Time evolution of a pair of distinguishable interacting spins subjected to controllable and noisy magnetic fields

The quantum dynamics of a Jˆ²=(jˆ₁+jˆ₂)²-conserving Hamiltonian model describing two coupled spins jˆ₁ and jˆ₂ under controllable and fluctuating time-dependent magnetic fields is investigated. Each eigenspace of Jˆ² is dynamically invariant and the Hamiltonian of the total system restricted to any one of such (j₁+j₂)−|j₁−j₂|+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 jˆ₁ and jˆ₂ 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.