Quantum Metrology of Newton's Constant with Levitated Mechanical Systems

Abstract

Levitated mechanical systems are emerging as powerful tools in precision sensing and tests of fundamental physics. By trapping small particles in near vacuum using optical or superconducting techniques, these systems provide a highly stable environment that enables force sensitivities at the atto-Newton scale. Their low dissipation, long coherence times, and compatibility with quantum state preparation make them promising candi- dates for probing weak forces at short length scales and for testing the interface between quantum mechanics and gravitation. In particular, they have been proposed for next- generation measurements of Newton’s gravitational constant G, which remains the least precisely known fundamental constant. This thesis investigates how magnetically levitated systems in superconducting traps can be used for quantum-enhanced estimation of G, for studying gravitationally induced entanglement between mesoscopic objects, and for improving our understanding of the magnetic trapping potentials employed in these setups. We present a quantum metrolog- ical scheme in which two magnetically levitated ferromagnetic spheres oscillate harmoni- cally in superconducting traps and interact only via their mutual gravitational potential. From the effective Hamiltonian we show that the gravitational interaction produces a phase shift in the system’s normal modes, giving rise to a mechanical interferometer whose phase encodes information about G. Using tools from Gaussian quantum information, we compute both the quantum and classical Fisher information to quantify the sensitiv- ity with which G can be estimated. We consider projective and continuous general-dyne measurement strategies and include realistic noise sources such as mechanical damping, thermal fluctuations, and Casimir interactions. Under experimentally motivated param- eters, our results indicate that the relative uncertainty δG/G can in principle be reduced by up to four orders of magnitude relative to current CODATA values, with around a two-day measurement time. We additionally analyse entanglement generated by gravitational coupling in this sys- tem using logarithmic negativity as a quantifier, and study how gravitationally induced correlations depend on the initial Gaussian states, damping, temperature, and measure- ment strategy. Upper bounds on achievable entanglement are identified for coherent and squeezed initial states, and these bounds are extended to continuous measurement schemes. This analysis clarifies which regimes of gravitationally mediated entanglement are likely to be experimentally accessible in the near term and informs the feasibility of proposed tabletop probes of quantum aspects of gravity. Finally, we derive an exact analytic expression for the magnetic potential experienced by a point dipole trapped between two infinite superconducting plates by summing an infi- nite series of image dipoles to satisfy the Meissner boundary conditions. The closed-form potential agrees with finite-element simulations in the near-infinite-plate geometry and provides a convenient benchmark for simulating more realistic trap geometries. The ana- lytic potential exhibits a double-well structure that is relevant for orientational tunnelling studies and for trap design in experiments aimed at dark-matter detection, quantum tunnelling, and ultra-sensitive force sensing.