Diamagnetic microchip traps for levitated nanoparticle entanglement experiments

TL;DR

This work proposes a microfabricated, wire-based diamagnetic trapping platform with an integrated superconducting shield to enable gravity-mediated entanglement experiments (QGEM) between two nanodiamonds. It combines fast, high-gradient cooling traps with a long, flat-direction trap to realize macroscopic spatial superpositions while screening electromagnetic interactions that would otherwise overwhelm gravity. The approach leverages diamagnetic forces, spin control in NV centers, and tailored chip geometry to create sub-μm separations and a shielded environment, aiming to observe gravity-driven entanglement and to explore short-range forces and macroscopic quantum coherence. Appendices provide a finite-size, beyond-point-dipole treatment of the diamagnetic interaction and quantify when dipole approximations hold, establishing a framework for accurate trap dynamics and frequency calculations in realistic nanoparticle regimes.

Abstract

The Quantum Gravity Mediated Entanglement (QGEM) protocol offers a novel method to probe the quantumness of gravitational interactions at non-relativistic scales. This protocol leverages the Stern-Gerlach effect to create $\mathcal{O}(\sim μm)$ spatial superpositions of two nanodiamonds (mass $\sim 10^{-15}$ kg) with NV spins, which are then allowed to interact and become entangled solely through the gravitational interaction. Since electromagnetic interactions such as Casimir-Polder and dipole-dipole interactions dominate at this scale, screening them to ensure the masses interact exclusively via gravity is crucial. In this paper, we propose using magnetic traps based on micro-fabricated wires, which provide strong gradients with relatively modest magnetic fields to trap nanoparticles for interferometric entanglement experiments. The design consists of a small trap to cool the center-of-mass motion of the nanodiamonds and a long trap with a weak direction suitable for creating macroscopic superpositions. In contrast to permanent-magnet-based long traps, the micro-fabricated wire-based approach allows fast switching of the magnetic trapping and state manipulation potentials and permits integrated superconducting shielding, which can screen both electrostatic and magnetic interactions between nanodiamonds in a gravitational entanglement experiment. The setup also provides a possible platform for other tests of quantum coherence in macroscopic systems and searches for novel short-range forces.

Figures (9)

• Figure 1: A schematic drawing of the two diamagnetically trapped nanodiamonds on either side of a microfabricated chip, which also functions as an electromagnetic screen, e.g. as studied in Ref. Schut2024. Electric current passing through wires on the chip surface generates high-gradient magnetic fields that can be used in combination with external bias fields to diamagnetically trap nanodiamonds within distances of order $\sim 10$$\,µm$ from the surface. The rainbow-colored shapes represent the trapping potentials. The shape of the potential will differ slightly depending on whether wires or permanent magnets create it. In the case of wires, the setup can also be rotated horizontally; this schematic is purely for illustrative purposes. Although the traps are illustrated to be flat in the $\hat{x}$-direction, there can be some confinement based on the exact profile. e.g., a pulsed magnetic field that can be used to create spatial superpositions from the embedded spin superposition in the diamond NVC via the Stern-Gerlach effect can produce confinement in the $\hat{x}$-direction.
• Figure 2: A comparison of the potentials given in eqs. \ref{['eq:casimir_SS']}-\ref{['eq:grav_SS']}. The lines show the Casimir-Polder (CP), electric dipole-dipole (DD), magnetic dipole-dipole (MM), and gravitational (GR) interactions between two spheres with center-of-mass separation $r$. The test masses are considered perfect diamond nanospheres ($R=500\,nm$, $\rho = 3513 \,kg\per m\cubed$, so $m\sim10^{-15}\,kg$) with a electric dipole moment of $0.1e\, µm$and a magnetic dipole moment with magnitude $\sim 10^{-18}\,J\per T$ (the induced dipole based on a $72\,mT$ field). The vertical grey line indicates a difference of approximately five orders of magnitude between the GR and MM interactions at a separation of $36\,µm$.
• Figure 3: Chip-based magnetic trap with micro-fabricated wires. The chip has three trap configurations, each made with wires of width w=5 $\,µm$. (1) Long Z-wire with dimensions $L= 5$ mm and $L_{z_1}=1$ mm. (2) Cooling z-wire with same $L$ and $L_{z_2}$= 30$\,µm$ and (3) Cooling u-wire with same dimensions as 2). Total thickness of the chip, including Nb thin film and Silicon Nitride substrate, "d"$\sim$ 20$\,µm$ (Origin of the coordinate system is defined w.r.t. the center of the Superconductor). The particle will be cooled to its motional ground state in the stiff trap with higher frequencies. Afterwards, this trap will be switched off, and the particle will be free to move in the flat direction of the long trap, where its motional wavepacket can expand, and the Stern Gerlach pulses can be applied to create spatial superpositions.
• Figure 4: A comparison of the potentials given in eqs. \ref{['eq:dip_pot']}-\ref{['eq:magdip_pot']} and eq. \ref{['eq:grav_SS']}. The lines show the Casimir-Polder (CP), electric dipole-dipole (DD), magnetic dipole-dipole (MM) and gravitational (GR) interactions between the sphere and the plate separated a distance $z_0$ (between center of mass). The test masses are considered perfect diamond nanospheres ($R=500\,nm$, $\rho = 3513 \,kg\per m\cubed$, so $m\sim10^{-15}\,kg$) with a electric dipole moment of $0.1e\, µm$ and a magnetic dipole moment of $\sim 10^{-18}\,J\per T$ (the induced dipole based on a $72\,mT$ field). The chip is taken to have $L=5\,mm$, and to consist of two Silicon Nitride substrates of thickness $9.5\,µm$ that squeeze a $1 \, µm$ superconducting Niobium film. The dotted gray line indicates $z_0 = 18\,µm$.
• Figure 5: Magnetic Field Profile of the Z-trap: By applying a uniform magnetic bias field perpendicular to a long current-carrying wire (parallel to the SC screen), the field above the wire cancels at the center line of a long magnetic trap. It takes the form of a 2-D quadrupole field. For a chip-based magnetic trap, the magnetic trapping gradient perpendicular to the trap center line is set by the wire current $I$ and applied bias field $B_{\rm bias}$ parallel to the chip surface. The plots show the total Magnetic Field Profile (|B|) along the longitudinal(x) and transverse for the long Z trap (1mm) with current 12A and bias fields 250 mT, including the perturbation from the Meissner Image (solid red line). Height of the particle from the screen shifts from 19.6$\,µm$ to 17 $\,µm$ due to this effect. The particles are free to move along the x-axis and tightly confined along the other two directions.