Constructing and manipulating quantum states in fast-rotating Bose-Einstein condensates (BECs) has long stood as a significant challenge as the rotating speed approached a critical velocity. Although a recent experiment [R. J. Fletcher et al., Science 372, 1318 (2021)] has realized the geometrically squeezed state of the guiding-center mode, the remaining degree of freedom, the cyclotron mode, remains unsqueezed due to the large energy gap of the Landau levels. To overcome this limitation, in this Letter, we propose a Floquet-based state-preparation protocol by periodically driving an anisotropic potential. This protocol not only facilitates single-cyclotron-mode squeezing, but also enables two-mode squeezing. Such two-mode squeezing offers a richer set of dynamics compared to single-mode squeezing and can achieve a wave-packet width well below the lowest-Landau-level limit. Our work provides a highly controllable knob for realizing diverse geometrically squeezed states in ultracold quantum gases within the quantum Hall regime.

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• Bose-Einstein condensates
• Cold atoms & matter waves