Two dissipative two-state level-crossing models conditionally integrable in terms of the Kummer functions

We present two dissipative level-crossing models of quantum time-dependent two-state problem. These are conditionally integrable models belonging to the single confluent Heun class. Each of the models is given by an exponentially varying Rabi frequency and a level-crossing detuning that starts from the exact resonance and exponentially diverges at the infinity. The models include irreversible losses from the excited state (without relaxation to the ground state). We show that the solution of the problem is written in terms of irreducible linear combinations of two Kummer confluent hypergeometric functions. Using this exact solution, we discuss the dynamics of states’ populations under different interaction regimes.