Role of phonons in the quantum chaos of Rydberg excitons

We investigate the transition from regular behavior towards chaos for Rydberg excitons in the presence of an external magnetic field. To this end, we develop a measure for chaos that is robust with respect to fluctuations in the density of states. We find that irrespective of whether or not the magnetic field is oriented along a high-symmetry direction of the crystal, all antiunitary symmetries of the system are broken for large field strengths. This result emphasizes the influence of phonons on the symmetry of Rydberg excitons. In addition, we find that the appearance of Landau levels above the band gap results in modified level spacing statistics that resemble systems without broken symmetry.

Figure 1

Black, red, and blue lines represent the binned and normalized two-gap correlation ratio distributions for the Poissonian, GOE and GUE ensemble at a bin width of 0.05, respectively. The dots show some experimentally determined two-gap distributions close to the ensemble predictions at energies of 2.162 99 eV and 1–5 T, 2.1775 eV and 4–8 T, and 2.1927 eV and 1–5 T.

Figure 2

Transmission spectrum of the yellow exciton series in ${\mathrm{Cu}}_2\mathrm{O}$ in a magnetic field up to 10 T. Small intensity steps arise due to the whole spectrum being composed of several partial spectra measured separately.

Figure 3

$\xi$ of the two-gap ratio distribution with the smallest $l^2\mathrm{distance}$ from the experimental data for different spectral and magnetic field ranges and the magnetic field oriented at ${30}^{\circ }$ to the [001] direction of the crystal. $\xi$ values of 0, 1, and 2 correspond to the Poissonian, GOE, and GUE ensembles, respectively, and are represented by dashed horizontal lines in black, red, and blue.

Figure 4

$\xi$ of the two-gap ratio distribution with the smallest $l^2\mathrm{distance}$ from the experimental data for different spectral and magnetic field ranges and the magnetic field oriented along the high-symmetry [001] direction of the crystal. $\xi$ values of 0, 1, and 2 correspond to the Poissonian, GOE and GUE ensembles, respectively, and are represented by dashed horizontal lines in black, red, and blue.