  We study the statistics of energy fluctuations in a three-level quantum
system subject to a sequence of projective quantum measurements. We check that,
as expected, the quantum Jarzynski equality holds provided that the initial
state is thermal. The latter condition is trivially satisfied for two-level
systems, while this is generally no longer true for $N$-level systems, with $N
> 2$. Focusing on three-level systems, we discuss the occurrence of a unique
energy scale factor $\beta_{\rm eff}$ that formally plays the role of an
effective inverse temperature in the Jarzynski equality. To this aim, we
introduce a suitable parametrization of the initial state in terms of a thermal
and a non-thermal component. We determine the value of $\beta_{\rm eff}$ for a
large number of measurements and study its dependence on the initial state. Our
predictions could be checked experimentally in quantum optics.
