On the origin of force sensitivity in tests of quantum gravity with delocalised mechanical systems

Abstract

The detection of the quantum nature of gravity in the low-energy limit hinges on achieving an unprecedented degree of force sensitivity with mechanical systems. Against this background we explore the relationship between the sensitivity of mechanical systems to external forces and properties of the quantum states they are prepared in. We establish that the main determinant of the force sensitivity in pure quantum states is their spatial delocalisation and we link the force sensitivity to the rate at which two mechanical systems become entangled under a quantum force. We exemplify this at the hand of two commonly considered configurations. One that involves gravitationally interacting objects prepared in non-Gaussian states such as Schrödinger-cat states, where the generation of entanglement is typically ascribed to the accumulation of a dynamical phase between components in superposition experiencing varying gravitational potentials. The other prepares particles in Gaussian states that are strongly squeezed in momentum and delocalised in position where entanglement generation is attributed to accelerations. We offer a unified description of these two arrangements using the phase-space representation of the interacting particles, and link their entangling rate to their force sensitivity, showing that both configurations get entangled at the same rate provided that they are equally delocalised in space. Our description in phase space and the established relation between force sensitivity and entanglement sheds light on the intricacies of why the equivalence between these two configurations holds, something that is not always evident in the literature, due to the distinct physical and analytical methods employed to study each of them. Notably, our findings demonstrate that while the conventional computation of entanglement via the dynamical phase remains accurate for systems in Schrödinger-cat states, it may yield erroneous estimations for systems prepared in squeezed cat states.

Keywords:

• Force sensing
• weak forces
• optomechanics
• quantumness of gravity
• gravitationally induced entanglement
• matter-wave interferometry
• macroscopic quantum superpositions

Acknowledgments

We would like to thank Carlo Rovelli and Kirill Streltsov for insightful discussions. With this manuscript, one of us, M.B.P., would like to express gratitude to Professor Sir Peter Knight FRS. Peter not only dedicated 30 years of his service as the Editor of Contemporary Physics but also generously offered his support throughout the same period at critical junctures of M.B.P.'s academic career.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A first-order approximation of the sentiment within the physics community could, perhaps, be gleaned from a recent bet involving Jonathan Oppenheim, Carlo Rovelli and Geoff Penington, who set the odds 5000 to 1 in favour of gravity being quantum [49].

2 To prepare the superposition of the ball, originally, Feynman describes a Stern-Gerlach-like apparatus where spin-1/2 particles are deflected to one of two counters. When a particle hits one of the counters it activates a mechanism that displaces the ball in a direction that is different for each counter. Provided that the mechanism that displaces the ball constitutes a quantum coherent interaction, this would prepare the ball in a superposition of two locations.

3 Notice that squeezing of a quadrature different from $\hat{X}$ or $\hat{P}$ would generate correlations between momentum and position and thus would not lead to a diagonal covariance matrix.

Additional information

Funding

This work was supported by the ERC Synergy grant HyperQ [grant number 856432] and the DFG via QuantERA project Lemaqume [grant number 500314265].

Notes on contributors

Julen S. Pedernales

Julen S. Pedernales earned his Ph.D. in theoretical physics at the University of the Basque Country in 2016. Following his doctoral studies, he joined the Institute of Theoretical Physics at Ulm University as an Alexander von Humboldt Fellow. Today, he remains at Ulm University as a senior postdoc under the guidance of Martin B. Plenio. His research spans a wide spectrum within the realms of quantum optics and the quantum control of quantum platforms, with applications in quantum simulation, quantum information, and quantum metrology. His most recent research interests are centred on leveraging massive quantum mechanical systems to explore the intricate realms of gravitational quantum mechanics.

Martin B. Plenio

Martin B. Plenio earned his Ph.D. in Physics from Georg-August-Universität Göttingen in 1994. In 1995, he joined the group of Professor Sir Peter Knight FRS with a Feodor-Lynen Fellowship from the Alexander von Humboldt Foundation. He remained at Imperial College for nearly 15 years, rising to the position of Full Professor in 2003. In 2009, he was honoured with an Alexander von Humboldt Professorship and relocated to Ulm University to assume the role of Director of the Institute of Theoretical Physics. He also held a part-time Professorship at Imperial College until 2015. Martin was awarded successive ERC Synergy grants in 2012 and 2019. In 2019, he founded the Center for Quantum BioSciences at Ulm University. He is known for his contributions to quantum information theory and its practical applications, as well as his work at the intersection of quantum technologies and the life sciences. This research has led to the development of quantum sensing and nuclear spin hyperpolarization technologies, along with foundational contributions to the field of quantum biology. Moreover, he has successfully translated his research into the real world through his involvement as a co-founder of two companies: NVision Imaging Technologies, which specialises in hyperpolarization technologies enabling metabolic MRI, and QCDesign, focussed on the development of scalable fault-tolerant architectures for quantum computers.