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

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. 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 $\mathrm{\Gamma }$ to be low enough that the final coherence $e^{-\mathrm{\Gamma }\tau }\le 0.1$ (where $\tau$ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: $\delta {\eta }_{\text{IHP}}\mathrm{/}{\eta }_{\text{IHP}}\lesssim {10}^{-13}$, where ${\eta }_{\text{IHP}}$ is the magnetic field curvature that gives rise to the inverted harmonic potential, and $\delta {\eta }_{\text{HP}}\mathrm{/}{\eta }_{\text{HP}}\lesssim {10}^{-6}$, 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.