Random Acceleration Noise on Stern-Gerlach Interferometry in a Harmonic Trap

We analyze decoherence in a one-loop Stern--Gerlach--type matter-wave interferometer for a massive nanoparticle embedded with a nitrogen vacancy (NV)-centred nanodiamond evolving under an effective harmonic-oscillator dynamics in a magnetic-field gradient. We assume that the Stern-Gerlach interferometer is subjected to a random acceleration noise external to the system. This could be along the direction of the superposition at an angle which can be varied. We quantify dephasing from two noise channels: fluctuations in the external acceleration $a(t)$ magnitude and direction as specified by the tilt angle $\theta_0(t)$ between the superposition axis and the acceleration. At the level of the action, we treat these two external noise as stochastic inputs, and compute the resulting stochastic arm-phase difference, and obtain the dephasing rate $\Gamma$ using the Wiener--Khinchin theorem. For a white noise and a coherence target $\Gamma \tau\leq 1$ and by assuming that we finish the one-loop interferometer within $\tau=2\pi/\omega_0\simeq 0.015~\mathrm{s}$, for a reasonable choice of the magnetic field gradient, $\eta_0=6\times 10^{3}~\mathrm{T\,m^{-1}}$ and mass of the nanodiamond, $m=10^{-15}~\mathrm{kg}$) to create a superposition size of $\Delta x\sim 1$nm. We find $\sqrt{\mathcal{S}_{aa}}\lesssim \mathcal{O}(10^{-11})~\mathrm{m\,s^{-2}\,Hz^{-1/2}}$ even if we take the external acceleration, $a=0~{\rm ms^{-2}}$ and $\theta_0=0^\circ$ (along the dirction of the superposition), and $\sqrt{\mathcal{S}_{\theta\theta}}\lesssim \mathcal{O}(10^{-10})~\mathrm{rad\,Hz^{-1/2}}$ for $a=g= 9.81~\mathrm{m\,s^{-2}}$ and $\theta_0=0^\circ$ (superposition direction is perpendicular to the Earth's gravity). We have also found an operating regime where the acceleration noise can be minimized by either varying $\theta_0$ or $a$ for a fixed set of other experimental parameters.