Coherent phonon control beyond amplitude saturation in a sliding ferroelectric

Abstract

The breakdown of Hooke's law marks the onset of nonlinear behaviour: when displacements become large, restoring forces weaken and conventional proportionality fails. In quantum materials, intense optical excitation can drive the crystal lattice into a similar regime, where established linear relations between light, electrons, and phonons no longer hold. Sliding ferroelectrics are particularly susceptible, as controlling their polarization requires large interlayer shifts. Displacive excitation of coherent phonons, the principal mechanism for launching structural motion, typically assumes that lattice-driving forces scale linearly with the photo-excited carrier density. Whether this linearity survives at high excitation, however, remains largely unexplored, and its breakdown can fundamentally limit accessible lattice displacements. Here we show that such nonlinear limitations can be surpassed in a sliding ferroelectric by timing, rather than strengthening the optical drive. Time-resolved second-harmonic generation reveals that the interlayer sliding phonon governing ferroelectricity saturates and even diminishes under single-pulse excitation. First-principles calculations attribute this nonlinearity to band-specific electron-phonon coupling that induces competing forces on the lattice. By splitting the optical energy into two well-timed pulses that avoid populating counteracting states, we achieve markedly larger phonon amplitudes at fixed total fluence. The resulting enhanced sliding motion exposes a regime of anharmonic phonon coupling that emerges only far from equilibrium. Our findings show that nonlinear limits in driven solids can be overcome, opening new pathways for steering lattice motion in quantum materials.

Figures (10)

