  One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian
skin effect, namely that the eigenstates are exponentially localized at the
boundary of the system. For open quantum systems, a short-time evolution can
often be well described by the effective non-Hermitian Hamiltonians, while
long-time dynamics calls for the Lindblad master equations, in which the
Liouvillian superoperators generate time evolution. In this Letter, we find
that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and
uncover its unexpected physical consequences. It is shown that the
non-Hermitian skin effect dramatically shapes the long-time dynamics, such that
the damping in a class of open quantum systems is algebraic under periodic
boundary condition but exponential under open boundary condition. Moreover, the
non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a
sharp wavefront. These phenomena are beyond the effective non-Hermitian
Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged
open quantum dynamics.
