Work statistics, quantum signatures, and enhanced work extraction in quadratic fermionic models

In quadratic fermionic models, we determine a quantum correction to the work statistics after both a sudden quench and a time-dependent driving. Such a correction lies in the noncommutativity of the initial quantum state and the time-dependent Hamiltonian, and is revealed via the Kirkwood-Dirac quasiprobability (KDQ) approach to two-times correlators. Thanks to the latter, one can assess the onset of nonclassical signatures in the KDQ distribution of work, in the form of negative and complex values that no classical theory can reveal. By applying these concepts on the one-dimensional transverse-field Ising model, we relate nonclassical behaviors of the KDQ statistics of work in correspondence of the critical points of the model. Finally, we also prove the enhancement of the extracted work in nonclassical regimes where the noncommutativity takes a role.