Many-body localized (MBL) systems are characterized by the absence of
transport and thermalization, and therefore cannot be described by conventional
statistical mechanics. In this paper, using analytic arguments and numerical
simulations, we study the behaviour of local observables in an isolated MBL
system following a quantum quench. For the case of a global quench, we find
that the local observables reach stationary, highly non-thermal values at long
times as a result of slow dephasing characteristic of the MBL phase. These
stationary values retain the local memory of the initial state due to the
existence of local integrals of motion in the MBL phase. The temporal
fluctuations around stationary values exhibit universal power-law decay in
time, with an exponent set by the localization length and the diagonal entropy
of the initial state. Such a power-law decay holds for any local observable and
is related to the logarithmic in time growth of entanglement in the MBL phase.
This behaviour distinguishes the MBL phase from both the Anderson insulator
(where no stationary state is reached), and from the ergodic phase (where
relaxation is expected to be exponential). For the case of a local quench, we
also find a power-law approach of local observables to their stationary values
when the system is prepared in a mixed state. Quench protocols considered in
this paper can be naturally implemented in systems of ultra cold atoms in
disordered optical lattices, and the behaviour of local observables provides a
direct experimental signature of many-body localization.Comment: 11 pages, 4 figure
