High-harmonic generation driven by temporal-mode quantum states of light

TL;DR

This work extends HHG theory to a realistic temporal-mode description, resolving conceptual issues from infinite plane-wave driving and aligning analytical results with finite-pulse simulations. By deriving a correction factor $C_\ell(\alpha_m)$, the authors show that for typical HHG intensities ($I_{\text{HHG}} \sim 10^{14}$ W/cm$^2$) the factor remains within $\sim 10^{-4}$ of unity, validating the diagonal (single-mode) approximation and the Husimi-distribution averaging as an exact description in free space. Consequently, HHG driven by any quantum state of light can be reproduced by averaging semi-classical simulations over $Q(\alpha)$, with no genuine quantum advantage in free space. The results imply that quantum fluctuations are negligible due to the large photon numbers required to reach HHG intensities, though nanophotonic environments with ultra-small mode volumes could reveal few-photon quantum effects in strong-field processes.

Abstract

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, 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 single-mode approximation and show that it remains below $10^{-4}$ for intensities typical of HHG ($\sim 10^{14}~$W/cm$^2$). This result confirms that free-space HHG driven by any quantum state of light is accurately described by averaging semi-classical calculations over the Husimi distribution, with no observable genuine quantum effects. The absence of such effects is attributed to the large photon numbers ($\sim 10^{11}$) 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.

Figures (4)

• Figure 1: Illustration of the electric field pulse shape $E(t)$ used, given by a single-frequency carrier wave modulated with a flat-top pulse envelope. The envelope rises linearly over 5 cycles, becomes constant for 15 cycles, and then decreases linearly over 5 cycles. $E(t)$ is given in atomic units for a peak intensity of $I_0 = 10^{14}$ W/cm$^2$ and $t$ in units of the period $T = \frac{2\pi}{\omega_0}$.
• Figure 2: Deviation of the frequency-independent overlap factor $f_{\text{ov}}$ from unity over a range of $\delta\alpha$ large enough to avoid complete Gaussian suppression.
• Figure 3: Evaluation of the difference $1 -|f_l(\alpha_m, \delta\alpha)|$, represented by the colorbar, as a function of $\omega$ and $\delta\alpha$. Vertical bands are a consequence of numerical noise.
• Figure 4: Harmonic spectrum of the different light distributions for a constant mean intensity. The peaks at odd multiples of the driving frequency $\omega_0$ arise because the electric field $E_{\alpha}(t)$ reaches a maximum (in absolute value) every half cycle.