• Figure 1: $\mid$ Ultrafast structural dynamics in ferroelectric Td-WTe$_2$.a, Calculated two-dimensional potential energy surface of WTe$_2$ spanned by interlayer sliding and intralayer shear displacements. b, Structural configurations of WTe$_2$ and lattice modes connecting the Td and 1T' polymorphs to the paraelectric high-symmetry phase. Dark grey circles, tungsten atoms; light brown circles, tellurium atoms; grey background structures denote the atomic configuration of the high-symmetry paraelectric phase, included for comparison. c, Polarization-dependent SHG intensity at selected pump–probe delays $\Delta t_{\text{p-pr}}$. The incident polarization of the fundamental is varied, with the analyzer fixed in the horizontal orientation. d, Normalized, delay-dependent SHG efficiency (polarizer: vertical; analyzer: horizontal) as a function of absorbed laser fluence $F$. Traces are vertically offset for clarity. e, Sliding-mode amplitude $A_\text{s}$ and lifetime $\tau_\text{s}$ as a function of absorbed fluence $F$. f, Sliding-mode frequency and quasi-static suppression of SHG intensity as functions of absorbed fluence $F$.
• Figure 2: $\mid$ Ab initio modeling of sliding-amplitude saturation.a, State-resolved electron-phonon coupling strength $g$ at the sliding-mode frequency at two electronic temperatures $T_{\text{elec}}=2000\,\text{K}$ and $T_{\text{elec}}=4000\,\text{K}$. Red (blue) indicates positive (negative) contributions to the sliding mode excitation with more saturated colors corresponding to stronger coupling. Along the energy axis, the coupling strength $g$ is weighted by the change in the Fermi–Dirac distribution $\Delta f_{\text{elec}}$ at the respective temperature. b, Calculated maximum amplitudes $Q_{\text{max}}$ of the sliding mode and of two higher-frequency modes of Td-WTe$_2$ as a function of absorbed fluence $F$. Triangular data points represent the sliding-mode amplitudes determined in our experiments for comparison.
• Figure 3: $\mid$ Coherent control and amplification of sliding motion.a, Schematic of amplitude control of displacively excited coherent phonons using double-pulse excitation. b, Normalized SHG intensity as a function of pump–probe delay $\Delta t_{\text{p-pr}}$ and double-pulse delay $\Delta t_{\text{p1-p2}}$ for a combined absorbed fluence of $2.1\,\text{mJ}\,\text{cm}^{-2}$. Red, yellow, and violet arrows indicate the double-pulse delays corresponding to the pump–probe traces shown in (c). c, Pump–probe SHG traces at selected double-pulse delays, demonstrating coherent control of the sliding-mode amplitude. d, Simulated sliding-mode amplitude $Q$ as a function of $\Delta t_{\text{p-pr}}$ and $\Delta t_{\text{p1-p2}}$, assuming an amplitude-dependent sliding frequency. e, Comparison of pump–probe SHG traces for $\Delta t_{\text{p1-p2}} = 0\,\text{ps}$ and $\Delta t_{\text{p1-p2}} = 8.3\,\text{ps}$. f, Sliding-mode amplitude after the second optical excitation as a function of $\Delta t_{\text{p1-p2}}$. Dashed black line, expected behaviour in coherent control experiments without amplitude saturation (see Methods); red line, fitted model incorporating amplitude suppression at short double-pulse delays; violet line, transient suppression of the sliding mode extracted from fits to the experimental data (see Methods for details). g, Comparison of vibrational amplitudes extracted from pump–probe (PP) and pump–pump–probe (PPP) measurements. Black line, estimated vibrational amplitude assuming a linear dependence of $A_\text{s}$ on the excited-carrier density.
• Figure 4: $\mid$ High-amplitude vibrational spectroscopy in targeted excited states.a, Sliding-mode amplitude $A_\text{s}$, damping $\gamma_{\text{s}}=\tau_{\text{s}}^{-1}$, and frequency $f_{\text{s}}$ as a function of the double-pulse delay $\Delta t_{\text{p1-p2}}$ for a combined absorbed fluence of $2.1\,\text{mJ}\,\text{cm}^{-2}$. Light-colored shades, $1\sigma$-confidence intervals of the fits used to extract parameters from the two-dimensional dataset shown in Fig. \ref{['fig:3']}b. b, Sliding-mode damping constant $\gamma_{\text{s}}$ as a function of the sliding-mode amplitude $A_{\text{s}}$ for two different combined fluences in double-pulse experiments. Solid red and violet lines, linear models fitted to the data to determine $(\Delta\gamma/\Delta A)_{\text{s}}$; errorbars, $1\sigma$-confidence intervals of the fits used to extract $A_{\text{s}}$ and $\gamma_{\text{s}}$. c, Low-energy phonon band structure of WTe$_2$. Red arrows indicate an exemplary pathway for the Klemens-like decay of the sliding phonon into two acoustic phonons of opposite momentum.
• Figure : Fig. S1 $\mid$ Polarization-dependent SHG signal in double-pulse and single-pulse experiments.a, Comparison of SHG signals from WTe$_2$ as a function of the delay between two pump pulses for parallel (violet) and cross-polarized (red) configurations (combined absorbed fluence $F=2.1\,\text{mJ}\,\text{cm}^{-2}$). The probe pulse interrogates the sample at a fixed delay of $\Delta t_{\text{p-pr}}= 100\,\text{ps}$, where coherent phonon oscillations are negligible. For parallel-polarized pump pulses, interference at the sample surface leads to a delay-dependent modulation of energy absorption, resulting in a corresponding variation of the SHG signal. In contrast, interference effects are strongly suppressed for cross-polarized pump pulses. Consequently, all double-pulse excitation experiments were performed using cross-polarized pump pulses. b, SHG intensity measured at $\Delta t_{\text{p-pr}}= 100\,\text{ps}$ for two different polarizer–analyzer configurations, shown as a function of the polarization angle of a single pump pulse (absorbed fluence $F=1.6\,\text{mJ}\,\text{cm}^{-2}$; $0^{\circ}: P\parallel a$-axis; $90^{\circ}: P\parallel b$-axis of the crystal). Only a weak anisotropy is observed, indicating that in double-pulse experiments both pump pulses, despite their differing polarizations, contribute similarly to the electronic excitation of the material.