Quantum Spin Stabilized Magnetic Levitation

We theoretically show that, despite Earnshaw’s theorem, a nonrotating single magnetic domain nanoparticle can be stably levitated in an external static magnetic field. The stabilization relies on the quantum spin origin of magnetization, namely, the gyromagnetic effect. We predict the existence of two stable phases related to the Einstein–de Haas effect and the Larmor precession. At a stable point, we derive a quadratic Hamiltonian that describes the quantum fluctuations of the degrees of freedom of the system. We show that, in the absence of thermal fluctuations, the quantum state of the nanomagnet at the equilibrium point contains entanglement and squeezing.

Figure 1

Relative equilibrium for a magnetically levitated nanomagnet. The nanomagnet is at the center of the trap and rotates about its magnetization axis at the constant frequency ${\omega }_S\equiv -\hslash ⟨{\widehat{L}}_3⟩/I$. The magnetic moment lies along the anisotropy axis and is antialigned to $\mathbf{B}(0)=B_0{\mathbf{e}}_z$.

Figure 2

(Top panel) Stability diagram for a nonrotating nanomagnet for ${\rho }_M={10}^4\mathrm{kg}/{\mathrm{m}}^3$, ${\rho }_{\mu }=[{\rho }_M{\mu }_B/(50\mathrm{amu})]\mathrm{J}/(\mathrm{T}{\mathrm{m}}^3)$ (${\mu }_B$ is the Bohr magneton and amu the atomic mass unit), $k_a={10}^4\mathrm{J}/{\mathrm{m}}^3$, $B^{\prime }={10}^4\mathrm{T}/\mathrm{m}$, and $B^{\prime \prime }={10}^6\mathrm{T}/{\mathrm{m}}^2$. The red dashed lines represent the approximate borders $B_{c1}=3[\hslash {\rho }_{\mu }{B}^{\prime 2}/(4{\mu }_B{\gamma }_0{\rho }_M){]}^{1/3}$, $R_c=\sqrt{5{\rho }_{\mu }/(8{\gamma }_0^2{B}_0{\rho }_M)}$, and $B_{c2}=2k_a{\mu }_B/(\hslash {\gamma }_0{\rho }_{\mu })$ of the stable EdH phase (the red region) and the $A$ phase (the blue region). (Bottom panel) Entanglement $\mathcal{P}$ (the black dashed line), and squeezing $\xi$ (the gray solid line) of the quantum state $|0⟩$ at the equilibrium as a function of $B_0$ in the stable regions for $R=2\mathrm{nm}$ (the green dashed lines in the stability diagram).