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We establish an equivalence between two different quantum quench problems,the joining local quantum quench and the M\"obius quench, in the context of$(1+1)$-dimensional conformal field theory (CFT). Here, in the former, twoinitially decoupled systems (CFTs) on finite intervals are joined at $t=0$. Inthe latter, we consider the system that is initially prepared in the groundstate of the regular homogeneous Hamiltonian on a finite interval and, after$t=0$, let it time-evolve by the so-called M\"obius Hamiltonian that isspatially inhomogeneous. The equivalence allows us to relate the time-dependentphysical observables in one of these problems to those in the other. As anapplication of the equivalence, we construct a holographic dual of the M\"obiusquench from that of the local quantum quench. The holographic geometry involvesan end-of-the-world brane whose profile exhibits non-trivial dynamics.

Bridging two quantum quench problems -- local joining quantum quench and M\"obius quench -- and their holographic dual descriptions