Squeezed states play a key role in diverse frontiers of quantum physics. Geometrically squeezed states, squeezed states in the orbital phase space of rotating Bose-Einstein condensates (BECs), have been conventionally generated by anisotropic-trapping potentials. In this work, we propose a different route to generate geometric squeezing via spin-orbit coupling (SOC) in a pseudospin-1/2 BEC. We show that the SOC enables effective two-phonon transitions within the lowest Landau level via virtual spin-flip processes, leading to exponential squeezing dynamics in both spin components. Furthermore, by applying a $\pi /2$ spin rotation, the two spin channels can be coherently coupled to produce two-mode geometric squeezing. We also investigate the influence of interatomic interactions on squeezing performance and identify parameters where robust squeezing can be achieved. Our work provides a viable pathway to realize and manipulate geometric squeezing in spinor quantum gases.

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• Cold atoms & matter waves
• Spin-orbit coupling
• Spinor Bose-Einstein condensates
• Squeezing of quantum noise