Lewis-Ermakov approach for the time-dependent two-level system

We construct an explicit Lewis-Ermakov-type dynamical invariant for time-dependent two-level systems by exploiting their algebraic correspondence with the time-dependent harmonic oscillator through the su(2) and su(1,1) algebras, which share the common complexification sl\(2,mathbb{C}\). This invariant provides a closed-form evolution operator and an exact propagator for arbitrary time-dependent coupling a(t) and detuning b(t). We illustrate the method on representative scenarios, including Landau-Zener transitions, non-Hermitian dissipative processes, and adiabatic rapid passage, and we obtain inverse-engineered shortcuts to adiabaticity with fully explicit control fields when the auxiliary scaling function is taken to be real.