Continuous-Variable Entanglement through Central Forces: Application to Gravity between Quantum Masses

Ankit Kumar1, Tanjung Krisnanda2,3, Paramasivan Arumugam1,4, and Tomasz Paterek5,6

1Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India
2School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
3Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
4Centre for Photonics and Quantum Communication Technology, Indian Institute of Technology Roorkee, Roorkee 247667, India
5Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-308 Gdańsk, Poland
6School of Mathematics and Physics, Xiamen University Malaysia, 43900 Sepang, Malaysia

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We describe a complete method for a precise study of gravitational interaction between two nearby quantum masses. 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.

The observation of entanglement between two massive objects is one of the most straightforward scenarios where quantum properties of gravity could be revealed. The origin of entanglement is depicted in the figure above, where Gaussian wave functions describe the particles. Since gravity decays with the distance, the parts of the two wave functions closer to each other are attracted more than the parts further apart. A moment later, higher momentum is developed at positions where the particles are closer, and these position-momentum correlations give rise to entanglement. On this basis, it is expected that the force gradient plays a dominant role in entanglement dynamics. We show this explicitly by developing methods to systematically treat arbitrary-order terms in the traditional expansion of gravitational potential. Among others, they additionally show that the entanglement is insensitive to the relative motion under the typical second-order approximation, whereas it is proportional to powers of relative momentum for higher-order terms. Any order can be treated by our methods, and they work for any number of central interactions.

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