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arXiv:2602.21288 · APS Open Science

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

Sneha Narasimha Moorthy, Andrew Geraci, Sougato Bose, Anupam Mazumdar

quant-phSubmitted Feb 24, 2026• Announced Feb 26, 2026• 0 citations
v1 · Feb 24, 2026v2 · Aug 12, 2026
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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.

DOI
10.1103/3jjv-vwmv
Comments
13 pages, 7 figures

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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)a(t)a(t) magnitude and direction as specified by the tilt angle θ0(t)θ_0(t)θ0​(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 ΓΓΓ using the Wiener--Khinchin theorem. For a white noise and a coherence target Γτ≤1Γτ\leq 1Γτ≤1 and by assuming that we finish the one-loop interferometer within τ=2π/ω0≃0.015 sτ=2π/ω_0\simeq 0.015~\mathrm{s}τ=2π/ω0​≃0.015 s, for a reasonable choice of the magnetic field gradient, η0=6×103 T m−1η_0=6\times 10^{3}~\mathrm{T\,m^{-1}}η0​=6×103 Tm−1 and mass of the nanodiamond, m=10−15 kgm=10^{-15}~\mathrm{kg}m=10−15 kg) to create a superposition size of Δx∼1Δx\sim 1Δx∼1nm. We find Saa≲O(10−11) m s−2 Hz−1/2\sqrt{\mathcal{S}_{aa}}\lesssim \mathcal{O}(10^{-11})~\mathrm{m\,s^{-2}\,Hz^{-1/2}}Saa​​≲O(10−11) ms−2Hz−1/2 even if we take the external acceleration, a=0 ms−2a=0~{\rm ms^{-2}}a=0 ms−2 and θ0=0∘θ_0=0^\circθ0​=0∘ (along the dirction of the superposition), and Sθθ≲O(10−10) rad Hz−1/2\sqrt{\mathcal{S}_{θθ}}\lesssim \mathcal{O}(10^{-10})~\mathrm{rad\,Hz^{-1/2}}Sθθ​​≲O(10−10) radHz−1/2 for a=g=9.81 m s−2a=g= 9.81~\mathrm{m\,s^{-2}}a=g=9.81 ms−2 and θ0=0∘θ_0=0^\circθ0​=0∘ (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 θ0θ_0θ0​ or aaa for a fixed set of other experimental parameters.
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\sim 10^{-15}∼10−15 kg to ∼1μ\sim 1 μ∼1μm. 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 ΓΓΓ to be low enough that the final coherence e−Γτ≤0.1e^{-Γτ}\leq 0.1e−Γτ≤0.1 (where τττ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: δηIHP/ηIHP≲10−13δη_\text{IHP}/η_\text{IHP}\lesssim 10^{-13}δηIHP​/ηIHP​≲10−13, where ηIHPη_\text{IHP}ηIHP​ is the magnetic field curvature that gives rise to the inverted harmonic potential, and δηHP/ηHP≲10−6δη_\text{HP}/η_\text{HP}\lesssim 10^{-6}δηHP​/ηHP​≲10−6, where ηHPη_\text{HP}ηHP​ 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.
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: δηIHP/ηIHP ≾ 10−13, where ηIHP is the magnetic-field curvature that gives rise to the inverted harmonic potential, and δηHP/ηHP ≾ 10−6, where ηHP 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.
We investigate a two-dimensional matter-wave interferometer where both spatial and rotational dynamics of a nanoparticle are intertwined in closing the one-loop interferometer in the Stern-Gerlach type setup. We consider the spin-contrast of the nitrogen-vacancy (NV) centred nanodiamond in combination with a two-dimensional magnetic field setup to extend the one-dimensional Stern--Gerlach interferometry. We analyse the dynamical motion along with the rigid rotation under the influence of the external magnetic field. Regarding rotation, we incorporate Euler-angle dynamics to analyse the stability of rotational degrees of freedom and their influence on the spin contrast to address the Humpty-Dumpty problem. We show that by imparting external rotation provides the gyroscopic stability to the liberating mode of the NV-spin and hence helps to improve the contrast. Our scheme creates a tiny spatial superposition of size ∼0.21 μm\sim 0.21~{\rm μm}∼0.21 μm for mass m=10−17m=10^{-17}m=10−17kg in less than t∼0.013t\sim 0.013t∼0.013s.
Stern-Gerlach interferometer (SGI) is a kind of matter-wave interferometer driven by magnetic field and has been proposed for various gravity experiments. Stochastic noises can lead to decoherence problems of SGI via various mechanisms. In this paper, I will theoretically study several mechanisms including dephasing, contrast loss, and position localization decoherence. I will first present a rigorous proof that the dephasing behaves as a linear response to the noise, with a transfer function given by the Fourier transform of the unperturbed classical trajectories. Then I will demonstrate that stochastic acceleration noise only induces dephasing to the witness operator constructed in spin space, while it does not lead to contrast loss or position localization decoherence due to common mode cancellation. In contrast, higher-order noise can induce both contrast loss and position localization decoherence, contributing a decay factor proportional to the noise power spectrum density at the intrinsic frequency, which can be physically interpreted as the resonance between the noise and test mass. Based on the result, I apply the framework to analyze two typical noise sources, magnetic field noise and quadratic noise.
Matter-wave interferometry is a striking quantum phenomenon that differs fundamentally from classical mechanics. In the quantum regime, matter behaves like a wave and can exist in spatial superpositions, allowing a single quantum object to move along multiple trajectories simultaneously. Similar to optical interferometers, such quantum matter systems act as interferometers themselves, with the differential phase that encodes information about external forces experienced during their evolution. This makes matter-wave interferometry a powerful platform for quantum sensing across various applications: (i) One of the most direct applications is precision measurement. Atomic interferometers can measure local acceleration and have already been implemented in devices such as accelerometers, seismometers, gravimeters, and gradiometers. (ii) Atomic interferometers are proposed as complementary tools for future gravitational-wave observatories, including ongoing projects MIGA, MAGIS-100, AION and ZAIGA. (iii) Matter-wave interferometry also