Attosecond electron motion control in dielectric

Attosecond science capitalizes on the extreme nonlinearity of strong fields—driven by few-cycle pulses—to attain attosecond temporal resolution and give access to the electron motion dynamics of matter in real time. Here we measured the relative electronic delay response of a dielectric system triggered by a strong field of few-cycle pulses to be of the order of a few hundred attoseconds. Moreover, we exploited the electronic response following the strong driver field to demonstrate all-optical light-field-sampling methodology with attosecond resolution. This methodology provides a direct connection between the driver field and induced ultrafast dynamics in matter. Also, we demonstrate control of electron motion in a dielectric using synthesized light waveforms. This on-demand control of electron motion paves the way for establishing long-anticipated ultrafast switches and quantum electronics. This advancement promises to increase the limiting speed of data processing and information encoding to rates that exceed one petabit per second, opening a new realm of information technology. Light-field-induced electron dynamics in a silicon dioxide dielectric system are exploited to directly measure the attosecond relative electronic delay response in the dielectric system, potentially extending the speed of data processing and information encoding into the petahertz realm.

In this work, we studied the light-field-induced electron dynamics in SiO 2 dielectric system by an all-optical methodology. In the presence of a strong field, electrons are injected from the valence band (VB) into the conduction band (CB) in the dielectric system. The electron injection mechanism depends on the Keldysh param- where ω is the carrier frequency of the driver field; Δ is the bandgap; F 0 is the peak field strength; and e and m are the electron charge and electron mass, respectively 15 . Electron injection can occur via electron tunnelling (at γ « 1) [22][23][24] or coherent multiphoton excitation (at γ » 1) 16,21 . In the presented experiments, we used field strengths in the range between 0.78 and 1.33 V Å -1 . In this case, the γ value (~2. 5-4.3) suggests that the electron injection mechanism is dominated by coherent multiphoton excitation 16 . Then, the electrons are accelerated and decelerated in the reciprocal space following the vector potential of the driver field in the CB (Fig. 1a,b). This electron motion is restricted by the values of the wavevector k within the first Brillouin zone, that is, −π/a < k < π/a where a is the lattice constant (Fig. 1a,b). At a certain critical field strength (Fig. 1b), the shift in electron momentum becomes greater than the Brillouin-zone extension (k = 2π/a), provoking electron Bragg reflection and Bloch oscillations. Thus, the dielectric constant (ε) and refractive index 16 are altered, causing the dielectric reflectivity to change in real time following the driver field 16,17,24 .
Here we exploited this light-field-driven electronic response and the related reflectivity modulation to directly measure the relative electronic delay response in the SiO 2 dielectric system. Moreover, we demonstrate an all-optical light-field-sampling methodology with attosecond resolution based on the same principle. Finally, we utilized light-field synthesis to control electron motion in the dielectric using complex synthesized waveforms.

Results and discussion
Electronic delay response in a dielectric. Previously, a strong light-field-induced current in a dielectric has been exploited to indirectly determine the carrier delay response by measuring the carrier-injection delay time in a dielectric nanocircuit 27 . In this work, we directly determined the relative light-field-induced electronic delay response by measuring the time-resolved reflectivity modulation of the SiO 2 sample at different field strengths (Methods). The pump beam modifies the optical properties and reflectivity of SiO 2 , which are probed by another weak probe beam. The integral of the reflected probe spectrum is calculated at each instant of time (Supplementary Section 1) to obtain the reflectivity modulation trace (normalized and shown in Fig. 2a). Then, we determine the relative electronic delay response by measuring the shift in phase delay between