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arXiv:2602.16454

Edge states and quantum optical high-harmonic generation from topological insulators

Christian Saugbjerg Lange, Lars Bojer Madsen

quant-phSubmitted Feb 18, 2026• Announced Feb 19, 2026
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Abstract

The strong-field process of high-harmonic generation (HHG) has, in recent years, been treated from a quantum optical perspective in the emerging research area of strong-field quantum optics. These investigations show that HHG radiation is, in general, in a nonclassical state of light. However, the quantum optical treatment of HHG from topological nontrivial materials is missing. Here, we aim to address this gap in current knowledge and consider the quantum optical HHG response from the Su-Schrieffer-Heeger model, a finite chain of atoms with both a topologically trivial and nontrivial insulating phase, the latter supporting edge states. We find that HHG from both topological phases is squeezed at the band-gap frequency. Interestingly, while the harmonic spectrum discriminates the two topological phases of the system, the degree of squeezing only discriminates the phases for smaller chain lengths. We attribute this difference to a relative increase in overlap between bulk and edge states in the topological nontrivial phase for smaller systems. Our findings reveal how the strength of dipole couplings governs the nonclassical HHG response and define new research questions on topologically protected generation of quantum light in strong-field physics.

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The strong-field process of high-harmonic generation (HHG) has, in recent years, been treated from a quantum optical perspective in the emerging research area of strong-field quantum optics. These investigations show that HHG radiation is, in general, in a nonclassical state of light. However, the quantum optical treatment of HHG from topological nontrivial materials is missing. Here, we aim to address this gap in current knowledge and consider the quantum optical HHG response from the Su-Schrieffer-Heeger model, a finite chain of atoms with both a topologically trivial and nontrivial insulating phase, the latter supporting edge states. We find that HHG from both topological phases is squeezed at the band-gap frequency. Interestingly, while the harmonic spectrum discriminates the two topological phases of the system, the degree of squeezing only discriminates the phases for smaller chain lengths. We attribute this difference to a relative increase in overlap between bulk and edge states in the topological nontrivial phase for smaller systems. Our findings reveal how the strength of dipole couplings governs the nonclassical HHG response and define new research questions on topologically protected generation of quantum light in strong-field physics.
The strong-field induced generation of nonclassical states of light is not only a subject of fundamental research but also has potential usage in quantum information science and technology. The emerging field of strong-field quantum optics has developed ways of generating nonclassical states of light from the process of high-harmonic generation (HHG) at much wider frequency ranges and intensities than is typical for quantum optics. Here, we study the response from an exciton in a Mott-insulating system, using the extended Hubbard model. We find that the exciton plays a key role in the nonclassical response and generates squeezing at the exciton energy, showing a clear mapping between the quantum nature of the electronic system and the quantum nature of emitted radiation. We relate this nonclassical response to the nonvanishing time correlations of the current operator in the system. Our work defines a direction for experimental work to search for squeezed light from HHG in a spectrally confined region defined by the exciton energy.
High-harmonic generation (HHG) has emerged as a pivotal process in strong-field physics, yielding extreme ultraviolet radiation and attosecond pulses for a wide range of applications. Furthermore, its emergent connection with the field of quantum optics has revealed its potential for generating nonclassical states of light. Here, we investigate the process of high-harmonic generation in semiconductors under a quantum optical perspective while using a Bloch-based solid-state description. Through the implementation of quantum operations based on the measurement of high-order harmonics, we demonstrate the generation of nonclassical light states similar to those found when driving atomic systems. These states are characterized using diverse quantum optical observables and quantum information measures, showing the influence of electron dynamics on their properties. Additionally, we analyze the dependence of their features on solid characteristics such as the dephasing time and crystal orientation, while also assessing their sensitivity to changes in driving field strength. This paper provides insights into HHG in semiconductors and its potential for generating nonclassical light sources.
