Thermodynamic Uncertainty Relations for Bosonic Otto Engines

We study a two-mode bosonic engine undergoing an Otto cycle. The energy exchange between the two bosonic systems is provided by a tunable unitary bilinear interaction in the mode operators modeling frequency conversion, whereas the cyclic operation is guaranteed by relaxation to two baths at different temperature after each interacting stage. We derive exact expressions for work and heat fluctuations, identities showing the interdependence among average extracted work, fluctuations and efficiency, along with thermodynamic uncertainty relations between the signal-to-noise ratio of observed work and heat and the entropy production. The full probability of the stochastic work and heat is then provided.