Quantum 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 electronic delay response of the dielectric system triggered by a strong field of few-cycle pulses to be in the order of 425 ± 98 as. Moreover, we exploited the electronic response following the strong driver field to demonstrate all-optical light field metrology with attosecond resolution. This field sampling methodology provides a direct connection between the driver field and the induced ultrafast dynamics in matter. Also, we demonstrate the quantum electron motion control in dielectric using synthesized light waveforms. This on-demand electron motion control realizes the long-anticipated ultrafast optical switches and quantum electronics. This advancement promises to increase the limiting speed of data processing and information encoding to rates that exceed 1 petabit/s, opening a new realm of information technology.

demonstrated elsewhere 18 and used for the Carrier-Envelope-Phase (CEP) and the waveform detection of the driver's few-cycle pulses 18,20-25. Also, the strongfield interaction with thin films of SiO 2 dielectric has been utilized to generate a wideband coherent EUV radiation extended up to 40 eV 9 . Based on these studies, the strong-field-induced electron dynamics in the dielectric can be explained by electron motion in the conduction band (illustration in Figure   1a&b). In a strong field (Figure 1a), the electron with initial wave vector ( ) is moving in the reciprocal space by acquiring a time-dependent wave vector $ ( , ) from the driving field, which can be expressed by 12,16 K ) (q, t) = q + . ℏ ∫ F 2 (z, t 4 )dt 4 6 78 (1) where : ( , 4 ) is the optical field strength, and is the electron charge.
Therefore, all electrons are shifted in the reciprocal space by the same wave vector ∆q(t) = . ℏ ∫ F 2 (z, t 4 )dt 4 6 78 (2) At a certain critical field strength value (Figure 1b), the shift (∆ ) becomes greater than the Brillouin zone extension k = 2π/a, causing electron Bragg reflection and Bloch oscillations. Thus, the dielectric constant ( ) and the dielectric material's optical properties are altered due to the strong polarizability. As a result, the dielectric system undergoes a semimetal-like phase transition 12,16 , and the reflectivity changes in real-time following the driver field. Hence, the dielectric time-resolved reflectivity measurement provides direct access to the induced electron motion dynamics in the system.
Here, we exploited this field-driven electronic response and the related dielectric reflectivity modulation to directly measure the SiO 2 dielectric system's electronic delay response in a strong few-cycle pulse. Also, we demonstrate all-optical light field sampling metrology with attosecond resolution based on the same principle. Finally, we utilized the light field synthesis to control the electron motion in the dielectric using complex synthesized waveforms.

Electronic delay response in dielectric
The strong-field-induced current in the dielectric has been exploited to indirectly determine the carrier delay response by measuring the carrier injection delay time in the dielectric nanocircuit 22 Figure 2b. The traces show relative phase delays of 425 ± 98as and 575 ± 45as. Note, the driver pulse CEP is passively stabilized (phase jittering during the measurements is on the order of 100 mrad). Thus, the CEP jittering contribution to the measured relative phase delay is reflected in the merged standard deviation (SD). The measured phase delay (Figure 2b) is attributed to the electronic delay response in a strong field 22 . The delay response increases at higher driver field strengths due to the increase of the system's polarizability and the excited carrier density. We calculated the number of excitation carriers at different field strengths using the driver pulse's measured electric field. The electric field presented in Figure 2c is retrieved from the derivative of the measured reflectivity modulation in Figure 2a, representing the driver field's vector potential, as explained in the SI Section I.
Remarkably, the retrieved temporal intensity profile of the driver field  Figure   2d, the excited carrier number F: ( ) is behaving as a function of the instantaneous electric field of the driver pulse (black dashed line in Figure 2d).
The excited carrier's F: ( ) has maximum values at the maxima, and minimum values at the minima of the pulsed electric field. Both the maximum population (at t ≈ 2 fs) and the residual CB population (for t > 9fs) monotonically increase with the excitation field amplitude, indicating high reversibility of the excitation, which happens at the high interband coupling matrix element [16][17][18][19] .
The reversible electronic dynamic response directly gives access to the triggering field of the pulse with high temporal resolution. The time-resolved measurement of the reflectivity modulation changes at different field strengths opening a direct window to the electronic response in the dielectric.

