Quantum communication through a spin chain with interaction determined by a Jacobi matrix

Abstract

We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearest-neighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case an arbitrary element of the correlation function is expressed in a simple closed form. Its asymptotic limit corresponds to the Jacobi matrix of the Charlier polynomial, and may be understood as a unitary evolution resulting from a Heisenberg group element. Correlation functions for Hamiltonians corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials are also studied. For the Hahn polynomials we obtain the general correlation function, some of its special cases, and the limit related to the Meixner polynomials, where the su(1,1) algebra describes the underlying symmetry. For the cases of dual Hahn and Racah polynomials the general expressions of the correlation functions contain summations which are not of hypergeometric type. Simplifications, however, occur in special cases.

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• Inhomogeneous XX spin chains and quasi-exactly solvable models
• Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results
• Birth and death processes and quantum spin chains
• Exact Fractional Revival in Spin Chains
• Time-crystalline behavior in an engineered spin chain
• Spin lattices, state transfer and bivariate Krawtchouk polynomials
• A classical model for perfect transfer and fractional revival based on $q$-Racah polynomials
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective
• Perfect state transfer in two dimensions and the bivariate dual-Hahn polynomials