Linear Stability Analysis of a Levitated Nanomagnet in a Static Magnetic Field: Quantum Spin Stabilized Magnetic Levitation

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by  Cosimo C. Rusconi

Linear Stability Analysis of a Levitated Nanomagnet in a Static Magnetic Field:
Quantum Spin Stabilized Magnetic Levitation
C. C. Rusconi,1,2 V. Pöchhacker,1,2 J. I. Cirac,3 and O. Romero-Isart1,2
1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.
2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.
3Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748, Garching, Germany.
Wetheoreticallystudythelevitationofasinglemagneticdomainnanosphereinanexternalstatic
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
7 magnetictrappingofaneutralatom. Forlowmagneticfields,themagneticanisotropystabilizesthe
1 systemviatheEinstein-deHaaseffect. Theseresultsareobtainedwithalinearstabilityanalysisof
0 a single magnetic domain rigid nanosphere with uniaxial anisotropy in a Ioffe-Pritchard magnetic
2 field.

r
p
A I. INTRODUCTION behaves as a soft magnet, namely its magnetization can
freelymovewithrespecttoitsorientation. TheEinstein-

6 de Haas (EdH) phase appears at sufficiently small mag-
According to the Earnshaw theorem [1, 2] a ferromag-
netic fields where the nanomagnet effectively behaves as
] netcanbestablylevitatedinastaticmagneticfieldonly
a hard magnet, namely the magnetization sticks to the
h when it is mechanically rotating about its magnetiza-
p crystal. The EdH phase requires the magnet to be suffi-
tion axis. Such a gyroscopic-based stabilization mech-
- ciently small. Furthermore, we also recover the Levitron
t anismcanbeneatlyobservedwithaLevitron[3–7]. The
n (L)phaseforalargerrotatingmagnet,whichcanbepre-
Earnshaw theorem does not account for the microscopic
a dictedwithoutaccountingforthequantumspinoriginof
u quantum origin of magnetization. For instance, a single
the magnetization. Such a rich stability phase diagram
q neutral magnetic atom can be stably trapped in a static
could be experimentally tested and opens the possibility
[ magnetic field by means of the Larmor precession of its
tocooltheseveraldegreesoffreedomofthenanomagnet
quantum magnetic moment [8, 9]. In both the Levitron
2 in the stable phases to the quantum regime [16].
andtheatom,themagnetization,initiallyanti-alignedto
v This article is structured as follows. In Sec. II, we
0 the magnetic field, adiabatically follows the local direc-
model a single magnetic domain nanoparticle in a static
1 tion of the magnetic field, thereby confining the center-
field. Both a quantum and a classical description of the
4 of-mass motion [6].
modelisgiven. InSec.IIIwederivethestabilitycriterion
5
0 Inthisarticle,westudythestabilityofalevitatedsin- asafunctionofthephysicalparametersofthesystem. In
. gle magnetic domain particle (nanomagnet) in a static Sec. IV we discuss the stability diagrams and the phys-
1
magnetic field. The magnetization of the nanomagnet ical origin of the different stable phases. We draw our
0
7 couples to its center-of-mass motion via the interaction conclusions and discuss further directions in Sec. V.
with the external inhomogeneous magnetic field, and
1
: to its orientation via the magnetocrystalline anisotropy
v
[10, 11]. The latter induces magnetic rigidity, namely its
i II. SINGLE MAGNETIC DOMAIN
X magnetic moment cannot freely move with respect to a NANOPARTICLE IN A STATIC MAGNETIC
r givenorientationofthecrystalstructureofthenanomag- FIELD
a net. Together with the quantum spin origin of the mag-
netization,givenbythegyromagneticrelation,thisleads
We consider a single magnetic domain nanoparticle in
to the well-known Einstein-de Haas effect [12]. That is,
an external static magnetic field B(r). The nanomag-
achangeofmagnetizationisaccompaniedbymechanical
net is modeled as a rigid sphere of radius R, mass M,
rotation in order to conserve total angular momentum.
and with a magnetic moment µ. B(r) is assumed to
The Einstein-de Haas effect is boosted at small scales
beapproximatelyhomogeneouswithinthevolumeofthe
due to the small moment-of-inertia-to-magnetic-moment
sphere such that the interaction energy between µ and
ratio [13–15].
B(r) can be expressed as V = −µ · B(r), where r is
b
Weshallarguethatthequantumspinoriginofmagne- the center-of-mass position of the sphere (point-dipole
tization opens the possibility to magnetically levitate a approximation). The exchange interaction between the
non-rotating nanomagnet in a static field configuration. magneticdipolesinsideamagneticdomainisassumedto
Indeed,weencountertwostablephasesofdifferentphys- be the strongest energy scale of the problem. Under this
ical origin. The atom (A) phase appears at sufficiently assumption, µ ≡ |µ| can be approximated to be a con-
large magnetic fields where the nanomagnet effectively stant. Thedegreesoffreedomofthesystemarehence: (i)
2
the center-of-mass motion (described by 6 parameters),
TABLE I. Physical parameters of the model. Whenever re-
(ii) the rotational motion (described by 6 parameters),
quired, the following values are used: ρ =104Kg/m3, ρ =
and(iii)themagnetizationdynamics(describedby2pa- M µ
[(cid:126)γ ρ /(50amu)]J/(Tm3)(whereγ =1.760×1011rad/(s T)
0 M 0
rameters) [17].
is the electronic gyromagnetic ratio and amu is atomic mass
Theorientationoftherigidsphereisrepresentedbythe unit), k =104J/m3, B0 =104T/m, and B00 =106T/m2.
