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...
A system of linearly coupled quantum harmonic oscillators can be diagonalized when the system is dynamically stable using a Bogoliubov canonical transformation. However, this is just a particular case of more general canonical transformations that can be performed even when the system is dynamically unstable. Specific canonical transformations can transform a quadratic Hamiltonian into a normal form, which greatly helps to elucidate the underlying physics of the system. Here, we provide a self-contained review of the normal form of a quadratic Hamiltonian as well as stepby-step instructions to construct the corresponding canonical transformation for the most general case. Among other examples, we show how the standard two-mode Hamiltonian with a quadratic position coupling presents, in the stability diagram, all the possible normal forms corresponding to different types of dynamical instabilities.
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