# Optimal protocols and universal time-energy bound in Brownian thermodynamics

## TL;DR

This paper introduces an optimal control strategy for stochastic thermodynamic systems, experimentally demonstrating a universal time-energy bound in Brownian thermodynamics that is achieved by optimized protocols.

## Contribution

It develops a variational approach to optimize work and transfer time in Brownian systems, revealing a universal time-energy bound and experimentally validating it.

## Key findings

- Universal relation between transfer duration and work expended
- Optimized protocols reach the theoretical lower bound
- Experimental verification with Brownian particles in optical traps

## Abstract

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 are either optimally cheap for a given duration or optimally fast for a given energetic cost. This approach unveils a universal relation $\Delta t\,\Delta W \ge (\Delta t\,\Delta W)_{\rm opt}$ between the transfer duration and the expended work. We verify experimentally that the lower bound is reached only with the optimized protocols.