Two-dimensional matter-wave interferometer, rotational dynamics, and spin contrast

Abstract

We investigate a two-dimensional matter-wave interferometer where both spatial and rotational dynamics of a nanoparticle are intertwined in closing the one-loop interferometer in the Stern-Gerlach type setup. We consider the spin-contrast of the nitrogen-vacancy (NV) centred nanodiamond in combination with a two-dimensional magnetic field setup to extend the one-dimensional Stern--Gerlach interferometry. We analyse the dynamical motion along with the rigid rotation under the influence of the external magnetic field. Regarding rotation, we incorporate Euler-angle dynamics to analyse the stability of rotational degrees of freedom and their influence on the spin contrast to address the Humpty-Dumpty problem. We show that by imparting external rotation provides the gyroscopic stability to the liberating mode of the NV-spin and hence helps to improve the contrast. Our scheme creates a tiny spatial superposition of size $\sim 0.21~{\rm μm}$ for mass $m=10^{-17}$kg in less than $t\sim 0.013$s.

Figures (6)

• Figure 1: The illustration of the Euler-angle rotations with the fixed frame $\{x,y,z\}$ and the reference frame $\{\hat{n}_1,\hat{n}_2,\hat{n}_3\}$ of a spherical nanodiamond with radius $R$, inside which an off-center NV spin is located at a distance $d$ from the center and have spin orientation $\hat{n}_s$ forms an angle $\alpha'$ and parallel to $\hat{n}_3$-axis ($\hat{n}_s \parallel \hat{n}_3$ ). There are three Euler angles: the precession $\alpha$, the nutation $\beta$, and the rotation $\gamma$. An initial rotational velocity $\omega_0$ is applied around the $\hat{n}_3$-axis for the gyroscopic stability, see the discussion in section-C. We also set initial position of $\beta$ as $\beta(0)=\beta_0 = 10^{-3}$ rad.
• Figure 2: Numerical solutions of Eqs. \ref{['eq:x_position']}, \ref{['eq:y_position']} and \ref{['eq:beta_eom_general']} for a spherical nanodiamond with mass $m = 10^{-17}\,\mathrm{kg}$ and radius $R = 50\,\mathrm{nm}$. For illustration, we take the center-of-mass motion is initialized at $x_0 = 0 \ \text{and} \ y_0 = 10^{-9}\,\mathrm{m}$ with $\dot x_0 = \dot y_0 = 0$, while the libration starts from $\beta(0)=\beta_0 = 10^{-3}\,\mathrm{rad}$ and $\dot\beta(0)=0$. In this plot we take the magnetic-field parameters $B_0 = 0.14\,\mathrm{T}$ and gradient $\eta = -7000\,\mathrm{T/m}$, and a libration frequency $\omega_0 = 2\pi\times 10^{4}\,\mathrm{s^{-1}}$. The evolution is shown up to $t_{\mathrm{close}} \simeq 0.01275\,\mathrm{s}$ (corresponding to $t_{\mathrm{close}}=t_{close}=2\pi/\Omega$ with $\Omega^2 = 12.08\,\mathrm{s^{-2}}$ ). Blue and orange curves correspond to the spin projections $s = +1$ and $s = -1$, respectively. (a) Three-dimensional spacetime trajectories $(x(t),y(t),t)$, showing a small spin-dependent splitting of the COM paths, with maximum superposition size $\Delta r_{max} = 0.215 \ \mu m$ at half loop time $t_{max} = \pi/\Omega$ (b) Time evolution of the libration angle $\beta(t)$ for the same parameters, illustrating that $\beta$ remains tightly confined around $\beta_0$ with only a small spin-dependent modulation, consistent with gyroscopic stabilization.
