Fig. 3. Detuning, fluence, and Fermi energy dependence of Fermi-polaron resonance shifts.

a The optical Stark shift ${\mathrm{\Delta }}_{\mathrm{X}}$ of excitons (black squares) measured at $E_{\mathrm{F},\mathrm{h}}=$0 meV is shown as a function of $F/{\delta }_{\mathrm{X}}$. Gray line represents $F/{\delta }_{\mathrm{X}}$ dependence of ${\mathrm{\Delta }}_{\mathrm{X}}$. Because our measurement is by far off-resonant, higher-order correction including interactions between virtual excitons or polaron-polaron interactions is negligible. b The pump-fluence $F$ dependence of resonance shifts ${\mathrm{\Delta }}_{\mathrm{P}}$ for attractive polarons (red circles) and repulsive polarons (black squares) is shown. The linear dependence is clearly seen for both polarons. The data have been taken at $E_{\mathrm{F},\mathrm{h}}$ of −12.4 meV (dotted line in Fig. 1c) with $\hslash {\omega }_{\mathrm{pump}}$ of 1.55 eV. At the same $F$ (120–360 $\mu \mathrm{J}{\mathrm{cm}}^{-2}$), ${\mathrm{\Delta }}_{\mathrm{P}}$ for the attractive polarons is much larger than that of the repulsive polarons. Due to the released holes from attractive interactions with excitons due to the pump excitation, the additional blueshift occurs in the attractive polarons. c Resonance shifts of attractive polarons (red circles) and repulsive polarons (black squares) at $F$ is 240 $\mu \mathrm{J}{cm}^{-2}$ and ${\delta }_{\mathrm{X}}$ is around $171$ meV are plotted as a function of the Fermi energy: −14.3$<E_{\mathrm{F},\mathrm{h}}<$−3.8 meV. The error bars are obtained from the fits. The red and black dashed lines are theoretical calculations of the spectral function obtained from the interacting Hamiltonian using Chevy ansatz (see the main text and Supplementary Note 7 for more details). Vertical error bars in a–c are obtained from the fits.