Magnetic levitation and spatial superposition of a nanodiamond with a current-carrying chip

TL;DR

The paper proposes a chip-based platform to generate spatial quantum superpositions of a diamagnetically levitated nanodiamond with an embedded NV center, aiming to test Quantum Gravity-induced Entanglement of Masses (QGEM) in a table-top setting. By combining two integrated quadrupole-field assemblies, the I-Cat Chip achieves strong confinement in $y$ and $z$ while enabling a spin-dependent, one-dimensional separation along $x$ via a bias field $B_0$, exploiting the NV spin–magnetic coupling. Numerical results show that for masses in the range $m\in[10^{-19},10^{-15}]$ kg, a spatial superposition with size up to ${\cal O}(10)\ \mu\mathrm{m}$ can be created within $t\le0.1$ s, with $\Delta x$ scaling approximately as $\Delta x\propto 1/m$. The framework provides a promising route to macroscopic Schrödinger-cat states and QGEM experiments, while acknowledging simplifications (e.g., neglecting rotation, finite-wire effects) and outlining directions for including rotational dynamics, decoherence, and improved loading/cooling in future work.

Abstract

We propose a current-carrying-chip scheme for generating spatial quantum superpositions using a levitating nanodiamond with a built-in nitrogen-vacancy (NV) centre defect. Our setup is quite versatile and we aim to create the superposition for a mass range of $10^{-19}~{\rm kg}< m< 10^{-15}~{\rm kg}$ and a superposition size ${\cal O}(10) {\rm μm} < Δx < {\cal O}(1){\rm nm}$, respectively, in $t\leq 0.1$s, depending on the position we launch from the center of the diamagnetic trap. We provide an in-depth analysis of two parallel chips that can create levitation and spatial superposition along the $x$-axis, while producing a very tight trap in the $y$ direction, and the direction of gravity, i.e., the $z$ direction. Numerical simulations demonstrate that our setup can create a one-dimensional spatial superposition state along the x-axis. Throughout this process, the particle is stably levitated in the z-direction, and its motion is effectively confined in the y-direction for a Gaussian initial condition. This setup presents a viable platform for a diamagnetically levitated nanoparticle for a table-top experiment exploring the possibility of creating a macroscopic Schrödinger Cat state to test the quantum gravity induced entanglement of masses (QGEM) protocol.

Figures (6)

