Quantum-heat fluctuation relations in $3$-level systems under projective measurements.

We study the statistics of energy fluctuations in a $3$-level quantum system subject to a sequence of projective quantum measurements. We check that, as expected, the quantum Jarzynski equality holds provided the initial state is thermal. The latter condition is trivially satisfied for $2$-level systems, while this is in general no longer true for $N$-level systems, with $N > 2$. Focusing on $3$-level systems, we discuss the occurrence of a unique energy scale factor $\beta_{\rm eff}$ formally playing 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.