The cost of speed: Time-optimal thermal control of trapped Brownian particles

Abstract

A thermal analogue of the classical brachistochrone problem, which minimizes the connection time between two equilibrium states of harmonically confined Brownian particles, has recently been solved theoretically. Here we report its experimental realization using two optically trapped microparticles subjected to a bang-bang effective temperature protocol. Despite their distinct relaxation times, both degrees of freedom are steered to their respective equilibrium states simultaneously in a finite minimal time. We provide a complete time-resolved characterization of the nonequilibrium dynamics through the evolution of the position variances and the entropy production within the framework of stochastic thermodynamics, enabling a quantitative comparison with direct relaxation and a suboptimal protocol. In addition, we employ information-geometric tools -- recently referred to as thermal kinematics -- to track the system's path in state space with a single dynamical quantity. Our results show that faster equilibration requires a larger entropy production and an increased thermodynamic length, revealing a direct trade-off between temporal optimality and thermodynamic cost in multidimensional stochastic systems driven by a single intensive control parameter.