Signatures of Exciton Orbits in Quantum Mechanical Recurrence Spectra of ${\mathrm{Cu}}_2\mathrm{O}$

The seminal work by Kazimierczuk et al. [Nature 514, 343 (2014)] has shown the existence of highly excited exciton states in a regime, where the correspondence principle is applicable and quantum mechanics turns into classical mechanics; however, any interpretation of exciton spectra based on a classical approach to excitons is still missing. Here, we close this gap by computing and comparing quantum mechanical and semiclassical recurrence spectra of cuprous oxide. We show that the quantum mechanical recurrence spectra exhibit peaks, which, by application of semiclassical theories and a scaling transformation, can be directly related to classical periodic exciton orbits. The application of semiclassical theories to exciton physics requires the detailed analysis of the classical exciton dynamics, including three-dimensional orbits, which strongly deviate from hydrogenlike Keplerian orbits. Our findings illuminate important aspects of excitons in semiconductors by directly relating the quantum mechanical band structure splittings of excitons to the corresponding classical exciton dynamics.

Figure 1

(a) Part of the scaled quantum mechanical density of states (QM) for $n_0=5$. (b) Quantum mechanical exciton recurrence spectrum (black solid line, with zero line shifted for better visibility) obtained by FT of the QM density of states shown in (a) and the semiclassical recurrence spectrum (colored bars). The peaks corresponding to one or multiple repetitions of nearly circular orbits are labeled with a single winding number $M_1$. Two winding numbers $M_1:M_2$ indicate planar orbits in one of the two different symmetry planes of the crystal, and fully three-dimensional orbits are marked by three winding numbers $M_1:M_2:M_3$. The observed structures agree very well with the semiclassical results.

Figure 2

(a) Nearly circular orbits with winding number $M_1=1$ and (b) planar orbits with winding numbers $M_1:M_2=7:1$ in the two different symmetry planes of the crystal. (c), (d) Two examples of fully three-dimensional orbits with winding numbers $M_1:M_2:M_3$. The colors are the same as in Fig. 1.

Figure 3

Actions ${\tilde{S}}_{\mathrm{PO}}$ of periodic orbits as function of the ratio of winding numbers $M_1/M_2$. The actions are normalized by the actions ${\tilde{S}}_{\perp [011]}$ of the corresponding orbits with the same winding numbers $M_1$ and $M_2$ in the plane perpendicular to the [011] axis. The two-dimensional orbits approach the action of the nearly circular orbit (indicated by rhombi) of the corresponding plane with increasing $M_1/M_2$. Some three-dimensional orbits with marked ratio $M_3:M_2$ are located in the area enclosed by orbits in the two different symmetry planes of the $O_h$ group. The orbits shown in Fig. 2 are highlighted by larger symbols.