Influence of quadratic Zeeman effect on spin waves in dipolar lattices
aa r X i v : . [ c ond - m a t . o t h e r] J un Influence of quadratic Zeeman effect on spin waves indipolar lattices
V.I. Yukalov , , ∗ and E.P. Yukalova Bogolubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia Instituto de Fisica de S˜ao Carlos, Universidade de S˜ao Paulo,CP 369, S˜ao Carlos 13560-970, S˜ao Paulo, Brazil Laboratory of Information Technologies,Joint Institute for Nuclear Research, Dubna 141980, Russia ∗ Corresponding author E-mail address: [email protected] (V.I. Yukalov)
Abstract
A lattice of particles with dipolar magnetic moments is considered under the pres-ence of quadratic Zeeman effect. Two types of this effect are taken into account, theeffect due to an external nonresonant magnetic field and the effect caused by an alternat-ing quasiresonance electromagnetic field. The presence of the alternating-field quadraticZeeman effect makes it possible to efficiently vary the sample characteristics. The mainattention is payed to the study of spin waves whose properties depend on the quadraticZeeman effect. By varying the quadratic Zeeman-effect parameter it is possible to eithersuppress or stabilize spin waves.
Keywords : Dipolar lattices, Quadratic Zeeman effect, Spin waves
Declaration of interests : none 1
Introduction
Spin waves is one of the important characteristics of magnetic materials, providing us informa-tion on the latter. Also, spin waves have been found to be an essential ingredient in magnonspintronics [1]. One usually studies spin waves in materials with magnetic exchange interac-tions, such as ferromagets, ferrimagnets, and antiferromagnets [2]. In these materials withself-organized magnetic order, spin waves can arise in the absence of external fields. Whilethe existence of spin waves in materials with dipolar interactions requires a sufficiently strongexternal magnetic field.There exist many materials, whose constituents interact through magnetic dipolar forces.Such lattices can be formed by magnetic nanomolecules [3–8], magnetic nanoclusters [9–11],magnetic particles inserted into a nonmagnetic matrix [12, 13]. Different dipolar atoms andmolecules can be arranged in self-assembled lattices or can form lattice structures with the helpof superimposed external fields [14–18]. Many biological systems contain molecules interactingthrough dipolar forces [19,20]. Numerous polymers are composed of nanomolecules with dipolarinteractions [21].In some cases, the existence of spin waves in dipolar systems can be due, in addition toa sufficiently strong external magnetic field, to the arising quadratic Zeeman effect. It isimportant to distinguish two types of the latter.One is the standard external-field quadratic Zeeman effect that appears, when atoms ormolecules possess hyperfine structure [22, 23]. This effect, that has been described in extensiveliterature [24–28], induces in an atom with a dipole magnetic moment m S , the quadratic Zeemanenergy shift ∓ ( B · m S ) ∆ W (1 + 2 I ) , proportional to the square of an external magnetic field B , where ∆ W is the hyperfine energysplitting, I is nuclear spin, and the sign minus or plus corresponds to the parallel or, respectively,antiparallel alignment of the nuclear and total electronic spin projections in the atom. The effectcan appear in any atom or molecule, whose nucleus possesses a nonzero spin. For example, inspinor atomic systems (not necessarily condensed) [18, 29] and in magnetic admixtures insidenonmagnetic matrix [13, 30, 31]. Quantum dots possess the properties similar to atoms andmolecules [32], because of which quantum dots can also display quadratic Zeeman effect [33].The other type of the effect is quasiresonance alternating-field quadratic Zeeman effect,due to the alternating-current Strak shift, induced either by applying off-resonance linearlypolarized light exerting the quadratic shift along the polarization axis [34–37] or by acting witha linearly polarized microwave driving field [38–40].A linearly polarized microwave driving field generates the quasiresonance quadratic Zeemaneffect by inducing hyperfine transitions in an atom [38–40]. While the alternating quasireso-nance polarized light generates quadratic Zeeman effect by inducing transitions between theinternal spin states and it exists even if