  We experimentally demonstrate a nano-scale stochastic Stirling heat engine
operating in the underdamped regime. The setup involves an optically levitated
silica particle that is subjected to a power-varying optical trap and
periodically coupled to a cold/hot reservoir via switching on/off active
feedback cooling. We conduct a systematic investigation of the engine's
performance and find that both the output work and efficiency approach their
theoretical limits under quasi-static conditions. Furthermore, we examine the
dependence of the output work fluctuation on the cycle time and temperature
difference between the hot and cold reservoirs. We observe that the
distribution has a Gaussian profile in the quasi-static regime, whereas it
becomes asymmetric and non-Gaussian as the cycle duration time decreases. This
non-Gaussianity is qualitatively attributed to the strong correlation of the
particle's position within a cycle in the non-equilibrium regime. Our
experiments provide valuable insights into stochastic thermodynamics in the
underdamped regime and open up new possibilities for the design of future
nano-machines.
