Linear stability analysis of a levitated nanomagnet in a static magnetic field: Quantum spin stabilized magnetic levitation

Abstract

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]

Theory of quantum acoustomagnonics and acoustomechanics with a micromagnet

“…As detailed in Ref. [1], the density matrixρ of the three-mode system described above obeys the dynamical equation (46) whereĤ(t) is given by Eq. (44), and the remaining three terms represent the dissipation of the magnon, the phonon, and the CM motion, respectively, through contact with thermal reservoirs at a common tempera-…”

“…Indeed, under the assumption of a cubic material undertaken in Sec. II, the magnetocrystalline anisotropy can be neglected (see Appendix A), and with it the main mechanism enabling the interaction between the micromagnet rotation and its internal degrees of freedom[45,46] …”

Quantum Spin Stabilized Magnetic Levitation

“…Let us first obtain the equilibrium configuration of the system. By writing the Heisenberg equations of motion for the nanomagnet in semiclassical approximation [20], the following relative equilibrium is found (see Fig. 1): (i) The center of mass is fixed at the center of the trap; (ii) The orientation is given by the body frame aligned to the laboratory frame (e 3 e z ) and rotating about e 3 at the frequency ω S ≡ − L 3 /I; (iii) The magnetic moment lies along the anisotropy axis e 3 and is anti-aligned to the magnetic field B(0) = B 0 e z .…”

“…The analysis of P Z (λ) and P T (λ) as functions of the physical parameters of the problem provides the stability diagram of a magnetically levitated nanomagnet, as discussed below. These polynomials can be alternatively obtained either by using classical equations of motion, or via the linearized Heisenberg equations of motion in semiclassical approximation without performing the bosonization, see [20]. Such methods allow to understand the results concerning the stability of a nanomagnet without the need to introduce the bosonization of the quantum mechanical operators.…”

“…Let us now analyze the linear stability of the system in the non-rotating case, see [20] for the ω S = 0 case. The stability phase diagram derived using Eq.…”