Phenomenologically broadened quasi-static Bogoliubov–de gennes framework for driven superconductivity in a two-dimensional attractive hubbard model

A self-consistent Bogoliubov–de Gennes (BdG) framework is developed to investigate the superconducting response of a two-dimensional attractive Hubbard model under externally applied electromagnetic driving. The electric field is incorporated through a gauge-invariant Peierls substitution, while finite quasiparticle spectral broadening is introduced phenomenologically through an effective spectral-lifetime parameter. The calculations examine the dependence of the superconducting order parameter, quasiparticle spectrum, and numerical convergence on the driving-field strength and phenomenological spectral broadening. The results show that increasing the field strength generally suppresses the superconducting order parameter through field-induced modifications of the electronic dispersion and hopping phases. Finite spectral broadening modifies the calculated quasiparticle spectrum and, in the parameter regimes investigated, can improve the numerical stability of the self-consistent solution. Importantly, the phenomenological lifetime parameter is interpreted as a spectral-lifetime parameter rather than as a microscopic quasiparticle relaxation time. The present formulation does not explicitly evolve the nonequilibrium quasiparticle distribution, calculate energy transfer to an external environment, or establish a genuine nonequilibrium steady state. Instead, the calculations provide a quasi-static, phenomenologically broadened description of the superconducting response under external driving. The proposed approach offers a computationally efficient framework for systematically studying field-dependent superconductivity and the sensitivity of self-consistent BdG solutions to finite quasiparticle spectral broadening. © 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.