  The treatment of quantum thermodynamic systems beyond weak coupling is of
increasing relevance, yet extremely challenging. The evaluation of
thermodynamic quantities in strong-coupling regimes requires a nonperturbative
knowledge of the bath dynamics, which in turn relies on heavy numerical
simulations. To tame these difficulties, considering thermal bosonic baths
linearly coupled to the open system, we derive expressions for heat, work, and
average system-bath interaction energy that only involve the autocorrelation
function of the bath and two-time expectation values of system operators. We
then exploit the pseudomode approach, which replaces the physical continuous
bosonic bath with a small finite number of damped, possibly interacting, modes,
to numerically evaluate these relevant thermodynamic quantities. We show in
particular that this method allows for an efficient numerical evaluation of
thermodynamic quantities in terms of one-time expectation values of the open
system and the pseudomodes. We apply this framework to the investigation of two
paradigmatic situations. In the first instance, we study the entropy production
for a two-level system coupled to an ohmic bath, simulated via interacting
pseudomodes, allowing for the presence of time-dependent driving. Secondly, we
consider a quantum thermal machine composed of a two-level system interacting
with two thermal baths at different temperatures, showing that an appropriate
sinusoidal modulation of the coupling with the cold bath only is enough to
obtain work extraction.
