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Reaching a target quantum state from an initial state within a finitetemporal window is a challenging problem due to non-adiabaticity. We study theoptimal protocol for swithcing on interactions to reach the ground state of aweakly interacting Luttinger liquid within a finite time $\tau$, starting fromthe non-interacting ground state. The protocol is optimized by minimizing theexcess energy at the end of the quench, or by maximizing the overlap with theinteracting ground state. We find that the optimal protocol is symmetric withrespect to $\tau/2$, and can be expressed as a functional of the occupationnumbers of the bosonic modes in the final state. For short quench durations,the optimal protocol exhibits fast oscillation and excites high energy modes.In the limit of large $\tau$, minimizing energy requires a smooth protocolwhile maximizing overlap requires a linear quench protocol. In this limit, theminimal energy and maximal overlap are both universal functions of the systemsize and the duration of the protocol.

Optimal protocols for finite-duration quantum quenches in the Luttinger model