Real-Time Reconstruction of the Strong-Field-Driven Dipole Response

The reconstruction of the full temporal dipole response of a strongly driven time-dependent system from a single absorption spectrum is demonstrated, only requiring that a sufficiently short pulse is employed to initialize the coherent excitation of the system. We apply this finding to the time-domain observation of Rabi cycling between doubly excited atomic states in the few-femtosecond regime. This allows us to pinpoint the breakdown of few-level quantum dynamics at the critical laser intensity near $2\mathrm{TW}/{\mathrm{cm}}^2$ in doubly excited helium. The present approach unlocks single-shot real-time-resolved signal reconstruction across timescales down to attoseconds for nonequilibrium states of matter. In contrast to conventional pump-probe schemes, there is no need for scanning time delays in order to access real-time information. The potential future applications of this technique range from testing fundamental quantum dynamics in strong fields to measuring and controlling ultrafast chemical and biological reaction processes when applied to traditional transient-absorption spectroscopy.

Figure 1

Nonequilibrium response of matter: illustration of the probing of a nonequilibrium state of matter induced by a time-dependent perturbation $V(t)$ using ultrashort laser pulses (dark blue at $t_0$) to trigger a response (purple), which is then modified, e.g., by a strong external time-dependent electric field (red). From the measured absorption spectrum [Eq. (1)], the strongly driven response can be fully reconstructed if the initial excitation pulse is much shorter than the system’s dynamics [Eq. (2)]. In this case, the initial excitation near $t_0=0$ produces a causal response, and the response can be reconstructed using the Fourier transform of the measured spectrum (see Supplemental Material, Sec. 3 [37]).

Figure 2

Test of the reconstruction method: (a) Amplitude (blue curve) and phase induced by the NIR field (modulo $2\pi$, orange curve) of the numerically simulated time-dependent response function using a few-level model including two excited states. The red curve shows the electric field of the nonlinearly interacting laser pulse. (b) Calculated optical density from (a) according to Eq. (1). (c) Test of the reconstruction method of amplitude and phase of the response function by selecting (Gaussian filter window with its FWHM indicated by green lines) the plotted spectral range and taking the inverse Fourier transform of the observable spectrum (b) restricted to the causal domain ($t\ge 0$). (d) Same as (c), using a broader filter window (orange lines). Here also the fast oscillations in the response can be reconstructed.

Figure 3

Experimental setup and data: (a) Illustration of the experimental apparatus. High-harmonic generation neon gas target (HHG), split-mirror setup for setting the XUV-NIR time delay (SM), toroidal mirror (TM), XUV flat-field spectrometer (XUV spec). (b) Level scheme of the relevant doubly excited states of helium. The NIR pulse induces resonant couplings ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$, ${\mathrm{\Omega }}_3$ between the autoionizing doubly excited states and also leads to resonantly enhanced strong-field multiphoton ionization into the $N=2$ continuum at the highest intensities of up to $20\mathrm{TW}/{\mathrm{cm}}^2$. (c) NIR intensity scan of Autler-Townes splitting [34] in doubly excited helium around the $2s2p$ state at 60.15 eV at fixed time delay $\tau =7.4\pm 0.1\mathrm{fs}$. The high spectrometer resolution allows the observation of the line shape in great detail. White dashed lines I–IV indicate the spectra used in the reconstruction discussed in Fig. 4.

Figure 4

TDDM $d_{2s2p}(t)$ of the doubly excited helium $2s2p$ state: (a),(b) Reconstructed TDDM amplitude and phase change (modulo $2\pi$, blue), showing the emerging departure from the simple exponential decay during the interaction with the central part of the NIR pulse (red shaded area) for the intensities $I_{\mathrm{NIR}}=0.5$, $2.0\mathrm{TW}/{\mathrm{cm}}^2$. (c),(d) The TDDM develops several minima and phase steps indicative of Rabi oscillations due to resonant coupling between the $2s2p$ and $2p^2$ states. The onset of strong-field ionization and the resulting depletion of the states during the NIR pulse for higher intensities at $I_{\mathrm{NIR}}=6.0$, $20.0\mathrm{TW}/{\mathrm{cm}}^2$ is visible. The error bars (blue shaded area) show the standard deviations of the reconstructed time-dependent response. The orange curves show the calculated amplitude and phase evolution of the TDDMs using a few-level model, including configuration interaction $V_{\mathrm{CI}}$ pathways and multiphoton ionization, where the latter becomes increasingly important at the highest laser intensities. The green lines represent ab initio calculations.

Figure 5

The onset of complex strong-field dynamics in a small quantum system: (a) Amplitude of the reconstructed response at time $t=7.5\mathrm{fs}$ during the interaction with the NIR pulse (at time delay $\tau =7.4\mathrm{fs}$). (b) Amplitude of the reconstructed response at real time $t=20\mathrm{fs}$, i.e., after the interaction with the NIR pulse has concluded. The few-level model starts to significantly deviate from experiment and ab initio simulation between 2.0 and $6.0\mathrm{TW}/{\mathrm{cm}}^2$ (c) The population ratio between the population in 24 excited states excluding the four states used in the few-level simulation and the population in these four states after the strong-field interaction at $t=20\mathrm{fs}$, extracted from the ab initio simulation. This explains the breakdown of the few-level model: an abrupt increase of the state space involved in the dynamics that sets in at intensities between 2.0 and $6.0\mathrm{TW}/{\mathrm{cm}}^2$. For even higher intensities, the interpretation in a few-state framework is no longer reliable.