apart from additional single-particle corrections to the local fields. The height of the two peaks of at aL and for Rb-Rb (green line in Fig. 3(d)) and of at for Rb–Cs (yellow line in Fig. 3(d)) can be tuned by varying the Rabi frequencies and detunings of the dressing laser. In particular, we choose Rabi frequencies Ω1=Ω2=2π × 35 MHz, ΩC=2π × 30 MHz and detunings Δ1=−2π × 618 MHz, Δ2=−2π × 373 MHz and ΔC=2π × 175 MHz which leads to light-shifts of for Rb–Rb and for Rb–Cs (Supplementary Note 2). We note that an external magnetic field and small vertical offset of the Cs atoms are used to obtain the final interaction strengths and precisely align the potential extrema with plaquette distances (Supplementary Note 2).