Experimental Observation of Resonance-Assisted Tunneling

We present the first experimental observation of resonance-assisted tunneling, a wave phenomenon, where regular-to-chaotic tunneling is strongly enhanced by the presence of a classical nonlinear resonance chain. For this we use a microwave cavity made of oxygen free copper with the shape of a desymmetrized cosine billiard designed with a large nonlinear resonance chain in the regular region. It is opened in a region, where only chaotic dynamics takes place, such that the tunneling rate of a regular mode to the chaotic region increases the line width of the mode. Resonance-assisted tunneling is demonstrated by (i) a parametric variation and (ii) the characteristic plateau and peak structure towards the semiclassical limit.

Figure 1

(a) Photograph of the cavity including a movable (white arrow) half disk, absorbers (gray), and antenna position (red). (b) Poincaré section along the upper boundary of the classical dynamics of the closed billiard with regular tori (red), a chaotic orbit (blue), and a $3:1$ nonlinear resonance chain (orange). The gray region indicates the absorbing part. (c) Frequency $\nu$ and (d),(e) width $\mathrm{\Gamma }$ of the regular modes $(n,m)=(1,10)$ (black pluses) and (4,9) (red diamonds) vs. half disk position $d$. Solid orange lines correspond to the $3\times 3$ matrix model, Eq. (3). Insets visualize numerically determined regular modes of the closed billiard.

Figure 2

(a) Frequency difference $\mathrm{\Delta }\nu ={\nu }_{(1,m)}-{\nu }_{(4,m-1)}$ of modes $(1,m)$ and $(4,m-1)$ for $m=8,\dots ,16$ and fixed half disk position $d=30.0\mathrm{cm}$. The dotted line indicates the value 0. (b) Experimental widths ${\mathrm{\Gamma }}_{(1,m)}$ (green pluses) of the inner mode for $m=5,\dots ,16$, the square root dependence of ${\mathrm{\Gamma }}_{\mathrm{wall}}$ (orange dotted line), and ${\mathrm{\Gamma }}_{\text{antenna}}$ (blue diamonds) with the dashed line showing the assumed linear dependence at the close encounter. (c) ${\mathrm{\Gamma }}_{\mathrm{RAT}}$ (black crosses) showing the initial exponential decay and the characteristic plateau and peak structure of RAT. (d) Numerical dimensionless tunneling rates ${\gamma }_{(1,m)}$ of the closed system (red open circles).