The time it takes a bound electron to respond to the electromagnetic force of light sets a fundamental speed limit on the dynamic control of matter and electromagnetic signal processing. Time-integrated measurements of the nonlinear refractive index1 of matter indicate that the nonlinear response of bound electrons to optical fields is not instantaneous; however, a complete spectral characterization of the nonlinear susceptibility tensors2—which is essential to deduce the temporal response of a medium to arbitrary driving forces using spectral measurements—has not yet been achieved. With the establishment of attosecond chronoscopy3, 4, 5, the impulsive response of positive-energy electrons to electromagnetic fields has been explored through ionization of atoms6 and solids7 by an extreme-ultraviolet attosecond pulse8 or by strong near-infrared fields9, 10, 11. However, none of the attosecond studies carried out so far have provided direct access to the nonlinear response of bound electrons. Here we demonstrate that intense optical attosecond pulses synthesized in the visible and nearby spectral ranges allow sub-femtosecond control and metrology of bound-electron dynamics. Vacuum ultraviolet spectra emanating from krypton atoms, exposed to intense waveform-controlled optical attosecond pulses, reveal a finite nonlinear response time of bound electrons of up to 115 attoseconds, which is sensitive to and controllable by the super-octave optical field. Our study could enable new spectroscopies of bound electrons in atomic, molecular or lattice potentials of solids12, as well as light-based electronics operating on sub-femtosecond timescales and at petahertz rates13, 14, 15.
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Extended data figures and tables
Extended Data Figures
- Extended Data Figure 1: Supercontinuum spectra and experimental set-up. (182 KB)
a, The supercontinuum spectra spans more than two octaves (1.1–4.6 eV), and is shown by the black line; the spectra of the individual channels ChNIR, Chvis, Chvis-UV and ChDUV are shown by the red, orange, blue and violet lines, respectively. b, Schematic illustration of the experiment set-up.
- Extended Data Figure 2: Spectral equalization by a metal–dielectric–metal (MDM) coating. (266 KB)
a, The designed reflectivity (red line) and spectral phase (black) of the MDM coating applied on the outer mirror of the double mirror module shown in Extended Data Fig. 1. b, Multi-octave spectrum at the exit of the synthesizer (red line) and upon reflection off the MDM mirror (black line).
- Extended Data Figure 3: Global phase and CEP difference in a sub-cycle regime. (134 KB)
a, Half-cycle, cosine-like (blue line) and sine-like (red line) waveforms, compared with the carrier wave (dotted blue and red lines) and envelope (dashed blue lines). b, Limits of CEP decomposition in the time domain. Retrieved CEP (solid lines) versus global-phase offset (dashed lines) as a function of the ratio between the FWHM of the intensity envelope and the period of the carrier wave for ϕG = π/4 rad (blue line), ϕG = π/2 rad (green line) and ϕG = 3π/4 rad (red).
- Extended Data Figure 4: Simulated global-phase spectrograms. (158 KB)
a, b, Calculated global-phase (ϕG) spectrograms using the adiabatic model (a) and the TDSE model (b) for experimentally sampled waveforms (optical attosecond pulse) and an intensity of about 4 × 1013 W cm−2. c, Simulated global-phase spectrograms for a sub-cycle pulse (about 2 fs; shown in inset). The polarization response is insensitive to global-phase variation of the waveform; see also experiments in Fig. 3. The colour scale represents the spectral intensity in arbitrary units.
- Extended Data Figure 5: Nonlinearity study of the interaction of an optical attosecond pulse with a Kr atom. (205 KB)
a–d, Data points show the spectrally integrated yield of the nonlinear polarization (0–8 eV) as a function of the intensity of the optical attosecond pulse for two global phases ϕG of the driving waveform, ϕG ≈ 0 (a, c) and ϕG ≈ π/2 rad (b, d), evaluated from TDSE simulations (a, b) and our experiments in Fig. 4 (c, d ). The error bars represent the standard error of the mean of the input field intensity and the calculated integrated spectral yield of the nonlinear polarization. Δ is the slope of the linear fitting of the log–log plots; the errors shown indicate the standard error of mean, with n = 3. Insets show temporal profiles of the driving electric fields in arbitrary units.
- Extended Data Figure 7: Synthetic global-phase spectrograms and nonlinear dipole fittings. (427 KB)
a–c, Synthetic global-phase (ϕG) spectrograms determined using the simple model (equation (1) with the same set of parameters (a, b, c) and delays dt = 0 as (a), dt = 20 as (b) and dt = 30 as (c). The colour scale represents spectral intensity in arbitrary units. d, e, Fits (red lines) of the spectrally filtered (d, 5.5–8 eV; e, 0–8 eV) nonlinear dipoles (ρ(t), black lines) with ϕG ≈ 0 (left column) and ϕG ≈ π/2 rad (right column) using equation (1) and 6 × 1013 W cm−2.
- Extended Data Figure 8: Reconstruction of the global-phase spectrogram calculated by the TDSE model. (338 KB)
a, Calculated spectrogram at different global-phase settings (ϕG = −1.2π−1.2π rad) of the optical attosecond pulse using the TDSE model. b, Reconstructed spectrogram using the model of equation (1). The colour bar represents spectral intensity in arbitrary units. c, d, Calculated nonlinear dipoles using the TDSE model at ϕG ≈ 0 (c) and ϕG ≈ π/2 rad (d) are plotted in black; the reconstructed nonlinear dipoles and the adiabatic dipoles are shown in red and blue, respectively.
- Extended Data Figure 9: Intensity dependence of the nonlinear dipole delay. (54 KB)
a, b, The delay between the adiabatic dipole and the calculated dipole using the TDSE model as a function of the peak intensity according to TDSE simulations (a) and evaluated from the experimental data (b; see Fig. 4).