  We analyze the dynamics and final populations in a Landau-Zener problem for a
two level system (or qubit) when this system interacts with one harmonic
oscillator mode that is initially set to a finite-temperature thermal
equilibrium state. The harmonic oscillator could represent an external mode
that is strongly coupled to the qubit, e.g. an ionic oscillation mode in a
molecule, or it could represent a prototypical uncontrolled environment. We
analyze the qubit's occupation probabilities at the final time in a number of
different regimes, varying the qubit and oscillator frequencies, their coupling
strength and the temperature. In particular we find some surprising
non-monotonic dependence on the coupling strength and temperature.
