Long-term stability of driven quantum systems and the time-dependent Bloch equation

This study looks at the finite-dimensional adiabatic evolution influenced by weak perturbations, extending the analysis to the asymptotic time limit. Beginning with the fundamentals of adiabatic transformations and time-dependent effective Hamiltonians, we intuitively derive the Bloch equation. Our investigation of the solutions of the Bloch equation underscores the critical role of initial conditions and the assured existence of solutions, revealing the intricate link between leakage phenomena and the Bloch transformation. Numerical and analytical evaluations demonstrate that the leakage can remain small eternally. That is, a system that starts in a particular eigenspace of the strong generator remains in the same respective eigenspace for arbitrary long times with an error of $\mathcal{O}(\gamma^{-1})$, where $\gamma$ describes the ratio between the strength of the system's strong Hamiltonian and the perturbation.