Superlattices with entangled modes and the Hopf bundle

Abstract

Superlattices with coupled modes are considered to analyze entangled multipartite quantum systems. The conservation of the probability current density leads to a natural connection with Hopf fibrations, allowing the application of methods of algebraic topology to the study of electron transport. A superlattice, with n channels and d terminals, each with incoming and outgoing wave contributions, will be seen as n coupled d level quantum systems after their interaction inside a potential region. As usual, the scattering matrix connects the incoming amplitudes on the terminals with the outgoing amplitudes and probability current conservation leads to unitarity of the S-matrix and this to hyperspheres. To compute the scattering amplitudes for two colinear leads, sectionally constant potentials in the longitudinal direction and arbitrary lateral dependency are studied, which allow the analytic calculation of the amplitudes. The amplitudes are given in terms of symmetric functions of the eigenvalues by applying the Sylvester theorem for matrix functions including degenerated eigenvalues. For symplectic transfer matrices, the associated Schur functions are expressed in terms of Chebyshev polynomials in several variables, using a method developed recently, probing to be a transparent analytical and fast numerical tool. The amplitudes can then be computed for explicit superlattices, with up to two coupled modes and displayed on the associated Bloch spheres and Clifford tori, showing curves parameterized by energy and the scattering potentials. For three and four coupled modes, all necessary analytic calculations will also be reported to make possible to perform the corresponding numerical explicit evaluations in the near future.