Potential-controlled filtering in quantum star graphs

Abstract

We study the scattering in a quantum star graph with a F\"ul\"op--Tsutsui coupling in its vertex and with external potentials on the lines. We find certain special couplings for which the probability of the transmission between two given lines of the graph is strongly influenced by the potential applied on another line. On the basis of this phenomenon we design a tunable quantum band-pass spectral filter. The transmission from the input to the output line is governed by a potential added on the controlling line. The strength of the potential directly determines the passband position, which allows to control the filter in a macroscopic manner. Generalization of this concept to quantum devices with multiple controlling lines proves possible. It enables the construction of spectral filters with more controllable parameters or with more operation modes. In particular, we design a band-pass filter with independently adjustable multiple passbands. We also address the problem of the physical realization of F\"ul\"op--Tsutsui couplings and demonstrate that the couplings needed for the construction of the proposed quantum devices can be approximated by simple graphs carrying only $\delta$ potentials.Comment: 41 pages, 17 figure

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

“…One example are fat graphs consisting of bonds with finite widths [16]. What is the effect of nonlinear interaction on quantum spectral filters modeled by star graphs [23,24]?…”

“…This implies that we implicitly know k n (N ). Note that as N → 0, all amplitudes ρ e,+ have to vanish and the condition (24) reduces to…”

Quantum graph as a quantum spectral filter

“…8 They define an interesting family of vertex couplings 9-11 with useful scattering properties. 3,4,12 One of the most natural singular interactions is the δ-coupling (also called "δ potential"), which is characterized by boundary conditions…”

“…18 Also graphs with infinite edges can be considered. 3,4 A natural application of the phenomenon arises in spectral filtering. Let io be constructed for α 1.…”

“…This construction has been generalized to quantum filters with multiple outputs and with multiple controllers. 4 There exist also different designs, e.g., a special trident filter. 5 In this paper we consider transmission characteristics of a graph built from four "components": an "input" half line, an "output" half line, a graph , and a certain special scale-invariant point interaction.…”