Quantum state engineering of light using intensity measurements and post-selection

Quantum state engineering of light is of great interest for quantum technologies, particularly generating non-classical states of light, and is often studied through quantum conditioning approaches. Recently, we demonstrated that such approaches can be applied in intense laser-atom interactions to generate optical “cat” states by using intensity measurements and classical post-selection of the measurement data. Post-processing of the sampled data set allows to select specific events corresponding to measurement statistics as if there would be non-classical states of light leading to these measurement outcomes. However, to fully realize the potential of this method for quantum state engineering, it is crucial to thoroughly investigate the role of the involved measurements and the specifications of the post-selection scheme. We illustrate this by analyzing post-selection schemes recently developed for the process of high harmonic generation, which enables generating optical cat states bright enough to induce non-linear phenomena. These findings provide significant guidance for quantum light engineering and the generation of high-quality, intense optical cat states for applications in non-linear optics and quantum information science.

💡 Research Summary

The paper presents a comprehensive theoretical investigation of a method for engineering intense, non‑classical optical “cat” states by exploiting high‑order harmonic generation (HHG) in a strong‑field laser‑atom interaction and applying intensity‑based measurements together with classical post‑selection of the recorded data. The authors begin by reviewing the recent surge of interest at the interface of quantum optics and strong‑field physics, emphasizing that while HHG has traditionally been described as a parametric process that maps an input coherent infrared (IR) driving field onto a coherent output field, this picture breaks down under conditions of high intensity, high gas density, or solid‑state targets where electron‑correlation effects, ground‑state depletion, and excitation become significant.

In the experimental configuration considered, a laser pulse described by a coherent state |√2 α⟩ is split by a 50:50 beam splitter. One arm drives HHG in an atomic gas, resulting in a depleted IR field |α + δ α⟩ and a set of harmonic modes initially in vacuum that become coherent states |χ_q⟩. After the interaction the IR beam passes through a second 50:50 beam splitter, producing a reflected mode (r) and a transmitted mode (t). Photodiodes record the photocurrents proportional to the photon numbers n_r(i) in the reflected arm and m_q(i) in each harmonic mode for each laser shot i.

The core contribution of the work is a rigorous formulation of the post‑selection (PS) scheme that respects total energy conservation across all optical modes. The authors point out that earlier experiments applied a simple condition n_r + ∑_q m_q = c, which accounts only for the measured reflected IR photons and the harmonics, neglecting the transmitted IR photons that also carry part of the energy. By introducing the exact photon‑number balance

 n_t(i) + n_r(i) + ∑_q m_q(i) = n_0 = |α|²,

they derive a PS operator

 Ô_PS = δ_{n_r+∑_q m_q, c} |n_t⟩⟨n_t| ⊗ Ô(n_r,{m_q}),

with c = n_0/2 reflecting the average split of photons between the reflected arm and the harmonics. This operator projects onto those shots in which the number of photons removed from the driving field (δ α) exactly matches the number of generated harmonic photons, thereby guaranteeing that the selected data set embodies the quantum correlations imposed by the HHG process.

Through extensive Monte‑Carlo simulations the authors compare three scenarios: (i) ideal coherent‑state statistics without HHG, (ii) HHG with the full energy‑conserving PS, and (iii) HHG with the simplified PS used in earlier work. They demonstrate that the full PS yields a reconstructed Wigner function with a significantly larger negative region, confirming genuine non‑classicality, whereas the simplified PS underestimates this effect. The simulations also explore the impact of realistic experimental imperfections such as laser‑pulse intensity jitter, ionization losses, and finite sampling in quantum tomography, showing how these factors broaden the photon‑number distributions and can mask the quantum signatures if not properly accounted for.

Finally, the authors argue that the presented framework is not limited to HHG. Any parametric nonlinear optical process that conserves photon number—such as optical parametric amplification, four‑wave mixing, or high‑order phase‑matched processes—can be treated with the same energy‑conserving PS formalism. This opens a pathway to generate high‑brightness, non‑classical states (cat, squeezed, or hybrid states) directly from intense laser sources, with the post‑selection step providing a flexible, experimentally accessible “knob” to tailor the quantum state without the need for low‑loss quantum‑optical components. The work thus offers both a deeper theoretical understanding of measurement‑conditioned state preparation in strong‑field regimes and practical guidelines for future experiments aiming at quantum‑enhanced nonlinear optics and quantum information processing with intense light.