Introduction

To date, high-speed signal processing is performed by electronic devices using semiconductor-based field-effect transistors driven by radio-frequency (RF) electric fields1. The speed of such electronic devices has reached a limit at the terahertz (1012 Hz: THz) regime because of the response time of band energy modulation with RF electric fields2. Lightwave-field control opens up a new opportunity to speed up the frequency into the petahertz (1015 Hz: PHz) regime because the ultrafast electric dipole variation with inter-band polarization activates an electronic device with instantaneous optical switching from an insulator to conductor or vice versa3.

A powerful way to monitor the temporally evolving coherent electronic motion in solid-state materials is to use modern material-sensing technology with an isolated attosecond pulse (IAP)4,5,6,7. In a previous study, we observed the electric dipole oscillation with a single petahertz frequency component using gallium-nitride (GaN) semiconductor6. To build ultrahigh-clock-rate logic operation systems, petahertz electron manipulation accompanied with multiple frequencies is the next challenge. Our approach for the signal manipulation is to use the interference provided by the multiple electron motions through material band engineering.

Here we study a petahertz interference constructed with near-infrared (NIR) lightwave-field-induced multiple electronic dipole oscillations in alumina with chromium dopant (Cr:Al2O3) and reveal by Fourier transform extreme ultraviolet attosecond spectroscopy (FTXUV) combined with an IAP.

Results

Experimental condition and properties of Cr:Al2O3

Trigonal (rhombohedral) α-Al2O3 is a typical electric insulator with a wide-bandgap8. It is commonly used in the manufacture of semiconductor epitaxial wafers, owing to its hardness, high thermal conductivity, and resistance to optical damage. In this experiment, the α-Al2O3 is doped with the Cr material during the single-crystalline α-Al2O3 crystal growth. The Cr3+ ions produce a donor-like intermediate level for the Al2O3 host material9,10. Figure 1a shows a schematic of the experimental setup for the FTXUV based on the transient absorption spectroscopy (for details, see the Methods section and Supplementary Note 1). The collinearly propagated IAP (44-eV center photon energy with 192-as duration11) and NIR pulse (1.55-eV center photon energy with 7-fs duration) are focused onto the target (for the IAP characterization, see the Supplementary Fig. 1). The timing jitter between IAP and NIR pulse is 23 as at the root mean square over 12 h in this pump-probe system6. The thin 36-nm-thick target without a substrate is manufactured from 400-μm-thick bulk target by mechanical polishing and ion beam milling. The electric fields of the IAP and NIR pulse are injected perpendicular to c-axis of the Al2O3. Figure 1b shows the measured absorption coefficient α in the target (red filled-circles and solid line) at room temperature. For comparison, that of a high-purity Al2O3 sample with the bandgap energy Eg of 8.7 eV8 is shown by a red dashed line. The Urbach tail12 region corresponding to the donor-like state is 5 to 8.7 eV. The measured absorption trace agrees with previous reports on Cr:Al2O3 solids10,13. The Al2O3 host material has the atomic number density of 2 × 1022 cm−3. The density of Cr dopant is 2 × 1017 cm−3, which was measured by the secondary ion mass spectrometry (SIMS) (see the Supplementary Fig. 2). The estimated doping level is approximately 1 × 10−3 at. % (10 ppm). Figure 1c shows the energy level diagram of the Al2O314. Here the intensity of the NIR pulse is 2 × 1012 W/cm2 on the target, which induces multiple inter-band polarizations from the valence band (VB) state to the Cr donor-like state and the Al2O3 conduction band (CB) state. Simultaneously, the IAP with high photon energy allows the excitation of electrons from the VB, donor-like, and CB states. The electrons are finally excited to the high-level CB states in the Al2O3, which have hyperfine states and behave as quasi-continuum states.

