  The many-body localization (MBL) transition is a quantum phase transition
involving highly excited eigenstates of a disordered quantum many-body
Hamiltonian, which evolve from "extended/ergodic" (exhibiting extensive
entanglement entropies and fluctuations) to "localized" (exhibiting area-law
scaling of entanglement and fluctuations). The MBL transition can be driven by
the strength of disorder in a given spectral range, or by the energy density at
fixed disorder - if the system possesses a many-body mobility edge. Here we
propose to explore the latter mechanism by using "quantum-quench spectroscopy",
namely via quantum quenches of variable width which prepare the state of the
system in a superposition of eigenstates of the Hamiltonian within a
controllable spectral region. Studying numerically a chain of interacting
spinless fermions in a quasi-periodic potential, we argue that this system has
a many-body mobility edge; and we show that its existence translates into a
clear dynamical transition in the time evolution immediately following a quench
in the strength of the quasi-periodic potential, as well as a transition in the
scaling properties of the quasi-stationary state at long times. Our results
suggest a practical scheme for the experimental observation of many-body
mobility edges using cold-atom setups.
