On-manifold localization in open quantum maps

A quantized chaotic map exhibiting localization of wave function intensity is opened. We investigate how such patterns as scars in the Husimi distributions are influenced by the losses through a number of numerical experiments. We find the scars to relocate on the stable or unstable manifolds depending on the position of the opening, and provide a classical argument to explain the observations. For asymmetrically introduced openings mode interaction contributes to determine the localization patterns. We finally show examples of similar localization in a simulated dielectric microcavity.

Figure 1

(a) The Husimi distribution of a scarred mode of the closed map (4). The Husimi distributions of (b) the left and (c) the right eigenfunctions of the open map (8) with $\Gamma =0.1$, where the dashed circles indicate the location of the opening. The unstable (positive sloped) and stable (negative sloped) manifolds of the classical map are also drawn. The densities are given in log-scale.

Figure 2

The Husimi distributions: (a) a scarred mode localized at the origin of the closed map, (b) the left and (c) the right eigenfunctions of the open map with $\Gamma =0.1$, and (d) the left and (e) the right eigenfunctions with $\Gamma =0.15$. The opening (dashed circle) is placed on the unstable manifold of the classical map at a distance $6\sqrt{\hslash }$ from the origin. Continuous and dashed arrows indicate the classical dynamics forward and backward in time, respectively. To avoid confusion it is reemphasized that the dashed arrows in the right eigenstates (c and e) describe time-reversed dynamics. Therefore, the classical trajectories first flow along the stable manifold on the left side and then escape into the opening via short path along the unstable manifolds on the right side.

Figure 3

The Husimi distributions of (a) and (b) two nearly degenerate scarred modes of the closed map, and (c) and (d) the right eigenfunctions evolved from (a) and (b) when the map is opened with $\Gamma =0.1$. The opening (the dashed circles) is provided in the same manner given in Fig. 2e. (e) The quasienergies and (f) the moduli of the complex eigenvalues of two interacting modes are given for several $\Gamma$'s.

Figure 4

(the upper) The quasienergies and (the lower) the moduli of the complex eigenvalues (16) as a function of $\Gamma$ with $c=1$.

Figure 5

Same as in Fig. 2, with the opening (dashed circle) placed on the stable manifold of the classical map at a distance $6\sqrt{\hslash }$ from the origin.

Figure 6

The Husimi distributions of (a) the scarred mode of the closed map, and (b) the left and (c) the right eigenfunctions of the open map with two openings, one from Fig. 2 and the other from Fig. 5.

Figure 7

(a) The Husimi distribution of the scarred mode of the closed map (20). (b) and (c) show Eqs. (21) and (24), respectively, with $n=3.5$. The Husimi distributions of the right eigenfunctions when the opening is given as (d) Eq. (21) and (e) Eq. (24). The red (blue) inclined line with a positive (negative) slope represents the unstable (stable) manifold.

Figure 8

(a) Intensity pattern of a quasibound state localized on the bowtie-shaped periodic orbit. (b) Husimi distribution of the mode (a). The red (blue) inclined line with a negative (positive) slope represents the unstable (stable) manifold. The horizontal dashed lines represent the critical lines of total internal reflection for $n=2$. The inset in (d) shows how the boundary length $s$ is defined. Here $s_h=\pi +2$. (c) The same as (a) for the horizontal bouncing ball type periodic orbit. (d) Husimi distribution of the mode (c).