The quantum-optical nature of high harmonic generation

High harmonic generation (HHG) occurs when matter in a strong laser field emits radiation at integer multiples of the driving frequency, enabling coherent EUV sources and attoscience. Classical and semiclassical theories (e.g., the three-step model and Lewenstein theory) accurately predict many features but treat the electromagnetic fields classically, leaving open fundamental questions about the quantum-optical nature of HHG. Specifically, it is unknown under what conditions the emitted spectrum and photon statistics significantly deviate from conventional HHG, whether genuinely quantum features (squeezing, bunching/antibunching, entanglement) arise, and whether an HHG photon should be viewed as a single-frequency particle or a superposition spanning the comb. The authors develop a fully quantum framework for extreme nonlinear optics (SFQED) to address these questions for single-atom and many-atom regimes, including beyond-dipole effects for the emitted field.

Semiclassical HHG models using a quantum electron with classical fields (e.g., three-step model; Lewenstein SFA) have been highly successful and extended with ab initio methods for atoms, molecules, and solids. Early QED approaches quantized the emitted field while keeping the drive classical, and later dressed-state formalisms treated emission as spontaneous transitions among laser-dressed states. Some recent studies quantized the driving field experimentally and theoretically. However, the conditions for strong deviations in spectral/statistical properties and clearly quantum features of HHG light remained unclear. Prior beyond-dipole analyses largely addressed the driving field, not the emitted field. The present work builds on and extends these by fully quantizing both drive and emission and incorporating beyond-dipole corrections for emitted radiation.

The authors construct a strong-field QED (SFQED) formalism where both the driving and emitted fields are quantized. Key steps: (1) Apply a unitary transformation to decompose the vector potential A into a classical time-dependent component A_c(t) corresponding to the coherent driving state and a quantum fluctuation A_q describing the emitted field. (2) Solve the time-dependent Schrödinger equation (TDSE) for the electronic system driven by A_c(t): iħ∂|ψ(t)⟩/∂t = H_TDSE|ψ(t)⟩ with H_TDSE = (p − qA_c(t))^2/2m + U (+ H_F as applicable). This step uses established analytical/numerical TDSE techniques. (3) Compute emission perturbatively to first order in A_q (weak coupling to emitted modes), yielding a combined electron–photon state where the electronic state |ψ(t)⟩ (laser-dressed) transitions to time-dependent final states |ψ_j(t)⟩ while creating photons |kσ⟩. In the dipole approximation, the first-order wavefunction includes time integrals of dipole matrix elements ⟨ψ(τ)|qr|ψ_j(τ)⟩ weighted by e^{iω_k τ}. Squaring and integrating over modes gives the spectrum dε/dω. This construction reveals entanglement between photonic and electronic degrees of freedom and that each emitted photon is a superposition over frequencies. Beyond-dipole emitted-field corrections: Starting from the full minimal-coupling Hamiltonian H = (p − eA/c)^2/2m + U + H_p without the dipole approximation in A for the emitted field, first-order perturbation in the many-atom regime yields a generalized spectral density de/(dω dΩ) ∝ (ω^2/16π^2ε_0 c^3 m^2) Σ_i |∫ P_i(t) e^{iω t} dt|^2, modifying the standard dipole formula. Numerical simulations: TDSE solved for a 1D helium-atom model with absorbing boundaries using a 3rd-order split-step method. Driving field: λ0 = 800 nm, intensity I = 2×10^14 W/cm^2, trapezoidal pulse with 15 optical cycles rise and fall and a 10-cycle plateau. Single-atom and many-atom emissions are computed from matrix-element contributions; many-atom coherent (∝ N^2) vs incoherent (∝ N) scaling is analyzed. Photon statistics: Compute Mandel parameter Q = ⟨n̂^2⟩/⟨n̂⟩ − 1 and squeezing η = 10 log10(4ΔX^2) per harmonic, with dependence on the number of phase-matched atoms N_p evaluated. Proposed experiment: Field autocorrelation measurement of attenuated HHG to the single-photon regime to test the prediction that each photon carries the full comb (normalized autocorrelation independent of photon number).

