  We calculate analytically the probabilities for intuitive and
counterintuitive transitions in a three-state system, in which two parallel
energies are crossed by a third, tilted energy. The state with the tilted
energy is coupled to the other two states in a chainwise linkage pattern with
constant couplings of finite duration. The probability for a counterintuitive
transition is found to increase with the square of the coupling and decrease
with the squares of the interaction duration, the energy splitting between the
parallel energies, and the tilt (chirp) rate. Physical examples of this model
can be found in coherent atomic excitation and optical shielding in cold atomic
collisions.
