Role of Equilibrium Fluctuations in Light-Induced Order

Engineering novel states of matter with light is at the forefront of materials research. An intensely studied direction is to realize broken-symmetry phases that are “hidden” under equilibrium conditions but can be unleashed by an ultrashort laser pulse. Despite a plethora of experimental discoveries, the nature of these orders and how they transiently appear remain unclear. To this end, we investigate a nonequilibrium charge density wave (CDW) in rare-earth tritellurides, which is suppressed in equilibrium but emerges after photoexcitation. Using a pump-pump-probe protocol implemented in ultrafast electron diffraction, we demonstrate that the light-induced CDW consists solely of order parameter fluctuations, which bear striking similarities to critical fluctuations in equilibrium despite differences in the length scale. By calculating the dynamics of CDW fluctuations in a nonperturbative model, we further show that the strength of the light-induced order is governed by the amplitude of equilibrium fluctuations. These findings highlight photoinduced fluctuations as an important ingredient for the emergence of transient orders out of equilibrium. Our results further suggest that materials with strong fluctuations in equilibrium are promising platforms to host hidden orders after laser excitation.

Figure 1

Competing charge density waves in rare-earth tritellurides. (a) Left: schematic of the layered structure of $R{\mathrm{Te}}_3$, where dashed lines indicate the primary unit cell. Right: enlarged view of the nearly square-shaped Te sheets that host the CDW instabilities. (b) Schematic phonon dispersion right above $T_c$ along $\mathrm{\Gamma }\text{−}X$ and $\mathrm{\Gamma }\text{−}Z$, featuring two Kohn anomalies at $q_{\mathrm{soft}}$. (c) Schematic of fluctuating CDWs right above $T_c$. (d),(e) Equilibrium electron diffractions of ${\mathrm{DyTe}}_3$ [$T_c=306(3)\mathrm{K}$ 24] taken at 100 K (d) and 307 K (e). (f),(g) Time-resolved diffractions of ${\mathrm{LaTe}}_3$ (f) before and (g) 1.6 ps after photoexcitation by an 80-fs, 800-nm laser pulse with an incident fluence of $2.1\mathrm{mJ}/{\mathrm{cm}}^2$, measured at 307 K. Blue and red arrows indicate the CDW peaks along the $c$ and $a$ axis, respectively. Difference in intensities of lattice Bragg peaks in (d) and (e) results from slight sample drift and tilt during the warm-up process.

Figure 2

Response of CDW fluctuations to photoexcitation. (a),(b) Schematic setups for ${\mathrm{DyTe}}_3$ and ${\mathrm{LaTe}}_3$. Both samples were kept at $T=307\mathrm{K}$. The incident fluence was $3.3\mathrm{mJ}/{\mathrm{cm}}^2$ in (a) and $1.0\mathrm{mJ}/{\mathrm{cm}}^2$ for each pump in (b). (c)–(h) Changes in the integrated intensities for thermal diffuse scattering [$I_{\mathrm{TDS}}$, (c),(d)], $a$-axis CDW peak [$I_a$, (e),(f)], and $c$-axis CDW peak [$I_c$, (g),(h)]. Integration areas are marked by solid circles in the insets. Traces are normalized by the average value of $I_c$ before photoexcitation. In (d),(f),(h), vertical lines indicate the arrival time of ${\mathrm{pump}}_2$ at $\mathrm{\Delta }t=1.3\mathrm{ps}$. For reference, dynamics in the absence of ${\mathrm{pump}}_2$ is shown in gray. (i),(j) Enlarged view of dashed rectangles in (e)–(h) after subtracting the respective thermal diffuse background [$I_{\mathrm{TDS}}$ in (c),(d)]. In (i), traces are normalized by the average values at $t<0$. In (j), traces are plotted as a function of the relative delay between the probe pulse and ${\mathrm{pump}}_2$ ($t^{\prime }$), where intensities are normalized by the average value in the interval $t^{\prime }\in [-0.7,-0.2]\mathrm{ps}$. A slightly larger reduction along the $c$ axis is attributed to a mismatch between pumped and probed volumes, leading to additional melting of residual long-range CDW by the second pulse. In (c)–(j), solid curves are fits to a phenomenological model in Eq. (S1) (see Supplemental Material [25]). The black fitted curve in (i) uses the averaged data along the $a$ and $c$ axes.

Figure 3

Simulated dynamics of photoinduced CDW and its relation to equilibrium fluctuations. (a) Evolution of integrated intensities of the $c$- (blue) and $a$-axis (red) CDW peak upon photoexciting the unidirectional CDW state. Triangle marks the maximum intensity of the light-induced CDW. The nonzero value of the $a$-axis peak at $t=0$ originates from thermal fluctuations. (b) Maximum intensity change of the photoinduced $a$-axis CDW peak [$\mathrm{\Delta }{I}_a(t)$] as a function of equilibrium diffuse intensity at a fixed temperature above $T_c$, the latter of which quantifies thermal fluctuations and is indistinguishable between the two axes. $\mathrm{\Delta }{I}_a(t)$ is normalized by $I_a(t=0)$ (see Fig. S8 [25]).