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

SERRA, PABLO; FERRÓN, ALEJANDRO; OSENDA, OMAR

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL

IOP PUBLISHING LTD

Año: 2022 vol. 55

1751-8113

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