the reflectivity modulation traces under the same experimental conditions at different field strengths: F1 = 0.78 V Å -1 ; F2 = 0.96 V Å -1 ; F3 = 1.10 V Å -1 . The intensity of the input beam was controlled by a variable metallic neutral density filter introduced in the beam path. Then, the reflectivity modulation was recorded (in a time window between −2.6 and 3.2 fs) as a function of the time delay between the pump and probe pulses with high resolution (delay steps of 25 as). The reflectivity modulation traces from the average of four recorded measurements at each field strength are plotted in Fig. 2b. The traces show relative phase delays of 425 ± 98 and 575 ± 45 as. We also calculated the reflectivity modulation traces by integrating the measured probe spectrum in different energy ranges, and the retrieved relative delays remain within the standard deviation (s.d.) values. Note that the CEP of the driver pulse is passively stabilized (phase jitter during the measurements is of the order of 100 mrad). The contribution of the CEP Attosecond science capitalizes on the extreme nonlinearity of strong fields-driven by few-cycle pulses-to attain attosecond temporal resolution and give access to the electron motion dynamics of matter in real time. Here we measured the relative electronic delay response of a dielectric system triggered by a strong field of few-cycle pulses to be of the order of a few hundred attoseconds. Moreover, we exploited the electronic response following the strong driver field to demonstrate all-optical light-field-sampling methodology with attosecond resolution. This methodology provides a direct connection between the driver field and induced ultrafast dynamics in matter. Also, we demonstrate control of electron motion in a dielectric using synthesized light waveforms. This on-demand control of electron motion paves the way for establishing long-anticipated ultrafast switches and quantum electronics. This advancement promises to increase the limiting speed of data processing and information encoding to rates that exceed one petabit per second, opening a new realm of information technology.
to jitter in the measured relative phase delays is reflected in the merged s.d.
The measured phase delay ( Fig. 2b) is attributed to the electronic delay response of the dielectric in a strong field 27 , which increases at higher driver-field strengths due to the increase in excitation carrier density, a.c. Stark effect 31 and system's polarizability 27,32,33 . Accordingly, we calculated the excitation carrier density at different field strengths using the driver pulse's electric field (Fig. 2c). This field is retrieved from the derivative of the measured reflectivity modulation trace (Fig. 2a), which represents the vector potential of the driver pulse, as explained in Supplementary Section 1. Remarkably, the temporal intensity profile of the driver field (τ FWHM = 10.5 fs) is in good agreement with the measured temporal profile obtained by the convenient transient-gating frequency-resolved optical gating measurement (τ FWHM = 10.3 fs), as shown in Supplementary Fig.  2. The calculated excitation carrier densities n ex (t) (Supplementary Section 2) at different field strengths are shown in Fig. 2d in contrast with the instantaneous intensity of the trigger field (plotted by the black dashed line). The total number of excited carriers at field strengths F2 and F3 is almost double (1.86 times) and triple (2.9 times) the number of excited carriers triggered by field strength F1. The number of excited carriers plays an important role in the electronic delay response due to electron-electron interactions and the screening effect 27 .
Moreover, in Fig. 2d, the excitation carrier densities n ex (t) behave as a function of the driver pulse (Fig. 2d, black dashed line), and they have maximum values at the maxima and minimum values at the minima of the instantaneous field intensity. Both maximum population (at t ≈ 2 fs) and residual CB population (for t > 9 fs) monotonically increase with the excitation-field amplitude, indicating the high reversibility of excitation, which happens at the high interband coupling matrix element [22][23][24][25] .
This reversible electronic response directly provides access to the waveform of the triggering pulse with high temporal resolution.
