Optimal protocols and universal time-energy bound in Brownian thermodynamics

We propose an optimization strategy to control the dynamics of a stochastic system transferred from one thermal equilibrium to another and apply it experimentally to a Brownian particle in an optical trap under compression. Based on a variational principle that treats the transfer duration and the expended work on an equal footing, our strategy leads to a family of protocols that is either optimally cheap for a given duration or optimally fast for a given energetic cost. This approach unveils a universal relation ΔtΔW≥(ΔtΔW)opt between the transfer duration and the expended work. We verify experimentally that the lower bound is reached only with the optimized protocols.