Exploring the extent of validity of quantum work fluctuation theorems in the presence of weak measurements

Work fluctuation theorems have been one of the important achievements in the field of nonequilibrium Statistical Physics, both in the classical and quantum regimes. Conventionally, the work done on a quantum system is defined by means of a two-point measurement scheme, where a projective measurement of the Hamiltonian is performed both at the beginning and at the end of the process. Recently, quantum work fluctuation theorems in the context of generalized measurements have received a lot of attention. Here, we define a weak value of work, within the broad frame-work of generalized measurements and show that the deviation from the exact work fluctuation theorems are much less in this formalism as compared to previous efforts in the literature, using a two-level system as the model. We find that the original form of Jarzynski equality (valid for projective two-point measurements) does not remain exact in this framework. Nevertheless, the deviations are in general small, so that an approximate effective temperature of the thermal bath can be deduced using our results. Further, in the limit of the measurements being projective, the exact form of the work fluctuation theorems is recovered.