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arXiv:2601.20949 · APS Open Science

Spatial superposition for a two-dimensional matter-wave interferometer in an inverted harmonic potential with gyroscopic rotational stability

Ryan Rizaldy, Tian Zhou, Run Zhou, Anupam Mazumdar

quant-phSubmitted Jan 28, 2026• Announced Jan 30, 2026• 0 citations
v1 · Jan 28, 2026v2 · Mar 2, 2026v3 · Sep 28, 2026
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Abstract

This study presents a mathematical model of the spatial and rotational motion of a nanodiamond in an inverted harmonic potential to create a macroscopic quantum spatial superposition. The model is based on the Stern-Gerlach Interferometer (SGI) scheme, which utilises linear and quadratic magnetic fields to generate a harmonic potential (linear magnetic field) and a non-linear potential (non-linear/quadratic magnetic field). By incorporating two-dimensional dynamics into the model, we provide a more realistic and accurate depiction of nanoparticle dynamics in linear and inverted harmonic potentials and explore the interaction between motion in a two-dimensional plane. Importantly, we derive the equations of motion for the rotational degrees of freedom, i.e. libration, precession, and rotation. The results show that adding a magnetic-field bias term to the magnetic-field profile in the linear stage affects the classical equations of motion but does not affect the width of the wave packet. Moreover, the libration mode always forms a harmonic potential at each stage because the applied initial angular velocity is dominated by the nanoparticle's defect axis, making it more stable in the presence of the trap frequency in the orthogonal direction along the axis that enables the creation of a macroscopic quantum superposition.

Comments
27 pages, 8 figures

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This study presents a mathematical model of the spatial and rotational motion of a nanodiamond in an inverted harmonic potential to create a macroscopic quantum spatial superposition. The model is based on the Stern-Gerlach Interferometer (SGI) scheme, which utilises linear and quadratic magnetic fields to generate a harmonic potential (linear magnetic field) and a non-linear potential (non-linear/quadratic magnetic field). By incorporating two-dimensional dynamics into the model, we provide a more realistic and accurate depiction of nanoparticle dynamics in linear and inverted harmonic potentials and explore the interaction between motion in a two-dimensional plane. Importantly, we derive the equations of motion for the rotational degrees of freedom, i.e. libration, precession, and rotation. The results show that adding a magnetic-field bias term to the magnetic-field profile in the linear stage affects the classical equations of motion but does not affect the width of the wave packet. Moreover, the libration mode always forms a harmonic potential at each stage because the applied initial angular velocity is dominated by the nanoparticle's defect axis, making it more stable in the presence of the trap frequency in the orthogonal direction along the axis that enables the creation of a macroscopic quantum superposition.
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.
Creating macroscopic spatial quantum superposition with a nanoparticle has a multitude of applications, ranging from testing the foundations of quantum mechanics, matter-wave interferometer for detecting gravitational waves and probing the electromagnetic vacuum, dark matter detection and quantum sensors to testing the quantum nature of gravity in a lab. In this paper, we investigate the role of rotation in a matter-wave interferometer, where we show that imparting angular momentum along the direction of a defect, such as one present in the nitrogen-vacancy centre of a nanodiamond can cause an enhancement in spin contrast for a wide-ranging value of the angular momentum, e.g. 103−10610^{3}-10^{6}103−106~Hz for a mass of order 10−14−10−1710^{-14}-10^{-17}10−14−10−17 Kg nanodiamond. Furthermore, the imparted angular momentum can enhance the spatial superposition by almost a factor of two and possibly average out any potential permanent dipoles in the nanodiamond.
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.
There is a severe impediment in Stern-Gerlach interferometry with a spin-embedded nanoparticle. This is because particle rotation creates a significant mismatch of the wave function of angular degrees of freedom during the creation and recombination of the spatial superposition, resulting in an inferior visibility contrast. We show that imparting angular momentum along the direction of a defect, such as the nitrogen-vacancy (NV) center, during the Stern-Gerlach interferometry provides excellent gyroscopic quantum stability to a nanodiamond, which opens up a way to maintain significant visibility in the quantum interference. The spin contrast improves by many folds for a wide-ranging value of the initial angular momentum, e.g., 10^3–10^6 Hz for a mass of order 10^{-14}–10^{-17} kg nanodiamond. Our scheme is independent of the material and is pertinent to any nanoparticle that can be used to create massive and large spatial superpositions with an embedded spin defect.
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.
The rotation of a nanoparticle in a quantum system has many applications, from theory to experiments. This paper will treat nanoparticle rotational dynamics for spin-embedded nanorotors. We will model it as a rigid body that properly treats the rotation in the coframe of the nanorotor in the presence of external fields. Besides rotation, we will further investigate how to create large spatial superpositions in the inhomogeneous external magnetic field, such as in the case of the Stern-Gerlach apparatus. The spin-embedded nanorotors play a crucial role in creating matter-wave interferometers through their spin and external magnetic field interaction Hamiltonian. We aim to provide a holistic interpretation of the dynamics of three Euler angles, their quantum evolution, and the nanorotor's spatial motion in a Stern-Gerlach-type setup, where we will consider one-full-loop interferometry. We will then study how the quantum evolution of all the Euler angles leads to a spin coherence loss upon interference and what manifests the Einstein-de Haas effect in an external magnetic field. In particular, we show that by imparting rotation along the direction of the magnetic field, we can stabilize the nanorotor's libration mode. We will also extend our analysis to a case where the initial state of the libration mode is thermal and discuss the contrast loss due to interference of the nanorotor upon one-loop completion.