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)-centered nanodiamond evolving under an effective harmonic-oscillator dynamics in a magnetic-field gradient. We assume that the Stern-Gerlach interferometer is subjected to an acceleration $\vec{a}$ external to the system, which is at an angle $\theta_0$ with respect to the direction of the superposition. For a one loop interferometer, we quantify dephasing from two noise channels: fluctuations in the external acceleration $\delta a(t)$ and fluctuations in the tilt angle $\delta \theta(t)$. At the level of the action, we treat these two external noise as stochastic inputs, compute the resulting stochastic phase difference between the interferometer arms, and obtain the dephasing rate $\Gamma$. We obtain the transfer function of the interferometer for each of the noise sources. We also show explicit results of constraints on the power spectral density of the noise sources for an interferometer that produces a superposition size of $\Delta x\sim 1$nm of a nanodiamond of mass $m=10^{-15}~\mathrm{kg}$ by considering white noise statistics and imposing a coherence target $\Gamma \tau\leq 1$, where $\tau\simeq 0.015~\mathrm{s}$. We find $\sqrt{\mathcal{S}_{aa}}\lesssim \mathcal{O}(10^{-11})~\mathrm{m\,s^{-2}\,Hz^{-1/2}}$ if we take the external acceleration, $a=0~{\rm ms^{-2}}$ and $\theta_0=0^\circ$ (along the direction 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 $\theta_0=90^\circ$ (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.