• Fully quantum emission state: The first-order SFQED wavefunction shows electron–photon entanglement and that each emitted HHG photon is a coherent superposition over all comb frequencies (“each photon is a comb”), retaining full spectral information (including cutoff).
• Multiple shifted combs: In the single-atom regime, HHG spectra contain multiple shifted frequency combs associated with transitions between different initial and final time-dependent electronic states |ψ_j(t)⟩.
• Transition to many-atom regime: Many-atom emission yields predominately the conventional comb via ground-to-ground dressed transitions, with coherent scaling ∝ N^2; other channels contribute incoherently ∝ N. Quantitatively, in simulations |d_21|^2 is about 10^4 times larger than |d_12|^2; thus for N < 10^4, incoherent channels can produce observable deviations (“quantum corrections”) from the standard HHG spectrum.
• Photon statistics: • Single-atom regime: Emission is weak (negligible squeezing), but shows both super-Poissonian (Q>0) for lower harmonics (ground→ground-like channels) and sub-Poissonian (Q<0) at higher harmonics linked to excited→ground transitions; a sharp negative Q peak occurs at the electronic transition energy ΔE ≈ 10.85 ω0 between the first excited and ground states. Example values (from Fig. 3): Q ≈ 1.2×10^−7 (1st), −4.3×10^−7 (11th), 4.4×10^−10 (25th), −2.6×10^−11 (41st). • Many-atom regime: Super-Poissonian statistics across harmonics with increasing Q and observable squeezing as the number of phase-matched atoms N_p increases. Example Q (Fig. 4): for N_a=10^3, Q ≈ 0.16 (5th), 0.13 (15th), 0.35 (25th), 0.23 (37th); similar values for N_a=10^4 and 5×10^4. Squeezing η up to ≈3.1 dB (19th harmonic for N_a=5×10^4); other examples include η ≈ 0.87 dB (1st, N_a=10^4) and ≈1.2 dB (13th, N_a=10^4).
• Beyond-dipole emitted-field effects: Predict conditions where the emitted field breaks the dipole approximation as harmonic order increases. A key parameter x = a/λ ≈ n λ/λ0 (with electron quiver radius a) grows with harmonic order; when x → 1 (e.g., several-hundredth harmonics with λ in the nanometer range), multipolar corrections become significant. Consequences include angular- and frequency-dependent spectral modifications and the appearance of even harmonics even for monochromatic drive, especially when emission is not unidirectional.
• Experimental signature: Proposed field autocorrelation measurement after strong attenuation down to <1 photon per pulse; the normalized autocorrelation is predicted to be independent of photon number, confirming that each photon carries the full comb.
• Applicability: The SFQED framework is general and applies to atoms, molecules, and solids, and to other extreme nonlinear processes.

The developed SFQED framework addresses the central questions on the quantum-optical nature of HHG by providing a fully quantum description of both drive and emission. It explains how macroscopic HHG can exhibit non-classical photon statistics (super-Poissonian and squeezing) even in many-atom ensembles and clarifies that each emitted photon is a frequency-comb superposition entangled with the emitter. The theory predicts observable deviations from semiclassical HHG when the number of emitters is mesoscopic (e.g., N < 10^4) and when emitted-field beyond-dipole effects become relevant at high harmonic orders. These findings have implications for attosecond metrology, quantum sensing, and the generation of entangled EUV/soft X-ray states. The proposed autocorrelation experiment offers a practical route to validate the single-photon comb picture and distinguish it from classical or alternative quantum states. Overall, the results bridge strong-field physics with quantum optics, opening paths to engineer quantum features in extreme nonlinear light sources.

The paper introduces a fully quantum strong-field electrodynamics framework for HHG that (i) predicts multiple shifted combs in single-atom emission, (ii) quantifies coherent vs incoherent channels and their scaling in many-atom HHG, (iii) reveals non-classical photon statistics including super-Poissonian behavior and measurable squeezing in the many-atom regime, (iv) demonstrates that each HHG photon coherently spans the entire comb spectrum, and (v) identifies beyond-dipole emitted-field effects at high harmonic orders, including even-harmonic generation. The authors propose an autocorrelation experiment to verify the single-photon comb nature. Future directions include extending the formalism to many-electron systems and solids, incorporating relativistic and magnetic-dipole corrections, exploring many-body HHG with entanglement and nontrivial photon statistics, and developing higher-order correlation measurements to further probe quantum features in extreme nonlinear optics.

• Many-body HHG: A complete many-body formulation remains open; current analysis treats single-atom and ensemble scaling without full many-body correlations.
• Numerical model: Simulations use a 1D helium atom and dipole approximation for most analyses; quantitative predictions may vary for full 3D, multi-electron, or solid-state systems.
• Experimental feasibility: Observing beyond-dipole emitted-field effects requires very high harmonic orders and specific emission geometries (non-unidirectional); current plasmonic HHG does not clearly exhibit such corrections for the drive.
• Photon statistics signals: Single-atom regime emissions are weak, making direct observation of sub-Poissonian features challenging; squeezing depends on the number of phase-matched atoms N_p (often much smaller than total N).
• Proposed autocorrelation: Single-photon-level measurements suffer from low signal-to-noise; optimization and higher-order correlation techniques may be needed.