  The Loschmidt echo, defined as the overlap between quantum wave function
evolved with different Hamiltonians, quantifies the sensitivity of quantum
dynamics to perturbations and is often used as a probe of quantum chaos. In
this work we consider the behavior of the Loschmidt echo in the many body
localized phase, which is characterized by emergent local integrals of motion,
and provides a generic example of non-ergodic dynamics. We demonstrate that the
fluctuations of the Loschmidt echo decay as a power law in time in the
many-body localized phase, in contrast to the exponential decay in few-body
ergodic systems. We consider the spin-echo generalization of the Loschmidt
echo, and argue that the corresponding correlation function saturates to a
finite value in localized systems. Slow, power-law decay of fluctuations of
such spin-echo-type overlap is related to the operator spreading and is present
only in the many-body localized phase, but not in a non-interacting Anderson
insulator. While most of the previously considered probes of dephasing dynamics
could be understood by approximating physical spin operators with local
integrals of motion, the Loschmidt echo and its generalizations crucially
depend on the full expansion of the physical operators via local integrals of
motion operators, as well as operators which flip local integrals of motion.
Hence, these probes allow to get insights into the relation between physical
operators and local integrals of motion, and access the operator spreading in
the many-body localized phase.
