Joint Estimation of a Two-Phase Spin Rotation beyond Classical Limit

Quantum metrology employs entanglement to enhance measurement precision [V. Giovannetti et al., Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004)SCIEAS0036-807510.1126/science.1104149, V. Giovannetti et al., Advances in quantum metrology, Nat. Photonics 5, 222 (2011)NPAHBY1749-488510.1038/nphoton.2011.35, and L. Pezzè et al., Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018)RMPHAT0034-686110.1103/RevModPhys.90.035005]. The focus and progress so far have primarily centered on estimating a single parameter. In diverse application scenarios, estimation of more than one single parameter is often required. Joint estimation of multiple parameters can benefit from additional advantages for further enhanced precision. Here we report quantum-enhanced estimation of simultaneous spin rotations around two orthogonal axes, making use of spin-nematic squeezing in an atomic Bose-Einstein condensate. Aided by the F=2 atomic ground hyperfine manifold coupled to the nematic-squeezed F=1 states as an auxiliary field through a sequence of microwave (MW) pulses, multiple spin-1 observables are simultaneously measured, reaching an enhancement factor 3.3 to 6.3 decibels (dB) beyond the classical limit over a wide range of rotation angles. Our work realizes the first quantum enhanced multiparameter estimation using entangled massive particles. The techniques developed and the protocols implemented also highlight the application of two-mode squeezed vacuum states in quantum-enhanced sensing of noncommuting spin rotations simultaneously.