Dynamical Slowing-Down in an Ultrafast Photoinduced Phase Transition

Complex systems, which consist of a large number of interacting constituents, often exhibit universal behavior near a phase transition. A slowdown of certain dynamical observables is one such recurring feature found in a vast array of contexts. This phenomenon, known as critical slowing-down, is well studied mostly in thermodynamic phase transitions. However, it is less understood in highly nonequilibrium settings, where the time it takes to traverse the phase boundary becomes comparable to the timescale of dynamical fluctuations. Using transient optical spectroscopy and femtosecond electron diffraction, we studied a photoinduced transition of a model charge-density-wave (CDW) compound ${\mathrm{LaTe}}_3$. We observed that it takes the longest time to suppress the order parameter at the threshold photoexcitation density, where the CDW transiently vanishes. This finding can be captured by generalizing the time-dependent Landau theory to a system far from equilibrium. The experimental observation and theoretical understanding of dynamical slowing-down may offer insight into other general principles behind nonequilibrium phase transitions in many-body systems.

Figure 1

Photoinduced CDW melting probed by multiple time-resolved techniques. (a) Schematics of time-resolved probes, including ultrafast electron diffraction (UED), transient optical spectroscopy (TOS), time- and angle-resolved photoemission spectroscopy (trARPES), and time-resolved x-ray diffraction (trXRD). The full electron diffraction pattern is shown in Fig. S1(b) in the Supplemental Material [28]. (b),(c) Schematics of the superlattice peak and density of states before (blue) and after (yellow) photoexcitation. (d) Transient response of superlattice peak intensity ($\mathrm{\Delta }{I}_{\mathrm{CDW}}$), in-gap spectral weight ($\mathrm{\Delta }\mathrm{SW}$), and reflectivity ($\mathrm{\Delta }R$) probed by corresponding time-resolved techniques. All traces are normalized between 0 and 1 and vertically offset for clarity. $\mathrm{\Delta }{I}_{\mathrm{CDW}}$ is inverted for easier comparison. All traces are measured in ${\mathrm{LaTe}}_3$ except for trXRD, which measures ${\mathrm{TbTe}}_3$, a similar compound in the same rare-earth tritelluride family with a lower $T_c$. The trace of trARPES is adapted from Ref. [18]. The trace of trXRD is adapted with permission from Ref. [37], copyrighted by the American Physical Society.

Figure 2

CDW suppression time at different excitation densities. (a) $\mathrm{\Delta }R/R$ traces at different excitation densities $F$ expressed in terms of absorbed photons per unit volume. A particular $\mathrm{\Delta }R/R$ cut is overlaid at the excitation density indicated by the dashed line. It is the same trace shown in Fig. 1. (b) $\mathrm{\Delta }R/R$ trace at $F=5.81\times {10}^{20}{\mathrm{cm}}^{-3}$, together with an example fit from Eq. (S1) in the Supplemental Material [28]. (c) Initial excited quasiparticle population $I_{\mathrm{peak},1}$ showing a superlinear dependence on $F$ below $F_{\mathrm{melt}}$ (arrow) and a plateau beyond $F_{\text{bleach}}$ (vertical dashed line). Black line is a linear fit with extrapolation (dashed) to zero. Blue curve is a fit to Eq. (S2) in the Supplemental Material [28]. (d) CDW suppression time $\tau$ as a function of excitation density $F$. Gray curve is a guide to eye. For the UED data, the corresponding trace of the dashed diamond is shown in Fig. 1 while the rest, measured on a separate sample, are shown in Fig. S2 in the Supplemental Material [28]. Error bars indicate uncertainties in curve fittings and in the instrumental temporal resolution [28].

Figure 3

Dynamical slowing-down in the generalized time-dependent Landau theory. (a) Schematic of CDW order parameter ($\psi$) dynamics in a Landau free-energy landscape. The solid circle represents $\psi$ before photoexcitation and dashed circles are nonequilibrium $\psi$ in an impulsively altered free energy, whose subsequent evolution, which is not drawn, is described by Eq. (S6) in the Supplemental Material [28]. Filled circles represent $\psi$ when CDW is transiently suppressed, either partially or completely. Colored curves are snapshots of transient free energies at different excitation densities right after laser pulse incidence. At $F_{\mathrm{melt}}$, $\psi$ first reaches zero in its temporal evolution. (b) Calculated CDW suppression time as a function of excitation density, based on the time-dependent Landau formalism [28].