  Two-level system strongly coupled to a single resonator mode (harmonic
oscillator) is a paradigmatic model in many subfields of physics. We study
theoretically the Landau-Zener transition in this model. Analytical solution
for the transition probability is possible when the oscillator is highly
excited, i.e. at high temperatures. Then the relative change of the excitation
level of the oscillator in the course of the transition is small. The physical
picture of the transition in the presence of coupling to the oscillator becomes
transparent in the limiting cases of slow and fast oscillator. Slow oscillator
effectively renormalizes the drive velocity. As a result, the transition
probability either increases or decreases depending on the oscillator phase.
The net effect is, however, the suppression of the transition probability. On
the contrary, fast oscillator renormalizes the matrix element of the transition
rather than the drive velocity. This renormalization makes the transition
probability a non-monotonic function of the coupling amplitude.
