Quasieigenstate evolution in open chaotic billiards

We experimentally studied the evolution of quasieigenmodes as classical dynamics undergoing a transition from being regular to chaotic in open quantum billiards. In a deformation-variable microcavity, we traced all high-$Q$ cavity modes in a wide range of frequency as the cavity deformation increased. By employing an internal parameter, we were able to obtain a mode-dynamics diagram at a given deformation, showing avoided crossings between different mode groups, and could directly observe the coupling strengths induced by ray chaos among encountering modes. We also show that the observed mode-dynamics diagrams reflect the underlying classical ray dynamics in the phase space.

Figure 1

(Color online) (a) For a COM with $\eta =18.7\%$, five uncoupled mode groups are identified. (b) Several ACs are observed as we vary $\eta$ when modes are followed adiabatically. Inset: the spectrum measured with a spectrometer with a higher resolution $\left(\sim 0.012\text{THz}\right)$, resolving AC of two adjacent modes.

Figure 2

(Color online) (a) Calculated relative frequencies of quasieigenmodes with radial mode order $l_0=1,2,\dots ,8$ for a circular cavity $\left(\eta =0\%\right)$ with the mode frequency of $l_0=3$ as a reference. (b) Observed relative frequencies of quasieigenmodes for $\eta =14.3\%$. Mode frequencies more or less follow diabatic lines (dotted straight lines) except for ACs with very small splittings. We employ shorthand notation $l^{l_0}$ as explained in the text. (c) The same for $\eta =18.7\%$. More pronounced ACs with decay-rate exchange as well as ordinary crossings are observed. The diameter of the circle drawn on each data point represents the half linewidth of the corresponding mode in THz. Filled circle indicates spectrometer resolution ${\gamma }_0\sim 0.05\text{THz}$. (d) The case of $\eta =22.3\%$. The splittings are much more larger than those of (c) and the mode frequencies deviate greatly from the diabatic lines.

Figure 3

(Color online) (a) Decay rates ${\gamma }_p$ of $p=1,2,4$ uncoupled states. (b) Coupling strength $C_{pq}$ between $p,q$ $\left(=1,2,4\right)$ uncoupled states restored from the observed sizes of AC and the decay rates of modes, assuming three-mode coupling. Vertical error bars indicate measurement errors. Systematic error in determining the exact deformation is indicated with a horizontal error bar in both (a) and (b). Solid lines in (a) and (b) are exponential and linear fits, respectively, with their extensions indicated by dotted lines. (c) PSOS’s for $\eta =17\%$ and (d) $\eta =19\%$. See the text for explanation.

Figure 4

(Color online) (a) Calculated relative frequencies of $l=1,2,3,4,5$ modes with respect to the reference frequency for $\eta =19\%$. (b) Husimi plots of the modes marked by arrows near 460 THz where $l=2,4$ modes undergo an AC.