High-harmonic generation as a tunneling delay probe

TL;DR

The paper investigates high-harmonic generation as a complementary diagnostic of tunneling delay in strong-field ionization by linking TDSE-based time-frequency HHG with classical trajectory analysis. An effective delay $\tau_d$ is extracted from barrier traversal, showing $\tau_d$ scales roughly as $\tau_d \propto 1/\sqrt{I_0}$ and collapses across atomic species when expressed via the Keldysh parameter $\gamma$, indicating a universal, barrier-geometry-driven dynamics in the tunneling regime. The approach provides internal consistency with attoclock trends, offering a robust cross-check rather than a direct tunneling-time measurement, and positions HHG as a complementary perspective on ultrafast electron dynamics. These findings highlight the central role of instantaneous field strength and barrier width in governing tunneling dynamics, with potential applicability to more complex targets and nonadiabatic regimes.

Abstract

We investigate the feasibility of using high-harmonic generation (HHG) as a complementary probe of tunneling delay in strong-field ionization. By combining time--frequency analysis of HHG spectra obtained from full time-dependent Schrödinger equation (TDSE) simulations with classical three-step-model (TSM) trajectories, we extract an effective tunneling delay associated with electron motion through the laser-suppressed Coulomb barrier. The analysis is carried out for Hydrogen, Helium, and Argon atoms over a range of laser wavelengths and peak intensities within the tunneling regime. The extracted delay exhibits a systematic dependence on the instantaneous field strength and barrier width at the ionization time, and follows the expected $τ_d \propto 1/\sqrt{I_0}$ scaling consistent with Keldysh--Rutherford-type models and attoclock observations. When recast in terms of the Keldysh parameter, the tunneling delay collapses onto a near-universal trend across different atomic species. While HHG does not provide a direct measurement of tunneling time, the present results demonstrate that it can serve as a robust, internally consistent diagnostic of tunneling dynamics, offering an independent and complementary perspective to established attoclock techniques.

Figures (6)

• Figure 1: The time-frequency analysis of HHG for the case of Helium atom is calculated along with the ionization and the recombination times as estimated using classical three-step-model (a), along with the HHG spectra in (b). Temporal profile of the laser pulse is shown in (c), such that $E(t_\text{ioni}) \equiv E_{\rm ioni}$ with $t_\text{ioni} \sim 1.8 \tau_0$ being the ionization time associated with the emission of the highest energy photon at recombination time $t_\text{reco} \sim 2.45 \tau_0$. The 1D effective potential $V_{\rm eff}(x) = V(x) + x E_{\rm ioni}$ is presented in (d), and the intersections with the ionization potential ($I_p \sim -0.9038$ a.u.) are marked as $x_1$ and $x_2$. The classical trajectory of the electron is shown in (e) after the ionization at $t_\text{ioni}$. The zoomed version of the (e) is presented in (f). The ionization time $t_\text{ioni}$ is shown by the vertical line in (a), (c), (e) and (f). The time ($\tau_d$) electron takes to cover $\ell_b$ is estimated from (f). Laser with pulse duration 4 cycle (sin$^2$ profile), 800 nm with peak intensity of $10^{15}$ W/cm$^2$ is used here.
• Figure 2: In (a), the ionization and the recombination times estimated using classical three-step-model are compared for the cases when the Couloumb potential is not included (NC) and when it is included (WC) while solving the classical equation of motion. All the physical parameters are same as considered in Fig. \ref{['fig1']}. The corresponding classical trajectory associated with the highest photon energy emission is also compared for these NC and WC cases (b). In (b), we have also presented the scaled up temporal profile of the dipole moment along the laser polarization direction, $d_z$, and also the expectation value of the radial coordinate i.e. $\braket{r}$. The $d_z$ and $\braket{r}$ is calculated though our TDSE solver. The $t_\text{ioni} \sim 1.8 \tau_0$ and $t_\text{reco} \sim 2.45 \tau_0$ are also shown by two vertical lines in both (a) and (b).
• Figure 3: The time-frequency and the classical trajectory analysis of the HHG for Helium is presented for wavelengths (intensity) as 800 nm ($1.5\times 10^{15}$ W/cm$^2$) in (a) and 1200 nm ($0.5\times 10^{15}$ W/cm$^2$) in (b). The temporal axis is plotted in terms of their respective optical cycles corresponding to 800 nm (a) and 1200 nm (b). The tunneling delay as a function of the peak intensity of the laser pulse is compared for 800 nm and 1200 nm wavelength cases. The laser pulse duration is considerd to be 4 cycles of respective wavelengths. The solid curve in (c) shows the $\propto 1/\sqrt{I_0}$ scaling for both the wavelength cases.
• Figure 4: The time-frequency and classical trajectory analysis of the HHG for Hydrogen with 1200 nm (1.5$\times 10^{14}$ W/cm$^2$) (a), and for the Argon with 800 nm (2$\times 10^{14}$ W/cm$^2$) laser (b) are presented. The variation of the tunneling delay with the peak intensity are compared for three different wavelengths for Hydrogen (c) and Argon (d). The solid lines in (c) and (d) represent the $\propto 1/\sqrt{I_0}$ scaling.
• Figure 5: Variation of the tunneling delay with the electric field strength at ionization ($|E_\text{ioni}|$) (a) and the barrier width (b) are compared for Hydrogen, Helium and Argon. For all the cases the wavelength is considered to be 1200 nm, and the peak intensity is varied from $0.5\times 10^{14} - 1.5\times 10^{14}$ W/cm$^2$, and accordingly the $|E_\text{ioni}|$ and so the $\ell_b$ is estimated.