Coherent spin rotation in the presence of a phonon-bottleneck effect

A characteristic of spin reversal in the presence of phonon-bottleneck is the deviation of the magnetization cycle from a reversible function into an opened hysterezis cycle. In recent experiments on molecular magnets (e.g. V$_{15}$ and Ru$_2$), the zero-field level repulsion was sufficiently large to ensure an otherwise adiabatic passage through zero-field and the magnetization curves can be described by using only a phonon-bottleneck model. Here, we generalize the phonon-bottleneck model into a model able to blend the non-adiabatic dynamics of spins with the presence of a non-equilibrium phonon bath. In this simple phenomenological model, Bloch equations are written in the eigenbasis of the effective spin Hamiltonian, considered to be a two-level system at low temperatures. The relaxation term is given by the phonon-bottleneck mechanism. To the expense of calculus time, the method can be generalized to multi-level systems, where the notion of Bloch sphere does not apply but the density matrix formalism is still applicable.