a, Angle-resolved white-light reflection spectra taken at 5 K on the second sample. White solid lines are fits using the coupled-oscillator model. Dashed white lines are fitted energies of the uncoupled cavity photon and excitons. b, Power-dependent reflection spectra for the lower polariton (left) and middle polariton (right). c, Shift of polariton energies versus carrier density (logarithmic scale) obtained from b. d, Extracted nonlinear coefficients for lower polariton (red circles) and the calculations using fitted polariton energies (solid line). The error bars on the energy data correspond to the 95% confidence interval of the Lorentzian fit. The error bars of g correspond to the 95% confidence interval of the fit using g(n) = |dE(n)/dn|.