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arXiv:2512.03987

Fully quantum theory of strong-field driven tunable entangled multi-photon states in HHG

Sebastián de-la-Peña, Heiko Appel, Angel Rubio, Ofer Neufeld

quant-phSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations• Revised Dec 12, 2025 (v2)
v1 · Dec 3, 2025v2 · Dec 12, 2025v3 · Mar 13, 2026
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Abstract

Quantum high-harmonic generation (HHG) is a growing field of research with capabilities of providing high photon-number entangled states of light. However, there is an open debate regarding the theory level required for correctly describing the quantum aspects of HHG emission, such as squeezing or entanglement. Previous approaches have employed non-interacting classical ensembles of trajectories, or perturbation theory utilizing the classical trajectories as a starting point, missing out key entanglement features. In this Letter, we develop a full quantum theory for entanglement measures in HHG solving exactly the light-matter interaction Hamiltonian and employ it for evaluating the entanglement between emitted photons of different harmonics. For the first time, we reach qualitative agreement of theory with recent experiments showing that the R entanglement parameter decreases with increasing laser power for below-threshold harmonics. Our results indicate that fine-tuning the laser power could enhance HHG entanglement features, which are observed to oscillate with the driving power and exhibit local non-classical maxima structures. Similarly, our theory predicts that the oscillatory behavior of entanglement observed for below-threshold harmonics also appears for entanglement involving above-threshold harmonics. We also show that the long-range behavior of driven electronic trajectories can qualitatively change the resulting entanglement. Lastly, we show that focal averaging over classical degrees of freedom, which has thus far been ignored in quantum HHG theories, plays a key role in entanglement measures and can change the qualitative behavior of observables. Our work establishes the state-of-the art in exploring entanglement features in HHG, and paves way for analysis and engineering of 'truly-quantum' multi-photon states in the XUV and ultrafast regime for more complex matter systems.

