Quantum states resembling classical periodic trajectories in mesoscopic elliptic billiards

Abstract

A quantum wave function with localization on classical periodic orbits in a mesoscopic elliptic billiard has been achieved by appropriately superposing nearly degenerate eigenstates expressed as products of Mathieu functions. We analyze and discuss the rotational and librational regimes of motion in the elliptic billiard. Simplified line equations corresponding to the classical trajectories can be extracted from the quantum state as an integral equation involving angular Mathieu functions. The phase factors appearing in the integrals are connected to the classical initial positions and velocity components. We analyze the probability current density, the phase maps, and the vortex distributions of the periodic orbit quantum states for both rotational and librational motions; furthermore, they may represent traveling and standing trajectories inside the elliptic billiard.

Figures (12)

• Figure 1: (a) $\mathcal{R}$-type periodic classic orbits for several indices $(p,n)$ in an elliptic billiard with eccentricity $\epsilon=0.5.$ Each trajectory is tangent to an elliptic caustic displayed with a dash-dotted red line. (b) $\mathcal{L}$-type trajectories with $\gamma_{p,n}<0$. All segments of the trajectory cross the $x$-axis through the interfocal line. Eccentricities are $\epsilon=0.924,$$\epsilon=0.714,$ and $\epsilon=0.848$. Dash-dotted red lines are the hyperbolic caustics of the librational orbits.
• Figure 2: Lower-order eigenstates in the elliptic billiard with eccentricity $\epsilon=0.5.$ (a) $|\psi_{r,m}^{e}(\xi,\eta)|^{2}$ of even standing-wave states. (b) $|\psi_{r,m}^{o}(\xi,\eta)|^{2}$ of odd standing-wave states. (c) $|\psi_{r,m}^{+}(\xi,\eta)|^{2}$ of positive traveling-wave states. (d) Phase distribution $\arg\psi_{r,m}^{+}(\xi,\eta)$ of positive traveling-wave states.
• Figure 3: Characteristic values of $q_{r,m}^{e}$ in function of the order $r$ for a billiard with eccentricity (a) (b) $\epsilon=0.5,$ and (c) $\epsilon=0.924.$ In each subplot, the horizontal solid line corresponds to the value of $q_{\mathrm{sc}}(p,n,r_{0},m_{0})$ and each marker defines a particular $m.$ The values $q_{r,m}^{e}$ closest to the horizontal line correspond to the degenerate states.
• Figure 4: POQS with localization on the classical periodic orbits shown in Fig. \ref{['F1_fig:tray_clas']} for some combinations of the parameters $(p,n,\phi_{0},S)$ using $r_{0}=50$ for $\mathcal{R}(3,1,3\pi/2,3)$, $r_{0}=120$ for $\mathcal{R}(5,2,3\pi/2,3),$$\mathcal{R}(7,2,\pi/2,1),$$\mathcal{R}(8,3,\pi,2)$ and $r_{0}=130$ for $\mathcal{L}(6,1,0,2),$$\mathcal{L}(6,2,\pi,2),$$\mathcal{L}(10,3,0,2),$ and $\mathcal{L}(14,3,0,2).$
• Figure 5: Effect of the phase factor $\phi_{0}$ in the $\mathcal{R}$-type POQS. (a) State $\mathcal{R}(8,3,\phi_{0},2)$. (b) State $\mathcal{R}(3,1,\phi_{0},3)$. Central quantum numbers $(r_{0},m_{0})$ are the same as Fig. \ref{['fig4:tray_quantum']} for the corresponding trajectories.