Bound states in the continuum in a two-channel Fano-Anderson model

In this article, we study the formation of the bound states in the continuum (BICs) in a two-channel Fano-Anderson model. We employ the Green's function formalism, together with the equation of motion method, to analyze the relevant observables, such as the transmission coefficient and the density of states. Most importantly, our results show that the system hosts true BICs for the case of a symmetric configuration with degenerate impurity levels, and a complete transmission channel is then suppressed. Finally, we argue that the proposed mechanism could be relevant for the realization of BICs in electronic and photonic systems.

Figure 1

Schematic view of the setup: a two-channel impurity coupled to left ($L$) and right ($R$) baths of excitations (bosonic or fermionic), each one featuring two degrees of freedom labeled by 1 and 2. The baths could be realized, for instance, as optical fibers or metallic leads, depending on the nature of the system considered. The parameter $V$ (blue dashed line) denotes the intrachannel coupling while the parameter $W$ (red dotted line) corresponds to the interchannel coupling.

Figure 2

Transmission coefficient $T$ as function of energy $\varepsilon$ for different values of the interchannel coupling $\lambda$, as given by Eq. (11). (a) $\lambda =0.0$; (b) $\lambda =0.5$; (c) $\lambda =0.8$; and (d) $\lambda =0.99$ (solid black line), $\lambda =1.0$ (dashed red line). Notice that the maximum value of the transmission coefficient is equal to 2 since two transmission channels are considered.

Figure 3

Map of transmission coefficient $T$ as function of the energy $\varepsilon$ and the parameter $\lambda$, which is given by Eq. (11). The section represented by the cyan dashed horizontal line for $\lambda =0.5$ ($\lambda =0.8$) corresponds to the plot in Fig. 2 [Fig. 2].

Figure 4

Density of states $\rho$ as function of energy $\varepsilon$, as given by Eq. (12). (a) $\lambda =0.0$; (b) $\lambda =0.5$; (c) $\lambda =0.8$; and (d) $\lambda =1.0$. In all panels, the gray dotted lines represent the value $\pi \gamma \rho =2$.

Figure 5

Map of the density of states $\rho$ as a function of the energy $\varepsilon$ and the parameter $\lambda$, which is given by Eq. (12). The section depicted by the cyan dashed horizontal line for $\lambda =0.5$ ($\lambda =0.8$) corresponds to the plot in Fig. 4 [Fig. 4]. The white zone observed at $\varepsilon =0$ for $\lambda \to 1$ corresponds to higher values than the color scale used.

Figure 6

Transmission coefficient $T$, given by Eq. (9), as a function of the energy $\varepsilon$ for fixed $\lambda =1.0$ and different values of the parameter $\delta$: (a) $\delta =0.0$, (b) $\delta =0.5$, (c) $\delta =1.0$, (d) $\delta =2.0$.

Figure 7

Schematic representation of the symmetric ($+$) and antisymmetric ($-$) channels: (a) the generic case; (b) the limiting case for $\delta =0$ and $\lambda \to 1$; (c) the shape of the resonances in the symmetric and antisymmetric channels.

Figure 8

Probability function to find the particle in the level (a) ${\varepsilon }_1$ or (b) ${\varepsilon }_2$ of the impurity given that in $t=0$ the particle is in ${\varepsilon }_1$. In both panels $\delta =0$, and the probability function is obtained from Eq. (18), together with Eq. (15). The green dashed and yellow dashed-dotted lines represent the values of ${\tau }_{+}$ and ${\tau }_{-}$ as a function of $\lambda$, respectively.

Figure 9

Integration contour in the $\varepsilon$-complex plane used to compute the integral in Eq. (A1).