a
threeEuleranglesΩ≡(α,β,γ)intheZYZparametriza-
tion [18], which specify the mutual orientation between Parameter Description [dimension SI]
the frame Oe1e2e3, fixed in the object and centered ρM ≡M/V mass density [Kg×m−3]
in its center of mass, and the frame Oexeyez, fixed in R radius [m]
the laboratory. According to this convention the co- ρ ≡µ/V magnetization [J×T−1×m−3]
µ
ordinate axes of the frame Oe e e and the coordi-
1 2 3 k magnetic anisotropy constant [J×m−3]
nate axes of the frame Oe e e are related through a
x y z B field bias [T]
(e ,e ,e )T = R(Ω)(e ,e ,e )T, where the transfor- 0
1 2 3 x y z B0 field gradient [T×m−1]
mation matrix reads R(Ω) ≡ R (γ)R (β)R (α), where
z y z
R (θ) is the clockwise rotation of the coordinate frame B00 field curvature [T×m−2]
n
(passive rotation) of an angle θ about the direction n
(see [18] for further details). Hereafter Latin indexes
i,j,k,...=1,2,3 label the body frame axes while Greek and point-dipole approximation. In Sec. V, we discuss
indexes µ,ν,λ... = x,y,z label the laboratory frame theseassumptionsandthepotentialgeneralizationofthe
axes. model.
Ferromagnetic materials exhibit magnetocrystalline GivenasetofvaluesinTableI,canthenanomagnetbe
anisotropy, namely they magnetize more easily in some stably levitated? To address this question, we first need
directions than others, due to the interaction between todescribethedynamicsofthesystem. Thiscanbedone
the magnetic moment and the crystal structure of the usingeitherquantummechanicsorclassicalmechanics2.
material [10]. This interaction determines preferred di-
rections along which the magnetic energy of the system
isminimized. Weconsideruniaxialanisotropy, forwhich A. Quantum description
the preferred direction is a single axis (easy axis) in the
crystal. By choosing e to be the easy axis, the uniax-
3 The degrees of freedom of the nanomagnet are de-
ial anisotropy energy is given by V ≡−k V (e ·µ/µ)2,
a a 3 scribed in quantum mechanics through the following
where k and V are, respectively, the anisotropy energy
a quantum operators. The center-of-mass motion by ˆr =
density and the volume of the nanomagnet. Va has two (xˆ,yˆ,zˆ) and pˆ =(pˆ ,pˆ ,pˆ ), where [rˆ ,pˆ ]=i(cid:126)δ . The
x y z ν λ νλ
minima corresponding to µ being aligned or anti-aligned
rotational motion by Ωˆ =(αˆ,βˆ,γˆ) and Lˆ =(Lˆ ,Lˆ ,Lˆ ),
to e . Note that e depends on Ω, and hence V couples x y z
3 3 a where the Euler angle operators commute with them-
the magnetization with the orientation of the nanomag-
selves, [Lˆ ,Lˆ ] = i(cid:15) Lˆ , and the commutators [Ωˆ,Lˆ],
net. ν λ νλρ ρ
which are more involved [18], are actually not required,
Regarding B(r), we consider the Ioffe-Pritchard field
see below. Regarding the magnetization dynamics, the
given by [19]
magnetic moment is given by µˆ =(cid:126)γ Fˆ, where γ is the
0 0
(cid:18) B00 (cid:19) (cid:18) B00 (cid:19) gyromagnetic ratio, and Fˆ is the total spin of the nano-
B(r)=ex B0x− 2 xz −ey B0y+ 2 zy magnet (macrospin), where [Fˆν,Fˆλ] = i(cid:15)νλρFˆρ. Fˆ is ob-
(1) tainedasthesumofthespinoftheN constituentsofthe
+ez(cid:20)B0+ B200 (cid:18)z2− x2+2 y2(cid:19)(cid:21), nanomagnet, Fˆ =PNi=1Fˆi. In the quantum description,
the constant magnetization assumption can be incorpo-
rated via the macrospin approximation: the total spin is
where B ,B0 and B00 are the three parameters charac-
0 projected into the subspace with Fˆ2 = Nf(Nf +1) ≡
terizing the Ioffe-Pritchard trap, namely the bias, the
F(F + 1), where f is the total spin of a single con-
gradient, and the curvature. This field, which is com-
stituent (assumed to be identical for simplicity). Under
monly used to trap magnetic atoms [19], is non-zero at
its center, i.e. B(0)=B e .
0 z
In summary, our model, whose physical parameters
are listed in Table I 1, assumes a: single magnetic do-
main, rigid body, spherical shape, constant magnetiza- 2 As shown below, the classical description is sufficient to obtain
the criterion for the stable magnetic levitation of the magnet.
tion, uniaxial anisotropy, Ioffe-Pritchard magnetic field,
However, we emphasize that the stable A and EdH phases cru-
cially depend on the quantum spin origin of the magnetization.
Intheclassicaldescription,thisisincludedwiththephenomeno-
logical Landau-Lifshitz-Gilbert (LLG) equations describing the
1 ThephysicalparameterslistedinTableIshouldnotbeconfused magnetizationdynamics[17]. Thequantumdescriptiondoesnot
withthe14dynamicalparametersdescribingthedegreesoffree- onlyincorporatethiskeyfactfromfirstprinciples,butwillalso
domofthenanomagnet. beusefulforfurtherresearchdirections,seeSec.V.
