  We study the non-equilibrium evolution of a one-dimensional quantum Ising
chain with spatially disordered, time-dependent, transverse fields
characterised by white noise correlation dynamics. We establish
pre-thermalization in this model, showing that the quench dynamics of the
on-site transverse magnetisation first approaches a metastable state unaffected
by noise fluctuations, and then relaxes exponentially fast towards an infinite
temperature state as a result of the noise. We also consider energy transport
in the model, starting from an inhomogeneous state with two domain walls which
separate regions characterised by spins with opposite transverse magnetization.
We observe at intermediate time scales a phenomenology akin to Anderson
localization: energy remains localised within the two domain walls, until the
Markovian noise destroys coherence and accordingly disorder-induced
localization, allowing the system to relax towards the late stages of its
non-equilibrium dynamics. We benchmark our results with the simpler case of a
noisy quantum Ising chain without disorder, and we find that the pre-thermal
plateau is a generic property of weakly noisy spin chains, while the phenomenon
of pre-thermal Anderson localisation is a specific feature arising from the
competition of noise and disorder in the real-time transport properties of the
system.
