Noisy Demkov-Kunike model

The Demkov-Kunike (DK) model, characterized by a time-dependent Rabi coupling $J~\text{sech}(t/T)$ and on-site detuning $\Delta_0+\Delta_1\tanh(t/T)$, has one of the most general forms of an exactly solvable two-state quantum system, and, therefore, it provides a paradigm for coherent manipulations of a qubit's quantum state. Despite its extensive applications in the noise-free cases, the exploration of the noisy DK model remains limited. Here, we extend the coherent DK model to take into account of a noisy coupling term $J\rightarrow J_{\text{noisy}}(t)$. We consider colored Markovian noise sources represented by the telegraph noise and Gaussian noise. We present exact solutions for the survival probability $Q^{\text{noisy}}_{\text{DK}}$ of the noisy DK model, namely the probability of the system to remain in its initial state. For the slow telegraph noise, we identify parameter regimes where the survival probability $Q^{\text{noisy}}_{\text{DK}}$ is suppressed rather than enhanced by noise. In contrast, for slow Gaussian noise, the noise always enhances the survival probability $Q^{\text{noisy}}_{\text{DK}}$, due to the absorption of noise quanta across the energy gap. This study not only complements the existing research on the noisy Landau-Zener model, but also provides valuable insights for the control of two-level quantum systems.