Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex Robin parameters

Abstract

Properties of the two-dimensional ring and three-dimensional infinitely long straight hollow waveguide with unit width and inner radius ρ0 in the superposition of the longitudinal uniform magnetic field B and the Aharonov–Bohm flux are analyzed within the framework of the scalar Helmholtz equation under the assumption that the Robin boundary conditions at the inner and outer confining walls contain extrapolation lengths Λin and Λout, respectively, with nonzero imaginary parts. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues E, in general, become complex too and the discrete bound states of the annulus, which are characteristic for the real Λ, turn into the corresponding quasibound states with their lifetime defined by the imaginary parts Ei of the eigenenergies while the current along the wire exponentially amplifies/attenuates with the distance depending on the sign of Ei. It is shown that, compared to the disk geometry, the annulus opens up additional possibilities of varying magnetization and currents by tuning imaginary components of the Robin parameters on each confining circumference; in particular, the possibility of restoring a lossless longitudinal flux by zeroing Ei is discussed from mathematical and physical points of view. For each ρ0, the energy E turns real under a special correlation between the imaginary parts of Λin and Λout with the opposite signs being what physically corresponds to the equal transverse fluxes through the inner and outer interfaces of the annulus. In the asymptotic case of the very large radius, ρ0→∞, simple expressions are derived and applied to the analysis of the dependence of the real energy E on Λin and Λout. New features also emerge in the magnetic field influence; for example, if, for the quantum disk, the imaginary energy Ei is quenched by the strong intensities B, then for the annulus this takes place only when the inner Robin distance Λin is real; otherwise, it almost quadratically depends on B with the corresponding enhancement of the reactive scattering. The closely related problem of the hole in the otherwise uniform medium is also addressed for the real and complex extrapolation lengths with the emphasis on the comparative analysis with its dot counterpart.