We theoretically show that, despite Earnshaw's theorem, a non-rotating 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.According to the Einstein-de Haas and the Barnett effect [1,2], a change in the magnetization of an object is accompanied by a change in its rotational motion. In particular, if the magnetic moment of a magnet is varied by a single Bohr magneton, it must rotate with an angular frequency /I about the magnetic moment axis to conserve angular momentum. Here I is its moment of inertia about the rotation axis. For a Cobalt sphere of radius R, this corresponds to a frequency /I ≈ 2π × 10 6 Hz/(R[nm]) 5 , where R[nm] is the radius in nanometers. This clear manifestation of the quantum spin origin of magnetization, as prescribed by the gyromagnetic relation, is hence boosted at the nanoscale [3][4][5].In this Letter, we explore the role of the quantum spin origin of magnetization in magnetic levitation. Earnshaw's theorem [6], very relevant in this context, prevents magnetic levitation of a non-rotating ferromagnet in a static magnetic field. The theorem can be circumvented by mechanically spinning the magnet, as neatly demonstrated by the Levitron [7][8][9][10], which is a magnetic top of a few centimeters. At the single atom level, magnetic trapping with static fields is also possible by exploiting the fast Larmor precession of its quantum spin [11,12]. In this case, the atom is, from the mechanics point of view, a point particle without rotational degrees of freedom. A magnetic nanoparticle lies in between the Levitron and the atom as both its rotational degrees of freedom and the quantum spin origin of magnetization have to be accounted for. Can a non-rotating magnetic nanoparticle, despite Earnshaw's theorem, be stably levitated with static magnetic fields?We show in this Letter that this is the case. In particular, we predict two stabilization mechanisms that crucially rely on the quantum spin origin of the magnetic moment. At low (large) magnetic fields, the Einstein-de Haas effect (the Larmor precession of its magnetic moment) stabilizes levitation. These results are obtained by deriving a quadratic Hamiltonian which describes the linearized dynamics of the degrees of freedom of the magnet (center-of-mass motion, rotation, and magnetization dynamics) around the equilibrium point. We further show that in the absence of thermal fluctuations, the equilibrium state exhibits both quantum entanglement and squeezing of its degrees of freedom. As discussed in the co...

We theoretically study the levitation of a single magnetic domain nanosphere in an external static magnetic field. We show that apart from the stability provided by the mechanical rotation of the nanomagnet (as in the classical Levitron), the quantum spin origin of its magnetization provides two additional mechanisms to stably levitate the system. Despite of the Earnshaw theorem, such stable phases are present even in the absence of mechanical rotation. For large magnetic fields, the Larmor precession of the quantum magnetic moment stabilizes the system in full analogy with magnetic trapping of a neutral atom. For low magnetic fields, the magnetic anisotropy stabilizes the system via the Einstein-de Haas effect. These results are obtained with a linear stability analysis of a single magnetic domain rigid nanosphere with uniaxial anisotropy in a Ioffe-Pritchard magnetic field. II. SINGLE MAGNETIC DOMAIN NANOPARTICLE IN A STATIC MAGNETIC FIELDWe consider a single magnetic domain nanoparticle in an external static magnetic field B(r). The nanomagnet is modeled as a rigid sphere of radius R, mass M , and with a magnetic moment µ. B(r) is assumed to be approximately homogeneous within the volume of the sphere such that the interaction energy between µ and B(r) can be expressed as V b = −µ · B(r), where r is the center-of-mass position of the sphere (point-dipole approximation). The exchange interaction between the magnetic dipoles inside a magnetic domain is assumed to be the strongest energy scale of the problem. Under this assumption, µ ≡ |µ| can be approximated to be a constant. The degrees of freedom of the system are hence: (i) arXiv:1701.05410v3 [quant-ph]