All-optical light-field sampling. The direct connection between the dielectric system's reflectivity modulation and the incident in reciprocal space at strong and critical field strengths, respectively. The main text provides the explanation. c, Experimental setup of the all-optical methodology. A mask splits the main beam into two beams: strong (pump) and weak (probe). These two beams are focused on a 100 µm dielectric (SiO 2 sample). One of these D-shaped mirrors is connected to a piezo-stage to control the relative delay between the pump and probe pulses with attosecond resolution. An optical spectrometer measures the reflectivity modulation of the spectrum of the probe beam reflected off the sample. A polarizer and a one-hole mask are introduced in the path of the probe beam before the spectrometer to enhance the signal-to-noise ratio of the reflectivity modulation measurements.  Fig. 2 | electronic delay response in Sio 2 dielectric system. a, normalized measured reflectivity modulation of the SiO 2 sample under the influence of a strong field of a few-cycle pulse. Each point represents the integral of the spectrum of the probe beam measured at each time delay between the pump and probe pulses. b, Measured reflectivity modulation traces in a time window between -2.6 to 3.2 fs at different field strengths of the driver, namely, 0.78, 0.96 and 1.10 V Å -1 , as shown by the blue, red and orange lines, respectively. The relative delay responses between these measured traces are 425 ± 98 and 575 ± 45 as. c, retrieved driver field from the derivative of the reflectivity modulation measurement in a. d, Calculated excitation carrier density n ex (t) triggered by different field strengths are shown in contrast to the driver's instantaneous intensity (E 2 ) plotted by the black dashed line. Delay (fs) driver field allows the establishment of a direct and simple all-optical light-field-sampling methodology. Accordingly, we conducted a numerical simulation to demonstrate the basic principle of this approach by calculating the field reflected from SiO 2 in a strong field of a single-cycle pulse (centred at 800 nm) at different field strengths (Supplementary Section 4) 18 . The reflected and incident fields are normalized, overlapped in time and plotted in Supplementary Fig.  3. The reflected field exactly follows the incident field at different intensities with a maximum s.d. of <1.5%. The results of these theoretical calculations show that the induced reflectivity change in the dielectric follows the driver-field waveform.
To experimentally demonstrate the viability of this methodology, we sampled an unknown synthesized waveform generated by a four-channel (250-1,000 nm) light-field synthesizer (LFS) apparatus 32-34 (Supplementary Section 4 and Supplementary Fig. 4a). The output beam from the LFS with an unknown waveform is divided into two separate beams by passing through the two-hole mask (Methods) using the same setup as the first experiment (Fig. 1c). The first high-intensity beam (pump) is used to alter the reflectivity of the dielectric. The field strength of the pump beam is estimated to be ~1.33 V Å -1 (well below the damage threshold of ~2.7 V Å -1 ) 23,27 . The second beam (probe) has a lower power (~10% of the pump beam) and ~13% of the field strength of the pump beam. The spectrum of the probe beam is recorded as a function of the time delay between the pump and probe pulses (with a delay step size of 100 as).
Then, the unknown synthesized waveform of the strong driver field (WF1) is retrieved from the reflectivity modulation trace (average of three scans are normalized; Fig. 3a) and plotted in Fig. 3b. Next, we changed the relative phase delay between one of the spectral channels (Ch VIS-UV ) with respect to the other channels in the LFS by −2 fs to generate a new waveform (WF2). The reflectivity modulation caused by WF2 is measured (Fig. 3c). The field of WF2 is obtained (Fig. 3d, solid red line). Subsequently, we calculated the expected waveform of WF2 from WF1 (Fig. 3d, black dashed line) in synchrony with the sampled WF2 field. This expected waveform is obtained by analysing WF1 to retrieve the fields and relative phase delays between the four individual spectral channels of the LFS forming WF1. Then, we mathematically introduced the delay of −2 fs in Ch VIS-UV and summed the fields of all the four channels (Supplementary Section 4). The expected and measured fields of WF2 (Fig. 3d) are in good agreement (s.d., ~10%). Moreover, we changed the relative phase of the deep-ultraviolet (DUV) spectral channel (Ch DUV ) in the LFS with respect to the other channels by −2 fs to generate a new waveform (WF3). The measured reflectivity modulation induced by WF3 is shown in Fig. 3e. The retrieved field of WF3 (red line) is plotted in Fig. 3f alongside the calculated expected waveform (black dashed line). The estimated s.d. between the measured and expected waveforms is <10%. The asymmetric shape of the reflectivity modulation (Fig. 3a,c,e) is due to the increase in the residual of the excitation carrier densities as it evolves in time depending on the field  strength and shape of the driver-field waveform. Notably, the changes in the relative amplitudes of the sub-cycles of the main field in WF1, WF2 and WF3 (Fig. 3b,d,f, small black arrows)-due to changes in the relative phases of the LFS channels-were successfully captured in our measurements, indicating the sub-cycle resolution of the demonstrated field-sampling approach. Furthermore, the constructed spectrum-calculated from the Fourier transform of the measured WF1 field-almost captures all the spectral components of the pump pulse ( Supplementary Fig. 5). Thus, the presented field-sampling methodology can resolve all the frequency components of a waveform spanning from 250 to 1,000 nm. These results show that the demonstrated all-optical field-sampling methodology exhibits sampling capability for a single broadband waveform spanning two octaves with attosecond temporal resolution, which has been beyond reach 22,23,27 .
