Geometric Phase of a Two-level System Driven by a Classical Field

We investigated the mixed state geometric phase of a two-level system which is coupled to a bosonic reservoir in both weak and strong coupling regimes, and driven by a classical field. Besides, we extended the direct-drive model to the indirect-drive model by incorporating an additional qubit. In the direct-drive model, the geometric phase undergoes corresponding alterations through the amplification of the classical field strength. Moreover, the combination of the classical driving and the strong coupling between the two-level system and the bosonic reservoir significantly changes the geometric phase. Furthermore, we elucidate the asymmetric effect of the detuning between the two-level system and the Lorentzian spectrum on the geometric phase, while achieving stability in the geometric phase regardless of the detuning value by appropriately adjusting the classical driving strength. In the indirect-drive model, intriguing phenomena arise, such as different behaviors of the geometric phase and the restoration of the symmetric effects of the detuning. The explicit dynamics of the geometric phase has been numerically analyzed by investigating the time-dependent factor of the reduced density matrix. Finally, the energy flow of the entire system was clarified by utilizing quasimode theory and visualizing the temporal evolution of the Bloch vector.