Gyroscopically stabilized quantum spin rotors

Abstract

Recent experiments demonstrate all-electric spinning of levitated nanodiamonds with embedded nitrogen-vacancy spins. Here, we argue that such gyroscopically stabilized spin rotors offer a promising platform for probing and exploiting quantum spin-rotation coupling of particles hosting a single spin degree of freedom. Specifically, we derive the effective Hamiltonian describing how an embedded spin affects the rotation of rapidly revolving quantum rotors due to the Einstein-de Haas and Barnett effects, which we use to devise experimental protocols for observing this coupling in state-of-the-art experiments. This will open the door for future exploitations of quantum spin rotors for superposition experiments with massive objects.

Figures (8)

• Figure 1: (a) Charged spheroidal nanodiamond with an embedded nitrogen-vacancy (NV) center is levitated in a Paul trap and revolves rapidly with frequency $\omega$ in the symmetry plane of the trap. The major principal axis $\mathbf{n}_1$ of the body is chosen to be aligned with the NV quantization axis so that the NV spin states $|S_1=\pm\hbar\rangle$ are split by the rotation-induced Barnett field as well as by an applied magnetic field $\mathbf{B}$. The particle rotation induces (b) an effective gyroscopic potential $I\omega^2/2\sin^2 \beta$ stabilizing the out-of-plane rotation angle $\beta$, up to small librations $\xi$, while the external magnetic field together with spin-rotation coupling induce (c) a spin state-dependent potential $V_{m}(\gamma)$ for $\gamma$ rotations around the body-fixed symmetry axis depending on the spin states $|S_1 = \hbar m\rangle$ with $m \in\{-1,0,1\}$.
• Figure 2: (a) Steady-state alignment $\langle \cos \gamma \rangle$ as a function of the applied magnetic field. The variance is indicated by the shaded region. (b) Potential energy surfaces $\Omega_+$ (solid) and $\Omega_-$ (dashed) for $B=-0.5\,\mathrm{mT}$ and $\omega/2\pi=1\,\mathrm{MHz}$. The inset shows the avoided crossing around $\gamma=\pi/2$ where the spin-rotor state $|\chi_+\rangle$ can be stably trapped in the $|\sigma_x = +1\rangle$ spin state. (c) Spin transition probability for an initial state in spin $|\sigma_x = +1\rangle$ (solid) and spin $|\sigma_x = -1\rangle$ (dashed) as a function of $\omega_\gamma t$ for different magnetic field strengths. The nanodiamond has an ellipsoidal shape with semiaxes $l_3 = 200$ nm and $l_1 = l_2 = 0.3 l_3$.
• Figure 3: (a) Pulse sequence and state evolution of the interference protocol. (b) Probability to measure $\left|\uparrow\right>$ at the end of the interferometer sequence as a function of interferometer time $\tau\omega_\gamma/\pi$ (evaluated at $\omega/2\pi=1\,\mathrm{MHz}$) for $B=-95\,\mathrm{mT}$ and different rotation frequencies $\omega/2\pi=1\,\mathrm{MHz}$ (gray, solid), $50\,\mathrm{MHz}$ (light blue, dotted), $100\,\mathrm{MHz}$ (dark blue, dashed). The inset shows the probability close to rephasing. (c) Duration of the full protocol (dark blue, solid) and recurrence probability (light blue, dotted) depending on $g/D_\mathrm{nv}$. The shaded region indicates when the interferometer time is exceeded by the $T_2$ time of the NV spin. The nanodiamond has an ellipsoidal shape with semiaxes $l_3 = 100$ nm and $l_1 = l_2 = 0.2 l_3$.
• Figure 4: Dynamics of $\gamma$ for the spin state $|S_1=-\hbar\rangle$ as a function of $\omega_0 t$, comparing the solution of (a) the gyroscopic Hamiltonian \ref{['eq:gyroham']} (light blue, solid) with the adiabatic Hamiltonian \ref{['eq:adiabaticham']} (dark blue, dashed) and (b) the adiabatic Hamiltonian \ref{['eq:adiabaticham']} with the dispersive Hamiltonian \ref{['eq:H_dispersive']} (light blue, solid). The nanodiamond has an ellipsoidal shape with semiaxes $l_3 = 200$ nm and $l_1 = l_2 = 0.4 l_3$, $B=-102\,\mathrm{mT}$, $\omega/2\pi=0.1\,\mathrm{MHz}$, and $\sigma_\gamma=\sqrt{\hbar/2I_\mathrm{eff}\omega_\gamma}$. We approximated the Fock space as a finite space of dimension $d = 40$ and chose for both the initial states of the $\beta$- and $\gamma$-oscillator a coherent state $|\alpha_{\beta,\gamma}=0.1\rangle$.
• Figure 5: Validity of the adiabatic elimination of the $\xi$-mode. Parameter region satisfying Eq. \ref{['eq:adiabatic_validity']} as a function of the rotation frequency $\omega$ and the major semiaxis length $l_3$, with $l_1=0.2 l_3$, $B=-100\,\mathrm{mT}$ and different temperatures $T$ determining the mean occupation $n_\gamma$.