  The non-equilibrium process where the system does not evolve to the
featureless state is one of the new central objects in the non-equilibrium
phenomena. In this paper, starting from the short-range entangled state in the
two-dimensional conformal field theories ($2$d CFTs), the boundary state with a
regularization, we evolve the system with the inhomogeneous Hamiltonians called
M\"obius/SSD ones. Regardless of the details of CFTs considered in this paper,
during the M\"obius evolution, the entanglement entropy exhibits the periodic
motion called quantum revival. During SSD time evolution, except for some
subsystems, in the large time regime, entanglement entropy and mutual
information are approximated by those for the vacuum state. We argue the time
regime for the subsystem to cool down to vacuum one is $t_1 \gg
\mathcal{O}(L\sqrt{l_A})$, where $t_1$, $L$, and $l_A$ are time, system, and
subsystem sizes. This finding suggests the inhomogeneous quench induced by the
SSD Hamiltonian may be used as the preparation for the approximately-vacuum
state. We propose the gravity dual of the systems considered in this paper,
furthermore, and generalize it. In addition to them, we discuss the relation
between the inhomogenous quenches and continuous multi-scale entanglement
renormalization ansatz (cMERA).
