  We develop a new method for computing the holographic retarded propagator in
generic (non-)equilibrium states using the state/geometry map. We check that
our method reproduces the thermal spectral function given by the Son-Starinets
prescription. The time-dependence of the spectral function of a relevant scalar
operator is studied in a class of non-equilibrium states. The latter are
represented by AdS-Vaidya geometries with an arbitrary parameter characterising
the timescale for the dual state to transit from an initial thermal equilibrium
to another due to a homogeneous quench. For long quench duration, the spectral
function indeed follows the thermal form at the instantaneous effective
temperature adiabatically, although with a slight initial time delay and a bit
premature thermalisation. At shorter quench durations, several new
non-adiabatic features appear: (i) time-dependence of the spectral function is
seen much before than that in the effective temperature (advanced
time-dependence), (ii) a big transfer of spectral weight to frequencies greater
than the initial temperature occurs at an intermediate time (kink formation)
and (iii) new peaks with decreasing amplitudes but in greater numbers appear
even after the effective temperature has stabilised (persistent oscillations).
We find four broad routes to thermalisation for lower values of spatial
momenta. At higher values of spatial momenta, kink formations and persistent
oscillations are suppressed, and thermalisation time decreases. The general
thermalisation pattern is globally top-down, but a closer look reveals
complexities.