3
the macrospin approximation the magnetization dynam- whoseexpectationvaluesdescribethedegreesoffreedom
icscanthusbedescribedbythetwospinladderoperators of the system in the semiclassical approximation. With
Fˆ ≡Fˆ ±iFˆ . The degrees of freedom of the nanomag- this approximation, the evolution of Eq. (7) as described
± x y
net can hence be represented by the 14 quantum opera- by Hˆ, Eq. (6), is used in Sec. III to analyze the linear
tors {ˆr,pˆ,Ωˆ,Lˆ,Fˆ±}. stability of the system for a given value of ωS and the
In the coordiante frame Oexeyez, the quantum me- physical parameters given in Table I.
chanicalHamiltonianofthenanomagnetintermsofthese
operators reads [18]
B. Classical description
pˆ2 (cid:126)2 h i2
Hˆ = + Lˆ2−(cid:126)γ Fˆ·B(ˆr)−(cid:126)2D e (Ωˆ)·Fˆ , (2)
2M 2I 0 3 Letusnowgiveaclassicaldescriptionofthesystemin
the Lagrangian formalism. The center-of-mass position
where I ≡ 2MR2/5 is the moment of inertia of a
of the nanomagnet is described by the coordinate vector
sphere, and D ≡ k V/((cid:126)F)2 parametrizes the uniaxial
a r =(x,y,z) and its orientation by the Euler angles Ω=
anisotropy strength.
(α,β,γ). The direction of the magnetic moment µ/µ
As discussed in [18], it is more convenient to express is described by (φ,θ), which represent, respectively, the
HtˆheinchtahnegecooofrvdainriaatbelefsraAˆmie(ΩˆO)e≡1eP2eν3.RiTν(hΩˆis)AˆisνdfoonreAˆv=ia Lpoalgarranagnidanazoifmtuhtehsaylsatenmglerseaindsthe frame Oexeyez. The
Lˆ,Fˆ,B(ˆr). The operators R (Ωˆ) can be written as a
iµ
function of the 9 D-matrix tensor operators Dˆ1 , where L=Tcm+Trot+Tmag−Va−Vb, (8)
mk
m,k = ±1,0 [18]. These 9 D-matrix operators are not
where T , which represents the kinetic energy of the
independent. They must satisfy the following relations cm
center-of-mass motion, reads
[20]
(−1)k−mDˆj =(cid:16)Dˆj (cid:17)†, (3) Tcm ≡ M2 (cid:0)x˙2+y˙2+z˙2(cid:1). (9)
mk −m−k
XDˆm1k(cid:16)Dˆm1k0(cid:17)† =δkk01, (4) Tbohdeyrfortaamtieoncaolorkdinineattice seynsetregmy Oofe1the2eer3i,grideadbsod[2y1]in the
m
X(cid:16)Dˆm1k(cid:17)†Dˆm10k =δmm01. (5) Trot ≡ I2(cid:0)α˙2+β˙2+γ˙2+2α˙γ˙ cosβ(cid:1). (10)
k
T accounts for the kinetic energy associated to the
mag
UsingtheD-matrixtensoroperators,theHamiltonianin
motion of the magnetic moment, namely [17]
the body frame reads [18]
µ
T ≡− φ˙cosθ. (11)
pˆ2 (cid:126)2 (cid:16) (cid:17) mag γ
Hˆ = + Jˆ2+2Sˆ Jˆ +JˆSˆ +JˆSˆ 0
2M 2I 3 3 ↑ ↓ ↓ ↑ (6)
We remark that Eq. (11) leads to the phenomenological
+(cid:126)γ Sˆ·B(ˆr,Ωˆ)−(cid:126)2DSˆ2
0 3 Landau-Lifshitz-Gilbert (LLG) describing the magneti-
zationdynamics[17,22]. Thequantumdescriptiongiven
by defining Jˆ ≡ Lˆ − Sˆ, Jˆ↑↓ ≡ Jˆ1 ∓ iJˆ2, and Sˆ↑↓ ≡ in Sec. IIA has the advantage to describe this from first
Sˆ ∓ iSˆ , where Sˆ ≡ −Fˆ for convenience. The D- principles. The classical uniaxial anisotropy interaction
1 2
matrixoperatorsfulfillthefollowingcommutationsrules: reads
[Dˆj ,Dˆj0 ] = 0, [Jˆ,Dˆj ] = kDˆj , and [Jˆ ,Dˆj ] =
p(mjk∓km)(0jk0±k+!)Dˆj3 m,ksee [18m] kfor furth↑e↓r dmetkails. Va ≡−kaV [sinβsinθcos(α−φ)+cosβcosθ]2, (12)
mk±1
The Hamiltonian Hˆ is invariant under a rotation about where we recall that e coincides with the anisotropy
3
the easy axis of the nanomagnet, namely [Hˆ,Lˆ ] = axis. Themagneticdipoleinteractionbetweentheexter-
3
[Hˆ,Jˆ −Sˆ ] = 0 [18]. Therefore it is convenient to de- nal field B(r) and the magnetic moment µ reads
3 3
fine ω ≡−(cid:126)hLˆ i/I, which represents the rotational fre-
S 3 (cid:2)
V ≡−µ B (r)cosφsinθ+B (r)sinφsinθ
quency of the nanomagnet about the easy axis e . Note b x y
3 (13)
that hJˆi for a given ω can then be written in terms +B (r)cosθ(cid:3).
3 S z
of hSˆ i and hSˆ i. Furthermore, using (3-5) one can ex-
↑ ↓
press hDˆ1 i ∈ C ∀m,k as a function of hDˆ1 i,hDˆ1 i Note that Va (Vb) couples the magnetization µ with the
mk 11 0−1 orientation Ω (center of mass r) of the nanomagnet.
and hDˆ−110i, which are given by three real independent TheLagrangianLisindependentonθ˙,therebyθisnot
parameters. Hence, we define the 13 operators
anindependentdynamicalvariable. Intheabsenceofdis-
(cid:16) (cid:17) sipation, the magnetic moment µ undergoes a constant
ξˆ≡ ˆr,pˆ,Jˆ↑,Jˆ↓,Dˆ111,Dˆ01−1,Dˆ−110,Sˆ↑,Sˆ↓ , (7) precessionaroundadirectiondeterminedbyEq.(12)and
4
Eq. (13), and thus can be described with a single preces- of the field (B(0)=B e ), and with µ/µ=−e =−e ,
0 z 3 z
sion angle [17]. namely magnetization parallel to the easy axis and anti-
Furthermore, L is independent on γ, and thus L ≡ aligned to the field at the center, see Fig. 1.a.
