  The quantum dynamics of a
$\hat{\mathbf{J}}^2=(\hat{\mathbf{j}}_1+\hat{\mathbf{j}}_2)^2$-conserving
Hamiltonian model describing two coupled spins $\hat{\mathbf{j}}_1$ and
$\hat{\mathbf{j}}_2$ under controllable and fluctuating time-dependent magnetic
fields is investigated. Each eigenspace of $\hat{\mathbf{J}}^2$ is dynamically
invariant and the Hamiltonian of the total system restricted to any one of such
$(j_1+j_2)-|j_1-j_2|+1$ eigenspaces, possesses the SU(2) structure of the
Hamiltonian of a single fictitious spin acted upon by the total magnetic field.
We show that such a reducibility holds regardless of the time dependence of the
externally applied field as well as of the statistical properties of the noise,
here represented as a classical fluctuating magnetic field. The time evolution
of the joint transition probabilities of the two spins $\hat{\mathbf{j}}_1$ and
$\hat{\mathbf{j}}_2$ between two prefixed factorized states is examined,
bringing to light peculiar dynamical properties of the system under scrutiny.
When the noise-induced non-unitary dynamics of the two coupled spins is
properly taken into account, analytical expressions for the joint Landau-Zener
transition probabilities are reported. The possibility of extending the
applicability of our results to other time-dependent spin models is pointed
out.
