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arXiv:0807.4985 · Journal of Physics A Mathematical and Theoretical

The characteristic polynomial of the next-nearest-neighbour qubit chain for single excitations

C J Mewton, Z. Ficek

Published Oct 7, 2008• 0 citations
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Abstract

The characteristic polynomial for a chain of dipole-dipole coupled two-level atoms with nearest-neighbour and next-nearest-neighbour interactions is developed for the study of eigenvalues and eigenvectors for single-photon excitations. We find the exact form of the polynomial in terms of the Chebyshev polynomials of the second kind that is valid for an arbitrary number of atoms and coupling strengths. We then propose a technique for expressing the roots of the polynomial as a power series in the coupling constants. The general properties of the solutions are also explored, to shed some light on the general properties that the exact, analytic form of the energy eigenvalues should have. A method for deriving the eigenvectors of the Hamiltonian is also outlined.

DOI
10.1088/1751-8113/41/44/445201

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We study a linear spin chain which was originally introduced by Shi et al. [Phys.Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site.Extending the model by a second parameter β, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form.The components of these eigenvectors turn out to be Hahn polynomials with parameters (α, β) and (α + 1, β -1).The construction of the eigenvectors relies on two new difference equations for Hahn polynomials.The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain.We also discuss some aspects of a q-extension of this model.