Many-body localization in Bose-Hubbard model: evidence for the mobility edge

Motivated by recent experiments on interacting bosons in quasi-one-dimensional optical lattice [Nature {\bf 573}, 385 (2019)] we analyse theoretically properties of the system in the crossover between delocalized and localized regimes. Comparison of time dynamics for uniform and density wave like initial states enables demonstration of the existence of the mobility edge. To this end we define a new observable, the mean speed of transport at long times. It gives us an efficient estimate of the critical disorder for the crossover. We also show that the mean velocity growth of occupation fluctuations close to the edges of the system carries the similar information. Using the quantum quench procedure we show that it is possible to probe the mobility edge for different energies.