Light waveforms whose spectra extend over more than two optical octaves in the visible and adjacent spectral range (about 1.1–4.6 eV) (Extended Data Fig. 1a) are generated by the nonlinear broadening of laser pulses (approximately 22 fs, 1 mJ, 790 nm) through a hollow-core fibre (HCF) filled with Ne gas (about 2.3 bar). The energy of the generated supercontinuum at the exit of the HCF is about 550 μJ. The spectra of these pulses are divided by dichroic beam-splitters into four almost equally wide spectral bands centred in the near infrared (NIR; about 1.1–1.75 eV), visible (vis; about 1.75–2.5 eV), visible-ultraviolet (vis-UV; about 2.5–3.5 eV) and deep ultraviolet (DUV; about 3.5–4.6 eV). The pulses in these bands are individually compressed by dispersive mirrors to durations of a few femtoseconds before they are spatially and temporally superimposed to yield a single beam/pulse at the exit of the apparatus. The pulses are temporally characterized by a transient-grating frequency-resolved optical gating (TG-FROG) apparatus. The durations T of the pulses in different channels of the synthesizer were measured to be T ≈ 8.5 fs, T ≈ 7 fs, T ≈ 6.5 fs and Τ ≈ 6.5 fs (see inset to Fig. 2a). The synthesizer apparatus transmits about 82% of the energy of the incoming supercontinuum. As a result, the pulse energy at the exit of the apparatus is about 320 μJ, and is distributed among the four channels as Ch ≈ 255 μJ, Ch ≈ 45 μJ, Ch ≈ 15 μJ and Ch ≈ 4 μJ, where Ch denotes the energy of channel i.
For the synthesis of optical attosecond pulses, both precise control of the relative delay between the constituent pulses in the synthesizer as well as an intensity control over the spectral channels is required. To this end, we followed a new approach, which effectively enables the passive spectral intensity control and facilitates the attosecond streaking characterization of the generated optical waveforms in the same set-up. The EUV attosecond probe is generated first (Extended Data Fig. 1b) by focusing the light transients from the synthesizer into a quasi-static Ne gas cell. The EUV radiation, which emerges collinearly to the driver waveform, is transmitted through a thin, round Zr foil, while the optical pulse, which is transmitted around the geometrical margins of this foil, forms an annular beam. A double-mirror module consisting of a concave, multilayer, coated inner mirror and a metal–dielectric–metal (MDM)-coated concave annular sector (outer mirror) (Extended Data Fig. 2) of the same focal length (f = 12.5 cm), focuses the light transients and the EUV attosecond probe into a second Ne gas nozzle placed near the entrance of a time-of-flight (TOF) spectrometer (Extended Data Fig. 1b). One of the essential characteristics of the MDM is that the imposed spectral control results in negligible phase distortions over the whole spectral range of the supercontinuum pulse. This was experimentally verified by FROG measurements of the pulses in the constituent channels upon reflection off the MDM unit. At the same time, because the EUV probe is generated before the spectral intensity control of the optical transient, it overcomes a fundamental limitation of a half-cycle field: the efficient generation of intense EUV-probe pulses. This is because a half-cycle field does not involve at least two intense field crests, which are required for ionization and subsequent acceleration of electrons27 leading to high harmonic generation.
Single-cycle optical pulses are generated by physically suppressing a part (>3 eV) of the high-frequency spectrum of the optical attosecond pulse.
We focus optical attosecond pulses into a quasi-static cell, which replaces the Ne gas jet used for performing attosecond-streaking characterization of their fields, filled with Kr atoms at a moderate pressure (about 80 mbar) (Extended Data Fig. 1b). We probe the nonlinear polarization of the system by recording VUV spectra that emerge collinearly with the driving optical field using a spectrometer placed downstream from the cell. The spectra are sampled at energies higher than about 5 eV, that is, beyond the constituent spectrum of the optical attosecond pulses, and extend no higher than the ionization threshold of neutral Kr (I ≈ 14 eV).
The concept of the carrier-envelope phase ϕ (CEP) is most readily understood in the time domain28. The carrier-envelope phase is the interval Δt between the maximum of the envelope and the maximum of the instantaneous field, translated in phase at the centroid frequency ω of the spectrum: ϕ = ω × Δt . Any electric field can be decomposed, according to Hilbert’s transform as
in which A(t) is the envelope (the modulus of the analytical field) and ϕ is the global (or absolute) phase.
The maximum of the field is determined by solving
On the basis of this equation and the estimation , we distinguish two different regimes:
(1) For pulses with durations longer than one cycle, and the solution of equation (2) yields ϕ = ω × Δt = ϕ , which confirms the equivalence between the global (ϕ ) and carrier-envelope (ϕ ) phases.
(2) For sub-cycle pulses, Δω/ω > 1 and so the solution of equation (2) is not trivial because, in a fraction of a cycle, the envelope varies substantially. Consequently, ω × Δt < ϕ and the global and carrier-envelope phases are no longer equivalent.