Comments
8 pages, 4 figures

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Quantum high-harmonic generation (HHG) is a growing field of research with capabilities of providing high photon-number entangled states of light. However, there is an open debate regarding the theory level required for correctly describing the quantum aspects of HHG emission, such as squeezing or entanglement. Previous approaches have employed non-interacting classical ensembles of trajectories, or perturbation theory utilizing the classical trajectories as a starting point, missing out key entanglement features. In this Letter, we develop a full quantum theory for entanglement measures in HHG solving exactly the light-matter interaction Hamiltonian and employ it for evaluating the entanglement between emitted photons of different harmonics. For the first time, we reach qualitative agreement of theory with recent experiments showing that the R entanglement parameter decreases with increasing laser power for below-threshold harmonics. Our results indicate that fine-tuning the laser power could enhance HHG entanglement features, which are observed to oscillate with the driving power and exhibit local non-classical maxima structures. Similarly, our theory predicts that the oscillatory behavior of entanglement observed for below-threshold harmonics also appears for entanglement involving above-threshold harmonics. We also show that the long-range behavior of driven electronic trajectories can qualitatively change the resulting entanglement. Lastly, we show that focal averaging over classical degrees of freedom, which has thus far been ignored in quantum HHG theories, plays a key role in entanglement measures and can change the qualitative behavior of observables. Our work establishes the state-of-the art in exploring entanglement features in HHG, and paves way for analysis and engineering of 'truly-quantum' multi-photon states in the XUV and ultrafast regime for more complex matter systems.
High-harmonic generation (HHG) is a prototypical strong-field process in which intense light drives matter to emit radiation at integer multiples of the driving frequency. Extending HHG into the quantum-optical regime offers new opportunities to probe and control strongly nonlinear light-matter interactions using nonclassical states of light. Yet describing this regime requires a fully quantum treatment of the correlated electron-photon dynamics, which becomes computationally challenging for broadband, strongly squeezed fields. Here we solve the quantum-electrodynamical dynamics of a two-level system driven by bright squeezed vacuum in a converged multimode Hilbert space. Both the driving field and emitted harmonics are fully quantized, with the light-matter interaction treated nonperturbatively. This enables direct access to the multimode quantum state and its higher-order correlations beyond semiclassical sampling or perturbative descriptions. We show that squeezed-vacuum driving produces harmonic emission with qualitatively distinct second- and third-order photon correlations compared with coherent excitation. Moreover, back-action from the driven emitter strongly reshapes the incident squeezed field, generating pronounced Wigner-function negativities that evolve on attosecond timescales. Our results establish a fully quantum framework for broadband strong-field dynamics with squeezed light and provide a route to predicting and interpreting quantum-HHG experiments and their extension to more complex emitters.
High harmonic generation (HHG) is a highly nonlinear optical process in which radiation from a strong driving field is up-converted into its high-order harmonics. In atomic systems, this nonlinearity manifests itself through the intensity scaling of the emitted harmonics with the driving field strength. Despite the highly nonlinear nature of HHG, when the driving field is prepared in a classical Gaussian state and atomic depletion remains negligible, the quantum statistical properties of the generated harmonics retains classical Gaussian quantum statistics. Driving HHG with bright squeezed vacuum (BSV) light challenges this paradigm, as its enhanced field fluctuations can modify the statistical properties of the generated harmonics. In this work, we investigate the conditions under which BSV-driven HHG gives rise to non-classical Gaussian states, and identify the regimes where this Gaussian description breaks down. For bichromatic driving by a strong coherent field at frequency ωωω and a perturbative BSV field at 2ω2ω2ω, the even-harmonic response is approximately linear in the BSV quadrature, leading to non-classical multimode Gaussian entanglement in the harmonic field. We show that this state can be described as a distributed collective squeezed mode over the even-harmonic manifold, and characterize its covariance matrix, entanglement structure, and quantum teleportation fidelity as an operational benchmark. Our results highlight the potential of non-classically driven HHG as a platform for engineering Gaussian and non-Gaussian states of light in the extreme ultraviolet regime.
We develop a theoretical framework for high-harmonic generation (HHG) driven by quantum states of light based on a temporal-mode expansion of the electromagnetic field. This approach extends previous single plane-wave mode treatments to realistic pulse configurations and arbitrary multi-mode states of light, resolving conceptual inconsistencies arising from non-normalizable infinite plane waves and establishing consistency between analytical and numerical methods. We derive a correction factor that quantifies deviations from the diagonal approximation (in which the yield becomes a statistical average over classical-field simulations) both for the response of a single atom and in the many-atom regime. Our results confirms that the HHG spectrum for atoms driven by any quantum state of light in free space is accurately described by averaging semi-classical calculations over the Husimi distribution, with no observable genuine quantum effects in the spectrum. We also demonstrate that in the many-atom regime, the mean-field coherent-state approximation underlying this treatment does not preserve probabilities, although unitarity is restored by in the diagonal approximation. The absence of genuine quantum effects in the HHG yield is attributed to the large photon numbers (∼1011\sim 10^{11}∼1011) required to reach HHG intensities in free space, which render quantum fluctuations negligible. We discuss nanophotonic environments with ultrasmall mode volumes as potential platforms where few-photon strong-field processes could exhibit genuine quantum signatures.
High-harmonic generation (HHG) driven by intense quantum light has recently emerged as a new frontier in strong-field physics. Yet, the mechanism underlying HHG with nonclassical fields such as bright squeezed vacuum (BSV) light—characterized by a vanishing mean electric field—remains elusive. In this work, we develop a fully quantum theory for HHG under arbitrary quantum driving fields. We show that the fluctuation component of the dipole correlation operator plays a central role in HHG with quantum light and is essential for BSV-driven HHG. Furthermore, our framework reveals the photon statistics distribution within individual harmonics and across different harmonics, demonstrating that the statistics inherit the statistical properties of the driving field. This theory overcomes key limitations of existing approaches and uncovers new quantum features of HHG in the presence of nonclassical light.
High-harmonic generation is a light up-conversion process occurring in a strong laser field, leading to coherent bursts of extreme ultrashort broadband radiation [1]. As a new perspective, we propose that ultrafast strong-field electronic or photonic processes such as high-harmonic generation can potentially generate non-classical states of light well before the decoherence of the system occurs [2, 3]. This could address fundamental challenges in quantum technology such as scalability, decoherence or the generation of massively entangled states [4]. Here, we report experimental evidence of the non-classical nature of the harmonic emission in several semiconductors excited by a femtosecond infrared laser. By investigating single- and double beam intensity cross-correlation [5], we measure characteristic, non-classical features in the single photon statistics. We observe two-mode squeezing in the generated harmonic radiation, which depends on the laser intensity that governs the transition from Super-Poissonian to Poissonian photon statistics. The measured violation of the Cauchy-Schwarz inequality realizes a direct test of multipartite entanglement in high-harmonic generation [6]. This result is supported by the theory of multimodal detection and the Hamiltonian from which the effective squeezing modes of the harmonics can be derived [7, 8]. With this work, we show experimentally that high-harmonic generation is a new quantum bosonic platform that intrinsically produces non-classical states of light with unique features such as multipartite broadband entanglement or multimode squeezing. The source operates at room temperature using standard semiconductors and a standard commercial fiber laser, opening new routes for the quantum industry, such as optical quantum computing, communication and imaging.
High harmonic generation (HHG) is an extremely nonlinear effect generating coherent broadband radiation and pulse durations reaching attosecond timescales. Conventional models of HHG that treat the driving and emitted fields classically are usually very successful but inherently cannot capture the quantum-optical nature of the process. Although prior work considered quantum HHG, it remains unknown in what conditions the spectral and statistical properties of the radiation depart considerably from the known phenomenology of HHG. The discovery of such conditions could lead to novel sources of attosecond light having squeezing and entanglement. Here, we present a fully-quantum theory of extreme nonlinear optics, predicting quantum effects that alter both the spectrum and photon statistics of HHG, thus departing from all previous approaches. We predict the emission of shifted frequency combs and identify spectral features arising from the breakdown of the dipole approximation for the emission. Our results show that each frequency component of HHG can be bunched and squeezed and that each emitted photon is a superposition of all frequencies in the spectrum, i.e., each photon is a comb. Our general approach is applicable to a wide range of nonlinear optical processes, paving the way towards novel quantum phenomena in extreme nonlinear optics.