Bound states in the continuum in open quantum billiards with a variable shape

We show the existence of bound states in the continuum (BICs) in quantum billiards (QBs) that are opened by attaching single-channel leads to them. They may be observed by varying an external parameter continuously, e.g., the shape of the QB. At some values of the parameter, resonance states with vanishing decay width (the BICs) occur. They are localized almost completely in the interior of the closed system. The phenomenon is shown analytically to exist in the simplest case of a two level QB and is complemented by numerical calculations for a real QB.

Figure 1

The log scale probability for the transmission through the rectangular billiard shown in the inset, versus energy $E$ and width $W$ of the billiard (in terms of the width of the lead). The dark areas correspond to low transmission probability. The length of the QB along the $x$ axis is four. The eigenenergies of the closed billiard are marked by crosses. The positions of the BICs are shown by bold circles. The patterns of the two BICs A and B are shown in Fig. 3.

Figure 2

The evolution of resonance width $\mathrm{Im}\left(z\right)$ and position $\mathrm{Re}\left(z\right)$ of one of the two resonance states in the vicinity of the BIC B in Fig. 1 for $W=[4.3,4.6]$. $\mathrm{Im}\left(z\right)$ vanishes at $W=4.45$ (marked by a cross). The widths of the other resonance states are much larger and are not shown here.

Figure 3

The patterns of the two BICs A and B marked in Fig. 1 by bold circles.

Figure 4

The same as in Fig. 1 but the QB is weakly connected to the leads by diaphragms. Due to the diaphragms the billiard becomes nonintegrable. As a result the eigenenergies of the closed QB slightly repel each other. They are marked by a line of small open circles, and the parity relative to the $y$ axis is indicated by a plus or minus.

Figure 5

(Color online) Left panel: The transmission probability $\mid T\mid$ through the two-state QB versus incident energy $E$ and $ϵ$, and $\mathrm{Re}\left(z_{\lambda }\right)$ (solid lines). The dark areas correspond to low transmission probability. $T$ and $z_{\lambda }$ are defined by (7, 8), respectively. The eigenenergies $\pm ϵ$ of the QB are shown by thin solid lines. Right panel: the resonant widths $\mathrm{Im}\left(z_{\lambda }\right)$ as a function of $ϵ$ (solid lines). (a) and (b) $u=0$, ${\Gamma }_1=0.1$, ${\Gamma }_2=0.05$. (c) and (d) $u=0.05$, ${\Gamma }_1={\Gamma }_2=0.1$. (e) and (f) $u=0.05$, ${\Gamma }_1=0.1$, ${\Gamma }_2=0.05$.