Fig. 1
figure 1

Experimental setup and properties of Cr:Al2O3. a Schematic view of experimental setup for the Fourier transform extreme ultraviolet attosecond spectroscopy (FTXUV) based on transient absorption spectroscopy. An IAP with 192-as duration (44-eV center photon energy) and a NIR pulse with 7-fs duration (1.55-eV center photon energy) were used. The target is 36-nm-thick bulk single-crystalline α-Al2O3 with Cr dopant. The transmitted IAP from the target was detected by an extreme ultraviolet (XUV) spectrometer. b Measured absorption coefficient α in the target (red filled circles and solid line). For comparison, the red dashed line shows that of the high-purity Al2O3 sample from ref. 8. The inset shows (αħω0)2 as a function of energy. The ħ and ω0 are the Dirac constant and the angular frequency of the injected light source in the absorption spectroscopy. The Urbach energy E0 of the initial donor-like state and bandgap energy Eg of the Al2O3 host material are 5 eV (orange arrow) and 8.7 eV (purple arrow), respectively. c Energy level diagram of the target. The green shaded area is the VB of the Al2O3 at K, Γ, A, and H points in wave vector14. The orange shaded area corresponds to the Cr donor-like state. The purple shaded area is the CB state of Al2O3 host material. Blue and red arrows show the IAP and NIR pulse, respectively

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Multi-petahertz interference in Cr:Al2O3

Figure 2a shows the measured multi-petahertz interferogram in the Cr:Al2O3. The trace shows a deviation of optical density (ΔOD) with and without the NIR pulse as a function of temporal delay (for the definition of ΔOD, see Methods section). The trace exhibits the characteristic temporal modulation in the whole energy region. The coherence of the superposition state created by electrons from the VB, donor-like, and CB states leads to quantum interference, which results in a temporal modulation of the IAP absorption spectrum in the whole photon energy region6. To confirm the electron transition process, we analyze the Keldysh parameter15 γ. The lightwave-field-induced electron tunneling is defined by the laser intensity, which corresponds to γ ≪ 1. In this experiment, the γ estimated from the minimum transition level of the Urbach energy (E0 = 5 eV) is 4.7, which is γ ≫ 1. Thus, the multiphoton process dominates the inter-band polarization, and the use of wide-bandgap materials makes it possible to induce the multiple petahertz oscillations via the multiphoton process6. In principle, such temporal modulation as shown in Fig. 2a is capable of producing the ultrafast conductivity variation in solid-state material3. Figure 2b shows the integrated line profiles for the photon energy regions of 38–46 eV in Fig. 2a. The profile has a variety of separations on an attosecond time scale, which are produced by the interference built on the superposition state of multiple electric dipole oscillations with different periodicities.

Fig. 2
figure 2

Multi-petahertz interferogram and energy components in Cr:Al2O3. a Measured interferogram based on transient absorption spectroscopy. The trace shows the deviation of optical density (ΔOD) with and without the NIR pulse as a function of the temporal delay. The trace is averaged over ten measurements. b Graph showing the integrated line profile (red dashed line) for the 38–46 eV photon energy region in a. The blue solid line corresponds to the interpolation process line profile. c Line profile spectrum (red solid line and shaded area) with Fourier transformation for the temporal delay axis in b. The dotted lines correspond to the photon energies of 6.2 (4ħω), 7.7 (5ħω), 9.2 (6ħω), and 10.8 eV (7ħω). The ω is the center angular frequency of the NIR pulse. The ħω corresponds to 1.55 eV. The blue dashed line shows the absorption coefficient α from our measurement (<8.7 eV) and ref. 33 (>8.7 eV). The orange and purple shaded areas correspond to the donor-like and CB states, respectively

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Figure 2c shows the energy components with Fourier transformation of the temporal delay axis in Fig. 2b. The dotted lines correspond to the 4ħω, 5ħω, 6ħω, and 7ħω photon energy components of the NIR pulse (ħω = 1.55 eV). The multiphoton process with the NIR pulse produces these specific energy components in the donor-like (orange shaded area) and the CB (purple shaded area) states. Here the 4ħω component has higher signal intensity than the 5ħω one because it is the first resonance state for a perturbative multiphoton process. However, the higher-order 6ħω and 7ħω components recover the signal intensity because the absorption coefficient drastically increases above the bandgap energy of Al2O3 (Eg = 8.7 eV). In principle, the absorption coefficient in the intermediate levels can be controlled with a variety of dopant materials9 and their doping level13. Consequently, this result strongly indicates that the interfered petahertz signal is manipulatable with the photon energy tuning of the driving laser and desirable with material band engineering in solid-state material.