the atom does not enjoy hyperfine structure [38–40].Both these alternating-current effects can be tailored with a high resolution and rapidlyadjusted, producing a quadratic Zeeman shift described by the ratio − ~ Ω / B that is non-resonant withrespect to internal atomic transitions, but depends only on the intensity and detuning of thequasi-resonance driving field. By employing either positive or negative detuning, the sign of2he above characteristic ratio can be varied. Below we accept that the linear polarization ofthe alternating quasiresonance field is chosen along the z -axis.Thus, the alternating-current quasiresonant quadratic Zeeman effect can be realized in anyatom, molecule, or quantum dot having nonzero spin. Therefore the parameters of the materialscan be varied in a wide range. The theory and experimental realization of the quasiresonantquadratic Zeeman effect have been expounded in many publications, e.g., in [34–40]. It isimportant that this effect can be generated in atoms without hyperfine structure, but merelyhaving nonzero total electronic spins [34, 36, 37].The quadratic Zeeman effect can be not very important for hard magnetic materials withexchange interactions. But it can essentially influences the properties of dipolar materials,whose constituents interact through dipolar forces that are much weaker than those due toexchange interactions. This is why we study here dipolar lattices, whose characteristics canbe regulated by quadratic Zeeman effect. The main attention is payed to the influence of thiseffect on spin waves. We show that by varying the parameters of the quadratic Zeeman effectspin waves can be either suppressed or stabilized. The system Hamiltonian is a sum of two terms corresponding to the Zeeman, ˆ H Z , and dipolar,ˆ H D , terms, ˆ H = ˆ H Z + ˆ H D . (1)These terms can be written in the following form [18, 29]. The Zeeman part includes the linearZeeman energy and quadratic Zeeman energy terms,ˆ H Z = X j (cid:2) − µ S B · S j + Q Z ( B · S j ) + q Z ( S zj ) (cid:3) . (2)Here µ S ≡ − g S µ B , with g S being the g -factor related to spin S and µ B is the Bohr magneton. B is a static external magnetic field along the z -axis, B = B e z . (3)The parameter of the non-resonant magnetic-field induced Zeeman effect is Q Z = ∓ µ S ∆ W (1 + 2 I ) , (4)where ∆ W is the hyperfine energy splitting and I is nuclear spin.Alternating quasiresonance fields generate quadratic Zeeman effect either by acting on theatom by a linearly polarized microwave driving field inducing hyperfine transitions [38–40] orby applying off-resonance linearly polarized light populating internal spin states of the atomand inducing the quadratic Zeeman shift along the polarization axis [34–37]. The last type ofthe effect exists even if the atom does not enjoy hyperfine structure. Both these methods canbe rapidly adjusted, producing a quadratic Zeeman shift described by the parameter q Z = − ~ Ω , (5)3here Ω is the driving Rabi frequency and ∆ is the detuning from an internal (spin or hyperfine)transition. This parameter does not depend on the external magnetic field B that is non-resonant with respect to internal atomic transitions, but depends only on the intensity anddetuning of the quasi-resonance driving field. By employing either positive or negative detuning,the sign of q Z can be varied. Here the linear polarization of the alternating quasiresonance fieldis chosen along the z -axis.Dipolar spin interactions are characterized by the Hamiltonianˆ H D = 12 X i = j X αβ D αβij S αi S βj , (6)in which D αβij is a dipolar interaction potential.In some cases, one needs to take into account finite sizes of molecules and their mutualcorrelations. Then, as suggested by Jonscher [41–43], it is possible to include the screeningthat can be characterized by an exponential function [41–45]. Therefore, for generality, wekeep in mind the regularized dipolar interaction potential D αβij = µ S r ij (cid:16) δ αβ − n αij n βij (cid:17) exp( − κ r ij ) , (7)where r ij ≡ | r ij | , n ij ≡ r ij r ij , r ij ≡ r i − r j . Although this regularization is not principal for what follows.If necessary, one can estimate the interaction screening parameter κ = 1 /r s , where r s isthe screening radius, from the equality of the effective energy of spin interactions and of theeffective kinetic