High-$$T_c$$ superconductivity with quadratic electron-phonon coupling

High-temperature superconductivity in cuprates remains a central challenge in condensed matter physics due to the complex interplay of lattice dynamics and electronic correlations. Traditional BCS theory fails to capture these effects, while competing models emphasize either phonons or correlations but rarely both. We propose a novel Hamiltonian that integrates linear and quadratic phonon-mediated interactions with strong electronic correlations in a two-dimensional cuprate lattice, representing the first unified framework incorporating quadratic electron-phonon coupling (QEPC) for enhanced pairing. This innovation addresses limitations in prior models by including multiple phonon modes, momentum-dependent coupling, and QEPC, which enables quantum bipolaron formation and significantly boosts [Formula: see text]. Through rigorous derivations using extended Eliashberg equations, we obtain analytical and numerical expressions for the critical temperature ([Formula: see text]) and superconducting gap (Δ), incorporating a frequency- and momentum-dependent pairing potential enhanced by QEPC, absent in prior models. Our proofs elucidate the synergy between linear/quadratic electron-phonon coupling (g, γ) and on-site repulsion (U), predicting enhanced [Formula: see text] up to [Formula: see text] K in cuprates, surpassing previous limits, along with a dome-shaped phase diagram peaking at optimal doping. Numerical simulations, with improved self-consistent solutions and larger grids, validate these results against experimental data, revealing non-monotonic trends in [Formula: see text] and Δ with respect to g, γ, and doping x, as well as a doping-dependent isotope coefficient that minimizes at optimal doping and increases in the underdoped and overdoped regimes. This unified framework bridges phonon- and correlation-driven mechanisms, offering novel insights for material design, such as engineered superlattices for QEPC, and resolving longstanding controversies in high-[Formula: see text] superconductivity by predicting higher [Formula: see text] through quantum effects.