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

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 $?_{eff}$ 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 $?_{eff}$ for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics.