  We study the equivalence between the recently proposed finite environment
quantum jump model and a master equation approach. We derive microscopically
the master equation for a qubit coupled to a finite bosonic environment and
show that the master equation is equivalent with the finite environment quantum
jump model. We analytically show that both the methods produce the same moments
of work when the work is defined through the two-measurement protocol excluding
the interaction energy. However, when compared to the work moments computed
using the power operator approach, we find a difference in the form of the work
moments. To numerically verify our results, we study a qubit coupled to an
environment consisting of ten two-level systems.
