  By making use of a recently proposed framework for the inference of
thermodynamic irreversibility in bosonic quantum systems, we experimentally
measure and characterize the entropy production rates in the non-equilibrium
steady state of two different physical systems -- a micro-mechanical resonator
and a Bose-Einstein condensate -- each coupled to a high finesse cavity and
hence also subject to optical loss. Key features of our setups, such as cooling
of the mechanical resonator and signatures of a structural quantum phase
transition in the condensate are reflected in the entropy production rates. Our
work demonstrates the possibility to explore irreversibility in driven
mesoscopic quantum systems and paves the way to a systematic experimental
assessment of entropy production beyond the microscopic limit.