This approach can be used under any experimental conditions, thus enabling the direct connection between the sub-femtosecond triggering field and the measured dynamics in potential timeresolved measurements, providing more insight into the physics of ultrafast dynamics in matter. Also, this simple field-sampling methodology promises a profound advancement in light-field synthesis technology and the attosecond control of electron motion in matter.
Attosecond control of electron motion in a dielectric. Lightfield-induced electron motion in a dielectric can be controlled on demand by tailoring the driver field's shape with attosecond resolution. To demonstrate ultimate control of electron motion in SiO 2 , we synthesized a few complex waveforms by changing the relative phases and intensities of the four channels in the LFS. Then, these complex synthesized waveforms were measured based on tracing the reflectivity modulation, driven by these fields, as explained earlier. These measured complex waveforms are illustrated in the left column of Fig. 4. The corresponding instantaneous intensity profiles of these waveforms-presented in blue lines in the right column in Fig. 4-reflect the triggered electronic response of the SiO 2 dielectric system in real time. Accordingly, the carrier densities (n ex (t)) triggered by these measured synthesized waveforms are calculated and depicted in contrast with the intensity profiles shown by the black lines in the right column of Fig. 4.
At the field strength (1 V Å -1 ) of the pump waveform used in this experiment, the carrier density n ex (t) follows the field intensity (E 2 ) profile and the number of triggered electrons is maximized at the highest field crests. For instance, using the optical attosecond pulse (Fig. 4a(i),(ii)), electron triggering is maximized at the highest field crest (shaded in red)-which has a full-width at half-maximum (FWHM) of ~400 as-as shown in the solid black line. In Fig. 4b(ii), the maximum electron-triggering signals occur at two time instants separated by 0.9 fs. The separation interval is controlled to be 2.7 fs (Fig. 4c(ii)) using the waveform displayed in Fig. 4c(i).
Alternatively, the measured complex synthesized waveforms presented in Figs. 4d(i) and 4e(i) show the ultimate control of electron Fig. 4 | Attosecond control of electron motion for ultrafast switching. a-e, Synthesized waveforms (red lines) and the corresponding instantaneous intensity profiles (blue lines) for controlling electron motion in the dielectric are shown in the left (a(i)-e(i)) and right (a(ii)-e(ii)) columns, respectively. The calculated carrier densities n ex (t) triggered by these waveforms are plotted in black lines (right column). Using the waveform in a(i), the number of triggered electrons is maximized in one instant of time at the highest field crest, as shaded in red (FWHM, ~400 as) in a(ii). The waveform in b(i) triggers the maximum number of electrons at two time instants, as shown in b(ii), and they are separated by 0.9 fs. By using the waveform in c(i), this time interval becomes 2.7 fs, as shown in c(ii). The highest triggering signals of electrons induced by the field in d(i) occur at three events that are equally separated in time (0.9 fs), as depicted in d(ii). e(i) shows a complex measured synthesized waveform that triggers the maximum number of electron signals at four-time instances, as shown in e(ii). The time separations between these signals are 0.9, 3.6 and 0.9 fs. The blue-shaded area in the right column represents a virtual threshold that would be potentially introduced in the dielectric nanocircuit to switch on/off the instantaneous light-induced current with attosecond resolution. 0.9 fs 0.9 fs 3.6 fs motion. In Fig. 4d(ii), the electron's maximum triggering occurs at three time instants with equal time intervals of 0.9 fs. In Fig. 4e(ii), the electron's highest triggering signals arise at four events where the first and second signals are separated by 0.9 fs, the second and third by 3.6 fs, and the third and fourth by 0.9 fs.
