Exact Work Characteristic Functions for Time-Dependent Bosonic Quadratic Hamiltonians

We derive an exact analytical expression for the characteristic function of work in closed many-body bosonic systems governed by general time-dependent quadratic Hamiltonians. This is made possible by adopting a unified prescription in which all operators appearing in the characteristic function are treated as elements of the inhomogeneous Metaplectic group. The esulting expression involves a single symplectic matrix, a single phase-space displacement, and a single accumulated phase. It holds for any number of degrees of freedom, arbitrary time dependence of the quadratic Hamiltonian, and any initial thermal equilibrium state of the system. The formalism trades the explicit time-ordering problem associated with the unitary evolution for the determination of a symplectic path in phase space, from which the corresponding driving Hamiltonian can be reconstructed. We also discuss the generalization to arbitrary nonequilibrium initial states within the two-projective-measurement scheme for quantum work. The same construction yields Loschmidt echoes and mixed-state total phases. Using the derived characteristic function, we obtain formulas for the Helmholtz free energy, the Jarzynski equality, the mean work, the work variance, and the nonequilibrium lag. Undisplaced Hamiltonians, sudden quenches, constant-Hessian protocols, and the adiabatic limit follow as special cases of the general result. As an illustration, we analyze a one-dimensional system constructed from an unconventional symplectic path and exhibiting physically relevant limiting cases. We then show how affine and purely symplectic contributions affect work statistics and irreversibility.