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

TL;DR

The paper develops a two-dimensional model for a diamagnetic nanodiamond in an inverted harmonic potential to enable macroscopic spatial superpositions via a five-stage Stern-Gerlach interferometer. It integrates translational motion in the x–y plane with rotational dynamics (libration, precession, rotation) and realistic bias magnetic fields, deriving 2D equations of motion for both HO and IHO regimes and for the rotational degrees of freedom. Key findings show that bias fields alter classical trajectories without changing wave-packet widths, libration remains harmonic due to gyroscopic stabilization from an initial rotation, and a y-direction trap stabilizes rotational motion, enabling trajectory closure within 0.3 s and a spatial separation around $\Delta x \sim 10\,\mu\mathrm{m}$ for $m \sim 10^{-15}$ kg. The framework provides a pathway to probe quantum features of gravity and spin–motion entanglement at mesoscopic scales, while highlighting practical constraints such as decoherence and gradient-induced contrast loss, and suggests avenues for experimental realization with realistic magnetic-field designs.

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.

Figures (8)

• Figure 1: (a) Illustration of the SGI protocol scheme (inspired by Zhou:2024vojBraccini2024ExponentialCats) in five stages (not in the real scale) corresponding to the time transitions $t_1 - t_5$ of two paths: $\ket{-1}\rightarrow\ket{0} \rightarrow\ket{-1} \rightarrow\ket{0} \rightarrow\ket{-1}$ and $\ket{+1}\rightarrow\ket{0} \rightarrow\ket{+1} \rightarrow\ket{0} \rightarrow\ket{+1}$. Each stage represents two types of Hamiltonians. The first is the Harmonic Oscillator ($H_l$), generated from a linear magnetic field profile, which governs the separation (stage 1), return (stage 3), where maximum superposition is achieved, and recombination (stage 5) of spin-up (blue line) and spin-down (red line) states. The second is the Inverted Harmonic Oscillator ($H_{nl}$), from a non-linear magnetic field profile, which functions as the enhancement and deceleration mode (stage 4). It should be noted that in this case a magnetic field bias is applied across stages 1 to 5, resulting in asymmetric trajectories of the two SGI arms, unlike the symmetric trajectories reported in Ref. Zhou:2024voj. (b) and (c) show the potential energy of the Harmonic Oscillator and the Inverted Harmonic Oscillator, respectively. The simulation uses NV-diamond mass $m=10^{-15}\,\text{kg}$, with a linear magnetic field bias $B_{0(l)} = 0.001\,\text{T}$ and a non-linear magnetic field bias $B_{0(nl)} = 0.1\,\text{T}$, under gradient field specifications as listed in Table \ref{['tab:magnetic_field_parameters']}.
• Figure 2: The numerical results from the five-stage SGI on the x-axis, where the NV nanodiamond used has a mass of $m=10^{-15}$ kg, using magnetic field parameters according to Table 1, the purple vertical lines represent the time transition of two paths: $\ket{-1}\rightarrow\ket{0} \rightarrow\ket{-1} \rightarrow\ket{0} \rightarrow\ket{-1}$ and $\ket{+1}\rightarrow\ket{0} \rightarrow\ket{+1} \rightarrow\ket{0} \rightarrow\ket{+1}$ with time durations: 0.0044601, 0.11436, 0.115484, 0.225385 and 0.230052 s, respectively in every stage. (a) The blue and red lines represent the Left and Right SGI arms, respectively. As we can see, the presence of a magnetic field bias in the harmonic oscillator stage ($B_{0(l)} = 0.001\,\text{T}$) results in asymmetric trajectories of the two arms (In contrast, if $B_{0(l)} = 0$, the trajectories of both arms become symmetric, as shown in Ref. Zhou:2024voj. This asymmetric trajectories similar with Ref. Marshman2022). Nevertheless, the presence of $B_{0(l)}$ does not affect the size of the superposition. (b) represents the superposition size (purple line) of the two SGI arms with a maximum separation value of $\Delta x \approx 10 \ \mu m$. (c) The green line represents the velocity difference between the two trajectories. The results have decreased significantly compared to Ref. Zhou:2024voj, which has a superposition size of around $50 \, \mu m$. This is because, in this study, the bias value of the non-linear magnetic field (in stages 3 and 4) was reduced to a more realistic value, $B_{0(nl)} \approx 10^{-1}$ T, which results in the SGI time until both arms close being twice as long as in Ref. Zhou:2024voj, approximately $0.23$ s.