3
∂L/∂γ˙ =I(γ˙ +α˙ cosβ), which represents the rotational
Thelinearstabilityofthissolutionisanalyzedthrough
angular momentum of the rigid sphere about the axis
the dynamics of the fluctuations δξ(t) ≡ hξˆ(t)i − ξ .
e3, is a constant of motion. Once ωS ≡ −L3/I is fixed, 0
To linear order in δξ(t), these are governed by the lin-
the state of the system can thus be described by the
13 independent parameters r,p,Ω,α˙,β˙,φ,φ˙. These are, ear equations δ˙ξ = C(t)δξ, where the matrix Cij(t) ≡
∂ G (ξ ) depends periodically on time with a period
roughly speaking, the classical analogs to the quantum j i 0
2π/ω . We remark that since ω is a constant of mo-
operators Eq. (7). S S
tion δD1 (t) = δD1 (0)exp(−iω t), which corresponds
11 11 S
to a trivial stable evolution. Hence we redefine δξ(t)
III. LINEAR STABILITY ANALYSIS as a 12 component vector by removing its δD111 com-
ponent. Physically these are the fluctuations of the
12 parameters describing the degrees of freedom of a
Letusnowdescribethecriterionwhichdeterminesthe
nanomagnet with constant rotational motion about the
linear stability of the system. While both the classical
anisotropy axis. The time dependence of C(t) can be
andthequantumdescriptionleadtothesameresults, as
removed with the following change of variables: δJr =
discussedattheendofthesection,wederivethecriterion ↑
usingthequantumdescription. TheHeisenbergequation (δJ↓r)† ≡ δJ↑exp[−i(ϕ − ωSt)] and δS↑r = (δS↓r)† ≡
fortheoperatorsξˆEq.(7)usingtheHamiltonianEq.(6) δS↑exp[−i(ϕ − ωSt)]. The linear system reduces then
can be written as ∂ ξˆ =[ξˆ,Hˆ]/i(cid:126)≡G(ξˆ). G is a vector to δξ˙r =Aδξr, where the matrix A is time independent
t
function of ξˆ that depends on the physical parameters and δξr is obtained replacing the old variables with the
given in Table I. These Heisenberg equations are a non- new ones, δS↑r↓ and δJ↑r↓. In the absence of dissipation,
linear system of differential equations for the operators linearstabilitycorrespondstotheeigenvaluesofAbeing
of the system. The stability of the system is studied in all purely imaginary [23].
the semiclassical approximation, namely the system is
The 12 × 12 complex matrix A can be block-
considered to be in a quantum state ρˆsuch that
diagonalized as A ⊕ A ⊕ A∗, where A is a 2 × 2
Z T T Z
h i matrix defined as
Tr ξˆξˆρˆ =hξˆξˆi’hξˆihξˆi ∀i,j, (14)
i j i j i j
whereξˆ isthei-thcomponentofξˆ. FurthermoresinceLˆ ! ! ! !
i 3 δz δz 0 1/M δz
isaconstantofmotion,weconsiderρˆtolieintheHilbert ∂ =A ≡ ,
subspace of eigenstates of Lˆ3 with eigenvalue −IωS/(cid:126). t δpz Z δpz −MωZ2 0 δpz
Within this subspace one can thus use Lˆ = −Iω 1/(cid:126). (17)
3 S
where ω is defined in Table II. A is a 5×5 matrix
TheHeisenbergequationsofmotionforagivenω canbe Z T
S
defined as
approximatedbytheclosedsetofsemiclassicalequations
∂ hξˆi=G(hξˆi) (15)
t
   
δp δp
+ +
by using Eq. (14) to approximate hG(ξˆ)i ’ G(hξˆi). A
 δρ   δρ 
solution of Eq. (15) is given by  +   + 
∂  δJr =A  δJr , (18)
t ↑  T  ↑ 
ξ (t)≡(cid:16)0,0,0,0,hDˆ1 i =ei(ϕ−ωSt),0,0,0,0(cid:17), (16)  δS↑r   δS↑r 
0 11 0
δD1 δD1
−10 −10
whereϕisaphasefactorfixedbytheinitialconditionon
hDˆ1 i . This solution3 corresponds to a nanomagnet ro-
11 0
tating at the frequency ω along e , at rest in the center where
S 3
3 Throughout this article we focus on the stability of the equi-
not exhaustively investigated the existence of other equilibrium
librium solution Eq. (16). However, we remark that we have
solutions.
5
a) b)
ez=e3 109 !L !D
108 !
! D !
S Hz] 107 !I !L
⇡[ 106 I
ey !/2 105 !T !T
e
e1 µ 2 104 !
e Z
x 103
0.1 1 10 102 103 5 10 15 20
B(r)
B [mT] R[nm]
0
FIG. 1. (a) Equilibrium solution for a levitated nanomagnet in a Ioffe-Pritchard magnetic field. The nanomagnet is at the
centerofthetrap,rotatingaboute withangularfrequencyω ,andthemagneticmomentisparalleltotheanisotropyaxisand
3 S
anti-aligned to B(0). (b) ω ,ω ,ω ,ω and ω , which are defined in Table II, as a function of the bias field B for R=4nm
T Z L I D 0
(left panel) and of theradius R for B =2mT (right panel). Other physical parametersare taken from the caption of Table I.
0
√
 0 −i(cid:126)ω B00S/(2B ) 0 i(cid:126)γ B0 i(cid:126)γ 2SB0
L 0 0 0
−i/M 0 0 0 0 
 √ 
A ≡i 0 ω SB0/B ω +ω ω +ω −ω −ω 2S , (19)
T  L 0 I S I S L L√ 
 0 −ωLSB0/B0 −√ωI ωL−ωI√−2ωD ωL 2S 
0 0 ω /( 2S) ω /( 2S) 0
I I
whose coefficients are given by
TABLE II. Relevant frequencies of the system.
