Recent experiments demonstrate that rapidly rotating Bose-Einstein condensates (BECs) near the lowest Landau level can self-organize into interaction-driven persistent quantum Hall droplet arrays. Inspired by this discovery, we investigate the formation and dynamics of single quantum Hall droplet and droplet arrays in rapidly rotating BECs. Guided by a rigorous theorem on localized many-body states for two-dimensional interacting systems in a magnetic field, we construct single quantum Hall droplet and droplet array states which are shown to be stationary solutions to the Gross-Pitaevskii equation in the rotating frame. The single quantum Hall droplet is shown to be dynamically stable, which underpins its role as the basic unit in a droplet array. The stability of the quantum Hall droplet arrays is demonstrated by their dynamic formation from a phase engineered initial condensate. Our study sheds light onto the nature of the quantum Hall droplet state in a rapidly rotating BEC and offers a new approach for generating and manipulating quantum Hall droplet arrays through designing the initial condensate phase.

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• Bose-Einstein condensates
• Cold atoms & matter waves
• Synthetic gauge fields