Quantum / CONCEPT

Spectrum and first-site amplitude of seven-site transfer chains with symmetric spectrum

Let a chain of seven sites with positive couplings, real potentials and a spectrum closed under negation have perfect state transfer between its end sites at a time T. Then its eigenvalues are 0, x, -x, y, -y, z, -z with 0 < x < y < z, exp(iTx) = -1, exp(iTy) = 1, exp(iTz) = -1, and its first-site amplitude at the time tau is P (1/(x^2 y^2 z^2) + cos(tau x)/(x^2 (y^2 - x^2)(z^2 - x^2)) + cos(tau y)/(y^2 (y^2 - x^2)(z^2 - y^2)) + cos(tau z)/(z^2 (z^2 - x^2)(z^2 - y^2))), P being the product of the couplings. For T > 0 this gives natural numbers a < b < c, with a and c odd and b even, such that the spectrum is (pi/T)(0, a, -a, b, -b, c, -c), the amplitude is a positive multiple of the cosine sum N(a, b, c; pi tau / T) of Escobar, Garcia and Minenkova (arXiv:2507.18767), and the chain has perfect state transfer at the time T/g for every common divisor g of a, b, c.