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The non-equilibrium process where the system does not evolve to thefeatureless state is one of the new central objects in the non-equilibriumphenomena. In this paper, starting from the short-range entangled state in thetwo-dimensional conformal field theories ($2$d CFTs), the boundary state with aregularization, we evolve the system with the inhomogeneous Hamiltonians calledM\"obius/SSD ones. Regardless of the details of CFTs considered in this paper,during the M\"obius evolution, the entanglement entropy exhibits the periodicmotion called quantum revival. During SSD time evolution, except for somesubsystems, in the large time regime, entanglement entropy and mutualinformation are approximated by those for the vacuum state. We argue the timeregime 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, andsubsystem sizes. This finding suggests the inhomogeneous quench induced by theSSD Hamiltonian may be used as the preparation for the approximately-vacuumstate. We propose the gravity dual of the systems considered in this paper,furthermore, and generalize it. In addition to them, we discuss the relationbetween the inhomogenous quenches and continuous multi-scale entanglementrenormalization ansatz (cMERA).

Inhomogeneous quenches as state preparation in two-dimensional conformal field theories