Extended Data Fig. 3a presents the case of a half-cycle pulse—an optical attosecond pulse. The sinusoidal waveform (red) reaches its maximum at 0.2 periods instead of 0.25, which is expected for a wave with a phase of π/2 rad. Extended Data Fig. 3b shows the CEP determined using the method of the field maximum depending on the pulse duration, for three settings of the global phase. This comparison reveals a considerable difference between ϕ and ϕ in the short pulse regime. In view of the above justification, it is clear that the CEP is an accurate description of the global phase only for pulses longer than one cycle.
To theoretically study the nonlinear dipole dynamics in Kr atoms exposed to intense optical attosecond pulses, we used two models. In the first, we solve the three-dimensional time-dependent Schrödinger equation (TDSE) within the single-active-electron approximation. To this end, we used a central potential for Kr, which was calculated using optimized effective-potential methods29. In the second, and in order to describe instantaneous response, we assumed an adiabatic model based on a two-level system12, 30, in which the dipole moment, in the quasi-static approximation, can be expressed as:
where d is chosen to match the nonlinear polarizability of Kr (refs 31, 32), ω is the excitation energy and E(t) denotes the electric field of the optical attosecond pulse. To access the nonlinear component of the induced electronic dipole moment at a given intensity of the driving field in both models, we perform a second calculation at a much lower (about six orders of magnitude) intensity. As a next step, we subtract the virtually linear dipole calculated at the lower intensity from the original one, after multiplying it by the corresponding ratio between the two intensities. The calculated global-phase spectrograms (spectral emission as a function of the global phase) using the adiabatic and TDSE models are shown in Extended Data Fig. 4a, b. In accordance with the discussion in the main text, the adiabatic model (Extended Data Fig. 4a) predicts uniform modulations of the spectral amplitude of the emitted spectral components as a function of the global phase ϕ . In contrast, the spectrogram calculated using the TDSE model embodies the signatures of the delayed electronic response (Extended Data Fig. 4b) in the form of asynchronous amplitude modulations between different frequencies/energies of the emitted dipole. These features are present in the entire emitted spectrum, not only close to the resonant area (10–14 eV). We show that these effects can be used to extract the dynamics of the nonlinear response by reconstructing the global-phase (ϕ ) spectrograms recorded in our experiments.
Extended Data Fig. 4c shows a global-phase spectrogram simulated for a single-cycle pulse using the TDSE Kr model. In this regime of single-cycle pulses, the global-phase spectrogram does not show discernible variation over ϕ , which is experimentally verified (see Fig. 3e–g).
An essential innovation introduced by using optical attosecond pulses is their unique capability to drive nonlinear dynamics in quantum systems without inducing a substantial degree of ionization or excitation, that is, without greatly altering the original system. In experiments where the polarization of the system is to be probed, both excessive ionization and excitations markedly modify the system, resulting in a considerable degree of ‘contamination’ in the emitted signal from the new atomic entities; such contamination is challenging to resolve both experimentally and theoretically.
This capability is unique to isolated sub-cycle pulse structures and is not observed for trains of sub-cycle field modulation33 because the nanosecond-long exposure of atoms to such fields yields a substantial degree of ionization, which is actually the means to trace the waveform in these experiments34. The ionization and excitation probability of Kr atoms calculated by the solution of the TDSE and the waveform used in our experiments for a range of intensities (summarized in Extended Data Table 1) verify this conjecture. Experimentally, we also verified this fact by using the previously established and highly sensitive technique of attosecond transient absorption spectroscopy18.
The interaction of optical attosecond pulses with matter, even at excessive intensities such as those used in our experiments and simulations, can be mostly considered as a scattering process. The system and the pulse virtually do not exchange energy; rather, the pulse probes the system via nonlinear scattering and the emission of coherent radiation.
Identifying the dominant nonlinearities in the interaction between the optical attosecond pulses and the Kr atoms, and understanding the dependence of these nonlinearities on the variation of the global phase (ϕ ) of the optical attosecond pulse—the key control point of the interaction in experiments with such waveforms—is essential for the development of intuitive models that can describe the nonlinear dynamics. As we show below, it allows the development of a robust methodology that permits the reconstruction of the dynamics of the response from global-phase spectrograms.
To this end, we theoretically (using TDSE simulations in Kr) studied the intensity dependence of the yield of the spectral emission (averaged over the range 0–8 eV) under optical attosecond pulses for the range of intensity settings used in our experiments. The corresponding nonlinearities can be evaluated from the slope of the linear fit of the data in the log–log diagram of Extended Data Fig. 5. A linear fitting over the entire range of intensities studied (2 × 1013 –10 × 1013 W cm−2) reveals a slope of about 4, and suggests the dominance and coexistence of the two most essential nonlinearities in centrosymmetric systems: the third- (E3) and the fifth- (E5) order nonlinearities—broadly know as bound-electronic nonlinearities.
The study presented in the previous paragraph highlights the dominance of bound-electron nonlinearities in the response and the low or negligible sensitivity of the nonlinearity to the global phase of our pulses.
