Limitations of an approximative phase-space description in strong-field quantum optics

TL;DR

This work analyzes the limits of the approximative positive P (APP) phase-space representation for driving HHG with nonclassical light. By formulating APP and benchmarking against a coherently driven one-band model with analytic solutions, the authors show that APP cannot reproduce quantum optical features such as quadrature squeezing below vacuum fluctuations or sub-Poissonian photon statistics, even when the HHG spectrum is reasonably accurate at high intensities. The results demonstrate that the driving field is effectively treated as a classical mixture, and this mischaracterization propagates to the emitted light, potentially misleading interpretations of nonclassicality. The study highlights the necessity of quantitative error assessment or alternative approaches when applying APP to predict quantum optical observables in strong-field systems.

Abstract

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.

Figures (2)

• Figure 1: Illustration of (a) a pulse of BSV and (b) multiple shots of a classical laser with stochastic amplitude and phase. Comparing the electric fields that would be measured for both fields, both have no mean field and an uncertainty that oscillates in time. The uncertainties are similar when intensity is large, but the minimum uncertainty in the stochastic laser (b) is bounded from below by $1/4$, whereas the BSV field (a) is not. As discussed in the main text, using the APP representation is equivalent to approximating (a) as (b).
• Figure 2: Relative error in quadrature variance in harmonic order $n=3$ introduced by the APP representation for a one-dimensional single-band solid as a function of the number of electrons in the material ($L$). Solid blue and dashed orange curves denote the error at the maximizing ($\theta_{\text{max}}$) and minimizing ($\theta_{\text{min}}$) angles, respectively, calculated from the highest-order terms in driving field intensity. The green curve is the second-highest-order term in the error at the maximizing angle $\theta_{\text{max}}$, and is seen not to contribute significantly.