All-optical light field sampling
The direct connection between the dielectric system's reflectivity modulation and the incident driver field shape allows the establishment of direct-simple all-optical light field metrology. First, we conducted a numerical simulation to demonstrate the basic principle of this approach by calculating the reflectivity change and the reflected field of the SiO 2 thin substrate in the strong field of a one cycle-pulse (spans over a broadband spectrum and centered at 800 nm) at different field strengths (the calculations is explained in the SI Section IV) 13 . The reflected and incident fields are normalized, overlapped in time, and plotted in Figure S3 (SI). The reflected field exactly follows the incident field at different intensities with a maximum SD < 1.5%. This calculation proves that the dielectric reflectivity modulation due to the strong-field interaction follows the driver field waveform shape.
We sampled an unknown synthesized waveform generated by four spectral channels (250-1000 nm) using a Light Field Synthesizer (LFS) apparatus mentioned above [26][27][28] (LFS is explained in the SI Section V and shown in Figure   S4 and) to prove the viability of this methodology experimentally. The output beam from the LFS with an unknown waveform is divided into two separate beams as explained in the previous section (setup is shown in Figure 1c). The first high-intensity beam (pump) is used to alter the dielectric reflectivity. The pump beam's field strength is ~1.33V/Å (well below the damage threshold ~2.7 V/Å) 18, 22 . In addition, the second beam (probe) has a lower intensity (~10% of the pump beam). The probe beam spectrum is recorded as a function of the time delay between the pump and probe pulses (with delay step size=100 as).
Afterward, the unknown synthesized waveform of the strong driver field The demonstrated all-optical field metrology exhibits field sampling capability with attosecond temporal resolution for a single broadband waveform spanning two octaves, which was beyond reach 17,18,22 . This approach can be used under any experimental conditions, enabling the direct connection between the triggering few femtosecond/attosecond field and the measured dynamics in potential time-resolved measurements, providing more insights into the ultrafast physics dynamics of matter. Also, this simple field sampling metrology promises a profound advancement in light field synthesis technology and the attosecond electron motion control in matter.

Quantum control of electron motion in dielectric
The light field-induced electron motion in the dielectric can be controlled ondemand by tailoring the driver field's shape with attosecond resolution. We with equal time intervals of 0.9 fs when utilizing these waveforms. In Figure   4EII, the electron's highest triggering signal arises at four events (illustrated by shaded red) where the four signals are separated in time. The first and second signals are separated by 0.9 fs, the second and third signals are separated by 3.6 fs, and 0.9 fs separates the third and fourth signals.
The presented waveforms can be used to induce and control current signalslasting ~ 400 as (Fig 4a (II) We exploited the dielectric's strong-field interaction to determine the attosecond electronic delay response in the dielectric. Also, we demonstrate an all-optical direct-simple approach to sample the light field spanning two octaves with attosecond resolution. This field sampling approach can be implemented in different environments and experiment setups to provide a real-time connection between the ultrafast dynamics in matter and its driver field. Consequently, using this realistic sampled field in simulations, calculations, and fitting algorithms related to the measured spectroscopic response of matter provides more accurate interpretation and insight into the underlying physics of these dynamics. Moreover, we utilized synthesized waveforms to exhibit full control of electron motion in the dielectric. This electron control can be used to develop quantum electronics, paving the way to extend the frontiers of modern electronics and data information processing technologies into the petahertz realm.

Method
Field -induced reflectivity modulation measurement of SiO 2 dielectric: in this experiment (setup is illustrated in Figure 1c), conducted in an ambient environment, the beam of few-cycle visible (500-700 nm, centered at = 600 , and p-polarized) laser pulses is split into two beams by passing the laser through a two-hole mask. The mask is designed to have two different hole diameters (3mm 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 the phase transition and alter the reflectivity of the SiO 2 substrate. The estimated pump beam's field strength is 0.78 V/Å (at a lower field strength ≤ 0.67 V/ Å; no significant reflectivity modulation signal was observed) which is lower than the damage threshold 18,22 . Note, the reflectivity signal disappears when the SiO 2 substrate is damaged, so all the presented measurements were collected at field intensity lower than the damage threshold. The second beam (probe beam) has a lower intensity (≤0.1 V/Å) than the threshold field strength required to induce any degree of phase transition in SiO 2 . The two beams deviate from the mask, are incident on two D-shape focusing mirrors (f=100 mm), and focused onto the 100 μm thick SiO 2 substrate (incident angle < 5º). An imaging system has been used to ensure perfect spatial overlapping between the two beams. One of these mirrors is attached to a piezo-stage device to control each beam's relative delay with attosecond resolution. The reflected probe beam (off the substrate's front surface) is tightly focused into an optical spectrometer entrance after propagating enough distance to be spatially isolated from the pump beam; a polarizer and a one-hole mask were introduced to filter out the pump beam.
The measured probe beam spectrum (in the presence of the reflected pump beam, as shown in Figure S1)  text for explanation. c, the electronic delay response and all-optical field sampling experiment setup. A mask splits the main beam into two strong (pump) and weak (probe) beams. The two beams are focused on a hundred microns dielectric (SiO2 substrate). One of these D-shape 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 reflected probe beam spectrum from the substrate.

Figures and Figure legends
A polarizer and a one-hole mask are introduced in the probe beam path before the spectrometer to enhance the signal-to-noise ratio of the reflectivity modulation measurements.