Magnetic noise in macroscopic quantum spatial superposition induced by inverted harmonic oscillator potential

Abstract: We investigate a Stern-Gerlach type matter-wave interferometer where an inhomogeneous magnetic field couples to an embedded spin in a nanoparticle to create spatial superpositions. Employing a sequence of harmonic and inverted harmonic oscillator potentials created by external magnetic fields, we aim to enhance the one-dimensional superposition of a nanodiamond with mass $\sim 10^{-15}$ kg to $\sim 1 \mu$m. Such matter-wave interferometers have diverse applications, including entanglement of adjacent interferometers mediated by Standard Model and beyond-Standard Model fields, precision tests of the equivalence principle, quantum sensing, and laboratory probes of the quantum nature of spacetime. However, random fluctuations of the magnetic field stochastically perturbs the interferometer paths and induce dephasing. We quantitatively estimate the susceptibility of the interferometer to white noise arising from magnetic-field fluctuations. Constraining the dephasing rate $\Gamma$ to be low enough that the final coherence $e^{-\Gamma \tau}\leq 0.1$ (where $\tau$ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: $\delta \eta_\text{IHP}/\eta_\text{IHP}\lesssim 10^{-13}$, where $\eta_\text{IHP}$ is the magnetic field curvature that gives rise to the inverted harmonic potential, and $\delta \eta_\text{HP}/\eta_\text{HP}\lesssim 10^{-6}$, where $\eta_\text{HP}$ is the linear magnetic field gradient that gives rise to the harmonic potential. For such tiny fluctuations, we demonstrate that the Humpty-Dumpty problem arising from a mismatch in position and momentum does not cause a loss in contrast of the interferometer. Further, we show that constraining the dephasing rate leads to stricter bounds on the noise parameters than enforcing a contrast threshold, indicating that good dephasing control ensures high interferometric contrast.