Time evolution of topological magneto-optics and superconducting qubits

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We study the nonlinear response to an external driving electric field E characterized by a conductivity tensor which depends on the magnitude of E. The quasiparticles (helical Dirac fermions) observed in topological insulators possess an important feature: the Fermi contours are circular for small acquire a snowflake shape at large values of the chemical potential. If a gap opens (e.g., by doping with magnetic impurities) the inversion symmetry is broken. We develop a nonlinear response theory based on a generalized Kubo formula to explain the frequency up-conversion in topological insulators.Flatbands, which may emerge as a consequence of symmetries or finetuning in certain tight-binding Hamiltonians, are characterized by a completely dispersionless single-particle energy spectrum. Remarkably, flat bands feature the existence of “compact localized states”, free of any dynamical evolution and with tailorable shape, the latter by judiciously superposing the entirely degenerate Bloch states. We start from the cross-stitch model that most simply describes a flat-band (FB) and a dispersive-band (DB). And consider the disorder effect by setting on-site energies with correlated random numbers. The disorder will induce decay from FB to DB together with dephasing in momentum space. This will limit the lifetime of the FB state.