  Artificial micro heat engines are prototypical models to explore and
elucidate the mechanisms of energy transduction in a regime that is dominated
by fluctuations [1-2]. Micro heat engines realized hitherto mimicked their
macroscopic counterparts and operated between reservoirs that were effectively
thermal [3-7]. For such reservoirs, temperature is a well-defined state
variable and stochastic thermodynamics provides a precise framework for
quantifying engine performance [8-9]. It remains unclear whether these concepts
readily carry over to situations where the reservoirs are out-of-equilibrium
[10], a scenario of particular importance to the functioning of synthetic
[11-12] and biological [13] micro engines and motors. Here we experimentally
realized a micrometer-sized active Stirling engine by periodically cycling a
colloidal particle in a time-varying harmonic optical potential across
bacterial baths at different activities. Unlike in equilibrium thermal
reservoirs, the displacement statistics of the trapped particle becomes
increasingly non-Gaussian with activity. We show that as much as $\approx$ 85\%
of the total power output and $\approx$ 50\% of the overall efficiency stems
from large non-Gaussian particle displacements alone. Most remarkably, at the
highest activities investigated, the efficiency of our quasi-static active heat
engines surpasses the equilibrium saturation limit of Stirling efficiency - the
maximum efficiency of a Stirling engine with the ratio of cold and hot
reservoir temperatures ${T_C\over T_H} \to 0$. Crucially, the failure of
effective temperature descriptions [14-16] for active reservoirs highlights the
dire need for theories that can better capture the physics of micro motors and
heat engines that operate in strongly non-thermal environments.

Statistical Mechanics