Extended Data Fig. 4: Understanding and controlling polariton spectra with the Floquet scheme.

a, Cavity transmission spectrum in the presence of the modulated atoms (see Extended Data Fig. 3), reproducing Fig. 2b. The spectrum was collected in three parts, corresponding to injection of photons into l = 3 (left, green), l = 6 (middle, black) and l = 9 (right, violet). Dark polaritons in the l = 6 mode are less photon-like than those in the other two modes, reducing their relative transmission (Supplementary Information section A1); we multiply the l = 6 transmission by four to improve visibility. The lower x axis indicates the frequency f of the probe laser relative to the l = 6 dark polariton resonance at f₆. The top x axis indicates the quasifrequency \(\tilde{f}\), proportional to the quasienergy of the polaritons from a treatment using Floquet theory; \(\tilde{f}\) is equal to f modulo the modulation frequency f_mod. The solid curves show three independent fits of the cavity electromagnetically induced transparency spectrum to the measured transmission in each angular momentum mode (see ref. ³², Supplementary equation (7)). b, c, Illustration of the theoretical dependence of the quasifrequencies of the three dark polariton features on the Rydberg beam detuning (b) and the modulation frequency f_mod (c). d, Example transmission spectra for the l = 6 (black, lower), l = 3 (green, middle) and l = 9 (violet, upper) dark polaritons as a function of quasifrequency. The scans are scaled to make their heights equal and have additional vertical offsets for clarity. Shortly before performing each of the experiments reported in the main text, we collect a sequence of plots similar to those displayed here and adjust the Rydberg detuning and modulation frequency to make all three dark polaritons have the same quasifrequency (right-most plot). The only experiments reported in the main text that did not use this sequence are those shown in Fig. 2c, where instead we varied δ_r to intentionally vary the energy mismatch between the polaritons. Throughout this figure, quasifrequencies \(\tilde{f}\) are reported relative to the l = 6 dark polariton resonance \({\tilde{f}}_{6}\). Solid curves provide a guide to the eye.