Multiple dipole oscillations and electron dephasing

Next, we analyze temporal components in each electron transition. Figure 3a shows the absorption spectra in Fig. 2a after the windowed Fourier transform. The center window energies of the Fourier filtering are 6.2 (4ħω), 7.7 (5ħω), 9.2 (6ħω), and 10.8 eV (7ħω). The window bandwidth is applied with ±0.35 eV with the NIR bandwidth taken into account. Figure 3b shows each integrated line profile for the photon energy regions of 38–46 eV in Fig. 3a. The oscillation periodicities are 667 (4ħω), 537 (5ħω), 450 (6ħω), and 383 as (7ħω). The converted driving frequencies correspond to 1.5, 1.9, 2.2, and 2.6 PHz, respectively. The electron dynamics of 383-as periodicity has the shortest electric dipole oscillation ever recorded in a direct time-dependent measurement based on a pump-and-probe scheme16. The result also indicates that the donor-like components (4ħω and 5ħω) have a relative time delay of approximately 2 fs from the CB components (6ħω and 7ħω). The relative time delay (2 fs) and the different oscillation periodicities (667–383 as) produce various temporal modulations on the interferogram, as shown in Fig. 2b.

Fig. 3
figure 3

Temporal structure of each energy component. a Temporal structures in Fig. 2a after windowed Fourier transform. The window energies of Fourier filtering are 6.2 (4ħω), 7.7 (5ħω), 9.2 (6ħω), and 10.8 eV (7ħω). The applied window energy width is ±0.35 eV, taking into account the NIR bandwidth. The 4ħω and 5ħω correspond to the donor-like state. The 6ħω and 7ħω correspond to the CB state. b Traces showing the integrated line profile for the photon energy regions of 38–46 eV in a. The electric dipole oscillations with periodicities of 667 (4ħω), 537 (5ħω), 450 (6ħω), and 383 as (7ħω) correspond to frequencies of 1.5, 1.9, 2.2, and 2.6 PHz, respectively

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For a better understanding of this time delay of approximately 2 fs, we numerically simulated the resonant high-order polarizations using the model in a two-level system17,18, which consists of the ground state and the excited state either resonant to the donor-like state (5ħω) or resonant to the CB state (7ħω) (for details, see Supplementary Note 4). The calculated results with the parameter value of μF/ħω = 0.59, where μ and F correspond to the strengths of the transition dipole moment and the applied NIR driving laser field (7-fs duration), are shown in Figs. 4a, b. Here, the parameter value is consistent with that estimated from the Rabi frequency under our experimental condition. We found that the time delay could be tuned via the polarization dephasing times τ parameterized in the donor-like and CB states. Assuming that the CB state (τCB = 0.2 fs) has the faster dephasing time of 2.8 fs relative to the donor-like state (τdonor = 3 fs), the observed time delay of approximately 2 fs is reproduced well, as shown in Figs. 4c, d. The value of the dephasing time at room temperature is comparable to those obtained in previous pump-probe experiments using silicon (Si)5 and gallium arsenide (GaAs)19 solid-state materials, where the ultrafast dephasing time of 5–14 fs observed originates from intra-band electron–electron scattering. Note that when the dephasing times in the donor-like and CB states are equivalent, the buildup times of the polarization are almost identical in this simulation even though the states have different transition energies. Generally, the dephasing time in the spatially localized energy state is much longer than that in the band energy state20,21. 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. Consequently, the delayed buildup of the polarization could be observed in the resonant polarization of the spatially localized Cr donor-like state. In addition, the dephasing time is commonly explained by the density- and energy-dependent damping rate22, which is proportional to cube root of the excited carrier density n; the value of n depends on the fluence of pump pulse and the absorption coefficient of the target23,24. The linear absorption coefficient α in the CB state drastically increases higher than the fourth order of magnitude compared with the donor-like state, as shown in Fig. 2c. Thus, the excited carrier density in the CB state might be higher than the donor-like state even though the nonlinear multiphoton excitation. The different carrier densities could produce individual dephasing times in the donor-like and CB states. In any case, the important issue is the tuneable time delay induced by polarization behavior, which can be designed by band-structure control through the kind of dopant material and doping level. This tuneability will further assist with the flexibility of the petahertz signal manipulation.