• Figure 1: Conception of the I-Cat chip design. In these diagrams (main, top and front views of the two chips), the black parts are Silicon wafers on which the wires are laid out. There are a total of four blue wires and four yellow wires on the carriers, which are made up of gold. According to their intended purposes, they are referred to here as levitation wires (blue) and separation (wave packet separation) wires (yellow), respectively. The coordinate system of the setup is defined at the geometric center of the entire device, where gravity is pointing in the negative $z$-direction, and the four levitation wires are parallel to the $x$-axis. These four wires are printed pairwise onto two "H"-shaped chips. The horizontal separation (spacing in the $y$-direction) between the two groups of levitation wires is denoted as $2a=18{\rm \mu m}$; the vertical separation (spacing in the $z$-direction) between the two levitation wires on the same chip is denoted as $2b=14 {\rm \mu m}$. Any two adjacent levitation wires carry currents of equal magnitude but opposite directions. A current flowing toward the positive $x$-direction is marked by a dot on the wire cross-section, while a current flowing toward the negative $x$-direction is marked by a cross on the cross-section (see also Tab. \ref{['levtab']} in Appendix). The positively defined (according to the drawn coordinate system) current strength is denoted as $I_{L}=24 {\rm A}$. Similarly, the four separation wires are printed pairwise onto two identical "H"-shaped structures and are parallel to the $z$-axis. The four separation wires form a square in the $x-y$ plane (see the top view in the first row) with a side length of $2L$. Likewise, any two adjacent separation wires carry currents of equal magnitude but opposite directions. The positively defined current strength is denoted as $I=10 {\rm A}$; a current flowing toward the positive $z$-direction is marked by a dot on the wire cross-section, while a current flowing toward the negative $z$-direction is marked by a cross on the cross-section (see also Tab. \ref{['splittab']} in Appendix). We consider that every wire has the same width and thickness, denoted as $w=10~\mu$m.
• Figure 2: Plots (a), (b), (c), and (d) present the results of the magnetic field and its components generated when both the separation assembly and levitation assembly are activated. Panels (a) and (b) display two-dimensional magnetic field magnitude distributions in the $y$-$z$ and $x$-$y$ planes, respectively. In panel (a), the parameters of the levitation wires ($2a=18~\mu\text{m}$, $2b=14~\mu\text{m}$, $w=10~\mu\text{m}$, $I_\text{L}=24~\text{A}$) are consistent with those in Fig. \ref{['figure2']}. Here, we show a slice taken at $x =0$. The result in (a) indicates that the diamagnetic force on the nanodiamond tends to confine the particle near $y = 0$, $z = 0$. Panel (b) illustrates the magnetic field at the slice $z = z_\text{L}$, where the field minimum occurs along the positive $x$-axis. The current magnitude $I=10$ A and the geometric parameter of the separation assembly is $L=200~\mu$m, the extension of four separation wires in the $z$-axis can be interpreted as infinitely long (400 $\mu$m in numerical calculation). This implies that the diamagnetic force acts to push the nanoparticle toward the positive $x$-direction. It should be noted that before activating the separation assembly, the nanodiamond must be initialised at the position $(-x_0, 0, z_\text{L})$. For the purpose of illustration, we take $x_0=-100{\rm \mu m}$ in the panel (b), as shown by $\star$. We propose that this can be achieved using a device analogous to a "Z"-wire, which is not depicted in Fig. \ref{['spdevice']}. Such a "Z"-wire structure would be positioned at the bottom of the setup, with its middle wire oriented perpendicular to the defined $x$-axis and fixed at $x = -x_0$, for details of Z-wire setup, see Elahi:2024dbb. The levitation height during initialisation can be controlled by adjusting the magnitude of $B_0$. A clear linear dependence can be observed in the vicinity of the region around $(0, 0, 0)$.
• Figure 3: Panels (a), (b), and (c) all present results generated by the levitation assembly shown in Fig. \ref{['spdevice']}, displaying the magnetic field magnitude, field components, and levitation height, respectively. The square cross-sections in panel (a) indicate the positions of the four wires of the levitation assembly in the $y$-$z$ plane, with horizontal and vertical separations of $2a=18~\mu\text{m}$ and $2b=14~\mu\text{m}$, respectively. The width and thickness of each wire is $w=10~\mu\text{m}$, resulting in a minimum inter-wire spacing of $4~\mu\text{m}$. The current passing through the levitation assembly is $I_\text{L}=24$ A. As shown in panels (a) and (b), the levitation assembly creates two harmonic potential wells along the $y$- and $z$-directions. When gravitational effects are neglected, the diamagnetic property of the nanodiamond confines the particle to the axis at $y=0$, $z=0$. This axis corresponds to the shaded region (centre) in panel (a), where the magnetic field of the levitation assembly can be approximated by the form given in Eq. (\ref{['Bfield2']}). Panel (c) shows that the particle's weight is balanced by the diamagnetic force, pulling the equilibrium levitation height downward. With our chosen parameters, the equilibrium height is $z_\text{L}\approx -0.0176~\mu\text{m}$.
• Figure 4: The figure shows the relation between geometric parameter $2a$ (see Fig. \ref{['spdevice']} for its definition) and magnetic gradient $\eta_\text{L}$ near the geometric centre of the assembly. The vertical spacing $2b=14~\mu$m is fixed. Curves represented by solid, dashed and dotted lines correspond to different levitation currents $I_\text{L}=24,~18,~12$ A, respectively.
• Figure 5: The figure shows the relation between geometric parameter $2a$ (see Fig. \ref{['spdevice']}) and magnetic field $z$ component experienced by the nanodiamond when levitated.