Multi-particle quantum walks and Fisher information in one-dimensional lattices.

Recent experiments on quantum walks (QWs) of a single and two particles demonstrated subtle quantum statistics-dependent walks in one-dimensional (1D) lattices. However the roles of interaction and quantum statistics in such a kind of walks are little known at a many-body level. In this letter, using time-evolving block decimation algorithm and many-body perturbation theory we rigorously study QWs, Bloch oscillations and quantum Fisher informations (FIs) for three indistinguishable bosons and fermions in 1D lattices. We show that such strongly correlated many-body QWs not only give rise to statistics-and-interaction-dependent ballistic transports of scattering states, two- and three-body bound states, but also present a quantum enhanced precision measurement of the gravitational force. It turns out that in contrast to the walks of the fermions, the QWs of three bosons exhibit richer dynamics of co-walkings and competitive Bloch oscillations, which remarkably present a surprising time scaling $t^3$ of FI below a characteristic time $t_0$ and saturate to the fundamental limit of $t^2$ for $t>t_0$.