This study investigates the quantum floating phase in Rydberg atom arrays, revealing its emergence in a two-leg ladder of up to 92 neutral-atom qubits. The quantum floating phase is an incommensurate phase that arises between crystalline ordered and disordered phases. The researchers observed the formation of domain walls in the commensurate ordered phase, which proliferate and lead to the floating phase with incommensurate quasi-long-range order. By analyzing the Fourier spectra of Rydberg density-density correlations, they identified clear signatures of the incommensurate wave order of the floating phase. As the system size increases, the wave vectors approach a continuum of values incommensurate with the lattice. The study highlights the importance of quantum phase transitions, particularly commensurate-incommensurate transitions, in condensed matter, atomic, and high energy physics. The floating phase is proposed as an intermediate phase between crystalline ordered and disordered phases. The researchers used a Rydberg atom ladder array to experimentally probe the quantum floating phase, demonstrating its existence through site-resolved measurements of Rydberg densities and correlation functions. The results show that the floating phase exhibits incommensurate wave vectors that continuously depend on physical parameters in the thermodynamic limit. The study also explores the critical behavior of the quantum floating phase, showing that its wave vectors continuously interpolate between crystalline orders with integer values of p. The researchers used numerical simulations and experimental measurements to confirm the existence of the floating phase and its unique properties. The results demonstrate that the quantum floating phase is a distinct phase with incommensurate wave vectors, and its behavior is consistent with theoretical predictions. The study provides insights into the nature of quantum phase transitions and their non-equilibrium physics, with potential applications in quantum simulation and lattice gauge theories.This study investigates the quantum floating phase in Rydberg atom arrays, revealing its emergence in a two-leg ladder of up to 92 neutral-atom qubits. The quantum floating phase is an incommensurate phase that arises between crystalline ordered and disordered phases. The researchers observed the formation of domain walls in the commensurate ordered phase, which proliferate and lead to the floating phase with incommensurate quasi-long-range order. By analyzing the Fourier spectra of Rydberg density-density correlations, they identified clear signatures of the incommensurate wave order of the floating phase. As the system size increases, the wave vectors approach a continuum of values incommensurate with the lattice. The study highlights the importance of quantum phase transitions, particularly commensurate-incommensurate transitions, in condensed matter, atomic, and high energy physics. The floating phase is proposed as an intermediate phase between crystalline ordered and disordered phases. The researchers used a Rydberg atom ladder array to experimentally probe the quantum floating phase, demonstrating its existence through site-resolved measurements of Rydberg densities and correlation functions. The results show that the floating phase exhibits incommensurate wave vectors that continuously depend on physical parameters in the thermodynamic limit. The study also explores the critical behavior of the quantum floating phase, showing that its wave vectors continuously interpolate between crystalline orders with integer values of p. The researchers used numerical simulations and experimental measurements to confirm the existence of the floating phase and its unique properties. The results demonstrate that the quantum floating phase is a distinct phase with incommensurate wave vectors, and its behavior is consistent with theoretical predictions. The study provides insights into the nature of quantum phase transitions and their non-equilibrium physics, with potential applications in quantum simulation and lattice gauge theories.