To account for the non-instantaneous response revealed in our TDSE simulations, and inspired by previous approaches in ultrafast spectroscopy35, in which the nonlinear response is decomposed into instantaneous and delayed components, we describe the nonlinear dipole moment as a sum over instantaneous third- and fifth-order nonlinearities, as well as the fifth-order delayed nonlinearity according to equation (1):
Here, a, b and c are coefficients, dt represents a delay of the fifth-order response and E(t, ϕ ) is the electric field of the optical attosecond pulse for a global phase ϕ .
One would generally expect delayed terms to be considered for all nonlinearities involved (including the third-order); the energy diagram of Extended Data Fig. 6 offers an intuitive explanation of our choice to limit the delayed terms to only fifth-order nonlinearities. Indeed, delayed response in the range 0–9 eV will primarily involve virtual transitions, which can coherently couple at least two electronic states of the system (ground and excited states or combinations of excited states). The diagram demonstrates this assuming virtual transition compatible with the energy spectrum of our optical attosecond pulse (1.1–4.6 eV). Such transitions can only occur within the fifth-order response (higher-order Kerr effects). The diagram also highlights—through the multiphoton picture—that at this extreme limit of pulse duration and corresponding bandwidth, non-resonant and resonant response are virtually inseparable. As a result, the coherent dynamics induced between two or more states of the system will be manifested at each nonlinearly emitted spectral component in the process.
Extended Data Fig. 7a–c shows representative, synthetic, global-phase spectrograms for three values of the parameter dt (dt = 0 (instantaneous response), dt = 20 as and dt = 30 as), generated by equation (1) in the spectral range of our experiments (see Fig. 4). The synthetic spectrograms highlight the capability of the model to capture key features of the experimental spectrograms (for example, Figs 1 or 4), such as the profoundly asynchronous modulation of the emission in the range 7–8 eV and the weakening of the amplitude in the range 6–6.5 eV—both are the result of dynamic nonlinear interference between delayed and instantaneous terms in equation (1).
To further verify the validity of our model, we used the TDSE simulations in Kr as a basis to explore how a dipole described by the above equations could represent the (below resonances) nonlinear response of the system.
Extended Data Fig. 7d shows the TDSE simulated dipoles (black line) and their fitting with equation (1). We investigated both the capability of the model to fit only a fraction (spectrally filtered from 5.5–8 eV; Extended Data Fig. 7d) of the dipole as well the entire dipole (0–8 eV) below resonances (Extended Data Fig. 7e). A single set of the parameters a, b, c and dt in equation (1), and a given peak intensity of the pulse, can precisely reproduce the nonlinear dipole (black lines) for all settings of the global phase. Representative examples for two (extreme) settings of the global phase ϕ ≈ 0 (left panels of Extended Data Fig. 7d and e) and ϕ ≈ π/2 rad (right panels of Extended Data Fig. 7d and e) are shown. Furthermore, the parameters a, b, c and dt extracted from fitting part of the nonlinear spectrum (5.5–8 eV) (Extended Data Fig. 7d) are identical to those required to fit the entire range (0–8 eV) (Extended Data Fig. 7e). As a conclusion, even a limited fraction of the nonlinear spectrum contains information about the interfering terms in equation (1) The model works well for a wide range of peak intensities (not shown); therefore, we conclude that it is adequate for reconstructing the response.
The most critical test of the capability of our model to reconstruct experimental data and to trace the time-domain nonlinear dipole dynamics, is to perform a numerical experiment. To do so, we create a theoretical spectrogram using the TDSE model (where the nonlinear dipole is known a priori) and attempt to reconstruct this nonlinear dipole using (i) equation (1), (ii) the spectrogram constructed by the TDSE simulation, and (iii) the a priori measured driver field waveform.
For both experimental and synthetic data, we used a quickly converging nonlinear algorithm to perform the reconstructions. It is based on a commercial, highly optimized numerical routine that uses the ‘trust-region’ method36, which is usually used for constrained problems. The root-mean-square deviation (r.m.s.d.) was determined, and defined as the main parameter to optimize our reconstruction via its minimization. The r.m.s.d. is calculated as:
Where and X are the computed and original (measured) spectra, respectively, at certain global-phase settings, ω = 5.5 eV, ω = 8 eV, is the number of global-phase settings of the measured VUV spectrogram, and n is the number of frequencies involved (from ω to ω ).
The results of this study are summarized in (Extended Data Fig. 8a, b), in which the original and the reconstructed spectrograms are shown. The reconstruction parameters in equation (1) are a = 0.180, b = 0.096, c = −0.177 and dt = 67 as. Extended Data Figure 8c, d compares the reconstructed (red line) and the a priori known nonlinear dipole (black line) for two different settings of the global phase. Blue lines show the instantaneous response for the same waveforms for comparison.
We study the delayed nonlinear bound-electronic response over a wide range of intensities ((2–8) × 1013 W cm−2). The evaluated delays between the instantaneous and the TDSE-simulated dipoles as function of the driver field intensity are shown in Extended Data Fig. 9a; the delays between the instantaneous dipoles and those reconstructed from our measured spectrograms (see Fig. 4) are shown in Extended Data Fig. 9b.