  We study the energy exchange between two bosonic systems that interact via
bilinear transformations in the mode operators. The first mode is considered as
the thermodynamic system, while the second is regarded as the bath. This work
finds its roots in a very recent formulation of quantum thermodynamics [1]
which allows to consider baths that are not described by the usual
Boltzmann-Gibbs canonical form. Baths can possess quantum properties, such as
squeezing or coherence, and can be initially correlated with the system, even
through entanglement. We focus mainly on the case of Gaussian states, by
quantifying the relation between their defining parameters, namely the mean
values of the quadratures and the covariance matrix, and relevant
thermodynamical quantities such as the heat exchanged and the work performed
during the interaction process. We fully solve the case of initially
uncorrelated Gaussian states and provide the most general form of the first law
of thermodynamics in this case. We also discuss the case of initially
correlated states by considering a number of relevant examples, studying how
correlations can assist some phenomena, e.g. work extraction or anomalous heat
flows. Finally, we present an information-theoretic approach based on the Renyi
entropy of order two for clarifying more generally the role of correlations on
heat exchanges.
