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

## TL;DR

This paper investigates optimal finite-time protocols for switching on interactions in a Luttinger liquid to reach its ground state efficiently, balancing energy minimization and state overlap, with results applicable to quantum control.

## Contribution

It introduces a method to determine optimal interaction switching protocols in a Luttinger liquid, highlighting the symmetry and mode occupation dependence of these protocols.

## Key findings

- Optimal protocols are symmetric around the midpoint of the quench duration.
- Fast oscillations excite high-energy modes in short quenches.
- Smooth protocols are optimal for large quench durations to minimize energy.

## Abstract

Reaching a target quantum state from an initial state within a finite temporal window is a challenging problem due to non-adiabaticity. We study the optimal protocol for swithcing on interactions to reach the ground state of a weakly interacting Luttinger liquid within a finite time $\tau$, starting from the non-interacting ground state. The protocol is optimized by minimizing the excess energy at the end of the quench, or by maximizing the overlap with the interacting ground state. We find that the optimal protocol is symmetric with respect to $\tau/2$, and can be expressed as a functional of the occupation numbers 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 protocol while maximizing overlap requires a linear quench protocol. In this limit, the minimal energy and maximal overlap are both universal functions of the system size and the duration of the protocol.