  Characteristic functions (CFs) provide a very efficient method for evaluating
the probability density functions of stochastic thermodynamic quantities and
investigating their statistical features in quantum master equations (QMEs). A
conventional procedure for obtaining these functions is to resort to a
first-principles approach; namely, the evolution equations of the CFs of the
combined system and its environment are obtained and then projected into the
degrees of freedom of the system. However, the QMEs can be unraveled by a
quantum jump trajectory. Thermodynamic quantities such as the heat, work, and
entropy production can be well defined along a trajectory. Hence, on the basis
of the notion of a trajectory, can we straightforwardly derive these CFs, e.g.,
their evolution equations? This is essential to establish the self-contained
stochastic thermodynamics of a QME. In this paper, we show that it is indeed
plausible and also simple. Particularly, these equations are fully consistent
with those obtained by the first-principles method. Our results have practical
significance; they indicate that the quantum fluctuation relations could be
verified by more realistic photocounting experiments.
