Fig. 1: Quantum phases of Rydberg atoms arranged in a two-leg ladder.

a Atoms are loaded into a two-leg ladder of optical tweezer traps generated using a spatial light modulator (SLM) and rearranged into defect-free patterns by a second set of moving tweezers using a pair of crossed acousto-optical deflectors (AODs). Coherent transitions are driven between the ground state \(\vert g\rangle=\vert 5{S}_{1/2}\rangle\) and the Rydberg state \(\vert r\rangle=\vert 70{S}_{1/2}\rangle\) in each atom with a two-photon transition induced by lasers at 420 nm and 1013 nm. The inset shows a linear detuning sweep Δ(t) at a constant Rabi frequency \({\Omega }_{\max }=2\pi \times 2.5\,\,{\mbox{MHz}}\,\) for preparing the ground states of the phase diagram via adiabatic evolution. Projection of the many-body quantum state into bitstrings of \(\left\vert g\right\rangle\) and \(\left\vert r\right\rangle\) for each atom can be detected on an Electron Multiplying Charge-Coupled Device (EMCCD) camera with the Rydberg state \(\left\vert r\right\rangle\) detected as loss of atom. b The ground-state phase diagram for the Rydberg Hamiltonian [Eq. (1)] in a two-leg ladder is shown with lattice spacings a_x = a and a_y = 2a. Structure factors S(k) are numerically computed for 1 ≤ R_b/a ≤ 3.5 using DMRG (Supplementary Note 2). The color map depicts the peak height S(k_o) at k_o = 2π/p with p being the wavelength in units of the lattice constant a, while contour lines show the constant-p lines. The \({{\mathbb{Z}}}_{p}\) orders have constant values of integer p, while the floating phase exhibits a continuously varying p. The black line cut corresponds to experimental parameters chosen in subsequent figures. c Experimentally measured Rydberg densities illustrate the \({{\mathbb{Z}}}_{p}\) orders and the floating phase. The radius of the dashed circle illustrates the Rydberg blockade radius R_b. The \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) and \({{\mathbb{Z}}}_{4}\) orders exhibit Rydberg density oscillations with periods of p = 3 and p = 4 lattice spacings, respectively. The incommensurate floating phase displays no discernible periodicity in density oscillations.