  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 photo-induced transition of a model charge-density-wave (CDW) compound,
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 quantitatively 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.