The demonstrated fine control of electron motion by the synthesized waveforms allows on/off switching of the average photoinduced current signal (Fig. 4a(ii) is on and Fig. 4b(ii) is off) in a dielectric-based nanocircuit demonstrated earlier 21,23,27 . In analogy to this previously presented current switching by CEP control, the ultimate capability of synthesizing complex waveforms via LFS enables a new level of switching by controlling the number of instantaneous current signals and their time intervals within the same pulse (Fig. 4c(ii), Fig. 4d(ii) and Fig. 4e(ii)), by setting a certain current signal threshold (shown by the blue-shaded area in the right column in Fig. 4) in the nanocircuits.
Remarkably, this demonstrated approach works under ambient conditions, and it is viable in miniaturized ultrafast switching devices based on implementing the well-established programmable pulse-shaping technology 35 into dielectric nanocircuits. This development promises an increase in the data-processing speed to rates that exceed one petabit per second-a million times faster than current technology.
We exploited the dielectric's strong light-field interaction to determine the attosecond relative electronic delay response in the dielectric. Moreover, we demonstrate an all-optical direct and simple methodology to sample the light field spanning two octaves with attosecond resolution. This field-sampling approach can be implemented in different environments and experimental setups to provide a real-time connection between the ultrafast dynamics in matter and its driver field. Consequently, the use of this realistic sampled field in simulations, calculations and fitting algorithms related to the measured spectroscopic response of matter provides a more accurate interpretation and insight into the underlying physics of these dynamics. Moreover, we used the synthesized waveforms to exhibit full control over the electron motion in the dielectric. This electron control can be used to establish ultrafast (femtosecond and attosecond) switches, paving the way to extend the frontiers of modern electronics and information-processing technologies into the petahertz realm.

online content
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Methods
Field-induced reflectivity modulation measurement of SiO 2 dielectric. In this experiment (setup shown in Fig. 1c) conducted in an ambient environment, the beam of few-cycle visible (400-700 nm centred at λ = 564 nm and p polarized) laser pulses is split into two beams by passing it through a two-hole mask with different hole diameters (3 and 1 mm). Therefore, the two beams that emerge through the mask have different intensities. The first beam has a high intensity (pump beam) to induce a phase transition and alter the reflectivity of the SiO 2 sample. At low field strengths of ≤0.67 V Å -1 , no notable reflectivity modulation signal is observed. This is the minimum threshold of the field strength required to excite enough carriers to cause an observable change in the reflectivity of the dielectric in the presence of the field. Therefore, the pump beam's field strength is set at ≥0.78 V Å -1 . Note that the reflectivity modulation signal disappears when the SiO 2 sample is damaged; therefore, all the presented measurements were collected at field strengths lower than the damage threshold (2.7 V Å -1 ) 23,27 . The second beam (probe beam) has a lower field strength (≤0.1 V Å -1 ) than the threshold strength required to induce any degree of phase transition in SiO 2 . The two beams deviating from the mask are incident on two D-shaped focusing mirrors (focus f = 100 mm) and focused onto a 100 μm SiO 2 sample (incident angle, <5°). An imaging system is used to ensure perfect spatial overlapping between these two beams. One of these mirrors is attached to a piezo-stage device to control the relative delay between the pump and probe beams with attosecond resolution. The probe beam reflected off the sample's front surface is tightly focused into an optical spectrometer entrance after propagating a distance sufficient for spatial isolation from the pump beam; a polarizer and a one-hole mask are introduced to filter out the pump beam. The measured spectrum of the probe beam (in the presence of the reflected pump beam, as shown in Supplementary  Fig. 1) is recorded as a function of the time delay between the pump and probe pulses with a delay step size of 100 as. Since the spectrum does not show any interference fringes, the reflected pump beam has no contribution to the reflectivity modulation measurements, providing a high signal-to-noise ratio. Optionally, the spectrometer can be replaced by a photodiode that can automatically perform the integration, thus simplifying the demonstrated all-optical field-sampling setup.

Data availability
Source data are provided with this paper. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

code availability
The analysis codes that support the findings of this study are available from the corresponding author upon reasonable request.