Extended Data Fig. 3: Scheme for forming the Landau level of Floquet polaritons.

a, The bare cavity modes are not degenerate in this work, but instead the length of the cavity is increased so that there is a f_cav ≈ 70 MHz splitting between every third angular momentum mode. b, To form polaritons in three modes, even though only the l = 6 mode is resonant with the un-modulated 5S_1/2 → 5P_3/2 transition, we utilize the Floquet scheme depicted in Extended Data Fig. 2¹⁸. Modulating the 5P_3/2 state at f_mod ≈ 70 MHz splits it into three bands (grey), each of which is resonant with one of the three chosen cavity modes. The coupling strengths g_l to each mode l are controlled by the modulation amplitude; in this work, g₃ = g₉ = 0.37(4)g₆. Note that each mode couples to a unique collective atomic excitation, as depicted at the top (blue atoms are included in the corresponding collective excitation, while grey atoms are not). c, This scheme produces polaritons in the l = 3, l = 6 and l = 9 modes. The dark polaritons can be made effectively degenerate (see Extended Data Fig. 4) without making the corresponding cavity modes degenerate, which protects the polaritons from intracavity aberrations (see Supplementary Information section B2).