energy, ρµ S S = ~ / mr s , where ρ is average spin density. This gives r s = ~ p mρµ S S . (8)For example, if the spin density is ρ ∼ cm − , hence the mean interspin distance is a ≈ ρ − / ∼ − cm, and µ S S ∼ µ B , so that the effective spin interaction energy is ρµ S S ∼ − erg, then the screening radius r s ∼ − cm is close to the mean distance a .But, as is stressed above, when one does not need to consider correlation effects, one canset κ to zero. So, for what follows the existence of screening is not important and is kept onlyfor generality.Thus the Zeeman Hamiltonian can be represented asˆ H Z = X j (cid:2) − µ S B S zj + Q ( S zj ) (cid:3) , (9)where the effective parameter of the quadratic Zeeman effect is Q ≡ Q Z B + q Z . (10)Invoking the ladder operators S ± j = S xj ± iS yj , the dipolar Hamiltonian can be written inthe formˆ H D = 12 X i = j (cid:20) a ij ( S zi S zj − S + i S − j ) + b ij S + i S + j + b ∗ ij S − i S − j + 2 c ij S + i S zj + 2 c ∗ ij S − i S zj (cid:21) , (11)4n which the notations are used: a ij ≡ D zzij = µ S r ij (cid:2) − n zij ) (cid:3) exp( − κ r ij ) ,b ij ≡ (cid:0) D xxij − D yyij − iD xyij (cid:1) = − µ S r ij (cid:0) n xij − in yij (cid:1) exp( − κ r ij ) ,c ij ≡ (cid:0) D xzij − iD yzij (cid:1) = − µ S r ij (cid:0) n xij − in yij (cid:1) n zij exp( − κ r ij ) . (12)At short distance, dipolar interactions enjoy the natural conditions excluding self-interactions, D αβjj ≡ , a jj ≡ b jj ≡ c jj ≡ . For a large lattice, where the boundary effects can be neglected, one has X j D αβij = 0 , (13)hence the interaction terms (12) satisfy the equalities X j a ij = X j b ij = X j c ij = 0 . The quantities ξ i ≡ ~ X j (cid:0) a ij S zj + c ij S + j + c ∗ ij S − j (cid:1) ,ϕ i ≡ ~ X j (cid:16) a ij S − j − b ij S + j − c ij S zj (cid:17) (14)play the role of the local fields acting on spins. For an ideal lattice, these quantities are zerocentered, so that h ξ j i = h ϕ j i = 0 , (15)which follows from equation (13). The equations of motion for the spin operators read as dS − j dt = − i ( ω + ξ j ) S − j − iϕ j S zj − i Q ~ (cid:0) S − j S zj + S zj S − j (cid:1) ,dS zj dt = − i (cid:0) ϕ + j S − j − S + j ϕ j (cid:1) , (16)where the Zeeman frequency is ω ≡ − µ S B ~ > . (17)5pin waves are defined as small spin fluctuations around the average spin values h S αj i . Thisaverage can correspond to a stationary or quasistationary state. Quasistationary are the stateswith lifetime longer then the oscillation time 2 π/ω . Following the standard technique [46], werepresent the spin operators in the form S αj = h S αj i + δS αj , (18)in which δS αj is a small deviation from the average h S αj i . In the stationary state, the spins areassumed to be directed along the field B , that is, along the z -axis, so that h S ± j i = 0 , h S zj i 6 = 0 . (19)Then the local fields (14) become ξ i ≡ ~ X j (cid:0) a ij δS zj + c ij δS + j + c ∗ ij δS − j (cid:1) ,ϕ i ≡ ~ X j (cid:16) a ij δS − j − b ij δS + j − c ij δS zj (cid:17) . (20)Representation (18) is substituted into equations (16) that are linearized with respect tothe deviations δS αj , taking into account that S − j = δS − j , according to equations (18) and (19).For the single-site expression in the right-hand site of the first of equations (16), we use theform S − j S zj + S zj S − j = (cid:18) − S (cid:19) h S zj i S − j (21)that is exact for spin one-half and is asymptotically exact for spin S → ∞ , as is explained inRefs. [6, 7, 47]. Thus we come to the equations ddt S − j = − ω s S − j − iϕ j h S zj i , ddt δS zj = 0 , (22)in which ω s ≡ ω + (cid:18) − S (cid:19) Q ~ h S zj i (23)is the effective frequency of spin rotation. With the initial condition δS zj (0) = 0, one has δS zj ( t ) = 0. Then the local fields (20) take the form ξ i ≡ ~ X j (cid:0) c ij S + j + c ∗ ij S − j (cid:1) , ϕ i ≡ ~ X j (cid:16) a ij S − j − b ij S + j (cid:17) . (24)Let us define the Fourier transforms for the spin operators, S ± j = X k S ± k exp( ∓ i k · r j ) , S ± k = 1 N X j S ± j exp( ± i k · r j ) (25)and for the interaction terms a ij = 1 N X k a k exp( i k · r ij ) , a k = X j a j exp( − i k · r ij ) . (26)6he Fourier transform for b ij is defined similarly to Eq. (26).In this way, we come to the equation ddt S − k = − iA k S − k + iB k S + k , (27)where A k ≡ ω s + a k ~ h S zj i , B k ≡ b k ~ h S zj i . (28)We