Recent experiments, in which Terahertz (THz) light has been used to induce nonequilibrium superconducting states, have raised a number of intriguing fundamental questions. Theoretically, these experiments are most often described within the Floquet formalism, which suffers a number of well-known limitations (e.g. Floquet heating). Alternative approaches rely on heavy numerical methods. In this Article we develop an analytical theory of nonequilibrium superconductivity that combines path integrals on the Kostantinov-Perel time contour with adiabatic perturbation theory [G. Rigolin, G. Ortiz, and V.H. Ponce, Phys. Rev. A~{bf 78}, 052508 (2008)]. We consider a general system of electrons and Raman phonons coupled by the Frohlich interaction, in the presence of a time-dependent external field which acts on the phonon subsystem. The latter is supposed to model the THz light-induced excitation of nonlinear interactions between infrared and Raman phonons. Assuming that the external field has a slow dependence on time, we derive equations for the dynamical superconducting gap, calculating the leading adiabatic term and the first non-adiabatic correction. Our nonequilibrium formulas can be solved numerically with a minimal increase of computational complexity with respect to that needed to calculate the superconducting gap at equilibrium.
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