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

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).