Optimal protocols for Hamiltonian and Schr\"odinger dynamics

TL;DR

This paper investigates optimal control protocols for classical and quantum systems to minimize work during finite-time transitions, revealing that harmonic potentials can achieve adiabatic work even in rapid processes.

Contribution

It demonstrates that for harmonic potentials, the minimal work equals the adiabatic work regardless of transition duration, and explores optimal protocols in anharmonic and quantum systems.

Findings

Optimal protocols for harmonic potentials match adiabatic work even at short times.

Numerical solutions for anharmonic potentials show feasible minimal work protocols.

Quantum two-level systems can reach adiabatic work in finite or very short times.

Abstract

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.