  We calculate Berry's phase when the driving field, to which a spin-1/2 is
coupled adiabatically, rather than the familiar classical magnetic field, is a
quantum vector operator, of noncommuting, in general, components, e.g., the
angular momentum of another particle, or another spin. The geometric phase of
the entire system, spin plus "quantum driving field", is first computed, and is
then subdivided into the two subsystems, using the Schmidt decomposition of the
total wave function -the resulting expression shows a marked, purely quantum
effect, involving the commutator of the field components. We also compute the
corresponding mean "classical" phase, involving a precessing magnetic field in
the presence of noise, up to terms quadratic in the noise amplitude -the
results are shown to be in excellent agreement with numerical simulations in
the literature. Subtleties in the relation between the quantum and classical
case are pointed out, while three concrete examples illustrate the scope and
internal consistency of our treatment.
