Signatures of Homoclinic Motion in Quantum Chaos

Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wave functions localized along periodic orbits we reveal the existence of an oscillatory behavior that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit.

Figure 1

Configuration space (a), phase space (b), and wavelength spectrum weights (c) representations of a scar wave function along the horizontal periodic orbit of a desymmetrized stadium billiard corresponding to BS wavelength $k_{\mathrm{B}\mathrm{S}}=211.665$. The unstable and stable manifolds of the orbit are also plotted in panel (b). In panel (a) the Birkhoff coordinates on the boundary of the billiard are shown.

Figure 2

Relative width for the scar states along the horizontal periodic orbit as a function of the Bohr-Sommerfeld quantized wavelength (inset), and its Fourier transform.

Figure 3

(a) Phase space portrait in Birkhoff coordinates relevant to our calculations. Label A indicates the position of the fixed point corresponding to the horizontal periodic orbit. The associated unstable and stable manifolds are represented in full and dashed line, respectively. They cross, with different topology, at the primary homoclinic points $h_1$ and $h_2$, which map into $h_1^{\prime }$, $h_1^{\prime \prime }$, $\dots$, and $h_2^{\prime }$, $h_2^{\prime \prime }$, $\dots$. (b) Primary homoclinic areas $S_{h_1}$ and $S_{h_2}$ associated with the horizontal orbit. (c) Satellite periodic orbits converging to the homoclinic points $h_1$ and $h_2$. See text for details.

Figure 4

Intensity of the peak at $S_{h_1}$ divided by $(\Delta k{)}^2$ as a function of $\Delta k$.