ωI (cid:126)S/I =5(cid:126)ρµ/(2µBρMR2) a0 ≡ −2ωDωIωLωT2,
ω (cid:126)DS =k µ /((cid:126)ρ ) a ≡i(cid:2)ω ω2(ω +ω )+ω ω ω2(cid:3),
D a B µ 1 D Z S I S L T
ω γ B 1
ωωLZ pp0(cid:126)(cid:126)γγ00BS0(0BS/02M−B B00/2)/MB a2 ≡ −(cid:20)2ωDωIωL− 2(2ωD−ωS)ω1Z2 −(cid:21)ωLωT2, (21)
T 0 0 0 a ≡i −2ω (ω +ω )+ω ω + ω2 ,
ω −(cid:126)hLˆ i/I 3 D S I S L 2 Z
S 3
a ≡2ω −ω −ω ,
4 D S L
a ≡ −i.
5
This is one of the main results of this article since the
roots of P (λ) and P (λ) allow to discern between sta-
Z T
ble and unstable levitation as a function of the physical
parameters of the system via Table I and Table II. In
with δρ ≡δx±iδy and δp ≡δp ±iδp . The relevant
± ± x y particular, stable levitation corresponds to the roots of
frequencies ω ,ω ,ω ,ω are defined in Table II. The
L I T D P (λ) and P (λ) being purely imaginary4.
eigenvaluesofA ,givenbytherootsofP (λ)≡λ2+ω2, Z T
Z Z Z Let us remark that at the transition between stabil-
are purely imaginary for B00 > 0. This leads to stable
ityandinstabilitythediscriminantsofP (λ)andP (λ),
harmonic oscillations of the center-of-mass motion along Z T
defined as ∆ and ∆ respectively, are zero. This hap-
the e direction with frequency ω . A accounts for Z T
z Z T pens whenever two distinct eigenvalues become degener-
the fluctuations of the remaining degrees of freedom and
ate(Krein’scollision)[23]. Theeigenvaluesofthematrix
its eigenvalues are given by the roots of the fifth order
associatedtoalinearsystemofdifferentialequationsde-
polynomial
scribing conservative Hamiltonian dynamics, as the ma-
4 OnecoulddefineA≡iA˜suchthatthecharacteristicpolynomial
of A˜ has real coefficients. Stability would require, in this case,
P (λ)=a +a λ+a λ2+a λ3+a λ4+a λ5, (20) realroots.
T 0 1 2 3 4 5
6
trix A ⊕A ⊕A∗ in our case, are always either com- A. Einstein-de Haas phase
Z T T
plexquadruplets,λ={a+ib,a−ib,−a+ib,−a−ib},real
pairsλ={a,−a},imaginarypairsλ={ib,−ib},orpairs InthesHMregime,whereω (cid:29)ω ,themagneticmo-
D L
of zero eigenvalues λ = {0,0}, where a,b ∈ R. There- ment can be considered, to a good approximation, fixed
fore, the transition from stability to instability, namely along the direction of the magnetic anisotropy. Due to
fromallimaginaryeigenvaluestohaveatleastacomplex the small dimension of the nanomagnet the spin angular
quadruplet or a real pair, happens at a Krein’s collision. momentumplaysasignificantroleinthedynamicsofthe
Note that this is a necessary but not sufficient condition system,namelyω (cid:29)ω (seeFig.1.b). 2ω isindeedthe
I L I
since the colliding eigenvalues could still remain on the frequency at which the nanomagnet would rotate if the
imaginary axis [23]. magneticmomentflippeddirection. ThisistheEinstein-
The polynomials PZ(λ) and PT(λ) have also been ob- de Haas effect, boosted at the lower scales due to the
tained via the classical description of the nanomagnet small moment-of-inertia-to-magnetic-moment ratio.
discussed in Sec. IIB. The procedure is very similar to The borders of the EdH stable phase in the non-
the one presented above, but care must be taken when rotatingcasecanbeanalyticallyapproximatedasfollows,
linearizingthesystemaroundthesolutionrepresentedin see Fig. 3. The upper border can be approximated by
Fig.1.asinceitcorrespondstoadegeneracypointofthe keepingtermsin∆ =0ofzeroorderinω /ω (cid:28)1and
T Z D
Euler angular coordinates in the ZYZ convention. That up to leading order in ω/ω (cid:28) 1 (for ω = ω ,ω ,ω ).
D T L I
is, for β = 0 it is not possible to distinguish between This is justified in the sHM regime, see Fig. 1.b. This
rotation of the angle α and γ. This is the so-called Gim- leads to the simple expression ω = 4ω , which using
I L
bal lock problem which can be circumvented either by Table II, reads
using an alternative definition of the Euler angles, which
s
moves the degeneracy point elsewhere, or by changing 5ρ
R ≡ µ . (22)
the parametrization of the Ioffe-Pritchard field Eq. (1), c 8γ2B ρ
namelybyaligningthebiasalongthee -ore -axis. The 0 0 M
x y
Gimbal lock problem is avoided in the quantum descrip- GivenB ,Eq.(22)approximatesthemaximumradiusto
0
tion in the frame Oe1e2e3 by the use of the D-matrices. allowstablelevitation. Suchanapproximatedexpression
is in good agreement with the exact upper border, see
Fig. 3. The left border can be approximated by keeping
IV. LINEAR STABILITY DIAGRAMS terms in ∆ = 0 of zero order in ω /ω (cid:28) 1 and of
T Z D
highest order in ω /ω (cid:29) 1, which is justified in the
I D
Using the criterion derived in Sec. III, let us now an- sH√M regime for R→0, see Fig. 1.b. This leads to ωL =
alyze the linear stability of the nanomagnet at the equi- 3 3ωT/2, which using Table II, reads
librium point illustrated in Fig. 1.a (nanomagnet at the
centerofthetrapanti-alignedtothelocalmagneticfield) B ≡3(cid:18) ρµB02 (cid:19)13 , (23)
as a function of the physical parameters given in Table I c1 4γ2ρ
0 M
and the rotation frequency ω .