• Figure 3: (a) The numerical results of the five-stage SGI in the y-direction with $y_0 = 1.1 \ \mu m$, where the NV nanodiamond used has a mass of $m=10^{-15}$ kg, the magnetic profile parameters according to Table 1. The gray vertical lines represent the time transition of two paths: $\ket{-1}\rightarrow\ket{0} \rightarrow\ket{-1} \rightarrow\ket{0} \rightarrow\ket{-1}$ and $\ket{+1}\rightarrow\ket{0} \rightarrow\ket{+1} \rightarrow\ket{0} \rightarrow\ket{+1}$ with time durations: 0.0044601, 0.11436, 0.115484, 0.225385 and 0.230052, respectively in every stage. the black solid line and dot represent without and with the given trap frequency ($\omega_y = 521$ Hz) from Ref. Elahi:2024dbb. In (b) and (c), the spatial two-dimensional motion diagrams of the nanodiamond are displayed, representing the conditions before and after the application of the trapping frequency along the y‑axis, respectively.
• Figure 4: The numerical solution of libration mode, Eq. (15), uses the initial angle $\beta_0 = 0.01^\text{o}$ and initial angular velocity $\Omega_0 = 2\pi \times 10^4 \ \text{Hz}$. The shape of the nanodiamond is assumed to be a perfect solid sphere with mass $m=10^{-15}$ kg and radius $R \approx 0.408 \ \mu\text{m}$. The position of the NV-spin is off-center by $d=10$ nm with a fixed angle $\alpha'=\pi/6$ from the center of mass of the nanodiamond. The magnetic field profile parameters can be found in Table 1. the purple vertical lines represent the time transition of two paths: $\ket{-1}\rightarrow\ket{0} \rightarrow\ket{-1} \rightarrow\ket{0} \rightarrow\ket{-1}$ and $\ket{+1}\rightarrow\ket{0} \rightarrow\ket{+1} \rightarrow\ket{0} \rightarrow\ket{+1}$ with time durations: 0044601, 0.11436, 0.115484, 0.225385 and 0.23005, respectively in every stage. The libration mode of the nanodiamond with the solid red and blue lines representing the Left and Right arms of the SGI, respectively. (a) represent without frequency trap at y-direction and (b) with frequency trap using specification of $\omega_y = 521$ Hz (Ref. Elahi:2024dbb). One can see that the libration angle is quite stable due to the relatively high mass of the nanodiamond, and both SGI arms become more stable if the trap frequency is included.
• Figure 5: The numerical solution of Eq. (16) - (17) uses the initial angle $\beta_0 = 0.01^\text{o}$ and initial angular velocity $\Omega_0 = 2\pi \times 10^4 \ \text{Hz}$. The shape of the nanodiamond is assumed to be a perfect solid sphere with mass $m=10^{-15}$ kg and radius $R \approx 0.408 \ \mu\text{m}$. The position of the NV-spin is off-center by $d=10$ nm with fix angle $\alpha'=\pi/6$ from the center of mass of the nanodiamond. The magnetic field profile parameters can be found in Table 1. The purple vertical lines represent the time transition of two paths: $\ket{-1}\rightarrow\ket{0} \rightarrow\ket{-1} \rightarrow\ket{0} \rightarrow\ket{-1}$ and $\ket{+1}\rightarrow\ket{0} \rightarrow\ket{+1} \rightarrow\ket{0} \rightarrow\ket{+1}$ with time durations: 0.00892, 0.11882, 0.11994, 0.22984 and 0.24094 s, respectively in every stage. The mismatch (between left and right SGI arms, $\delta q = q_L - q_R, \ q=\{\alpha,\gamma\}$) of the precession angle ($\alpha$) in brown solid line and the rotation angle ($\gamma$) in purple solid line, where both have a symmetrical all over the stages ($\delta\alpha \approx -\delta\gamma$) for both cases, (a) represent without frequency trap at y-direction and (b) with frequency trap using spesification of $\omega_y = 521$ Hz (Ref. Elahi:2024dbb).