Light exhibits a dual nature as a photon and a light field. In light-matter interactions, the border of the photon and the light field is known as the Keldysh parameter1,5,23,24 \(\gamma (=\omega \sqrt{mΔ}/\left|e\right|E)\), where \(\omega\) is the frequency of light, \(m\) is the mass of the electron, Δ is the band gap energy, \(e\) is the electron charge, and \(E\) is the electric field strength of the light. For \(\gamma >1\), light acts on matter as a photon, which can be attributed to a traditional particle (photon) picture. For \(\gamma <1\), light cannot be described as a photon as it behaves as an electric field, namely a light field. Typical examples are high-order harmonics generation (HHG)25–30, the Landau–Zener transition1,5,21,22, and the Floquet state31–34. Non-equilibrium phenomena induced by a light field have attracted significant attention for exploring novel material functionalities and quasi-equilibrium phases. Hence, the investigation of light-field-driven phenomena is currently necessary. In addition to the Keldysh parameter lower than 1 (\(\gamma <1\)), coherence between the light field and electrons is essential for observing light-field-driven phenomena. Coherence is easily broken by electron-electron and/or electron-phonon scattering. As the electron scattering time in ordinary materials is very short (in the order of 10 fs), ultrashort pulses of less than 10 fs have been used to coherently drive electrons in materials5,6. To obtain ultrashort pulses of 10 fs or less, light with energies in the mid-infrared range (~ 100 THz) or higher is required. However, high-energy light can cause real electron excitations and prevent coherent driving. To realize coherent driving of electrons in materials, it is ideal to drive electrons in materials with a long electron scattering time using a low-energy light field35, i.e., a THz electric field. According to the Keldysh parameter \(\gamma\), ultra-intense light (~ 109 V/m) is required to achieve a light field picture at high frequencies, such as near-infrared5; however, using longer wavelengths, such as the THz wave (~ 1 ps), the light field picture (\(\gamma <1\)) can be easily achieved even in a weak electric field (~ 105 V/m). Therefore, we performed a THz-field drive in a Weyl semimetal, with a long electron scattering due to spin-momentum rocking, maintaining a long coherence between light and electrons. Therefore, it is expected that a light-field drive can be realized using a THz waves. Furthermore, Weyl semimetals, which have a Dirac point only tens meV away from the Fermi energy12, have a long coherence time and inhibit real excitation; therefore, they are ideal for light-field drives. Therefore, the THz wave is the ideal light source for observing of Weyl fermion dynamics driven by light fields.

This study shows the light-field-driven electrodynamics of the Weyl semimetal Co3Sn2S2 (CSS)13–20 in the THz frequency (~ 1ps). CSS has a relatively long electron scattering time of 0.1– 10 ps, a considerably slower timescale than the previously reported graphene5 and organic material6. As the time scale is comparable to a monocycle THz pulse, carrier dynamics by coherent acceleration can be observed. Consequently, we demonstrate THz-pulse-driven direct current (DC) generation in a CSS via coherent acceleration and non-adiabatic excitation (Landau–Zener transition1,5,21,22), which is completely different from Ohm's law. The nonlinearity of the DC with respect to the THz-electric-field strength indicates that the Landau–Zener transition, which is specific to a light field, becomes more pronounced as the electric field strength increases. This indicates a Keldysh crossover23 with increasing electric field strength of the THz pulse.

First, the carrier dynamics and electronic structure of CSS are described. Figures 1(a) and 1(b) show the temperature dependence of the reflectivity (R) and optical conductivity (σ) spectrum, derived from the Kramers–Kronig analysis of the R spectrum, in the c-plane of CSS, respectively. The R spectrum exhibits a strong temperature dependence representing electronic structure modulation with temperature due to the phase transition from a normal semimetal to a Weyl semimetal with decreasing temperature across TC=177 K13–20. In the Weyl semi-metallic phase, a pseudogap appears at a photon energy of approximately 30 meV, and a renormalized Drude component emerges below 15 meV. The Drude component has a relaxation time as long as 1 ps, which satisfies the condition for coherent acceleration by a THz pulse. Subsequently, the coherently acceleration was measured using a THz pulse, as shown in Fig. 1(c). THz pulse was irradiated onto a single crystalline CSS in the c-plane, and the reflection/emission spectra from CSS were measured using THz-time-domain-spectroscopy. The THz pulse has a linear polarization (x polarization) with a maximum field strength of 36 kV/cm and a center frequency of 0.8 THz. In this study, we performed the experiments at a temperature of approximately 7 K, which is significantly lower than the Curie temperature.

