  We consider the Landau-Zener problem for a multilevel quantum system that is
coupled to an external environment. In particular, we consider a number of
cases of three-level systems coupled to a harmonic oscillator that represents
the external environment. We find that, similar to the case of the Landau-Zener
problem with a two-level system, when the quantum system and the environment
are both initially in their ground states the probability that the system
remains in the same quantum state is not affected by the coupling to the
environment. The final occupation probabilities of the other states are well
described by a common general principle: the coupling to the environment turns
each Landau-Zener transition process in the closed system into a sequence of
smaller transitions in the combined Hilbert space of the system and
environment, and this sequence of transitions lasts a total duration that
increases with increasing system-environment coupling strength. These results
provide an intuitive understanding of Landau-Zener transitions in open
multilevel quantum systems.
