Using a reverse-engineering approach on the time-distorted solution in a reference potential, we work out the external driving potential to be applied to a Brownian system in order to slow or accelerate the dynamics, or even to invert the arrow of time. By welding a direct and time-reversed evolution toward a well chosen common intermediate state, we analytically derive a smooth protocol to connect two arbitrary states in an arbitrarily short amount of time. Not only does the reverse-engineering approach proposed in this Letter contain the current—rather limited—catalog of explicit protocols, but it also provides a systematic strategy to build the connection between arbitrary states with a physically admissible driving. Optimization and further generalizations are also discussed.

• Received 21 May 2021
• Revised 1 September 2021
• Accepted 6 October 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.190605