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

Abstract

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 $θ_0(t)$ 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 $Γ$ using the Wiener--Khinchin theorem. For a white noise and a coherence target $Γτ\leq 1$ and by assuming that we finish the one-loop interferometer within $τ=2π/ω_0\simeq 0.015~\mathrm{s}$, for a reasonable choice of the magnetic field gradient, $η_0=6\times 10^{3}~\mathrm{T\,m^{-1}}$ and mass of the nanodiamond, $m=10^{-15}~\mathrm{kg}$) to create a superposition size of $Δx\sim 1$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 $θ_0=0^\circ$ (along the dirction of the superposition), and $\sqrt{\mathcal{S}_{θθ}}\lesssim \mathcal{O}(10^{-10})~\mathrm{rad\,Hz^{-1/2}}$ for $a=g= 9.81~\mathrm{m\,s^{-2}}$ and $θ_0=0^\circ$ (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 $θ_0$ or $a$ for a fixed set of other experimental parameters.

Figures (7)

• Figure 1: Schematic of a nanoparticle interferometer in a tilted spin-dependent harmonic potential. The magnetic-field gradient produces spin-dependent harmonic oscillator potentials. External random acceleration is acting at an angle $\theta_0$ to the axis of spatial superposition. The gaussian wavepackets represent the states constituting the two interfermeter arms.
• Figure 2: The graph depicts the variation of the maximum superposition size achievable in a harmonic potential with variation of the mass of the nanoparticle in a Stern-Gerlach-type interferometer for a set of parameters given by: $~\eta_0=6\times 10^{3}{\rm Tm^{-1}}, ~m=10^{-15}{\rm kg}$.
• Figure 3: The plot shows the frequency response of the interferometer to acceleration fluctuations. The horizontal axis is the dimensionless quantity $\xi = \omega/\omega_0$, where $\omega$ is the angular frequency of the noise source and $\omega_0$ is the characteristic angular frequency of the harmonic oscillator. The vertical axis shows $f_{aa}(\xi)$, the frequency-dependent weighting factor appearing in the acceleration-noise transfer function eq.\ref{['eq.eff_transfun_a']}. Larger $f_{aa}(\xi)$ indicates greater sensitivity to perturbations at that normalised frequency.
• Figure 4: The contour plots map the dephasing rate $\Gamma$ as a function of the white noise amplitude spectral density $\sqrt{\mathcal{S}_{aa}}$ and the external acceleration magnitude $a$ with all other parameters fixed to the values: $\theta_0 = 0^\circ, ~\eta_0=6\times 10^{3}{\rm Tm^{-1}}, m=10^{-15}{\rm kg},~ \tau=2\pi/\omega_0\simeq0.015$s, which yields $\Delta x=1$nm. The region below the black dashed contour corresponds to the experimentally desirable operating regime satisfying $\Gamma \tau < 1$. The graph shows that the behaviour of the dephasing rate and the tolerable square root of PSD, $\sqrt{\mathcal{S}_{aa}}$ is finite and smooth at $a_m\sim 0.03 \ \rm m s^{-2}$.
• Figure 5: The contour plot maps the dephasing rate $\Gamma$ as a function of the white noise amplitude spectral density $\sqrt{\mathcal{S}_{aa}}$ and the tilt angle $\theta_0$ with all other parameters fixed to the values: (no external acceleration) $a = 0 {\rm ms^{-2}}$, $\eta_0=6\times 10^{3}{\rm Tm^{-1}}$, $m=10^{-15}{\rm kg}$, $\Delta x=1$nm, $\tau=2\pi/\omega_0\simeq0.015$s. The region below the black dashed contour corresponds to the experimentally desirable operating regime satisfying $\Gamma \tau < 1$.