Recent experiments in quantum simulators have provided evidence for the Many-Body Localized (MBL) phase in 1D and 2D bosonic quantum matter. The theoretical study of such bosonic MBL, however, is a daunting task due to the unbounded nature of its Hilbert space. In this work, we introduce a method to compute the long-time real-time evolution of 1D and 2D bosonic systems in an MBL phase at strong disorder and weak interactions. We focus on local dynamical indicators that are able to distinguish an MBL phase from an Anderson localized one. In particular, we consider the temporal fluctuations of local observables, the spatiotemporal behavior of two-time correlators and Out-Of-Time-Correlators (OTOCs). We show that these few-body observables can be computed with a computational effort that depends only polynomially on system size but is independent of the target time, by extending a recently proposed numerical method [Phys. Rev. B 99, 241114 (2019)] to mixed states and bosons. Our method also allows us to surrogate our numerical study with analytical considerations of the time-dependent behavior of the studied quantities.
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