  The floating phase, a critical incommensurate phase, has been theoretically
predicted as a potential intermediate phase between crystalline ordered and
disordered phases. In this study, we investigate the different quantum phases
that arise in ladder arrays comprising up to 92 neutral-atom qubits and
experimentally observe the emergence of the quantum floating phase. We analyze
the site-resolved Rydberg state densities and the distribution of state
occurrences. The site-resolved measurement reveals the formation of domain
walls within the commensurate ordered phase, which subsequently proliferate and
give rise to the floating phase with incommensurate quasi-long-range order. By
analyzing the Fourier spectra of the Rydberg density-density correlations, we
observe clear signatures of the incommensurate wave order of the floating
phase. Furthermore, as the experimental system sizes increase, we show that the
wave vectors approach a continuum of values incommensurate with the lattice.
Our work motivates future studies to further explore the nature of
commensurate-incommensurate phase transitions and their non-equilibrium
physics.