S where we neglected the contribution B00 in ω , since
As shown below the stability of the system depends T
B00B /B02 (cid:28)1. This approximates the minimum B for
very much on the size of the magnet, parametrized by 0 0
stable levitation in the EdH phase. As shown in Fig. 3,
ω , the local magnetic field strength, parametrized by
I
this gives a good estimation of the left border. Plugging
ω , and the magnetic rigidity given by the magnetic
L
Eq. (23) into Eq. (22) one obtains an approximated ex-
anisotropy energy, parametrized by ω . In particular,
D pressionfortheradiusR? ofthelargestnanomagnetthat
we distinguish the following three regimes: (i) the small
can be stably levitated in the non-rotating EdH phase.
hard magnet (sHM) regime, ω ,ω (cid:29) ω , (ii) the soft
I D L
Note that these expressions explain the dependence of
magnet regime (SM), ω (cid:28)ω , and (iii) the large hard
D L the EdH phase on the field gradient B0 and the uniaxial
magnet (lHM) regime, ω (cid:29)ω (cid:29)ω .
D L I
anisotropy strength k shown in Fig. 4.
We present the results in a two-dimensional phase di- a
In particular, note that the EdH phase is nearly inde-
agram with the x-axis given by the bias field B and the
0 pendent of k provided the condition ω (cid:29) ω holds.
y-axis given by the radius of the nanomagnet R. Results a D L
Therefore, one can describe this regime with a simpli-
areshowninFig.2. NotethatinthesHM(lowerleftcor-
fied model assuming k → ∞ (perfect hard magnet),
ner) and SM (right part) regimes two stable phases are a
which corresponds to the magnetic moment frozen along
present for a non-rotating nanomagnet (ω = 0) (cen-
S e (rightest panel in Fig. 4.a). In this limit, the Hamil-
tral panel). In the lHM regime (upper left corner) on 3
tonian of the system reads
the other hand, stable levitation is possible only for a
mechanically rotating nanomagnet (ωS 6= 0). As argued pˆ2 (cid:126)2 (cid:16) (cid:17)
below these three stable phases have a different physical HˆsHM = 2M + 2I Jˆ2+2SJˆ3 +(cid:126)γ0SB3(ˆr,Ωˆ). (24)
originandrepresentthreedifferentloopholesintheEarn-
shawtheorem, theEinstein-deHaas(EdH)loophole, the Eq. (24) is obtained from Eq. (6) projecting the spin de-
atom (A) loophole, and Levitron (L) loophole. grees of freedom on the eigenstate |Si of Sˆ , where S
3
7
50
40
30
m]
n 20
[
R
10
0.1 1 10 102 103 0.1 1 10 102 103 0.1 1 10 102 103 0.1 1 10 102 103 0.1 1 10 102 103
B [mT] B [mT] B [mT] B [mT] B [mT]
0 0 0 0 0
FIG. 2. From left to right stability diagrams for ω /ω = −0.2, −0.02, 0, 0.02, 0.2, where ω /(2π) ≈ 5×108 Hz is ω for a
S 0 0 I
nanomagnetofradiusR=1nm. OtherphysicalparametersaretakenfromthecaptionofTableI.Stablephasesareillustrated
inred(EdHphase),blue(Aphase),andwhite(Lphase). ω >0(ω <0)correspondstoclockwiserotation(counterclockwise
S S
rotation).
B B
c1 c2
20 TABLE III. Coefficients of the stability polynomial P (λ) in
T
the sHM, lHM and SM regime.
sHM (asHM) SM (aSM) lHM (alHM)
i i i
15 a0 −ωIωLωT2 −ωLωT2 −ωIωLωT2
a iω2(ω +ω )/2 iω2/2 iω2ω /2
1 Z S I Z Z S
a −ω ω −ω2/2 −ω −ω ω −ω2/2
] 2 I L Z L I L Z
m
a −i(ω +ω ) −i −iω
n 3 S I S
[ 10 a 1 0 1
R 4
5 R? case) but to a simplified PT(λ) given by
Rc
PsHM(λ)=asHM+asHMλ+asHMλ2+asHMλ3+asHMλ4,
T 0 1 2 3 4
(26)
0.1 1 10 102 103 whosecoefficientsaregiveninTableIII.Thisleadstothe
stability diagram shown in the rightest panel of Fig. 4.a.
B [mT]
0 Note that PsHM(λ) is of fourth order since the magne-
T
tization is frozen along e and hence there are only 10
3
FIG. 3. Stability diagram for a non-rotating nanomagnet
independent parameters.
(ω = 0). Other physical parameters are given in the cap-
S
tion of Table I. The approximated borders of the red EdH
phase (blue A phase) are illustrated with red (blue) dashed
lines. B. Atom phase
In the SM regime, where ω (cid:29) ω (see Fig. 1.b), the
is the largest value for the spin projection along e . In L D
3
coupling between the magnetization and the anisotropy
theclassicaldescription,thislimitcorrespondstotheLa-
isnegligible. Inthisregime,themagneticmomentunder-
grangian
goes a free Larmor precession about the local magnetic
I h i field. This stabilizes the system in full analogy to mag-
L = α˙2+β˙2+(γ˙ +ω )2+2(γ˙ +ω )α˙ cosβ
sHM 2 I I netic trapping of neutral atoms [8, 9].
