplified two-band model, consisting of valence and conduction bands described by the Bloch Hamiltonian, effectively captures the essential theoretical framework. A more rigorous analysis, employing time-dependent density functional theory, extends beyond the scope of this work. The high-harmonic spectra are derived by Fourier transforming the electron velocity, with sampling across the Brillouin zone that accounts for topological phase transitions. These transitions induce the closing and reopening of the band gap, influencing the nonlinear response.

Christian Hünecke, Ivan Gonoskov, Stefanie Gräfe Institute of Physical Chemistry, Friedrich Schiller University Jena, Helmholtzweg 4, 07743 Jena, Germany A recently developed theoretical framework [1] allows the calculation of non-classical light by combining quantum-mechanical electronic dynamics driven by classically treated fields and the quantized harmonic radiation that arises from them. Solving these coupled equations becomes computationally infeasible especially for multiple emitters and high photon numbers of the generated non-classical light. We present a scheme which circumvents the numerical limitations using the time-dependent parametricalized electronic response from atomic and periodic systems and translates them in a form which can be used to directly obtain the corresponding non-classical radiation, similar to the specific scheme proposed in [2]. The electronic response is obtained within different regimes and various quantum systems, both within the single-electron approximation and multi-electron systems or semiconductors by solving the TDSE. We will also discuss limitations of scheme and possible extensions. References [1] Ivan Gonoskov and Stefanie Gräfe, Light–matter quantum dynamics of complex laser-driven systems, J. Chem. Phys. 154, 234106, (2021). [2] Ivan Gonoskov, René Sondenheimer, Christian Hünecke, Daniil Kartashov, Ulf Peschel, and Stefanie Gräfe, Nonclassical light generation and control from laser-driven semiconductor intraband excitations, PRB 109, 125110, (2024).

Quantum light --- specifically light in non-coherent quantum states --- can now be generated with intensities high enough to induce nonlinear responses in atoms, leading to strong-field phenomena such as high harmonic generation [1-3] and above threshold ionization [4,5]. In our study, we explore the influence of quantum states of light on one of the key processes in this field: atomic double ionization. We have developed a theoretical framework to model the interaction between a two-electron atom and light in arbitrary quantum states, such as phase-squeezed coherent states or bright squeezed vacuum states. Our results reveal that the quantum state of light significantly impacts the atomic double ionization process, leading to substantial changes in ionization probability and correlated electron momentum distribution. We also find the sequential and nonsequential double ionization channels respond differently to quantum light [6]. These findings are expected to lay the groundwork for future experiments exploring double or multiple ionization processes driven by quantum light. [1] A. Gorlach et al., Nat. Phys. 19, 1689 (2023). [2] M. E. Tzur et al., Nat. Photonics 17, 501 (2023). [3] A. Rasputnyi et al., Nat. Phys. 20, 1960 (2024). [4] Y. Fang et al., Phys. Rev. Lett. 130, 253201 (2023). [5] S. Wang et al., Phys. Rev. A 108, 063101 (2023). [6] H. Liu, H. Zhang, X. Wang, and J. Yuan, Phys. Rev. Lett. 134, 123202 (2025).

We theoretically investigate the prospect of inducing a non-trivial, non-linear Hall response in Dirac materials obeying both inversion and time-reversal symmetries, specifically pristine graphene. This is possible by creating a non-thermal electronic distribution in the system by driving it with a finite duration ultrafast sub-cycle laser pulse. The resultant non-equilibrium state, generated by non-adiabatic transitions induced by the laser, can be made to break the general trigonal ($C3$) symmetry of the Hamiltonian and hence allow for a second-order Hall response to a weak electric probe, closely related to the quantum geometry of the system.

The dynamics of light-matter interactions in the realm of strong-field ionization has been a focal point and has attracted widespread interest. We present the \c{eTraj.jl} program package, designed to implement established classical/semiclassical trajectory-based methods to determine the photoelectron momentum distribution resulting from strong-field ionization of both atoms and molecules. The program operates within a unified theoretical framework that separates the trajectory-based computation into two stages: initial-condition preparation and trajectory evolution. For initial-condition preparation, we provide several methods, including the Strong-Field Approximation with Saddle-Point Approximation (SFA-SPA), SFA-SPA with Non-adiabatic Expansion (SFA-SPANE), and the Ammosov-Delone-Krainov theory (ADK), with atomic and molecular variants, as well as the Weak-Field Asymptotic Theory (WFAT) for molecules. For trajectory evolution, available options are Classical Trajectory Monte-Carlo (CTMC), which employs purely classical electron trajectories, and the Quantum Trajectory Monte-Carlo (QTMC) and Semi-Classical Two-Step model (SCTS), which include the quantum phase during trajectory evolution. The program is a versatile, efficient, flexible, and out-of-the-box solution for trajectory-based simulations for strong-field ionization. It is designed with user-friendliness in mind and is expected to serve as a valuable and powerful tool for the community of strong-field physics.

The random walk—both classical and quantum—has laid the foundation for broad applications across computer, macromolecule science and technology and financial stock market. To date, photonic implementations of cascade random walks have been expanded from real space to orbital angular momentum (OAM) space. In this work, we demonstrate a novel high-dimensional random walk in the OAM space of light using solid-state high-harmonic spectroscopy. This discovery could establish a promising platform based on a novel quantum protocol for exploring quantum computations.