offers a promising platform for probing quantum aspects of gravity. Numerous tabletop proposals aim to test quantum gravity models, investigate collapse mechanisms, study gravity-induced decoherence, and search for gravity-mediated quantum entanglement. The Stern-Gerlach interferometer (SGI) is a type of matter-wave interferometer that manipulates spatial superpositions using magnetic-field gradients. The witness of an SGI can be designed via its spin, enabling more precise measurements of the differential phase. Recent developments in atomic chip technology have enabled the generation of highly controlled magnetic fields on compact platforms, allowing for precise manipulation of spin states and making SGIs experimentally realizable. Robustness is a critical challenge for matter-wave interferometers. In real experiments, various sources of classical noises arise from mechanical vibrations, seismic motions, inertial forces, tidal forces, magnetic field fluctuations and so on. These noises can disturb the spin and spatial degrees of freedom of the test mass, resulting in witness loss of the interferometer. While spin decoherence is relatively well understood and can be formulated by the Bloch-Redfield master equation, the impact on spatial degrees of freedom still requires systematic investigation. This thesis therefore aims to explore witness loss in an SGI induced by stochastic noises acting on spatial degrees of freedom. As investigated by this thesis, the witness gets lost via several mechanisms: (i) Dephasing. The spatial phase is fluctuated by stochastic noise, resulting in a random phase factor eiδϕ, whose ensemble average contributes a decay factor e−Γ to the spin density matrix and leads to a loss of witness. As is demonstrated by the thesis, this decay factor is a linear response to the external noise, with a transfer function as the Fourier transform of the unperturbed classical trajectories. (ii) Humpty-Dumpty problem. The classical trajectories are also disturbed by stochastic noise, resulting in a non-closure problem of the interferometer, known as the Humpty-Dumpty problem. The resulting witness loss is usually characterized by the contrast of the final states ⟨ψ1|ψ2⟩, which equals the phase-space distance |α1 − α2|2 when the test mass is in coherent states. (iii) Position localization decoherence. Due to the randomness of the noise, the spatial wavefunction of the test mass is thus also stochastic, of which the ensemble average experiences decoherence characterized by the power spectrum density of external noise. However, as illustrated by this thesis, this type of decoherence does not affect spin-based witnesses in most practical scenarios. The general theoretical framework is mainly constructed in chapter 3 of this thesis. The subsequent chapters analyze several representative noise sources: (i) Gravitational and electrostatic noise. Molecules in the surrounding gas move randomly due to thermal motion, generating stochastic gravitational accelerations and electrostatic forces that can be expanded in multipole series. Since the random motion follows Maxwell’s distribution and approximately has a cylindrical symmetry, the resulting noise power spectrum density takes the form of Bessel functions. (ii) Inertial acceleration noise. The mechanical support of the experimental apparatus is always fluctuated by vibrational noise, resulting in random inertial force on the test mass. In practice, dephasing can be mitigated through destructive interference by introducing appropriate coupling between different directional components. (iii) Inertial torsion noise. In suspended experiments, the experimental apparatus is fluctuated by torsional noise, leading to another inertial noise on the test mass. The power spectrum density of the inertial torsion noise is formulated by the selfconvolution of the original noise, resulting in the peak frequencies that are doubled. Taken together, these analyses build a comprehensive picture of how stochastic noise affects the spatial dynamics of an SGI and leads to the loss of spin-based witnesses. Future work may include more realistic noise sources such as magnetic noise and trap-potential noise. These insights will be valuable for future experimental implementations, particularly in precision sensing for quantum gravity, where preserving coherence and maintaining reliable witness signals are essential.
In this paper, we will show how random fluctuations in the magnetic field will jitter the paths of a matter-wave interferometer randomly, hence, decohere the quantum superposition. To create a large spatial superposition with nanoparticles, we envisage embedding a spin in a nanoparticle as a defect and applying an inhomogeneous magnetic field as in a Stern-Gerlach type experiment to create a macroscopic quantum superposition. Such matter-wave interferometers are the cornerstone for many new fundamental advancements in physics; particularly, adjacent matter-wave interferometers can use entanglement features to test physics beyond the Standard Model, test the equivalence principle, improve quantum sensors, and test the quantum nature of spacetime in a lab. In particular, we will use white and flicker noise to study the decoherence and constrain the parameters keeping in mind ambient temperatures suitable for superconducting wires embedded on a chip. We will show that to obtain a tiny spatial superposition of a nanometer separation, Δx∼O(10−9)Δx \sim {\cal O} (10^{-9})Δx∼O(10−9)m and to minimize decoherence, Γ≤O(ω02π)Γ\leq {\cal O}(\frac{ω_0}{2π})Γ≤O(2πω0​​), where ΓΓΓ is the decoherence and ω0ω_0ω0​ is the frequency of the oscillator, we will need current fluctuations to be δI/I≤O(10−8)δI/I\leq {\cal O}(10^{-8})δI/I≤O(10−8), which is not impossible to obtain in superconducting wire arrangements. For such tiny fluctuations, we demonstrate that the Humpty-Dumpty problem in a matter-wave interferometer arising from a mismatch in position and momentum does not cause a loss in contrast.