Optically-Induced Faraday-Goldstone Waves

TL;DR

The paper demonstrates that ultrafast optical pulses can induce Faraday-like spatiotemporal order in symmetry-broken quantum materials through nonlinear coupling between Higgs and Goldstone modes, yielding Faraday-Goldstone waves at finite momentum and half-frequency phasons. A dynamical Landau-type theory is developed, revealing a threshold-driven parametric instability with clear signatures in both amplitude (Higgs) and phase (Goldstone) sectors, including Higgs–Goldstone beating and frequency softening. The theory quantitatively explains pump–probe measurements in K$_{0.3}$MnO$_3$, including reflectivity changes, ARPES responses, and the emergence of $q_0$-centered phase dynamics, validating the mechanism. The work suggests a route to engineer non-thermal, optically induced periodic structures in quantum materials and highlights the robustness of driven spatiotemporal order against thermal fluctuations.

Abstract

Faraday waves, typically observed in driven fluids, result from the confluence of nonlinearity and parametric amplification. Here we show that optical pulses can generate analogous phenomena that persist much longer than the pump time-scales in ordered quantum solids. We present a theory of ultrafast light-matter interactions within a symmetry-broken state; dynamical nonlinear coupling between the Higgs (amplitude) and the Goldstone (phase) modes drives an emergent phason texture that oscillates in space and in time: Faraday-Goldstone waves. Calculated signatures of this spatiotemporal order compare well with measurements on K$_{0.3}$MnO$_{3}$; Higgs-Goldstone beating, associated with coherent energy exchange between these two modes, is also predicted. We show this light-generated crystalline state is robust to thermal noise, even when the original Goldstone mode is not. Our results offer a new pathway for the design of periodic structures in quantum materials with ultrafast light pulses.

Figures (8)

• Figure 1: Spatial and temporal dependence of the order parameter's amplitude and phase after a pulse excitation, Eq. \ref{['eq:init']}. (a,c) Spatial and temporal profile of $\Delta(x,t)$, below (a) and above (c) the the threshold $\delta$ for parametric excitations. Note that in below the threshold in (b), the phase mode does not respond at all and remains fixed at $\phi =0$. Above the threshold, in (d), the phase mode exhibits coherent spatial and temporal oscillations. Here, $\eta =0.5 m$.
• Figure 2: Onset of parametric oscillations in the out-of-equilibrium state. (a) shows temporal oscillations at the $x=0$ cut, for two regimes, above and below the threshold $\delta$ discussed in Sec. \ref{['sec:param']}. (b) As in (a), but for the phase $\phi(x,t)$. While no significant change is seen for the amplitude mode $\Delta(x,t)$ (a), above a threshold a parametric instability sets in, manifest in a transient exponential growth in time. Inset shows the dependence of the spectral weight of the amplitude mode at $\omega=m$ on $\delta$ remaining linear for several dissipation values $\eta$ (green, orange, blue curves). (c) Fourier spectrum of the amplitude and phase modes (blue, orange, respectively). Modes shows a clear resonance at the $\omega=m$ for the amplitude, and the parametrically driven mode $m/2$ for the phase. (d) Fourier spectrum $\phi(q,\omega)$ for the phase mode. Clearly visible are the linear dispersion of the phase mode, the resonances at $\pm m/2, q=\pm q_0\approx \pm m/(2v)$ and higher-order replicas driven by nonlinearities.
• Figure 3: Threshold properties and momentum space structure of the Goldstone-Faraday waves. (a) Log of the Fourier amplitude $\phi(q_0,m/2)$ as a function of drive intensity for three different initial conditions $\phi(0) = 10^{-5}, 10^{-4},10^{-3}$. Colored dashed lines correspond to the growth rate $\lambda$ (Eq. \ref{['eq:grate']}), off set only by the initial value of $\phi$. The different regimes $I,II,III$ (main text) are delineated. (b) Peak structure near $q_0$ at maximum gain $t_{\max}$, for different initial displacements $\delta$ on a log scale. In region $I$, no peak is visible, since it is below the fundamental threshold. The peaks intensity clearly grows with displacement $\delta$, showing that parametric amplification occurs for a range of $q$ modes. In region $III$ nonlinearities cause deviation from Gaussianity for the peak structure (red curve) (c) Same as (b), but normalized on a linear scale. Even though a range of $q$ modes are parametrically excited, the overall similarity of peaks in region $II$ suggests that spatial coherence is maintained up to a minimal width of $\eta/v$ (black scale bar). In region $III$, broadening increases.
• Figure 4: Properties of the Faraday-Goldstone state. (a) Fourier spectrum of $\Delta$ at $\omega=m$, $q=0$, as a function of displacement. The colors -- from blue to red -- indicate increasing dissipation $\eta$. Increasing dissipation leads a low saturation value. (b) Fourier spectrum of $\phi$ at $\omega=m/2$, $q=m/(2v)$ as a function of displacement. A pronounced increase onsets concomitantly with saturation of $\Delta(m,q=0)$. The dashed line follows the numerical inversion of Eq. \ref{['eq:thresh_gam']}. (c) Displacement threshold vs dissipation. Black dashed line denoted the threshold obtained from an inversion of Eq. \ref{['eq:thresh_gam']}. The red dashed is the equivalent "continuous driving" threshold Kaplan2025$\delta_c \propto \eta$. (d) Frequency softening as a function of displacement. The peak response frequency of the amplitude mode $\omega_{max}/m$ depends on $\delta$ via $\delta^{2}$, approximately, due to intrinsic nonlinearity of the amplitude mode, Eq. \ref{['eq:soft']}. Colors (blue to red) indicate increasing dissipation $\eta$.
• Figure 5: Signatures of Higgs-Goldstone oscillations in time dependence of $\Delta(t,x=0)$ and $\phi(t,x=0)$ for two values of the drive $\delta$ and small $\eta = 0.07m$. (a) Spatiotemporal order parameter and phase mode oscillations (blue and orange curves), for an initial displacement of $\delta/\Delta(0) = 0.07$. (b) Same as (a), but $\delta/\Delta(0) = 0.10$. Here, the amplitude and frequency of beating is directly controlled by the initial displacement $\delta$. Coherent oscillation of spectral intensity between amplitude and phase modes are clearly observed for stronger deviation from equilibrium, shown in (b).