Fig. 4
figure 4

Calculated time-dependent density matrix element and polarization. a The applied electric field waveform of the pulse with 7-fs duration. b Populations of the excited state (ρ11) in the donor-like (5ħω) and CB (7ħω) resonances with parameter values of μF/ħω = 0.59, τCB = 0.2 fs and τDonor = 3 fs. Fourier filtered polarizations in c donor-like and d CB resonances. The pink dotted lines and arrows show the time delay between the donor-like and CB components

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Discussion

Meanwhile, the intra-band polarization is another possible source of the inducement of ultrafast dipole oscillation. The intra-band polarization could already be contained in the measured dynamics, which makes it difficult to discuss intra-band and inter-band electric motions separately25. Here, since the measured dynamics already has high-order nonlinear processes (4–7ħω photon energies), the ponderomotive energy26 Up might be a useful parameter for qualitatively discussing the intra-band motion because it has been well used in terms of high-order harmonic generation (HHG) from solid-state material27,28,29,30. The Up is proportional to the intensity and square of the wavelength of the driving laser. High harmonics are typically generated by a low-intensity (approximately 1 × 1012 W/cm2), long-wavelength (middle-infrared and far-infrared) driving laser27,28,29 or by a high-intensity (approximately 1 × 1013 W/cm2), short-wavelength NIR one30. In this experiment, the estimated Up is 0.12 eV, which is extremely low compared to the Up = 1 to 23 eV in HHG experiments27,28,29,30. Therefore, in the measured dynamics, the inter-band polarization could have a large contribution compared with the intra-band polarization. Note that, the concept of the ponderomotive energy Up contains the effective mass approximation corresponding to the parabolic band in the HHG theory.

Previously, the petahertz electric dipole oscillations have been discussed in terms of the electric-field reconstruction of high-order harmonics emitted from polycrystalline quartz (SiO2)30 in solid-state materials. In contrast, the present study with the FTXUV method provides the direct time-domain observation of electronic dipole oscillations with nonlinear polarization. This property will provide sensitive detection and ultrafast manipulation. Since the dipole oscillation is the origin of light-matter interaction, the benefits are related to the control of various optical phenomena through the dielectric polarization. In addition, the directly identified time dependence in the solid reveals electric dipole oscillations with transition energy of up to 10.8 eV (7ħω) in this experiment. The energy region covers the bandgap energies for almost all semiconductor and insulator materials31. Therefore, this study lays the essential groundwork for exploring the band states in solid-state material, and the controllable time dependence resulting from the material band engineering will be important for developing petahertz digital electronics in the future.

Methods

Definition of deviation of optical density (ΔOD)

In the FTXUV material sensing, the transient absorption spectrum at temporal delay t between the IAP and NIR pulse is given by: ΔOD(ωIAP,t) = log[INIRout(ωIAP,t)/INIRin(ωIAP,t)], where ωIAP is the laser frequency of the IAP, INIRout(ωIAP,t) is the absorption spectrum of the IAP without the NIR pulse, and INIRin(ωIAP,t) is that with the NIR pulse added. Consequently, the ΔOD(ωIAP,t) monitors the absorbance deviation by the NIR pulse.

FTXUV experimental setup

In this experiment, a few-cycle pulse (1.55-eV center photon energy with 7-fs duration) from a Ti:sapphire laser was used for high-harmonic generation and as a pump-NIR pulse for the FTXUV (see Supplementary Note 1). The probe-IAP (44-eV center photon energy with 192-as duration) is generated by the double optical gating (DOG) technique32 using argon atom (see Supplementary Note 2). In this pump-probe system, the timing jitter is 23 as at the root mean square over 12 h, which is monitored by a helium neon laser6. The IAP spectrum transmitted from the Cr:Al2O3 target is detected by a regular extreme ultraviolet spectrometer (see Supplementary Note 3). The spectrometer resolution is 120 meV at 45.5-eV photon energy11.

Data availability

The data that support the findings of this study are available from the corresponding authors.