Coherence of dipole-forbidden Rydberg excitons in Cu$_2$O measured by polarization- and time-resolved multi-photon spectroscopy

Quantum applications of solid state systems base upon generation and control of coherent electronic excitations. Prominent examples are exciton states in semiconductors excitable by photons. The high oscillator strength of electric-dipole (ED) allowed exciton states favors their efficient coherent generation, but limits also their lifetime. ED-forbidden exciton states with long recombination times might maintain long-lived coherence, especially in highly-quality crystals with suppressed exciton scattering. Here, we propose a multi-photon technique combining two-photon excitation with difference frequency generation (2PE-DFG) for time-resolved measurements of exciton coherence. The technique utilizes polarization tomography for state-selective control in both the pump and probe processes. Its potential is demonstrated by measuring the coherent dynamics of the ED-forbidden $S$ and $D$ excitons in Cu$_2$O crystals. The excited states of the Rydberg excitons with principal quantum number $n=2$, $3$, and $4$ have short dephasing times of a few picoseconds, limited by their relaxation to lower lying states. The dephasing time reaches 3 ns for the $1S$ state. In an external magnetic field up to 10 T, the $1S$ exciton splits into a triplet so that quantum beats are observed after coherent excitation, for which three distinct regimes are found depending on the chosen polarization tomography scheme. These results establish the 2PE-DFG technique as a powerful tool to assess the coherent dynamics of ED-forbidden excitons.