  We study Landau-Zener transitions in a qubit coupled to a bath at zero
temperature. A general formula is derived that is applicable to models with a
non-degenerate ground state. We calculate exact transition probabilities for a
qubit coupled to either a bosonic or a spin bath. The nature of the baths and
the qubit-bath coupling is reflected in the transition probabilities. For
diagonal coupling, when the bath causes energy fluctuations of the diabatic
qubit states but no transitions between them, the transition probability
coincides with the standard LZ probability of an isolated qubit. This result is
universal as it does not depend on the specific type of bath. For pure
off-diagonal coupling, by contrast, the tunneling probability is sensitive to
the coupling strength. We discuss the relevance of our results for experiments
on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of
superconducting phase qubits.
