Strong-field Driven Sub-cycle Band Structure Modulation Measured with Ultrafast Electric Field Observables

TL;DR

The study demonstrates sub-cycle, strong-field–driven modulation of MgO's band structure by measuring complete electric-field observables in degenerate four-wave mixing. Using spectral interferometry, the authors show amplitude and chirp oscillations with sub-cycle delays that cannot be explained by field-free perturbation theory alone; incorporating ultrafast band-structure modulation into a perturbative framework reproduces the observations. Complementary time-dependent perturbation theory and Semiconductor Bloch Equation simulations reveal that field-dressed bands drive the nonlinear response, with intra- and inter-band currents contributing comparably. The work establishes electric-field observables as direct probes of attosecond electron dynamics in solids and suggests avenues for ultrafast control of nonlinear processes and potential generation of non-classical light via field-induced squeezing ($χ^{(3)}$ modulation) at attosecond timescales.

Abstract

Over the past decade, ultrafast electron dynamics in the solid state have been extensively studied using various strong light-matter interaction techniques, such as high-harmonic generation. These studies lead to multiple interpretations of light-matter interaction in the strong-field regime, with exact mechanisms not yet fully understood. It is known that strong-field interaction with a crystalline solid leads to significant modification of its band structure and hence its optical properties on ultrafast timescales. In this work, we present measurements with ultrafast electric field observables in magnesium oxide from a non-resonant nonlinear optical interaction. Using field observables, we show that the ultrafast, strong-field light-matter interaction modulates the band structure on sub-cycle time scales, resulting in a modulation of the nonlinear optical response of the material. We perform time-dependent perturbation theory calculations with a field-dependent dispersion relation and non-perturbative semiconductor Bloch equation calculations, which agree with experimental observations. Our work offers a new perspective on strong-field-driven electron dynamics in solids through the lens of electric field observables. The demonstrated attosecond modulation of the nonlinear response could have important implications for quantum light generation using nonlinear optical processes.

Figures (5)

• Figure 1: (a) Ultrafast Band modulation: The net electric field resulting from the superposition of three ultrafast femtosecond pulses interacts with MgO and significantly modifiesthe band structure of the material. This interaction induces a modulation of the band structure (represented by opaque curves) and causes a shift (transparent curves) from its original position on an ultrafast timescale. The measured output electric field from the nonlinear process can reveal and capture this modulation, providing insights into the strong-field interaction dynamics in a solid. (b) Schematic diagram of the spectral interferometry measurement for degenerate four-wave mixing (DFWM): Experimental setup illustrates the interaction of three ultrafast pulses (Gate 1, Gate 2, and Probe) with the magnesium oxide crystal and the generation of a DFWM signal pulse. This signal interferes with a reference pulse at a spectrometer resulting in a spectral interference measurement. R - Reference pulse, S - Signal pulse and BS - beam splitter.
• Figure 2: Experimentally Measured Amplitude and Phase: (a) Amplitude $|E_0(t,\tau)|$ and (b) Phase $\varphi(t,\tau)$ of the Degenerate Four-Wave Mixing (DFWM) electric field signal in Magnesium Oxide (MgO). (c) Time integrated temporal amplitude and (d) Temporal Chirp as a function of Time Delay $\tau$ with a $\tau$ delay step of 200 as extracted from (a) and (b) respectively. The shaded region represents the statistical error in the measurement.
• Figure 3: Fourier Transformation in Time Delay: The Fourier transforms ($\mathcal{FT}$) with respect to the time delay of the experimentally measured (a) time-integrated amplitude and (b) temporal chirp parameter are shown. These results are compared with the corresponding (c) time-integrated amplitude and (d) chirp parameter obtained from the electric field calculated using time-dependent (TD) perturbation theory, which includes the band-structure modulation induced by the strong field (see the main text for details). The shaded regions in panels (a) and (b) represent the standard deviation errors in the experimental measurements. The calculation parameters are: IR pulse intensity $I_0 = 0.2\ \mathrm{TW/cm^2}$, photon energy $\hbar\omega_0 = 1.56\ \mathrm{eV}$, and MgO bandgap $\Delta = 7.78\ \mathrm{eV}$. The (e) time-integrated amplitude and (f) temporal chirp are also extracted from the calculated output electric field using the Semiconductor Bloch Equations (SBE), where the contributions from intraband and interband currents are shown separately.
• Figure 4: The 1$\omega_0$ component extracted from the Fourier transformation of the temporal chip is isolated, and the inverse Fourier transformation is performed. (a) The oscillation of the 1$\omega_0$ component of the chirp parameter extracted from the experimentally measured DFWM electric field (green) is compared with the nonlinear susceptibility modulation model (magenta) as a function of time delay. (b) The time-dependent unwrapped phases of the oscillations in (a) are also presented, showing that the susceptibility modulation model accurately reproduces the $\pi$-phase jump near $\tau = 20~\mathrm{fs}$, consistent with the experimental observations.
• Figure 5: (a) Time-integrated amplitude and (b) temporal chirp are calculated using time-dependent perturbation theory with and without band structure modulation. Their corresponding Fourier transformations in the time delay domain are shown in (c) and (d), respectively.