Assignment of quasi-Landau levels in magneto-oscillatory spectra in cuprous oxide to classical unstable trajectories of hydrogen type atoms

Abstract

It is found, for the first time in the field of solid state spectroscopy, that quasi-Landau levels in magneto-oscillatory spectra in cuprous oxide reflect “classical non-integrability.” Cuprous oxide is well known to exhibit excitonic absorption spectra of typical wannier type near the absorption edge in yellow spectral region. Magneto-oscillatory spectra are observed in the region above the limiting energy of the exciton series in magnetic fields. The spectra are measured at liquid helium temperature with right and left circularly polarized light in magnetic fields up to 4.5 T generated by a superconducting magnet. The observed spectra look like “Landau levels” corresponding to the optical transitions between states of the hole in a valence band and those of the electron in a conduction band without Coulomb attraction between them qualitatively, but never coincide with the Landau levels quantitatively. By calculating inverse Fourier transform of the observed spectra (IFFT spectra), three peaks are found in the auto-correlation function of the excited excitonic state. These spectra are interpreted as those of a hydrogen type atom with the effective masses of the electron–hole pair in homogeneous magnetic field, which is known to be a typical non-integrable system in classical mechanics. Instead of obtaining the quantum mechanical motions of wave packet, the classical trajectories are numerically calculated. First peak of the IFFT spectra is assigned to the trajectories on which the wave packet circulates and returns to approximately to the starting point after the duration corresponding to the first peak. Immediately after that, the trajectories are rapidly apart from the starting point on account of their instability, which reflects the classical non-integrability of the system.