  A calorimetric measurement has recently been proposed as a promising
technique to measure thermodynamic quantities in a dissipative superconducting
qubit. These measurements rely on the fact that the system is projected into
energy eigenstates whenever energy is exchanged with the environment. This
requirement imposes a restriction on the class of systems that can be measured
in this way. Here we extend the calorimetric protocol to the measurement of a
driven quantum harmonic oscillator. We employ a scheme based on the two-level
approximation to define a new work quantity and show how its statistics relates
to the standard two-measurement protocol. We find that for the average work the
two-level approximation holds in the underdamped regime for short driving times
and, in the overdamped regime, for any driving time. However, this
approximation fails for the variance and higher moments of work at finite
temperatures. Furthermore, we show how to relate the work statistics obtained
through this scheme to the work statistics given by the two-measurement
protocol.
