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

TL;DR

The study tackles the challenge of generating macroscopic spatial quantum superpositions in a spin-1 NV-centered nanodiamond while mitigating magnetic-noise–induced dephasing. It introduces a five-stage protocol that couples harmonic and inverted-harmonic potentials to accelerate separation and then binds the analysis with a transfer-function framework that relates magnetic-field fluctuations to phase noise. The authors derive stringent bounds on noise amplitudes for the inverted-harmonic-stage curvature and harmonic-stage gradient, finding approximately 10^-13 and 10^-6 as the permissible noise-to-signal ratios, respectively, and a total dephasing rate below about 7 Hz for a ~0.31 s protocol. They also examine Humpty-Dumpty constraints and show that, with controlled trajectory fluctuations and near-closed interferometer paths, high-contrast interference remains feasible, supporting the potential use of such setups in quantum-gravity–motivated tests. Together, these results provide quantitative guidelines for designing macroscopic quantum experiments with NV-diamond systems and inverted-potential dynamics.

Abstract

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 $\sim 1 μ$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 \(Γ\) to be low enough that the final coherence \(e^{-Γτ}\leq 0.1\) (where \(τ\) is the experimental time duration), we obtain the following bounds on the noise to signal ratios: $δη_\text{IHP}/η_\text{IHP}\lesssim 10^{-13}$, where $η_\text{IHP}$ is the magnetic field curvature that gives rise to the inverted harmonic potential, and $δη_\text{HP}/η_\text{HP}\lesssim 10^{-6}$, where $η_\text{HP}$ 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.

Figures (3)

• Figure 1: Spin-dependent trajectories in a five-stage interferometric protocol for a spin-1 particle. This schematic represents the trajectories of the two spin states composing the superposition, in their center of mass frame. The trajectories have been obtained by plotting the EOM obtained in Sec.\ref{['sec.IHP-5stage']} and using the parameter values mentioned in Table.\ref{['tab:stage_params']}. The two arms (blue and orange) correspond to $S_x = \pm 1$ states. Stage 1 (spin-dependent harmonic potential) generates spin-position entanglement and ends at maximum velocity, in contrast with the previous scheme PhysRevA.111.052207 that stopped at maximum spatial separation. This change enables faster and greater separation in Stage 2 (spin-independent inverted harmonic potential); Stage 3 (harmonic potential) reverses motion at peak separation; Stage 4 (inverted harmonic potential) decelerates the arms; and Stage 5 (spin-dependent harmonic potential) recombines them. Red dashed lines indicate stage boundaries. The zoom-in confirms a smooth trajectory even in Stage 3. Field strengths remain below niobium's lower critical field $H_{c1}=135$ mT elahi2024alekseevskiy2025Hudson1971, minimising flux noise in the Nb superconductors used.
• Figure 2: Evolution of the width of a minimum uncertainty Gaussian wavepacket: The blue graph depicts the evolution under an IHP potential: $\eta_I = 1\times 10^6$ Tm$^{-2}$ and $B_{0I}=0.1$ T, and the orange graph depicts evolution under free evolution.
• Figure 3: Comparing Model I and Model II: In Model I, the end of the harmonic stage is when the two spin states have maximum spatial superposition achievable in the harmonic potential. In Model II, the end of the harmonic stage is when the two spin states have maximum velocity, achievable in the harmonic potential. The latter helps give an initial velocity at the beginning of the inverted harmonic potential stage. We see that the second model gives us a superposition size higher than the first model by one order of magnitude, within a similar time duration.