Localisation in space and time in disordered-lattice open quantum dynamics

We study a two-dimensional tight-binding lattice for excitons with on-site disorder, coupled to a thermal environment at infinite temperature. The disorder acts to localise an exciton spatially, while the environment generates dynamics which enable exploration of the lattice. Although the steady state of the system is trivially uniform, we observe a rich dynamics and uncover a dynamical phase transition in the space of temporal trajectories. This transition is identified as a localisation in the dynamics generated by the bath. We explore spatial features in the dynamics and employ a generalisation of the inverse participation ratio to deduce an ergodic timescale for the lattice.