High-order harmonic generation (HHG) is a significant nonlinear phenomenon triggered by the interaction of intense laser fields with matter. As a high-quality coherent light source, it has found extensive applications in areas such as attosecond pulse generation and ultrafast spectroscopy. Recent studies have revealed the prevalent existence of nonclassical effects in high-harmonic radiation, including nonclassical photon statistics, intermodal correlations, and entanglement. It holds broad development prospects in directions including the generation of nonclassical states with high photon numbers, precise control of entanglement in solid-state systems, and integration with cavity quantum electrodynamics. This review systematically summarizes the research progress concerning the nonclassical effects in HHG. It begins with an overview of the semiclassical model and strong-field quantum theory, clarifying the differences in the quantum properties of harmonic fields when driven by coherent states versus nonclassical states. Subsequently, it focuses on various physical schemes for generating nonclassical optical field states, such as optical cat states and squeezed states, via this process. Research on strong-field HHG not only deepens our understanding of the fundamental nature of intense laser physics but also opens new avenues for the generation and application of quantum light sources in the extreme ultraviolet to X-ray spectral regions.
In recent years, strong-field processes such as high-order harmonic generation (HHG) and above-threshold ionization driven by nonclassical states of light have become an increasingly popular field of study. The theoretical modeling of these processes often applies an approximate phase-space expansion of the nonclassical driving field in terms of coherent states, which has been shown to accurately predict the harmonic spectrum. However, its accuracy for the computation of quantum optical observables like the degree of squeezing and photon statistics has not been thoroughly considered. In this work, we introduce this approximative phase-space description and discuss its accuracy, and we find that it mischaracterizes the quantum optical properties of the driving laser by making it an incoherent mixture of classical states. We further show that this error in the driving field description maps onto the light emitted from HHG, as neither sub-Poissonian photon statistics nor quadrature squeezing below vacuum fluctuations can be captured by the approximative phase-space description. Lastly, to benchmark the approximative phase-space description, we consider the quantum HHG from a one-band model, which yields an exact analytical solution. Using the approximative phase-space representation with this specific model, we find a small quantitative error in the quadrature variance of the emitted field that scales with pulse duration and emitter density. Our results show that using this approximative phase-space description can mischaracterize quantum optical observables. Attributing physical meaning to such results should therefore be accompanied by a quantitative analysis of the error.
In recent years, the study of phenomena pertaining to non-linear properties of materials has seen a major upswing - especially in the context of High Harmonic Generation (HHG). In this thesis, an account of the current state of research in this area will be given. In addition, we seek to combine two seemingly disparate areas of research, both of which have been the subject of numerous years of intense research - namely those of HHG in solid state materials, and the optical response of strongly correlated quantum systems. To this end we will rely on solving the Fermi-Hubbard model coupled to an electromagnetic field under time propagation. With the aid of both numerical work and simple energy level analyses, we are able to explain most of the characteristics of our calculated spectra. High field strength will induce a Mott insulator to metal phase transition whose imprints will be established through time resolved high harmonic emission and doublon and spin correlation functions. The various HHG spectra, together with said correlation functions will be studied through an Exact diagonalization algorithm in conjunction with an Arnoldi-Lanczos time propagator. The parameter space considered is comprised of the electron-electron repulsion energy, U, and peak electric field strength. Additionally, the study is not restricted to half-filling, and results for doped materials will be presented. We find that the high U-induced energy gap in the Fermi-Hubbard model plays the same role as the valence-conduction bandgap in models like for instance the Semiconductor Bloch Equations when it comes to generating high harmonic radiation through multiphoton processes. The central part of this thesis aims at providing insight to and making further contributions to an article published in Nature Photonics in March 2018. [1]
It has been found previously that the presence or absence of topological edge states in the Su-Schrieffer-Heeger (SSH) model has a huge impact on harmonic generation spectra. More specifically, the yield of harmonics for harmonic orders that correspond to photon energies below the band gap is many orders of magnitude different in the trivial and topological phases. It is shown in this work that this effect is still present if electron-electron interaction is taken into account, i.e., if a Hubbard term is added to the SSH Hamiltonian. To that end, finite SSH-Hubbard chains at half filling are considered that are short enough to be accessible to exact diagonalization but already show edge states in the topological phase. We show that the huge difference in the harmonic yield between the trivial and topological phases can be reproduced with few-level models employing only the many-body ground state and a few excited many-body states.