Recent advances in experimental control of colloidal systems have spurred a revolution in the production of mesoscale thermodynamic devices. Functional “textbook” engines, such as the Stirling and Carnot cycles, have been produced in colloidal systems where they operate far from equilibrium. Simultaneously, significant theoretical advances have been made in the design and analysis of such devices. Here, we use methods from thermodynamic geometry to characterize the optimal finite-time nonequilibrium cyclic operation of the parametric harmonic oscillator in contact with a time-varying heat bath with particular focus on the Brownian Carnot cycle. We derive the optimally parametrized Carnot cycle, along with two other new cycles and compare their dissipated energy, efficiency, and steady-state power production against each other and a previously tested experimental protocol for the Carnot cycle. We demonstrate a 20% improvement in dissipated energy over previous experimentally tested protocols and a $\sim 50\%$ improvement under other conditions for one of our engines, whereas our final engine is more efficient and powerful than the others we considered. Our results provide the means for experimentally realizing optimal mesoscale heat engines.

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• Nonequilibrium & irreversible thermodynamics
• Nonequilibrium statistical mechanics
• Heat engines