  We consider a generalization of the spin-boson model in which two different
two-level systems are coupled to an oscillator, under conditions where the
oscillator energy is much less than the two-level system energies, and where
the oscillator is highly excited. We find that the two-level system transition
energy is shifted, producing a Bloch-Siegert shift in each two-level system
similar to what would be obtained if the other were absent. At resonances
associated with energy exchange between a two-level system and the oscillator,
the level splitting is about the same as would be obtained in the spin-boson
model at a Bloch-Siegert resonance. However, there occur resonances associated
with the transfer of excitation between one two-level system and the other, an
effect not present in the spin-boson model. We use a unitary transformation
leading to a rotated system in which terms responsible for the shift and
splittings can be identified. The level splittings at the anticrossings
associated with both energy exchange and excitation transfer resonances are
accounted for with simple two-state models and degenerate perturbation theory
using operators that appear in the rotated Hamiltonian.