may notice that A k is real, since a ij = a ji = a ∗ ij .Looking for the solution in the form S − k = u k e − iω k t + v ∗ k e iω k t , we get the eigenvalue equations A k u k − B k v k = ω k u k , B ∗ k u k − A k v k = ω k v k . From here we find the spectrum of spin waves ω k = q A k − | B k | . (29)In the long-wave limit, the spectrum is quadratic, ω k ≃ | ω s | − h S zj i ~ ω s X h ij i a ij ( k · r ij ) , (30)where k → For concreteness, let us consider a cubic lattice with the side a . Then for each lattice site thereare six nearest neighbors, so that the unit vector n ij for six values of j , corresponding to thenearest neighbors to a site i , has the following components n xij = { , − , , , , } , n yij = { , , , − , , } ,n zij = { , , , , , − } . (31)Then the Fourier transforms for the interaction terms, defined in Eq. (26), become a k = 2 ρµ S [ cos( k x a ) + cos( k y a ) − k z a ) ] ,b k = − ρµ S [ cos( k x a ) − cos( k y a ) ] , (32)and c k = 0.It is convenient to pass to the dimensionless expression of the spin-wave spectrum ω ( p ) ≡ ω k ω (33)7hat is a function of the dimensionless momentum p ≡ k a ( p α ≡ k α a ) . (34)Also, let us introduce the dimensionless parameter of the quadratic Zeeman effect ζ ≡ Q ~ ω (35)and the dimensionless strength of dipolar interactions γ D ≡ ρµ S ~ ω . (36)For the spin-rotation frequency (23), we have ω s ω = 1 + (cid:18) − S (cid:19) ζ S , (37)where S ≡ h S zj i . (38)And instead of A k and B k , we introduce the dimensionless quantities α p ≡ A k ω = 1 + (cid:18) − S (cid:19) ζ S + γ D S (cos p x + cos p y − p z ) ,β p ≡ B k ω = − γ D S (cos p x − cos p y ) . (39)The eigenvalue equations for the spin-wave spectrum take the form α p u p − β p v p = ω ( p ) u p , β p u p − α p v p = ω ( p ) v p , (40)which gives the spectrum ω ( p ) = q α p − β p . (41)When the momentum p is along the external magnetic field, such that p = p z e z , then β p = 0 and the eigenvalue equations (40) reduce to α p u p = ω ( p ) u p , α p v p = − ω ( p ) v p . These equations do not possess nontrivial solutions for u p and v p , which implies that spin wavesdo not propagate along the direction of the external magnetic field B .For the transverse propagation with respect to the external magnetic field, we can set p x = p , p y = p z = 0 . (42)Then functions (39) read as α p = 1 + (cid:18) − S (cid:19) ζ S − γ D S (1 − cos p ) , β p = 3 γ D S (1 − cos p ) . (43)The long-wave limit of the spin-wave spectrum (41) is ω ( p ) ≃ | C | (cid:18) − γ D S C p (cid:19) ( p → , (44)where C ≡ (cid:18) − S (cid:19) ζ S . (45)8 Stability conditions
A well defined spectrum of stable spin waves presupposes that it is non-negative: ω ( p ) ≥ . (46)Otherwise, when it is complex, spin waves are not stable, but decay. Thus, if the expressionunder the square root in Eq. (41) is negative, then the spectrum becomes imaginary, such that ω k = i | ω k | . Then spin waves are described by the operator S − k = u k e | ω k | t + v ∗ k e −| ω k | t showing that the spin-wave stability is lost after the time 2 π/ω k .To be defined as a real quantity, spectrum (41) requires that the expression under the squareroot be non-negative, which implies the stability condition (cid:18) C − β p (cid:19) (cid:18) C + 23 β p (cid:19) ≥ . (47)For a more detailed investigation of stability, we need to specify the average spin S . Inthe state of absolute equilibrium, the latter is defined from the minimization of the system freeenergy. More generally, S can be prepared by polarizing the system at the initial momentof time and then considering the system behavior. Such a setup with a prepared polarizationis very important for studying spin dynamics from an initially prepared state [6, 7, 47]. Thespin motion from a prepared initial state is triggered by spin waves, because of which theexistence of the latter plays a crucial role for spin dynamics. There are two opposite cases ofinitial polarization. One corresponds to an initially polarized state with a positive polarization S >
0, while the second, to an equilibrium state with a negative polarization S < O . Weshall consider both these cases.If the average spin polarization is positive, hence β p is non-negative, then the stabilitycondition (47) is valid when either C ≥ β p ( β p ≥ , (48)or when C ≤ − β p ( β p ≥ . (49)Since these inequalities have to be valid for all p ∈ [ − π, π ], they reduce to the conditionsrequiring that either C ≥ γ D S ( S > , (50)or C ≤ − γ D S ( S > . (51)In dimensional units, this means that either ~ ω + (cid:18) − S (cid:19) QS ≥ ρµ S S , (52)9r ~ ω + (cid:18) − S (cid:19) QS ≤ − ρµ S S , (53)where S >
