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

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.

arXiv comment19 pages, 8 figures; minor corrections

References in corpus (14)

• Quantum Computation with Quantum Dots
• Quantum state engineering with Josephson-junction devices
• Coupled quantum dots as quantum gates
• Quantum Communication through Spin Chain Dynamics: an Introductory Overview
• Entangled states and collective nonclassical effects in two-atom systems
• Implementation of Spin Hamiltonians in Optical Lattices
• Entangling strings of neutral atoms in 1D atomic pipeline structures
• Frustrated quantum Heisenberg ferrimagnetic chains
• DMRG study of scaling exponents in spin-1/2 Heisenberg chains with dimerization and frustration
• Comparison between disordered quantum spin 1/2 chains
• Complete eigenstates of identical qubits arranged in regular polygons
• Multi-quantum eigenstates of a linear chain of coupled qubits
• Crossover behavior of the J1-J2 model in a staggered magnetic field
• Scaling of excitations in dimerized and frustrated spin-1/2 chains