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We study the effect of spatially correlated classical noise on both Andersonand many-body localization of a disordered fermionic chain. By analyzing theevolution of the particle density imbalance following a quench from an initialcharge density wave state, we find prominent signatures of localization also inthe presence of the time-dependent noise, even though the system eventuallyrelaxes to the infinite temperature state. In particular, for sufficientlystrong static disorder we observe the onset of metastability, which becomesmore prominent the stronger the spatial correlations of the noise. In thisregime we find that the imbalance decays as a stretched-exponential - abehavior characteristic of glassy systems. We identify a simple scalingbehavior of the relevant relaxation times in terms of the static disorder andof the noise correlation length. We discuss how our results could be exploitedto extract information about the localization length in experimental setups.

Anderson and many-body localization in the presence of spatially correlated classical noise