0. This should be compared with the condition of stability for the case when thequadratic Zeeman effect is absent, ~ ω ≥ ρµ S S ( Q = 0 , S > . (54)The latter condition means that a sufficiently strong external magnetic field, that is much largerthan the effective strength of dipolar interactions, stabilizes spin waves. These cannot exist indipolar systems without such a strong external field.The existence of the quadratic Zeeman effect extends the region of the magnetic-fieldstrength, where spin waves are stable. The external magnetic field can be very small, althoughspin waves perfectly exist, provided that the quadratic Zeeman parameter Q is sufficiently largeand positive in case (52) or sufficiently large by its magnitude and negative in case (53). Inthat sense, the quadratic Zeeman effect stabilizes spin waves. For example, it may happen thatcondition (54) does not hold, hence spin waves do not arise in the absence of the quadraticZeeman effect. But switching on the quadratic Zeeman effect condition (52) may become valid.Then spin waves can exist as stable collective excitations. On the contrary, when condition(54) holds true, spin waves exist without the quadratic Zeeman effect. Then switching on thiseffect corresponding to a negative Q can lead to the situation when neither condition (52) nor(53) are satisfied. This means that spin waves become suppressed.The other situation occurs, if the stationary spin polarization is negative, S <
0, hence β p is nonpositive. In such a case, spin waves are stable if either C ≥ | β p | ( β p ≤
0) (55)or C ≤ − | β p | ( β p ≤ . (56)To be valid for all p ∈ [ − π, π ], these conditions result in the validity of the inequality C ≥ γ D | S | ( S <
0) (57)or, respectively, C ≤ − γ D | S | ( S < . (58)In dimensional units, we have either condition ~ ω − (cid:18) − S (cid:19) Q | S | ≥ ρµ S | S | (59)or ~ ω − (cid:18) − S (cid:19) Q | S | ≤ − ρµ S | S | . (60)And if the quadratic Zeeman effect is absent, then spin waves are stable if ~ ω ≥ ρµ S | S | ( Q = 0 , S < . (61)10gain we see that, depending on the values of the Zeeman frequency ω and the quadraticZeeman effect parameter Q , this effect can either stabilize of suppress spin waves.The conditions of stability for spin waves with respect to the value of the quadratic Zeemaneffect parameter Q are summarized as follows: For a positive polarization, spin waves are stableprovided that either Q ≥ − ~ ω − ρµ S S S − (cid:18) SS (cid:19) ( S >
0) (62)or Q ≤ − ~ ω + 4 ρµ S S S − (cid:18) SS (cid:19) ( S > . (63)And in the case of a negative polarization, spin waves are stable when either Q ≤ ~ ω − ρµ S | S | S − (cid:18) S | S | (cid:19) ( S <
0) (64)or Q ≥ ~ ω + 8 ρµ S | S | S − (cid:18) S | S | (cid:19) ( S < . (65)Recall that the quadratic Zeeman effect parameter Q , as defined in Eq. (10), consists ofa nonresonant field term and of an alternating quasiresonance field term. The latter can bevaried in a rather wide range, because of which the parameter Q is also changeable. In thatway, by varying this parameter Q , one can either stabilize spin waves or suppress them.To be more specific, let us consider the case of spin S = 1, when notation (45) reduces to C = 1 + ζ S ( S = 1) . (66)Setting the spin polarization to be positive S = 1, we find that spin waves are stable wheneither ζ ≥ − γ D ( S = 1) (67)or ζ ≤ − − γ D ( S = 1) . (68)While in the case of the negative polarization S = −
1, the conditions of spin wave stabilitybecome either ζ ≤ − γ D ( S = −
1) (69)or ζ ≥ γ D ( S = − . (70)Here ζ is the dimensionless quadratic Zeeman effect parameter (35) and γ D is the dimensionlessstrength of dipolar interactions (36). To illustrate how the spin-wave spectrum is influenced by the quadratic Zeeman effect, wepresent below numerical calculations for spectrum (41) corresponding to the transverse propa-gation of spin waves, with the dimensionless momentum (42).11igure 1 demonstrates the spin-wave spectrum ω ( p ), under the positive polarization of theaverage spin S = 1 and the dipolar parameter γ D = 0 .
