Multi-petahertz electron interference in Cr:Al2O3 solid-state material

Lightwave-field-induced ultrafast electric dipole oscillation is promising for realizing petahertz (1015 Hz: PHz) signal processing in the future. In building the ultrahigh-clock-rate logic operation system, one of the major challenges will be petahertz electron manipulation accompanied with multiple frequencies. Here we study multi-petahertz interference with electronic dipole oscillations in alumina with chromium dopant (Cr:Al2O3). An intense near-infrared lightwave-field induces multiple electric inter-band polarizations, which are characterized by Fourier transform extreme ultraviolet attosecond spectroscopy. The interference results from the superposition state of periodic dipole oscillations of 667 to 383 attosecond (frequency of 1.5 to 2.6 PHz) measured by direct time-dependent spectroscopy and consists of various modulations on attosecond time scale through individual electron dephasing times of the Cr donor-like and Al2O3 conduction band states. The results indicate the possible manipulation of petahertz interference signal with multiple dipole oscillations using material band engineering and such a control will contribute to the study of ultrahigh-speed signal operation.


Supplementary note 1: Experimental setup
A few-cycle near-infrared (NIR) pulse (1.55-eV centre photon energy with 7-fs duration) from a Ti:sapphire laser was used for the pump-NIR pulse in the Fourier transform extreme ultraviolet attosecond spectroscopy (FTXUV) based on transient absorption. The NIR pulse is also used to generate the isolated attosecond pulse (IAP) via the double optical gating (DOG) method 1 . The IAP is used as the probe-pulse in the FTXUV method. The pump-andprobe system is described in refs. (2) and (3). The stability of the pump-probe system is 23-as timing jitter at the root mean square over 12 hours, which is monitored by a co-propagated continuous-wave laser (633-nm wavelength) 2 . The target intensity of the pump-NIR pulse is approximately 2×10 12 W/cm 2 , which is estimated from the photoelectron energy shift with the intensity dependence of the attosecond streak 4 . The collinearly propagated IAP and NIR pulse are focused onto the target of alumina with chromium dopant (Cr:Al 2 O 3 ). After the target, the transmitted IAP is sent to an extreme ultraviolet (XUV) spectrometer equipped with a micro-channel plate and a cooled charge-coupled device camera. The spectral resolution is 120 meV at 45.5-eV photon energy 3 .

Supplementary note 2: Temporal characterisation of IAP
To confirm the IAP duration, we use the frequency-resolved optical gating for complete reconstruction of attosecond bursts (FROG-CRAB) method 5 based on an attosecond streak camera 4 . The collinearly propagated IAP and NIR pulse are focused to the gas jet with helium atom (50-µm interaction length; 740-mbar backing pressure). The estimated target intensity of the NIR pulse is approximately 2×10 12 W/cm 2 in this measurement. The ionized photoelectrons induced by the IAP are detected by a regular time-of-flight system. Supplementary Figures 1(a) and (b) show the experimental and retrieved FROG-CRAB traces. Supplementary Figure 1(c) shows the reconstructed temporal profile and phase of the IAP pulse. The duration is 192 as at the full width at half maximum (FWHM). The IAP spectrum reconstructed by the FROG-CRAB method (red solid line) agrees well with the measured spectrum (blue dashed line), as shown in Supplementary Figure 1(d).

Supplementary note 3: Cr:Al 2 O 3 target
In this experiment, trigonal (rhombohedral) a-Al 2 O 3 is used for the host material, which has the band-gap energy of 8.7 eV 6 . The a-Al 2 O 3 is doped with the Cr material during the single-crystalline a-Al 2 O 3 crystal growth. The Cr 3+ ions produce a donor-like intermediate level for the Al 2 O 3 host material. Supplementary Figure 2 shows the atomic number densities of the Cr dopant and the Al 2 O 3 host material. The density of Al 2 O 3 is 2×10 22 cm -3 . The density of Cr is estimated to be 2×10 17 cm -3 by secondary ion mass spectrometry (SIMS), which gives rise to the doping level of approximately 1×10 -3 at. % (10 ppm).
The thin 36-nm-thick target without a substrate was manufactured from a 400-µm-thick bulk target by NTT-AT Inc. 7 by mechanical polishing and ion beam milling. Commonly, the technology is used for the process in transmission electron microscopy. The target has thickness graduation from a few nanometers to a few hundred micrometers. It was mounted on ring holder equipped with a linear electronic actuator to select the proper thickness.
Supplementary Figure 3 shows the optical density (OD) using the IAP without the NIR pulse. Here, we defined the OD at laser frequency w as OD(w)=log[I IAPin (w)/I IAPout (w)], where I IAPin (w) is the spectrum of the input IAP. The I IAPout (w) is the absorption spectrum with the target, and it also corresponds to the transmitted spectrum from it. The OD(w) is proportional to the regular absorption coefficient. Consequently, the OD(w) monitors the spectral deviation with the target. The effective thickness of the target is 36 nm, which is directly estimated from the IAP absorption using the extinction coefficient from ref. (8). The target was used for the transient absorption spectroscopy with the NIR pulse.