Uncertainty products play a central role in quantum mechanics as simple expressions that encode fundamental limits imposed in measurements of conjugate observables. For example, the classical Fourier limit in non-sequential two-photon ionization constraints the spectral information probed by a coherent pulse as short delay times are enforced [1]. On the other hand, violations of certain types of joint uncertainty products are used to certify entanglement [2,3], which is critical in many quantum information protocols and have recently captured attention for its role in strong-field phenomena [4,5]. In this work, we derive general expression for the relation between the two-photon bandwidth and delay and study the conditions of the violation of its classical limit for arbitrary multimode quantum states of light. We show that there is an overall trend of the nonclassical correction being inversely proportional to the average number of photons. Moreover, when considering only the one-photon bandwidth, arbitrary resolution on the uncertainty relation can still be achieved at high intensity, however not due to time-energy entanglement. Our work sheds light on the long-standing notion of intensity scaling of quantum light advantages, demonstrating the statistical nature of such effects. References: [1] Phys. Rev. Lett. 103, 063002 (2009). [2] Phys. Rev. Lett. 62, 2205 (1989). [3] Phys. Rev. Lett. 92, 210403 (2004). [4] Nat. Phys. 20, 1960–1965 (2024). [5] Phys. Rev. X 15 (1), 011023 (2024).

Quantum optics and strong-field physics are joining forces in bringing a revolution to attosecond physics, particularly to high harmonic generation. Here we demonstrate a novel route to generating harmonics in non-classical quantum states carrying massive amount of photons. Starting with a strong laser pulse in a coherent state at a fundamental frequency, $\omega$, and a quantum matter in an uncorrelated ground state (e.g., atomic system), the excitation of the matter by the generated harmonic light at a particular harmonic, $n_{0}\omega$, leads to a variety of non-classical effects, from squeezing of every harmonic to entanglement between harmonics. Building on above results, we will describe the cases involving different quantum matter dynamics, including solid-state systems driven by two-color fields, paving the way to the generation of non-classical bright states of light in solid-state devices.

High bright x-ray free electron laser (XFEL) with photon energy of hundreds to thousands eV can directly ionize the inner shell electrons of atoms, inducing sequential ionization processes such as Auger decay, electron impact excitation, ionization, and electron attachment. Dozens of electrons can be ionized from complex atoms within a few femtoseconds to tens of femtosec-onds, resulting in a non-equilibrium plasma con-taining dozens of ionization stages of ions with exotic electronic configurations and hundreds of eV temperature free electrons. One of the most interesting physical processes following the highly ionization and excitation of the target is the possible atomic X-ray lasing between the energy levels of the highly charged ions in the plasma. By using a one-dimensional Maxwell-Bloch equation, the propagation of the emitted x-rays and the time- and space-dependent atomic kinet-ics driven by XFEL are coupled together to simulate the lasing and amplifying processes of the atomic photon emissions of highly ionized ions in the plasma. The population evolutions of the charged ions, the emission spectra and the gain curves of some lasing tran-sitions are given for references. One can find the changes of emission spectra with propagation distances and the saturation of some lasing lines.

Limited by vacuum ultraviolet laser technology and the extremely small transition moment of atomic nuclei, the interaction between light and atomic nuclei is typically very weak, confined to linear and perturbative regimes. This hampers the ability to achieve significant nuclear excitation probabilities, greatly restricting effective control over atomic nuclei and impeding potential applications such as nuclear clocks[1], nuclear lasers[2], and nuclear energy storage[3]. Therefore, it is imperative to find ways to fully influence atomic nuclei using light. Research has revealed a peculiar property in hydrogen-like thorium-229 ions (229Th89+), where the presence of outer 1s electron significantly shortens the nuclear γ decay half-life for 5-6 orders of magnitude [4,5]. Leveraging this property, in our study, we combined 229Th89+ ions with intense lasers currently obtainable, pushing the interaction of light with atomic nuclei into a highly nonlinear and non-perturbative regime, implying that strong lasers can profoundly influence atomic nuclei[6,7]. By utilizing this high degree of nonlinear interaction, excitation probabilities exceeding 10% per single atomic nucleus can be achieved with a single femtosecond laser pulse. Moreover, under strong laser driving, 229Th89+ ions emit high-order harmonics at multiples of the laser frequency. Such radiation not only substantiates the highly nonlinear interaction between light and atomic nuclei but also holds promise in applications such as probing nuclear parameters and obtaining ultra-short pulses. These outcomes have opened up a new realm of research into light-matter interactions, providing a potent means for effective control over atomic nuclei and paving the way for novel approaches to nuclear coherent light emission. [1] E. Peik, and C. Tamm, Europhys. Lett. 61, 181 (2003). [2] E. V. Tkalya, Phys. Rev. Lett. 106, 162501 (2011). [3] P. Walker, and G. Dracoulis, Nature 399, 35-40 (1999). [4] S. Wycech, and J. Źylicz, Acta Phys. Pol. B 24, 637 (1993). [5] V. M. Shabaev et al., Phys. Rev. Lett. 128, 043001 (2022). [6] H. Zhang, T. Li, and X. Wang, Phys. Rev. Lett. 133, 152503 (2024). [7] H. Zhang, T. Li, and X. Wang, Phys. Rev. C 111, 044614 (2025).