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We study a single photon transmission through 1D N- qubit chain. The qubitsare supposed to be identical with equal distance between neighbors. We expressthe transfer matrix of N- qubit chain in terms of Chebyshev polynomials, whichallows us to obtain simple expressions for the transmission and reflectionamplitudes for arbitrarily large N. If the distance between neighbor qubits isequal to half wavelength, the transmission spectrum exhibits a flat bandgapstructure with very steep walls. We show that for odd N the tuning of theexcitation frequency of a central qubit gives rise to the appearance within abandgap of a narrow resonance with a full transmission. The position of theresonance and its width can be controlled by the frequency of a central qubit.We show that the formation of the bandgap and of the transmission resonance isconditioned by the overlapping the widths of individual qubits which resultsfrom the strong coupling between qubits and waveguide photons.

Waveguide bandgap N-qubit array with a tunable transparency resonance