Lie algebras for some specific dissipative Landau–Zener problems

• A finite temperature result for dissipative Landau–Zener transitions in a qubit coupled to an environment is proposed.
• The quantum noises for bosonic and spin baths are considered.
• Lie algebras reduction method coupled to the separation method and the fast driving approximation is proposed.
• Jaynes–Cummings and a double quantum dots models are studied as illustrations of the algebra.

We demonstrate that some specific problems of Landau–Zener transitions in a qubit coupled to an environment (problems designed as dissipative) can be matched onto the frame of the original problem without dissipation, providing an appropriate Lie algebra. Focusing on the origin of quantum noises, the cases of bosonic and spin baths are considered and presented. Finally, making use of the algebra framework, the logic is shown in action for respectively two important additional quantum models, namely the Jaynes–Cummings and an isolated double quantum dots models.