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

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.

💡 Research Summary

In this paper the authors critically examine the “approximate positive‑P” (APP) phase‑space representation that has become a standard tool for modelling strong‑field processes driven by non‑classical light, such as high‑order harmonic generation (HHG) and above‑threshold ionization (ATI). The APP method expands an arbitrary quantum state of the driving laser in terms of coherent states, replaces the exact positive‑P distribution by a Gaussian kernel, and then approximates that kernel by a two‑dimensional Dirac delta. The result is a diagonal representation in which the laser is described by a smooth, positive Husimi‑Q function multiplied by coherent‑state projectors. While this construction makes numerical calculations tractable and has been shown to reproduce HHG spectra with high accuracy, the authors ask whether it can also capture genuinely quantum‑optical observables such as quadrature squeezing below the vacuum level or sub‑Poissonian photon statistics.

The paper begins with a general Hamiltonian for an electronic system interacting with a quantised field in the dipole approximation. By moving to a displaced interaction picture the authors separate the semiclassical laser field from the quantum fluctuations and derive equations of motion for the photonic wave packets associated with each electronic eigenstate. They then discuss two exact phase‑space representations: the Glauber‑Sudarshan (GS) P‑function, which is diagonal but often highly singular for non‑classical states, and the full positive‑P (PP) representation, which is always positive and smooth but non‑diagonal, leading to a high‑dimensional integral that is usually intractable.

The APP representation is introduced as a practical compromise: the PP distribution is expressed in terms of the Husimi Q‑function via a Gaussian convolution, and the Gaussian is replaced by a delta function. This yields the simple form

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