Figures 2(a) and 2(b) show the electric field waveform \({E}_{\text{S}}\) of a THz pulse reflected/emitted by the CSS and its Fourier transform (FT) spectrum, respectively, depending on the incident electric field strength. Figure 2(b) shows a peak structure centered at 0.8 THz, similar to the reference (Cu plate) spectrum, and, a Drude-like spectral structure below 0.2 THz only at high electric field incidence, indicating that high light fields generate a DC. In the general case of THz-TDS, as the THz pulse has a symmetric amplitude, the reflected THz pulse \({E}_{\text{S}}\) is symmetric, and the time integral of its waveform is zero. However, in this case, the time integral of \({E}_{\text{S}}\) at high light fields is not zero as shown in Fig. 2(c), and, is proportional to the square of the electric field strength (\(\propto {E}^{2}\)). \({E}_{\text{S}}\) can be decomposed into two components, \({E}_{\text{DC}}\) for the DC component and \({E}_{\text{AC}}\) for the remaining current (AC) component, in which the shape and intensity of the \({E}_{\text{AC}}\) component is evaluated as a waveform obtained at the lowest laser intensity and the multiplication of the incident laser intensity relative to the lowest one, respectively. The electric field waveforms of \({E}_{\text{DC}}\) and \({E}_{\text{AC}}\) are shown in Figs. 2(d) and 2(f), respectively, and their FT spectra are shown in Figs. 2(e) and 2(g), respectively. The FT spectra of \({E}_{\text{AC}}\) shown in Fig. 2(e) are similar to the reflection spectrum from a reference material (Cu plate) originating from a simple reflection, indicating high reflectivity due to the metallic character shown in Fig. 1(a).

On the other hand, the DC component has a positively biased asymmetric-electric-field waveform, as shown in Fig. 2(f). The FT spectrum of the DC component shown in Fig. 2(g) appears to be a broadband Drude-like spectrum below 3 THz. The integrated intensity is proportional to the square of the incident electric field amplitude (\({E}_{\text{DC}}\propto {E}^{2}\)), which is consistent with the behavior of \({E}_{\text{S}}\). This result suggests that the asymmetric emitted electric waveform originates from the DC component, i.e., the observed asymmetric emission and DC generation are driven by a monocycle-intense THz pulse (depicted in Fig. 2(h)).

Next, the origin of DC generation is discussed. Assuming that the Ohm’s law (\(j\left(t\right)=\sigma E\left(t\right)\), where \(j\left(t\right)\) is the current and \(\sigma\) the conductivity) is dominant, when a symmetric THz pulse waveform isotropically accelerates the electrons in a material, no DC components appear. However, the DC components are visible as a THz pulse, as shown in Fig. 2(g). If no scattering occurs within the THz pulse acceleration time, the DC component \(j\left(t\right)\) can be generated by the incident THz pulse \(E\left(t\right)\), as expressed by Eq. (1).

$$\begin{array}{c}j\left(t\right)=\frac{n{e}^{2}}{m}\int E\left(t\right)dt,\#\left(1\right)\end{array}$$

where \(n\), \(e\), and \(m\) denote the carrier density, electron charge, and electron mass, respectively. The differential current generates THz emissions as follows:

$$\begin{array}{c}{E}_{\text{DC}}\left(t\right)\propto \frac{d\varvec{j}}{dt}.\#\left(2\right)\end{array}$$

Therefore, if carrier density is preserved, the DC electric field of the THz emission is proportional to the incident THz pulse (\({E}_{\text{DC}}\left(t\right)\propto E\left(t\right)\)); however, this is not consistent with the experimental results, i.e., \({E}_{\text{DC}}\left(t\right)\) does not follow \(E\left(t\right)\) but follows \(\int E\left(t\right)dt\), as shown in Fig. 3(a), indicating a coherent acceleration6. To understand this inconsistency, the carrier density is changed with time, i.e., Eq. (1) can be rewritten as follows:

$$\begin{array}{c}\frac{dj}{dt}=\frac{{e}^{2}}{m}\left(\frac{\partial n}{\partial t}\right)\int E\left(t\right)dt+\frac{{e}^{2}}{m}nE\left(t\right).\#\left(3\right)\end{array}$$

As shown in Fig. 3(a), the second term may only make a minor contribution; however, the first term is dominant because \({E}_{\text{DC}}\left(t\right)\) almost follows \(\int E\left(t\right)dt\), i.e., both \(\int \varvec{E}\left(t\right)dt\) and \({E}_{\text{DC}}\) have asymmetric positively biased waveforms with well-matched amplitude signs. This suggests that carriers are generated by the high electric field. To investigate the origin of the carrier excitation, we focus on the electric-field-strength dependence of \({E}_{\text{DC}}\left(t\right)\), as shown in Fig. 3(b). \({E}_{\text{AC}}\) exhibits linear dependence with the incident electric field amplitude owing to a simple metallic reflection at the sample. In contrast, the area of the FT spectra of \({E}_{\text{DC}}\) is proportional to the square of the incident electric field. One possible origin of such nonlinear phenomena is two-photon absorption; however, the energy of the Weyl points from the Fermi level of CSS (several tens of millielectronvolts) cannot be excited by a two-photon process of approximately 4 meV. Multiphoton excitation is possible; however, multiphoton absorption is unlikely because of the small transition probability. In addition, the experimental \({E}^{2}\)-law does not fit the multiphoton process.

