Single-Spin Magnetomechanics with Levitated Micromagnets

We demonstrate a new mechanical transduction platform for individual spin qubits. In our approach, single micromagnets are trapped using a type-II superconductor in proximity of spin qubits, enabling direct magnetic coupling between the two systems. Controlling the distance between the magnet and the superconductor during cooldown, we demonstrate three-dimensional trapping with quality factors around $1\times {10}^6$ and kHz trapping frequencies. We further exploit the large magnetic moment to mass ratio of this mechanical oscillator to couple its motion to the spin degrees of freedom of an individual nitrogen vacancy center in diamond. Our approach provides a new path towards interfacing individual spin qubits with mechanical motion for testing quantum mechanics with mesoscopic objects, realization of quantum networks, and ultrasensitive metrology.

Figure 1

Concept of the experiment. (a) Individual microscopic hard ferromagnets are isolated in microfabricated pockets and levitated with a type II superconductor, exploiting its flux trapping properties. The magnet’s stray field allows for efficient coupling between the magnet’s motional degrees of freedom and bulk nitrogen vacancy centers in the nearby diamond, which generally feature long spin coherence times. (b) The coupling $\lambda$ depends on the relative orientation of the NV center ${\mathbf{n}}_s$, the magnetic moment ${\mathbf{n}}_{\mu }$, the direction of motion of the magnet ${\mathbf{n}}_m$, the distance between the magnet and the NV center $r^{\prime }$, the magnet radius $a$, and the frequencies of the motion.

Figure 2

Mechanical properties of levitated micromagnets. (a) To levitate the magnets, we adjust the magnet-sc distance above the critical temperature $T_c$. Then we cool the sc below $T_c$ to freeze in the magnetic flux from the magnet. After cooldown, we reduce the distance until the magnet begins to levitate. (b) We observe the magnet motion through a microscope objective and record its motion with a video camera. (c) Power spectral density extracted from video analysis of the magnet motion in the vertical (horizontal) direction, corresponding to the data shown as circles in (e). The inset shows a typical ringdown measurement of the $y$ mode (${\omega }_y=0.839\mathrm{kHz}$) from which we extract the $Q$ factor. (d) Frequencies of center-of-mass motion as a function of the levitation height normalized to magnet radius. Dashed (solid) lines show the three center-of-mass modes for magnet 1 (2). (e) $Q$ factor as a function of trap frequency for magnet 2. Each symbol corresponds to a different levitation experiment and the different colors correspond to the three different center-of-mass modes.

Figure 3

Measurement of spin-motional coupling. (a) Schematic of the experiment. (b) Nitrogen-vacancy ground state spin levels. Resonant microwaves drive transitions between the $m_s=0$ and $m_s=\pm 1$ states. The transition frequency depends on the magnetic field, which depends on the magnet-NV distance. The $m_s=0$ is brighter than the $m_s=\pm 1$ states. Hence, for near-resonant MW driving, the fluorescence intensity depends on the magnet position. (c) ODMR spectrum. The magnet motion is measured by recording a time trace of the fluorescence photons while applying a MW tone at the steepest slope of the transition (2.918 GHz) (solid red line). We measure the slope by frequency modulating the MW tone (red dashed lines). (d) Power spectral density of the video time trace and (f) fluorescence time trace. The gray area corresponds to the numerical integration of the variance $⟨x^2⟩$, $⟨c_{\mathrm{NV}}^2⟩$, and $⟨c_{\mathrm{cal}}^2⟩$. The second sharp peak is the calibration peak due to the MW frequency modulation. The black dashed line represents the photon shot noise. Histograms of the variances reveal the thermal character of the motional state during both the (e) camera and (g) NV measurement which we fit (red line) to extract the variances $⟨x^2⟩$ and $⟨\delta {\omega }_{\mathrm{NV}}^2⟩$, respectively.

Figure 4

Prospect for frequencies and couplings. (a) Mode frequency following dipole model [39] for $h_{\mathrm{lev}}/a=3$. The lowest (solid) lying modes correspond to the center of mass and the highest (dashed) lying modes to rotational motion. Data points are the experimental frequencies for particle 1 Fig. 2. MHz frequencies are predicted for sub-$\mu \mathrm{m}$ particles. (b) Gradient (solid) and dipole-dipole (dashed) couplings, respectively. Straight lines show couplings for constant gap-particle size ratio $\overline{d}=d/a=5.5$ (color) and $\overline{d}=1$ (black). Curved lines (gray) show coupling for constant gap $d=250\mathrm{nm}$ with a maximum coupling at $a=d$. The black data point is the experimental gradient coupling for particle 3.