Quantum Trajectory Separation and Attosecond Mapping in Liquid High-Harmonic Generation

Abstract

High-harmonic generation (HHG) from liquids offers a potential pathway to attosecond spectroscopy in chemically complex and disordered environments, yet fundamental questions remain open: whether liquid harmonic emission preserves well-defined attosecond synchronization, and whether harmonic emission can involve simultaneous contributions from multiple quantum trajectories with distinct excursion times despite strong disorder and scattering. Here, we address these issues experimentally by resolving the trajectory-dependent temporal structure of liquid HHG. By optimizing the laser focusing geometry, we achieve clear spatial discrimination of short- and long-trajectory contributions, providing direct evidence for the existence of multiple quantum trajectories in liquids. Using a phase-controlled two-color driving field, we independently retrieve the attochirp associated with each trajectory and demonstrate opposite energy-time correlations for short and long trajectories, establishing a trajectory-resolved energy-time mapping in liquid HHG. All observations are well reproduced by semiclassical recollision simulations. These results place liquid HHG on the same conceptual footing as gas- and solid-phase HHG and establish a robust foundation for attosecond-resolved spectroscopy of ultrafast electronic and chemical dynamics in liquid environments.

Figures (4)

• Figure 1: (a) Experimentally measured laser focal spot profile at the position $z = 5$ mm. (b) Corresponding spatial distribution of the generated liquid high-order harmonics. (c)-(d) Same measurements performed at $z =$$-$1.5 mm. Here, $z>0$ and $z<0$ denote the liquid sheet placed behind and before the focal plane, respectively.
• Figure 2: (a) Far-field spatially resolved harmonic spectrum simulated for the large beam size shown in Fig. \ref{['fig1']}(a). (b) Time–frequency spectrogram of the on-axis harmonic emission corresponding to panel (a). (c) Same as (a), but for the small beam size shown in Fig. \ref{['fig1']}(c). (d) Time–frequency spectrogram of the off-axis harmonic emission at $\theta = 5$ mrad in (c).
• Figure 3: (a) Schematic of the experimental setup of the two-color driving scheme. (b) The spatial distribution of the 8th harmonic (H8) measured at the large beam spot in Fig. \ref{['fig1']}(a) as a function of the time delays of the two-color field. (c) Same as (b), but for the results measured at small beam spot in Fig. \ref{['fig1']}(c). (d) Spatially integrated harmonic intensities of even harmonics (H6-H10) from the short trajectory measured as a function of the two-color time delays. (e) Same as (d), but for the long trajectory. In (d)-(e), the solid lines represent the Fourier fitting of the experimental data.
• Figure 4: (a) Comparison between the experimental and simulated time–frequency mapping of liquid HHG. Open circles with error bars denote the experimental data for the short (green) and long (blue) quantum trajectories. Solid lines represent linear fits to the experimental data, yielding attochirps of $53.2~\mathrm{as/eV}$ and $-45.6~\mathrm{as/eV}$, respectively. Solid symbols indicate the calculated trajectories corresponding to three microscopic mechanisms: mean-free-path-limited (MFP-limited, triangles), nearest-neighbor (NN, squares), and next-nearest-neighbor (NNN, dots) recombination. (b) Schematic illustration of the microscopic electron trajectories in the liquid environment. The arrows depict the recollision pathways associated with the MFP-limited (dark blue solid line), NN (orange dashed line), and NNN (purple dotted line) mechanisms shown in (a). The inset displays the oxygen–oxygen radial distribution function of liquid water, with the first and second solvation shells indicated.