Following the coherent acceleration by the intense THz pulse, another possible mechanism for carrier excitation is the Landau–Zener transition1,5,21,22 caused by the light-field drive. As shown in Fig. 4(a), the Landau–Zener transition is caused by quantum tunneling, in which the electron is accelerated by a light field, resulting in a large momentum change. The current associated with the Landau–Zener transitions in band structures with linear dispersion, such as graphene and Weyl semimetals, is reportedly proportional to the \(d-1\) power (\(d\): dimension) of the electric field amplitude36,37. Owing to the three-dimensional Weyl semimetal CSS, the current intensity associated with the Landau–Zener transition in the material is expected to be proportional to the square of the electric field strength (\(\propto {E}^{2}\))37. As shown in Fig. 3(b), the electric field strength dependence of \({E}_{\text{DC}}\) obeys the \({E}^{2}\)-law, which agrees well with theoretical predictions. The DC is the origin of the coherent acceleration, and the DC intensity is proportional to the square of the THz electric field intensity. Therefore, the carrier excitation is attributed to the Landau–Zener transition. The Landau–Zener transition is a phenomenon specific to light-field pictures, therefore, as shown in Fig. 4(b), the crossover from a photon picture at a lower electric field to a light-field picture at a higher electric field can be observed by increasing the electric field strength of the incident light. The Keldysh parameter can be described by the following Eq. 5.

$$\begin{array}{c}\gamma =\frac{Δ }{2\hslash {{\Omega }}_{\text{R}}}.\#\left(4\right)\end{array}$$

$$\begin{array}{c}{{\Omega }}_{\text{R}}=\frac{{v}_{\text{F}}\left|e\right|E}{\hslash \omega },\#\left(5\right)\end{array}$$

where \({{\Omega }}_{\text{R}}\) is the Rabi frequency, \(\omega\) is the frequency of light, Δ is the band gap energy, \({v}_{\text{F}}\) is the Fermi velocity of the electron, \(e\) is the electron charge, and \(E\) is the electric field strength of light. The band gap energy Δ corresponds to twice the Fermi energy due to the gapless structure of the Weyl band. Assuming \(Δ =\)30 meV, \({v}_{\text{F}}={1.0\times 10}^{6}\) m/s and \(\hslash \omega =\) 4 meV, the Keldysh parameter is 1 at \(E=\)6 kV/cm. As shown in Fig. 4(b), a light-field (photon) picture is dominant for \(\gamma <1\), and a photon picture is dominant for \(\gamma >1\). The electric field strength changes from 1 to 36 kV/cm, for which the dependence is measured in this experiment, corresponding to a change in the Keldysh parameter \(\gamma\) from 4 to 0.1. At electric fields higher than \(E=\)6 kV/cm (\(\gamma \sim1\)), the DC component \({E}_{\text{DC}}\) increases significantly, as shown in Fig. 2(c). The appearance of the light-field picture observed in this study suggests a Keldysh crossover23 in a Weyl semimetal.

As shown in Fig. 2(g), the DC generation also suggests that the THz pulse breaks the spatial-inversion symmetry of the electron system in a non-equilibrium state38–40. As a change in the system symmetry leads to a change in topology, the possibility of ultrafast topology control31,33,41,42 by an intense THz pulse may be feasible.

In this study, we observed DC generation from a Weyl semimetal Co3Sn2S2 by irradiating it with an intense monocycle THz pulse. The DC amplitude proportionally increases to the square of the incident electric field amplitude, suggesting that the intense THz pulse generates a non-Ohmic current in the Weyl semimetal and that its origin is the coherent acceleration by the light field drive and electronic excitation by the Landau–Zener transition. The nonlinear DC generation driven by a monocycle intense THz pulse suggests the realization of a light-field picture across a Keldysh crossover from a photon picture. By combining a THz pulse with a low frequency and Weyl semimetals with a gapless structure, we observed the phenomenon of a light-field picture (\(\gamma <1\)) even at an easily achievable light-field strength of several tens of kilovolts per centimeters. The long electron scattering time in Weyl semimetals allows the occurrence of the light-field-driven phenomena even on a picosecond time scale. These results show that the THz-pulse drive of Weyl fermions is a good platform for investigating the initial process of light-field-driven phenomena. Furthermore, the generation of DC by an intense THz pulse indicates a light-induced system symmetry change and is expected to enable the establishment of a new approach for ultrafast topological phase transitions. The proposed approach is expected to significantly contribute to the development and exploitation of novel non-equilibrium phases and functionalities of materials.