1, for different parameters (35) of thequadratic Zeeman effect. The spectrum is stable if either ζ ≥ − . ζ ≤ − . ζ shift the spectrum up, while negative values of ζ move it down. At a value of ζ = − .
2, the gap disappears. If the value of ζ is diminished further below − .
2, the spectrumbecomes imaginary, hence spin waves become unstable. But for ζ ≤ − . ζ = − . ζ below − . S = −
1, for the strength of dipolar interactions γ D = 0 .
1, and different quadratic Zeeman-effect parameters ζ . The spectrum is stable for ζ ≥ . ζ ≤ . ζ one can make the spectrum gapful or gapless.The feasibility of influencing the properties of spin waves by the quadratic Zeeman effectcan be used for regulating spin dynamics. From equation (27), one sees that the rotation speedof the average spin is influenced by the quadratic Zeeman-effect parameter Q . When the spinsystem is prepared in an initial nonequilibrium (or quasiequilibrium) state, then the velocityof spin motion essentially depends on the strength and the oscillation frequency of spin wavesthat serve as a trigger for starting the spin dynamics [48, 49]. The possibility of regulating spindynamics can be employed in spintronics and in quantum information processing. We have considered a dipolar lattice subject to the action of the usual linear Zeeman effectand also of the quadratic Zeeman effect. The latter can be of two types, the constant-fieldquadratic Zeeman effect and the alternating-current quadratic Zeeman effect. Both these casesare taken into account. The existence of the quadratic Zeeman effect can strongly influencethe properties of spin waves. The feasibility of regulating the strength of this influence makesit possible to vary the spectrum of spin waves and their stability. Since spin waves serve asa triggering mechanism initiating spin rotation in spin systems prepared in a nonequilibriumstate, the regulation of spin-wave properties can be used as a tool for governing spin dynamicsin spintronics and in quantum information processing. This problem of spin dynamics requiresa separate investigation and will be done in a separate paper.As is explained in the Introduction, there exists plenty of atoms or molecules interactingthrough dipolar forces and possessing quadratic Zeeman effect. It is therefore possible to varythe system parameters in a very wide range. In order to illustrate by a particular example thatthe quadratic Zeeman effect can really be sufficiently large, such that it would be feasible to useit for regulating the properties of spin waves, caused by dipolar interactions, let us consider thecase of Cr. This case is interesting, since the nuclear spin of this atom is zero, so that Crdoes not have hyperfine structure, because of which the stationary-field parameter defined inEq. (4), is Q Z = 0. And the alternating-field parameter of the quadratic Zeeman effect, definedin Eq. (5) can be made [35] as large as q Z ∼ ~ / s. Taking for typical dipolar lattices [18, 50]the density of atoms ρ ∼ (10 − ) cm − , and the dipolar magnetic moment µ S ∼ (1 − µ B ,where µ B is the Bohr magneton, we get ρµ S ∼ (1 − ) ~ / s. Then the ratio of the quadratic12eeman parameter (35) to the strength of dipolar interactions (36) is ζγ D = q Z ρµ S ∼ − . Hence the quadratic Zeeman effect can essentially influence the properties of spin waves, eithersuppressing or stabilizing them. 13 eferences [1] H. Yu, J. Xiao, P. Pirro, Magnon spintronics, J. Magn. Magn. Mater. 450 (2018) 1–2.[2] A.I. Akhiezer, V.G. Bariakhtar, S.V. Peletminsky, Spin Waves, Academic, New York, 1967.