+ M (cid:0)x˙2+y˙2+z˙2(cid:1)+µ(cid:2)B (r)cosαsinβ The borders of the A phase are approximately inde-
2 x pendent on the rotational state of the nanomagnet ωS,
+B (r)sinαsinβ+B (r)cosβ(cid:3), as shown in Fig. 2. Therefore, considering the case of a
y z
non-rotating nanomagnet, they can be analytically ap-
(25)
proximated as follows, see Fig. 3. The left border at
where we set θ = β and φ = α. The linear stability lowmagneticfieldscanbeapproximatedbykeepingonly
analysis in this limit leads to P (λ) (as in the general terms in ∆ =0 up to zero order in ω /ω (cid:28)1 and up
Z T Z L
8
(a) 20
15
m]
n 10
[
R
5
(b) 20
15
m]
n 10
[
R
5
0.1 1 10 102 103 0.1 1 10 102 103 0.1 1 10 102 103 0.1 1 10 102 103 0.1 1 10 102 103
B [mT] B [mT] B [mT] B [mT] B [mT]
0 0 0 0 0
FIG. 4. From left to right stability diagrams for a non-rotating nanomagnet (ω = 0) for (a) k [J/m3] = 0,103,104,105,∞
S a
and for (b) B0[T/m]=102,103,104,105,106. Other physical parameters are given in the caption of Table I. Stable phases are
illustrated in red (EdH phase) and blue (A phase).
to leading order in ω /ω (cid:28)1 and ω /ω (cid:28)1, which is coincideswithEq.(23). NotethatPSM(λ)isonlyathird
I L T L T
well justified in the SM regime at R → ∞, see Fig. 1.b. orderpolynomialbecausetherotationaldynamicsdonot
This leads to the condition ω = 2ω , which using Ta- affect the stability of the system. The only relevant de-
L D
ble II reads grees of freedom for the stability are thus the magnetic
moment and the center-of-mass motion (8 independent
2k
B ≡ a. (27) parameters).
c2 ρ
µ
B approximates the lowest field bias for which stable
c2
C. Levitron phase
levitation is possible in the A phase, see Fig. 3. The A
phaseextendsuptothefieldbiasB =2B02/B00, above
c3
which ω becomes imaginary. This is shown in the left- In the lHM regime, the magnetic moment can be con-
T
most diagram in Fig. 4.b, while in the remaining panels sidered to be fixed along the easy axis (ωD (cid:29) ωL) and
it falls out of the B interval shown. Note that there is the contribution of the spin to the total angular momen-
0
noupperlimitinR fortheAphase. Howeverrecallthat tum can be neglected due to the large dimension of the
our model assumes a single magnetic domain nanomag- nanomagnet (ωL (cid:29) ωI), see Fig. 1.b. In this respect,
net, which most materials can only sustain for sizes up the nanomagnet behaves in good approximation like a
to hundred nanometers [24]. Note that the dependence classical Levitron. The dynamics in this regime can be
of B and B on the field gradient B0 and the uniaxial approximately described by the Hamiltonian
c2 c3
anisotropystrengthk explainsthequalitativebehaviour
a
of the A phase in Fig. 4. Hˆ = pˆ2 + (cid:126)2Lˆ2−µe (Ωˆ)·B(ˆr), (29)
Inthelimitofavanishingmagneticanisotropy,k =0, lHM 2M 2I 3
a
theHamiltonianofthenanomagnetreadsHˆ =Hˆ +
SM AT
(cid:126)2Lˆ2/2I, where HˆAT =pˆ2/2M −(cid:126)γ0Fˆ ·B(r) represents which is obtained from Hˆ by taking the limit ka → ∞
the Hamiltonian describing a single magnetic atom of (magnetization frozen along the anisotropy axis) and by
mass M and spin F in the external field B(r) [8, 9]. In using µˆ = µe (Ωˆ). The latter treats the magnetization
3
the same limit, the system is described classically by the classicaly,namelyµisascalarquantityinsteadofaquan-
Lagrangian L obtained from L by setting V = 0. In tum spin operator. The classical description is given in
SM a
this limit, the linear stability analysis leads to P (λ) (as this limit by the Lagrangian
Z
in the general case) and to
L =I (cid:0)α˙2+β˙2+γ˙2+2α˙γ˙ cosβ(cid:1)
PSM(λ)=aSM+aSMλ+aSMλ2+aSMλ3, (28) lHM 2
T 0 1 2 3 (30)
+ M (cid:0)x˙2+y˙2+z˙2(cid:1)+µ·B(r),
whose coefficients given in Table III are, as expected, in-
2
dependent on ω and ω , namely on the rotational state
S I
of the nanomagnet. This leads to the stability diagram where µ=µ(cosαsinβ,sinαsinβ,cosβ) for a magnetic
shownintheleftmostpanelinFig.4.a,whoseleftborder moment frozen along the anisotropy axis.
9
The linear stability analysis applied to this limit leads along the local direction of the magnetic field. The sta-
to the polynomials P (λ) (as in the general case) and bility mechanism is thus fully analogous to the magnetic
Z
PlHM(λ)=alHM+alHMλ+alHMλ2+alHMλ3+alHMλ4, trapping of neutral atoms [8, 9]. The Einstein-de Haas
T 0 1 2 3 4
where its coefficients are defined in Table III. The lin- EdHphasearisesatalowmagneticbiasfield(ω (cid:28)ω ),
L D
ear stability diagram derived from P (λ) and PlHM(λ) wheretheuniaxialmagneticanisotropyinteractiondom-
Z T
corresponds to the L phase of the lHM regime in Fig. 2, inates the magnetization’s dynamics. The magnetic mo-
thusshowingthatstablelevitationinthisregimerequires ment is thus frozen along the easy axis and can be mod-
mechanical rotation. Furthermore, in this limit the sta- eled as µˆ =−(cid:126)γ [Fˆ·e (Ωˆ)]e (Ωˆ). In this case the quan-
0 3 3
bilityregionis symmetric withrespect toclock-orcoun- tumspinoriginofµiscrucialtostabilizethelevitationof
terclockwise rotation, as in the classical Levitron [4–6]. asmallnanomagnetthroughtheEinstein-deHaaseffect.
