The ability to create and harness entanglement is crucial to the fields of quantum sensing and simulation, and ultracold-atom–cavity systems offer pristine platforms for this undertaking. Here, we present a method for creating and controlling entanglement between solely the motional states of atoms in a cavity without the need for electronic interactions. We show this interaction arises from a general atom-cavity model and discuss the role of the cavity frequency shift in response to atomic motion. This cavity response leads to many different squeezing interactions between the atomic momentum states. Furthermore, we show that when the atoms form a density grating, the collective motion leads to one-axis twisting, a many-body energy gap, and metrologically useful entanglement even in the presence of noise. Notably, an experiment recently demonstrated this regime leads to an effective momentum-exchange interaction between atoms in a common cavity mode [Luo et al., Science 384, 551 (2024)]. This system offers a highly tunable, many-body quantum sensor and simulator.

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• Atom interferometry
• Cold atoms & matter waves
• Collective effects in quantum optics
• Quantum optics
• Quantum state engineering
• Quantum cavities