Spin dynamical decoupling for generating macroscopic superpositions of a free-falling nanodiamond

Levitated nanodiamonds containing negatively charged nitrogen-vacancy centers $\left({\text{NV}}^{-}\right)$ have been proposed as a platform to generate macroscopic spatial superpositions. Requirements for this include having a long ${\text{NV}}^{-}$ spin coherence time, which necessitates formulating a dynamical decoupling strategy in which the regular spin flips do not cancel the growth of the superposition through the Stern-Gerlach effect in an inhomogeneous magnetic field. Here, we propose a scheme to place a $250\text{−}\mathrm{n}\mathrm{m}$-diameter diamond in a superposition with spatial separation of over $250\mathrm{n}\mathrm{m}$, while incorporating dynamical decoupling. We achieve this by letting a diamond fall for $2.4\mathrm{m}$ through a magnetic structure, including $1.13\mathrm{m}$ in an inhomogeneous region generated by magnetic teeth.

Figure 1

(a) Schematic of the proposed experiment for testing macroscopic superpositions. A nanodiamond should be cryogenically cooled to $5\mathrm{K}$ on a cold surface and then shaken off with ultrasound to be levitated in a magnetic trap or Paul trap in ultrahigh vacuum. The diamond is then dropped through a magnetic field that includes homogeneous and inhomogeneous regions. The diamond lands on a glass slide held at $5\mathrm{K}$. Magnets are shown schematically with their north and south poles in red (dark gray) and blue (light gray), respectively. The schematic is not to scale. (b) Schematic of the paths taken by the two superposition components, $A$ in solid blue (solid light gray), $B$ in solid red (solid dark gray), and dashed purple (dashed gray) for presuperposition. $T$ is the period of oscillation of the separation and $s$ the maximum separation reached.

Figure 2

Finite element simulations, using comsol multiphysics software, of a shorter length of the proposed magnetic tooth structure to produce an alternating inhomogeneous magnetic field, generating a spatial superposition in the $x$ direction. Neodymium magnets are considered, both magnetized in the $+x$ direction. (a) A $y=0$ plane plot of the magnitude of the magnetic field. The flux concentrates in the teeth, producing large field gradients across the gap as the opposing teeth are offset from each other by the tooth width. (b) The gradient of the $x$ component of the magnetic field in $x$ direction, simulated along the $z$ axis of the magnetic geometry. The noise present is due to the finite size of mesh elements in comsol. The gradient of the $y$ and $z$ components of the magnetic field are plotted in Fig. 4 in Appendix pp1.

Figure 3

(a) Schematic showing the coordinate axes for the spatial superposition generation scheme. $x$, $y$ (not shown), and $z$ axes are defined in the reference frame of the magnets. Any tilt to the magnets creates an angle $\varphi$ with the vertical defined by gravity. The orientation of the axis of the ${\text{NV}}^{-}$ center in the falling nanodiamond is given by $z^{\prime }$. Magnets are shown schematically with their north and south poles in red (dark gray) and blue (light gray), respectively. The magnitude of $\varphi$ is exaggerated. (b) An annotated and scaled copy of Fig. 1, depicting the paths taken by the two spatial superposition components, one in solid red (solid dark gray), the other in solid blue (solid light gray), and dashed purple (dashed gray) for presuperposition. $s$ is the maximum separation reached. Key microwave pulses are indicated, but the spin dynamical decoupling spin flips that coincide with crossing magnetic teeth are not shown for clarity. $\mathrm{\Delta }{x}_{\text{eq}}$ indicates the separation of equilibrium positions about which the paths oscillate.

Figure 4

Gradient of the $y$, in green (light gray), and $z$, in black (black), components of the magnetic field as well as the $x$ component, in blue (dark gray) displayed in Fig. 2. Away from edge effects, along the $z$ axis of the magnet geometry, the only field gradient in the $x$ direction is from the $x$ component of the magnetic field. As is the case in Fig. 2, the noise present is due to the finite size of mesh elements in comsol.

Figure 5

Simulations of the paths taken by the two superposition components in the presence of magnetic teeth, as described by the Hamiltonian of Eq. (A1). Along with an extra spin flip at the first recombination of paths, 9800 teeth and the associated spin flips for each tooth crossing are considered with a magnetic field of the form $\mathbf{B}\left(\widehat{\mathbf{r}}\right)=(\pm B^{\prime }\widehat{x}+B_0){\mathbf{e}}_x$, where $B^{\prime }=940\mathrm{T}{\mathrm{m}}^{-1}$ and $B_0=420\mathrm{m}\mathrm{T}$. (a), (b) Paths taken in a geometry where all teeth are the same width. Here, $x$ is the absolute position along the $x$ axis shown in Fig. 3. The diamagnetic oscillation of both superposition components is clear. (b) Despite the shared diamagnetic oscillation, the magnitude of the separation between the two components is as described analytically by the Hamiltonian of Eq. (A3) [19]. Here, “Sim.” labels the simulated separation and “Ana.” the analytical. (c), (d) Paths taken in a geometry where the first tooth is half the width of the rest. Here, $\left|\mathrm{\Delta }x\right|$ is the absolute value of the separation along the $x$ axis between superposition components $A$ and $B$. The shared diamagnetic oscillation is well canceled, without changing the relative motion of the two components.