• Figure 3: The maximum superposition size from Eq. \ref{['eq:superposition_max_size']}, as a function of mass and variations in the magnetic field gradient, calculated using the approximation of libration mode $\beta \rightarrow \beta_0 = 10^{-3}$ rad and evaluated when the time reaches half a loop (at $t = \pi / \Omega$, with $\Omega = \sqrt{|\chi_\rho| \, \eta^2 / \mu_0}$).
• Figure 4: (a) Time evolution of the libration angle $\beta(t)$ for spin projections $s=\pm1$ and three different nanodiamond masses $m=10^{-16},10^{-17},10^{-18}\,\mathrm{kg}$ (top to bottom). The trajectories are obtained by numerically solving Eqs. (18), (19), and (40) with initial conditions $x_0 = y_0 = 10^{-9}\,\mathrm{m}$, $\dot x_0 = \dot y_0 = 0$, $\beta(0)=\beta_0 = 10^{-3}\,\mathrm{rad}$, and $\dot\beta(0)=0$, for a two–dimensional magnetic field characterized by $B_0 = 0.14\,\mathrm{T}$ and gradient $\eta = -7000\,\mathrm{T/m}$. The evolution is shown from $t=0$ to $t=t_{\mathrm{close}} = 2\pi/\Omega=0.01275$s, corresponding to two periods of the slow libration mode. For each mass, the spin states $s=\pm1$ produce slightly shifted mean values of $\beta(t)$ but the motion remains tightly confined around $\beta_0$, illustrating the strong gyroscopic stabilization of the NV axis. (b) Absolute angular mismatch $|\delta\beta(t)| = |\beta_{+}(t)-\beta_{-}(t)|$ for the same three masses and parameters as in Fig. \ref{['fig:traj_beta']} (b). The mismatch remains well below a milliradian for $m=10^{-16}\,\mathrm{kg}$ and grows to a few milliradians as the mass is reduced to $10^{-18}\,\mathrm{kg}$, reflecting the $1/I$ scaling of the spin–torque–induced motion. Despite this increase, $|\delta\beta(t)|$ stays small over the whole interferometric time $0\le t\le t_{\mathrm{close}}$, indicating that the rotational dynamics of a fast-spinning, nearly spherical nanodiamond only weakly degrade the spin contrast in the regime considered.
• Figure 5: (a) Time evolution of the precession angle $\alpha(t)$ for spin states $s=\pm1$ and masses $m=10^{-16},10^{-17},5\times10^{-18}\,\mathrm{kg}$, obtained by integrating Eq. \ref{['eq:beta_eom_general']} for $\beta(t)$ and inserting the result into Eq. \ref{['Euler_angle-1']}. Lighter particles show stronger spin-dependent precession due to their smaller rotational inertia. (b) Time evolution of the rotation angle $\gamma(t)$ from Eq. \ref{['euler_angle-2']}, using the same $\beta(t)$ as in panel (a). Note that at initial time $t=0$, $\gamma(0)=\omega_0$. The nearly linear growth reflects rigid-body rotation at frequency $\omega_0$, with a small spin-dependent splitting from the $\mu\cdot B$ contribution in the Hamiltonian. (c) Precession and rotation-angle mismatch $\delta\alpha(t)=|\alpha_{+}(t)-\alpha_{-}(t)|$ and $\delta\gamma(t)=|\gamma_{+}(t)-\gamma_{-}(t)|$ for the same three masses. The mismatch increases for smaller masses, consistent with the scaling $\delta\alpha\propto I^{-1}$. Common parameters for all panels: $\beta(0)=\beta_0=10^{-3}\,\mathrm{rad}$, $\dot{\beta}(0)=0$, $\alpha(0)=0$, $\dot{\alpha}(0)=0$, $\gamma(0)=0$, $\dot{\gamma}(0)=\omega_0$, $x_0=0,y_0=10^{-9}\,\mathrm{m}$, $\dot{x}(0)=\dot{y}(0)=0$, $B_0=0.14\,\mathrm{T}$, $\eta=-7000\,\mathrm{T/m}$, and closure time $t_{\mathrm{close}}\simeq 2\pi/\Omega\,\mathrm{s}=0.01275$s.