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Since the displacements of these masses are much smaller than the initial separation between their centers, the displacement-to-separation ratio is a natural parameter in which the gravitational potential can be expanded. We show that entanglement in such experiments is sensitive to initial relative momentum only when the system evolves into non-Gaussian states, i.e., when the potential is expanded at least up to the cubic term. A pivotal role of force gradient as the dominant contributor to position-momentum correlations is demonstrated. We establish a closed-form expression for the entanglement gain, which shows that the contribution from the cubic term is proportional to momentum and from the quartic term is proportional to momentum squared. From a quantum information perspective, the results find applications as a momentum witness of non-Gaussian entanglement. Our methods are versatile and apply to any number of central interactions expanded to any order.<\/jats:p>","DOI":"10.22331\/q-2023-05-15-1008","type":"journal-article","created":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T13:51:47Z","timestamp":1684158707000},"page":"1008","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Continuous-Variable Entanglement through Central Forces: Application to Gravity between Quantum Masses"],"prefix":"10.22331","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3639-6468","authenticated-orcid":false,"given":"Ankit","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6360-5627","authenticated-orcid":false,"given":"Tanjung","family":"Krisnanda","sequence":"additional","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore"},{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9624-8024","authenticated-orcid":false,"given":"Paramasivan","family":"Arumugam","sequence":"additional","affiliation":[{"name":"Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India"},{"name":"Centre for Photonics and Quantum Communication Technology, Indian Institute of Technology Roorkee, Roorkee 247667, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8490-3156","authenticated-orcid":false,"given":"Tomasz","family":"Paterek","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-308 Gda\u0144sk, Poland"},{"name":"School of Mathematics and Physics, Xiamen University Malaysia, 43900 Sepang, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2023,5,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. 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