To conclude this section, let us compare the descrip- Asthesizeofthenanomagnetincreases,thecontribution
tion of the magnetic moment in the approximated mod- of the spin angular momentum becomes negligible due
elsofthelHMandthesHMregimes. ThelHMandsHM totheincreasingmoment-of-inertia-to-magnetic-moment
bothdescribeananomagnetwithalargemagneticrigid- ratio and the classical Levitron behavior is recovered.
ity whose magnetic moment can be approximated to be To derive these results, we assumed a: (i) single mag-
frozen along the easy axis e3. In the lHM regime, due to netic domain, (ii) macrospin approximation, (iii) rigid
the negligible role played by the macrospin angular mo- body, (iv) sphere, (v) uniaxial anisotropy, (vi) Ioffe-
mentum (ωI (cid:28) ωL), the magnetic moment is modeled Pritchard magnetic field, (vii) point-dipole approxima-
as µˆ = µe3(Ωˆ), where µ is a classical scalar quantity. tion, (viii) absence of gravity, and (ix) dissipation-free
In the sHM regime, on the other hand, the role of the dynamicsforthesystem. Whilenotaddressedinthisar-
spin angular momentum is crucial (ωI (cid:29) ωL) , and the ticle, it would be very interesting to relax some of these
quantum origin of the nanomagnet’s magnetic moment assumptions and study their impact on the stability di-
has to be taken into account. The magnetic moment agrams. For instance, levitating a multi-domain magnet
is thus given by µˆ = (cid:126)γ0[Fˆ ·e3(Ωˆ)]e3(Ωˆ). This crucial could allow to study the effects of the interactions be-
difference is manifested in the coefficients of the char- tweendifferentdomainsonthestabilityofthesystem. It
acteristic polynomial, see Table III. In the sHM regime wouldbeparticularlyinterestingtoexploreiftheAphase
the rotational frequency ωS is shifted by ωI, thus retain- persists for a macroscopic multi-domain magnet at suffi-
ing the contribution of the spin angular momentum Fˆ to cientlyhighmagneticfields. Inthisscenarioanddepend-
the total angular momentum of the system. In essence, ingonthesizeofthemagnet,notonlyassumption(i)and
the quantum spin origin of the magnetization plays the (ii),butalso(iii),(v),(vii),(viii)and(ix)shouldbecare-
same role as mechanical rotation, a manifestation of the fullyrevisited. Onecouldusetheexquisiteisolationfrom
Einstein-de Haas effect. the environment obtained in levitation in high vacuum
tostudyin-domainspindynamicsbeyondthemacrospin
approximation. Generalization to different shapes and
V. CONCLUSIONS magnetocrystalline anisotropies would allow to investi-
gate the shape-dependence of the stable phases, as done
for the Levitron [4]. In particular, one could explore the
In conclusion, we discussed the linear stability of a
presenceofmulti-stabilitywithothermagnetocrystalline
single magnetic domain nanosphere in a static external
anisotropies that contain more than a single easy axis.
Ioffe-Pritchardmagneticfieldattheequilibriumpointil-
Levitationindifferentmagneticfieldconfigurations,such
lustrated in Fig. 1.a. This corresponds to a nanomag-
as quadrupole fields, might be used to further study the
net at the center of the field, with the magnetic mo-
role of B (crucial for the levitation of neutral magnetic
ment parallel to the anisotropy axis, anti-aligned to the 0
atoms [8, 9]) in the levitation of a nanomagnet. In par-
magnetic field, and mechanically spinning with a fre-
ticular, to discern whether stable levitation can occur in
quency ω . We derived a stability criterion given by
S
a position where the local magnetic field is zero. The
the roots of both a second order polynomial P (λ) and
Z
effect of noise and dissipation on the stability of the sys-
of a fifth order polynomial P (λ). Eigenvalues with
T
temmightnotonlyenrichthestabilitydiagram,butalso
zero (non-zero) real component correspond to stability
play a crucial role in any experiments aiming at control-
(instability). This stability criterion is derived both
ling the dynamics of a levitated nanomagnet [16]. We
with a quantum description and a (phenomenological)
remark that linear stability is a necessary but not suffi-
classical description of the nanomagnet. Apart from
cientconditionforthestabilityofthesystematlongtime
the known gyroscopic-stabilized levitation (Levitron L
scales. Athoroughanalysisofthestabilityofananomag-
phase), we found two additional stable phases, arising
net in a magnetic field under realistic conditions might
from the quantum mechanical origin of the magnetiza-
tion, µˆ = (cid:126)γ Fˆ, which surprisingly (according to Earn- demand to consider non-linear dynamics.
0
shaw’s theorem) allow to stably levitate a non-rotating Toconclude,weremarkthatonecouldconsidertocool
magnet. The atom A phase appears at a high magnetic thefluctuationsofthesysteminthestablephasestothe
bias field (ω (cid:29) ω ), where despite the magnetocrys- quantum regime [16]. The degrees of freedom of the sys-
L D
talline anisotropy the magnetic moment freely precesses tem could then be described as coupled quantum har-
10
monic oscillators using the bosonization tools given in quantum regime.
[18]. One could then study the quantum properties (en- We thank K. Kustura for carefully reading the
tanglement and squeezing) of the relevant eigenstates of manuscript. This work is supported by the European
the quadratic bosonic Hamiltonian [25], and exploit the Research Council (ERC-2013-StG 335489 QSuperMag)
richphysicsofmagneticallylevitatednanomagnetsinthe and the Austrian Federal Ministry of Science, Research,
and Economy (BMWFW).
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