Fig. 2: Histogram of the measured bitstring occurrence frequency.

Bottom row: In each of the \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\), floating, and \({{\mathbb{Z}}}_{4}\) phases, the histograms of many-body state occurrence frequency are shown. Top row shows fluorescence images of the \(\left\vert g\right\rangle\) and \(\left\vert r\right\rangle\) bitstrings with the largest occurrence for (b, c, d), and a typical bitstring for (a), where atoms excited to \(\left\vert r\right\rangle\) are detected as atom loss and marked with red circles. The experimental parameters for these phases are along the black line shown in Fig. 1b with Δ/Ω = 3.5. The atoms are arranged in a (2 × 21 + 2) ladder array for all phases with L = 21. a The disordered phase at Δ/Ω = 0, R_b/a = 2.4 is sampled 573 times, with each measured bitstring occurring only once. b The \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) order at R_b/a = 2.1 exhibits a perfect \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) state occurring 24 times out of 588 samples, significantly more frequent than all other states. Blue boxes encircle the \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) unit cells. c The floating phase at R_b/a = 2.35 is measured 600 times, with \({{\mathbb{Z}}}_{4}\) (orange boxes) and \({{\mathbb{Z}}}_{3}\times {{\mathbb{Z}}}_{2}\) (blue boxes) cells appearing alternatively in the most frequent state. Right arrows point to the domain walls where the \({{\mathbb{Z}}}_{4}\) domain changes sublattices (the index i modulo 4 changes for the rungs containing Rydberg states). d The \({{\mathbb{Z}}}_{4}\) phase at R_b/a = 2.7 is sampled 599 times, with perfect \({{\mathbb{Z}}}_{4}\) states occurring most frequently at 33 times. Orange boxes encircle the \({{\mathbb{Z}}}_{4}\) unit cells.