Decay rates and decoherence of an interstitial two-level spin impurity in a ferromagnetic lattice

Abstract

The decay rate of an interstitial two-level spin impurity, located in the center of a unit cell of an anisotropic ferromagnetic lattice subjected to an external magnetic field, is derived. The impurity is coupled to nearest-neighbor spins through a Heisenberg $XY$ interaction. By mapping the lattice spin operators using the Holstein-Primakoff transformation, we establish the similarity with the Fano-Anderson model at low temperatures, and we calculate the retarded Green's function in one and two dimensions analytically for arbitrary coupling strength. It is shown that the reduced density matrix of the impurity satisfies an exact master equation in Lindblad form, from which the decay rate and the Lamb shift are deduced. The evolution in time of the latter together with the excited-state occupation probability is investigated and its dependence on the applied magnetic field is discussed. It is found that there exists a critical resonance-like value of the magnetic field around which the behavior of the decay rate and the density matrix changes drastically. The Markovian decay law, as given by the Fermi golden rule, does not hold in the weak-coupling regime unless the magnetic field is weak, typically less than the critical value. The weak-coupling regime is further treated perturbatively up to second order, and the obtained results are compared with the exact solution. We also discuss the Zeno regime of the dynamics, where it is shown that, at short times, the effective decay rate is twice as small as the exact decay rate, and that when the impurity energy lies outside the lattice continuum, the measurement speeds up the decay of the survival probability.