Plasma-Enhanced Interaction and Optical Nonlinearities of ${\mathrm{Cu}}_2\mathrm{O}$ Rydberg Excitons

We theoretically investigate the nonlinear optical transmission through a cuprous oxide crystal for wavelengths that cover the series of highly excited excitons, observed in recent experiments. Since such Rydberg excitons have strong van der Waals interactions, they can dynamically break the conditions for resonant exciton creation and dramatically modify the refractive index of the material in a nonlinear manner. We explore this mechanism theoretically and determine its effects on the optical properties of a semiconductor for the case of degenerate pair-state asymptotes of Rydberg excitons in ${\mathrm{Cu}}_2\mathrm{O}$. Upon analyzing the additional effects of a dilute residual electron-hole plasma, we find quantitative agreement with previous transmission measurements, which provides strong indications for the enhancement of Rydberg-induced nonlinearities by surrounding free charges.

Figure 1

(a) The transmission of a laser field with incident intensity $I_0$ through a ${\mathrm{Cu}}_2\mathrm{O}$ crystal of length $L$ can be strongly affected by the resonant optical response of exciton states $|k⟩$. Hereby, the excitons are generated with light-matter couplings $g_k$, and their spectral properties are determined by the corresponding frequency detunings ${\mathrm{\Delta }}_k$ and decay rates ${\gamma }_k$. A large optical nonlinearity arises from the strong van der Waals interaction between excitons in highly excited Rydberg states, which inhibits the simultaneous creation of excitons with a blockade radius $R_{\mathrm{bl}}$ and thereby profoundly alters their absorptive response to the incident light. (b) The calculated optical nonlinearities based on this mechanism are in quantitative agreement with existing measurements [11], upon including the effects of plasma screening on high lying Rydberg exciton states.

Figure 2

Nonlinear and nonlocal optical susceptibilities ${\chi }^{(3)}$ for resonant excitation of the $15p-15p$-asymptote as a function of the exciton-exciton distance (radial coordinate in $\mu \mathrm{m}$) and angle $\theta$ between the distance vector and the polarization axis of the applied light laser field for linear (a)–(b) and circular polarization (c)–(d). Nonlinear refraction is represented by the real part $\Re ({\chi }^{(3)})$ (a),(c), while the nonlinear absorption coefficient, $\Im ({\chi }^{(3)})$, is shown in panels (b),(d). Symmetry under exciton exchange leads to the depicted fourfold mirror symmetry.

Figure 3

Experimentally measured [11] optical density OD on excitonic resonances with different principal quantum numbers $n$ obtained for a crystal of length $L=34\mu \mathrm{m}$. Reflections off the crystal surface as well as a phonon-assisted background have been accounted for as described in Ref. [25]. The solid lines indicate linear fits used to extract the linear and nonlinear absorption coefficients.

Figure 4

Debye screening by a dilute residual plasma in the semiconductor tends to weaken the electron-hole attraction that binds the Rydberg exciton. (a) This effect enhances the extent of the excitonic wave function, as shown exemplarily for the $20p$ state. The corresponding enhancement factor $\nu =⟨r{⟩}_D/⟨r{⟩}_0$ is shown in panel (b) as a function of the principal Rydberg-state quantum number. The depicted growth of the orbital size of the exciton implies a drop of its oscillator strength. As shown in panel (c), the correspondingly calculated oscillator strength [25], is in excellent agreement with the experimentally observed drop [11] of the light-matter coupling strength $g$, as extracted from the linear absorption coefficient. All calculations have been performed for a Debye screening length ${\lambda }_D=0.4\mu \mathrm{m}$.