Supplementary note 4: Simulation of resonant polarizations
To calculate the resonant high-order polarizations in the Cr:Al 2 O 3 , we simply consider the time-dependent density matrix formalism with a two level system 9,10 whose energy separation is either resonant to the donor-like state corresponding to the fifth-order component (5ħw) or resonant to the conduction band (CB) corresponding to the seventhorder component (7ħw). The Hamiltonian system is expressed as in Ref. (10): Here, µ and µ are the dipole moment and its magnitude projected along the applied electric laser field, respectively. F and F are the field vector and its strength of the applied laser pulse, which is given by F(t) = Ae −t 2 /2σ sin(ωt) , where A is the field amplitude, 2s (=7 fs) is the pulse duration, and w (=2p×0.375 PHz) is the centre angler frequency of the laser pulses. By numerically solving -.
with a phenomenological dephasing time t, we could obtain the time-dependent matrix elements r ij (i, j=0, 1) of the density matrix r and the polarization P=Tr(µr). Here, the longitudinal relaxation is neglected, the initial state of the system was assumed to be the ground state, and the matrix element of the dipole moment µ was assumed to be a real number.
The calculated results with the parameter value of µF/ħw=0.59 are shown in Fig. 4 in the main text. In the simulation, we used t donor =3 fs for the donor-like state and t CB =0.2 fs for the CB state. The result corresponds to the NIR-field-induced polarization under the perturbative regime, where the polarization amplitude decreases with the increasing order of the harmonics. The resonant polarization can be extracted by Fourier filtering, and the inverse Fourier transform of the filtered polarization gives the time-domain waveform as shown in Fig. 4(b) in the manuscript. Here, if the dephasing time is shorter than the pulse duration of the NIR pulse, the polarization almost follows the harmonics of the NIR electric field. On the contrary, if the dephasing time is much longer than the pulse duration, the resonant polarization builds up until the NIR pulse passes by. Therefore, the peak position of the timedomain polarization will be delayed if the dephasing time is long compared with the pulse duration. This is exactly what we observed in the experiment, where the low-order harmonics (4th and 5th) are delayed by 2 fs compared with the high-order harmonics (6th and 7th).
The measured relative time delay of approximately 2 fs, as shown in Fig. 3, could originate from the difference in an intra-band electron-electron scattering in the donor-like and CB states. Generally, the dephasing time in the spatially localized energy state is much longer than that in the band energy state 11,12 . Thus, the spatially localized Cr donor-like state with the low doping level has many fewer relaxation channels in the unoccupied state compared with the CB state. In addition, the dephasing time is commonly explained by the density-and energy-dependent damping rate, which is proportional to cube root of the excited carrier-density 13 n. The value of n is given by where ħw is the photon energy of the pump pulse and the n is the repetition rate of laser 14 .
The w 0 is beam spot size. The R and a are the reflectivity at normal incidence and the linear absorption coefficient, respectively. The term of P/(npw 0 2 ) corresponds to the incident pump fluence. Here, the estimated fluence of the NIR pulse is 7 mJ/cm 2 on the target. The reflectivity R is from ref. (8). Here, we use the linear absorption coefficient a, as shown in Fig. 2(c), because each nonlinear absorption coefficient of the 4-7th orders is difficult to directly determined in the Cr:Al 2 O 3 target. The estimated excited carrier densities are n 4ħw =5.5×10 16 cm -3 (6.2 eV), n 5ħw =5.2×10 16 cm -3 (7.7 eV), n 6ħw =1.8×10 21 cm -3 (9.2 eV), and n 7ħw =2.6×10 21 cm -3 (10.8 eV). Actually, the values should be much lower with a perturbative multiphoton process. However, the largely different carrier densities could produce the individual dephasing times in the donor-like and CB states. Thanks to the reduced pulse duration of the high-order harmonics as well as the sub-cycle time-resolution of our setup, we could clearly visualize the ultrafast time delay due to the ultrafast dephasing. Consequently, even using the simple time-dependent density matrix formalism, we could reproduce the remarkable characteristics of the observed multi-petahertz polarizations in the Cr:Al 2 O 3 .