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We derive the formula of energy flux for the hierarchical equations of motion(HEOM) method with the help of stochastic decoupling technique. The resultingexpression is a combination of the terms in the first two layers of thehierarchy. The formula is applied to a three-level "heat engine" coupledrespectively to three baths, of which two for heat sources and one for workdump. All the three baths are modeled by collections of harmonic oscillators.We illustrate the proper parameterizing to converge the third bath to the"work-dump" limit. As an example, the effect of the engine parameters onworking efficiency is studied.

Energy Flux in Hierarchical Equations of Motion Method and Its Application to a Three-Level Heat Engine