Optimal protocols for Hamiltonian and Schrödinger dynamics

Abstract

For systems in an externally controllable time dependent potential, the optimal protocolminimizes the mean work spent in a finite time transition between given initial and finalvalues of a control parameter. For an initially thermalized ensemble, we consider bothHamiltonian evolution for classical systems and Schrödinger evolution for quantum systems.In both cases, we show that for harmonic potentials, the optimal work is givenby the adiabatic work even in the limit of short transition times. This result iscounter-intuitive because the adiabatic work is substantially smaller than the work for aninstantaneous jump. We also perform numerical calculations for the optimal protocol forHamiltonian dynamics in an anharmonic quartic potential. For a two-level spinsystem, we give examples where the adiabatic work can be reached in either afinite or an arbitrarily short transition time depending on the allowed parameterspace.