[3] O. Kahn, Molecular Magnetism, VCH, New York, 1995.[4] B. Barbara, L. Thomas, F. Lionti, I. Chiorescu, A. Sulpice, Macroscopic quantum tunnelingin molecular magnets, J. Magn. Magn. Mater. 200 (1999) 167–181.[5] A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorace, A. Cornia, M.A. Novak,C. Paulsen, W. Wernsdorfer, The molecular approach to nanoscale magnetism, J. Magn.Magn. Mater. 200 (1999) 182–201[6] V.I. Yukalov, Superradiant operation of spin masers, Laser Phys. 12 (2002) 1089–1103.[7] V.I. Yukalov and E.P. Yukalova, Coherent nuclear radiation, Phys. Part. Nucl. 35 (2004)348–382.[8] V.I. Yukalov, V.K. Henner, P.V. Kharebov, Coherent spin relaxation in moleculat magnets,Phys. Rev. B 77 (2008) 134427[9] R.H. Kodama, Magnetic nanoparticles, J. Magn. Magn. Mater. 200 (1999) 359–372.[10] G.C. Hadjipanayis, Nanophase hard magnets, J. Magn. Magn. Mater. 200 (1999) 373–391.[11] V.I. Yukalov, E.P. Yukalova, Possibility of superradiance by magnetic nanoclusters, LaserPhys. Lett. 8 (2011) 804–813.[12] V.I. Yukalov, Nonlinear spin dynamics in nuclear magnets, Phys. Rev. B 53 (1996) 9232–9250.[13] G.L. Viali, G.R. Gon¸calves, E.C. Passamani, J.C.C. Freitas, M.A. Schettino, A.Y.Takeuchi, C. Larica, Magnetic and hyperfine properties of Fe P nanoparticles dispersedin a porous carbon matrix, J. Magn. Magn. Mater. 401 (2016) 173–179.[14] A. Griesmaier, Generation of a dipolar Bose-Einstein condensate, J. Phys. B 40 (2007)R91–R134.[15] M.A. Baranov, Theoretical progress in many-body physics with ultracold dipolar gases,Phys. Rep. 464 (2008) 71–111.[16] M.A. Baranov, M. Dalmonte, G. Pupillo, P. Zoller, Condensed matter theory of dipolarquantum gases, Chem. Rev. 112 (2012) 5012–5061.[17] B. Gadway, B. Yan, Strongly interacting ultracold polar molecules, J. Phys. B 49 (2016)152002.[18] V.I. Yukalov, Dipolar and spinor bosonic systems, Laser Phys. 28 (2018) 053001.1419] L.F. Cameretti, Modeling of Thermodynamic Properties in Biological Solutions, Cuviller,G¨ottingen, 2009.[20] T.A. Waigh, The Physics of Living Processes, Wiley, Chichester, 2014.[21] W. Barford, Electronic and Optical Properties of Conjugated Polymers, Oxford University,Oxford, 2013.[22] G.K. Woodgate, Elementary Atomic Structure, Oxford University, Oxford, 1999.[23] W. Demtr¨oder, Molecular Physics, Wiley, Berlin, 2005.[24] F. A. Jenkins, E. Segre, The quadratic Zeeman effect, Phys. Rev. 59 (1939) 52–58.[25] L. I. Schiff, H. Snyder, Theory of the quadratic Zeeman effect, Phys. Rev. 59 (1939) 59–62.[26] J. Killingbeck, The quadratic Zeeman effect, J. Phys. B 12 (1979) 25–30.[27] S.L. Coffey, A. Deprit, B. Miller, C.A. Williams, The quadratic Zeeman effect in moderatelystrong magnetic fields, New York Acad. Sci. 497 (1987) 22–36.[28] K.T. Taylor, M.H. Nayfeh, C.W. Clark, eds., Atomic Spectra and Collisions in ExternalFields, Plenum, New York, 1988.[29] D.M. Stamper-Kurn, M. Ueda, Spinor Bose gases: symmetries, magnetism, and quantumdynamics, Rev. Mod. Phys. 85 (2013) 1191–1244.[30] B. Pajot, F. Merlet , G. Taravella, P. Arcas, Quadratic Zeeman effect of donor lines inSilicon and Germanium, Can. J. Phys. 50 (1972) 1106–1113.[31] L. Veissier, C.W. Thiel, T. Lutz, P.E. Barclay, W. Tittel, R.L. Cone, Quadratic Zeemaneffect and spin-lattice relaxation of Tm : YAG at high magnetic fields, Phys. Rev. B 94(2016) 205133.[32] J.L. Birman, R.G. Nazmitdinov, V.I. Yukalov, Effects of symmetry breaking in finite quan-tum systems, Phys. Rep. 526 (2013) 1–91.[33] S.J. Prado, C. Trallero-Giner, A.M. Alcalde, V. Lopez–Richard, G.E. Marques, Magneto-optical properties of nanocrystals: Zeeman splitting, Phys. Rev. B 67 (2003) 165306.