Avoided level crossings in an elliptic billiard

In an elliptic billiard, we find avoided level crossings taking place over wide ranges, which are of a Demkov type for generations of eigenfunctions localized on an islands chain and its pair unstable periodic orbit. For a proof of the existence of avoided level crossings, first, we show that the quantized eigenvalue of the unstable periodic orbit, obtained by the Einstein-Brillouin-Keller quantization rule, passes the eigenvalues of bouncing-ball modes localized on the unstable periodic orbit after Demkov type avoided level crossings so that pairs of bouncing-ball modes are sequentially generated. Next, by using a perturbed Hamiltonian, we show that off-diagonal elements in Hamiltonian are nonzero, which give rise to an interaction between two eigenfunctions. Last, we verify that the observed phenomenon is Fermi resonance: that is, the quantum number difference of two normal modes equals the periodic orbits, where eigenfunctions are localized after an avoided level crossing.

Figure 1

Eigenvalue spectra in an elliptic billiard as a function of $\epsilon$ in the region $0.0<\epsilon <0.93$ and $18<k<32$. The eigenvalues are developed from the $(10,0), (9,2), (8,4), (7,6), (6,8), (5,10), (4,12), (3,14), (2,16)$, and $(1,18)$ normal mode, respectively. $(l,m)$ denotes the normal mode of a circular billiard, where $l$ and $m$ are the radial and the angular quantum number, respectively, of normal modes in a circular billiard. In region A, the $(10,0)$ and $(9,2)$ normal mode interact with each other through an ALC, and a vertical and a horizontal BBM are generated. In regions from B through I, horizontal BBMs interact with normal modes through ALCs, and new vertical and horizontal BBMs are generated. The thick black solid line is the quantized eigenvalue of the period-2 unstable periodic orbit obtained by the EBK quantization rule.

Figure 2

The left and the right column is an example of ALC and a crossing between two neighboring eigenvalues, respectively. The first row is two neighboring eigenvalues, and the second row the difference between eigenvalues. The third row is the matrix element of perturbed Hamiltonian between two neighboring modes.

Figure 3

Intensity plots of eigenfunctions around region A. (a, b) The $(10,0)$ and $(9,2)$ normal mode at $\epsilon =0.0$. (c, d) The vertical and the horizontal BBM at $\epsilon =0.09$ developed from the $(10,0)$ and $(9,2)$ normal mode, respectively. (e, f) The $[(10,0)-(9,2)]/\sqrt{2}$ and $[(10,0)+(9,2)]/\sqrt{2}$ state, which are the superposed states of the $(10,0)$ and $(9,2)$ normal mode.

Figure 4

Husimi functions depending on $\epsilon$. (a, b) The Husimi functions of the $(10,0)$ and $(9,2)$ normal mode at $\epsilon =0.0$. (c, d) The vertical and the horizontal BBM at $\epsilon =0.09$, which are localized on the period-2 islands chain and and its pair unstable periodic orbit.

Figure 5

Intensity plots of eigenfunctions when interactions take place. (a, b) The horizontal BBM developed from the $(9,2)$ normal mode and $(8,4)$ normal mode at $\epsilon =0.1$ before the interaction around region B. (c, d) The vertical and the horizontal BBM developed from the two normal modes due to the interaction at $\epsilon =0.5$ and 0.28, respectively. After the ALC, the $(9,2)$ and $(8,4)$ normal mode develop to a vertical BBM with three column and a horizontal one, respectively. (e, f) The vertical BBM with five columns and the horizontal one developed from the $(8,4)$ and $(7,6)$ normal mode at $\epsilon =0.6$ and 0.38 after an ALC, respectively. (g, h) The vertical BBM with seven columns and the horizontal one developed from the $(7,6)$ and $(6,8)$ normal mode at $\epsilon =0.7$ and 0.48 after an ALC, respectively. (i, j) The vertical BBM with nine columns and a horizontal one developed from the $(6,8)$ and $(5,10)$ normal mode at $\epsilon =0.8$ and 0.58 after an ALC, respectively.

Figure 6

Eigenvalue spectra in a quadrupole billiard as a function of $\epsilon$ in the region $0.0<\epsilon <0.09$ and $20<k<26$. In the regions of A, B, C, and D, sequential interactions take place to generate BBMs as in an elliptic billiard. The inset shows an interaction of the $(1,20)$ normal mode with the $(2,16)$ to generate a pair of eigenfunctions localized on a period-4 islands chain and its pair period-4 unstable periodic orbit.

Figure 7

Intensity plots of eigenfunctions. (a, b) The $(1,20)$ and $(2,16)$ normal mode at $\epsilon =0.0$. (c, d) The eigenfunctions localized on the unstable period-4 orbit and period-4 islands chain at $\epsilon =0.01$, respectively.