  For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between given initial and final values of a control parameter. For an initially thermalized ensemble, we consider both Hamiltonian evolution for classical systems and Schr\"odinger evolution for quantum systems. In both cases, we show that for harmonic potentials, the optimal work is given by the adiabatic work even in the limit of short transition times. This result is counter-intuitive because the adiabatic work is substantially smaller than the work for an instantaneous jump. We also perform numerical calculations of the optimal protocol for Hamiltonian dynamics in an anharmonic quartic potential. For a two-level spin system, we give examples where the adiabatic work can be reached in either a finite or an arbitrarily short transition time depending on the allowed parameter space. 
Optimal protocols for Hamiltonian and Schr\"odinger dynamics