[34] C. Cohen-Tannoudji, J. Dupon-Roc, Experimental study of Zeeman light shifts in weakmagnetic fields, Phys. Rev. A 5 (1972) 968–984.[35] L. Santos, M. Fattori, J. Stuhler, T. Pfau, Spinor condensates with a laser-inducedquadratic Zeeman effect, Phys. Rev. A 75 (2007) 053606.[36] K. Jensen, V.M. Acosta, J.M. Higbie, M.P. Ledbetter, S.M. Rochester, D. Budker Cancel-lation of nonlinear Zeeman shifts with light shifts, Phys. Rev. A 79 (2009) 023406.[37] A. de Paz, A. Sharma, A. Chotia, E. Marechal, J. Huckans, P. Pedri, L. Santos, O. Gorceix,L. Vernac, B. Laburthe-Tolra, Nonequilibrium quantum magnetism in a dipolar lattice gas,Phys. Rev. Lett. 111 (2013) 185305. 1538] F. Gerbier, A. Widera, S. Folling, O. Mandel, I. Bloch, Resonant control of spin dynamicsin ultracold quantum gases by microwave dressing, Phys. Rev. A 73 (2006) 041602.[39] S.R. Leslie, J. Guzman, M. Vengalattore, J.D. Sau, M.L. Cohen, D.M. Stamper-Kurn,Amplification of fluctuations in a spinor Bose-Einstein condensate, Phys. Rev. A 79 (2009)043631.[40] E.M. Bookjans, A. Vinit, C. Raman, Quantum phase transition in an antiferromagneticspinor Bose-Einstein condensate, Phys. Rev. Lett. 107 (2011) 195306.[41] A.K. Jonscher, Universal Relaxation Rate, Chelsea Dielectrics, London, 1996.[42] A.K. Jonscher, Dielectric relaxation with dipolar screening, J. Mater. Sci. 32 (1997) 6409–6414.[43] A.K. Jonscher, Low–loss dielectrics, J. Mater. Sci. 34 (1999) 3071–3082.[44] V.E. Tarasov, Universal electromagnetic waves in dielectric J. Phys. Condens. Matter 20(2008) 175223.[45] V.I. Yukalov, Bose-condensed atomic systems with nonlocal interaction potentials, LaserPhys. 26 (2016) 045501.[46] S.V. Tyablikov, Methods in Quantum Theory of Magnetism, Springer, Berlin, 1995.[47] V.I. Yukalov, Nonlinear spin relaxation in strongly nonequilibrium magnets, Phys. Rev. B71 (2005) 184432.[48] V.I. Yukalov, Origin of pure spin superradiance. Phys. Rev. Lett. 75 (1995) 3000–3003.[49] V.I. Yukalov, E.P. Yukalova, Processing information by punctuated spin superradiance.Phys. Rev. Lett. 88 (2002) 257601.[50] B. Gadway, B. Yan, Srongly interacting ultracold polar molecules, J. Phys. B 49 (2016)152002. 16 igure Captions Figure 1 . Dimensionless spectrum of spin waves ω ( p ) as a function of the dimensionlesstransverse momentum p , under the positive average spin polarization S = 1 and the dimen-sionless strength of dipolar interactions γ D = 0 .
1, for different dimensionless parameters of thequadratic Zeeman effect ζ . The spectrum is stable for ζ ≥ − . ζ ≤ − . Figure 2 . Dimensionless spectrum of spin waves as a function of the dimensionless trans-verse momentum, under the negative spin polarization S = − γ D = 0 .
1, for differentquadratic Zeeman-effect parameters ζ . The spectrum is stable for either ζ ≥ . ζ ≤ . p (p) (a) = - 0.2 = 0.2 = 0 = 0.5 = 1 -3 -2 -1 0 1 2 3 p (p) = - 1.4 = - 2 (b) Figure 1: Dimensionless spectrum of spin waves ω ( p ) as a function of the dimensionless trans-verse momentum p , under the positive average spin polarization S = 1 and the dimensionlessstrength of dipolar interactions γ D = 0 .
1, for different dimensionless parameters of the quadraticZeeman effect ζ . The spectrum is stable for ζ ≥ − . ζ ≤ − . p (p) = 1.8 = 3 (a) -3 -2 -1 0 1 2 3 p (p) = - 1 = 0.6 = 0 (b) Figure 2: Dimensionless spectrum of spin waves as a function of the dimensionless transversemomentum, under the negative spin polarization S = − γ D = 0 .
1, for different quadraticZeeman-effect parameters ζ . The spectrum is stable for either ζ ≥ . ζ ≤ ..