Self-Kerr Effect across the Yellow Rydberg Series of Excitons in ${\mathrm{Cu}}_2\mathrm{O}$

We investigate the nonlinear refraction induced by Rydberg excitons in ${\mathrm{Cu}}_2\mathrm{O}$. Using a high-precision interferometry imaging technique that spatially resolves the nonlinear phase shift, we observe significant shifts at extremely low laser intensity near each exciton resonance. From this, we derive the nonlinear index ${\mathrm{n}}_2$, present the ${\mathrm{n}}_2$ spectrum for principal quantum numbers $n\ge 5$, and report large ${\mathrm{n}}_2$ values of order ${10}^{-3}{\mathrm{mm}}^2/\mathrm{mW}$. Moreover, we observe a rapid saturation of the Kerr nonlinearity and find that the saturation intensity ${\mathrm{I}}_{\mathrm{sat}}$ decreases as $n^{-7}$. We explain this with the Rydberg blockade mechanism, whereby giant Rydberg interactions limit the exciton density, resulting in a maximum phase shift of 0.5 rad in our setup.

Figure 1

(a) Simplified view of the experimental apparatus. Acousto-optic modulator (AOM), photodiode (PD). (b) Examples of nonlinear phase shifts reconstructed from interferograms (see Supplemental Material [36]): red-detuned (up) and blue-detuned (down) from the $n=10$ resonance. The pattern matches the input intensity profile (shown in [36]). (c) Intensity dependence of the phase shift extracted from the intensity and phase pictures. We fit the data with the saturable function $f(I)=[\alpha I/(1+I/I_{\mathrm{sat}})]$.

Figure 2

(a) Optical density spectrum of the yellow Rydberg series, plotted versus the energy difference between the probe light and the gap $\mathrm{\Delta }E=E_{\mathrm{gap}}-\hslash \omega$. Blue dots: experimental data (incl. error bars), red line: theory. Different powers were used for different states. (b) Color map of the nonlinear phase shift, plotted against $\mathrm{\Delta }E$ and the intensity. (c) Maximum (saturated signal) nonlinear phase shift. Blue dots: experimental data (incl. error bars), red line: theory.

Figure 3

(a) Nonlinear index ${\mathrm{n}}_2$ extracted form the low-power derivative of $\mathrm{\Delta }\varphi (I)$. Blue dots: experimental data (including error bars), red line: theory. Note that here the state $n=14$ is visible. (b) The power law fit (red line) of $I_{\mathrm{sat}}(n)$ (blue dots) indicates an exponent of $-6.9\pm 0.2$.