Magnetic noise in macroscopic quantum spatial superposition induced by an 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 ≈10⁻¹⁵ kg to ≈1 µm. 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 Γ to be low enough that the final coherence e^−Γτ > 0.1 (where τ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: δη_IHP/η_IHP ≾ 10⁻¹³, where ηIHP is the magnetic-field curvature that gives rise to the inverted harmonic potential, and δη_HP/η_HP ≾ 10⁻⁶, where η_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. Furthermore, 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.