  Anderson localization predicts that wave spreading in disordered lattices can
come to a complete halt, providing a universal mechanism for {dynamical
localization}. In the one-dimensional Hermitian Anderson model with
uncorrelated diagonal disorder, there is a one-to-one correspondence between
dynamical localization and spectral localization, i.e. the exponential
localization of all the Hamiltonian eigenfunctions. This correspondence can be
broken when dealing with disordered dissipative lattices. Recently, it has been
shown that when the system exchanges particles with the surrounding environment
and random fluctuations of the dissipation are introduced, spectral
localization is observed but without dynamical localization. Such previous
studies considered lattices with mixed conservative (Hamiltonian) and
dissipative dynamics, and were restricted to a semiclassical analysis. However,
Anderson localization in purely dissipative lattices, displaying an entirely
Lindbladian dynamics, remains largely unexplored. Here we consider the
purely-dissipative Anderson model in the framework of a Lindblad master
equation and show that, akin to the semiclassical models with conservative
hopping and random dissipation, one observes dynamical delocalization in spite
of strong spectral localization of all eigenstates of the Liouvillian
superoperator. This result is very distinct than delocalization observed in the
Anderson model with dephasing effects, where dynamical delocalization arises
from the delocalization of the stationary state of the Liouvillian
superoperator.
