Uncertainty-Limited Turnstile Transport in Deformed Microcavities

We present both experimental and theoretical evidence for uncertainty-limited turnstile transport in deformed microcavities. As the degree of cavity deformation was increased, a secondary peak gradually emerged in the far-field emission patterns to form a double-peak structure. Our observation can be explained in terms of the interplay between turnstile transport and its suppression by the quantum mechanical uncertainty principle.

Figure 1

(a)–(d) Far-field emission patterns observed in experiment and (e)–(h) those in wave calculation. Degree of cavity deformation $\eta$ was 0.09 for (a) and (e), 0.10 for (b) and (f), 0.12 for (c) and (g), and 0.16 for (d) and (h). The wave calculation was done for the actual cavity (shown in the positive half) and the equivalent quadrupole cavity (in the negative half). Both cavities show very similar emission patterns: As the chaotic transition progresses, a secondary narrow peak emerges around $\theta \sim \pm 55°$ in the presence of a broad peak around $\theta \sim \pm 40°$. All plots are normalized.

Figure 2

Phase-space structure of stable (blue/dark gray) and unstable (red/gray) manifolds of the unstable period-4 orbit with fixed points $F_n$ ($n=1,2,3,4$) for $\eta =0.12$, presented in Birkhoff’s coordinates, in which a ray collides with the cavity boundary at polar angle $\varphi$ with an incidence angle $\chi$. Turnstile transport from the region above the chained areas into the chained areas (I, II, III, IV) occurs by lobe dynamics $a\to b\to c\to d\to e\to f$.

Figure 3

Results of ray simulations for (a) $\eta =0.09$, (b) 0.10, (c) 0.12, and (d) 0.16. In (a)–(d), it is shown that lobe $d^{\prime }$ comes down below the critical line, while the already existing lobe $f$ moves further downwards. Plots in (e)–(h) show the corresponding output angular distributions histogramatically calculated in the ray simulation. A narrow peak around 55° originating from lobe $f$ is ever present, while a broad peak from lobe $d^{\prime }$ gradually grows as $\eta$ increases.

Figure 4

Phase-space distributions of quasieigenstates or Husimi distributions (blue/dark gray) for (a) $\eta =0.09$ and (b) $\eta =0.16$. The tangled structure of stable and unstable manifolds of the period-4 orbit are represented as green (gray) curves. The horizontal line at $\sin \chi =0.73$ represents the critical angle of refraction. In (a), the cantori represented as red dots blocks the distribution from overlapping with the tangled structure. In (b), the distribution has a considerable overlap with the manifold structure. (c) The ratio of the lobe area to the resolution limit ${\hslash }_{\mathrm{eff}}/2$ under our experimental condition as a function of the deformation parameter.

Figure 5

Configurational plots of quasieigenmodes. (a) $\eta =0.09$ and $x=221.07-0.005i$; (b) $\eta =0.16$ and $x=227.65-0.024i$. In (a), the emission, coming from evanescent leakage, has its direction tangent to the boundary. In (b), the emission comes from turnstile transport, and thus it is asymmetric and refractive. Dual strong emissions to $\theta \sim -45°,-60°$ directions are visible.