Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer

We construct the exact solution for a family of one-half spin chains explicitly. The Hamiltonian corresponds to an isotropic Heisenberg Hamiltonian, with staggered exchange couplings that take only two different values. work out solutions in one-excitation subspace. Regarding problem quantum state transfer, we use and some theorems concerning approximation irrational numbers, show appearance conclusive pretty good transmission particular lengths. present numerical evidence is achieved by whose length not power two. set shows subset solution. Using perturbation theory, thoroughly analyze case when one coupling strengths orders magnitude larger than other. This strong limit allows us study, simple way, transmission. analytical closed expressions eigenvalues, eigenvectors, probabilities obtain precise asymptotic behavior time where observed. Moreover, this scales as law exponent increasing function chain length. also discuss crossover obtained between regimes observed are same order magnitude.