Regulating spin reversal in dipolar systems by the quadratic Zeeman effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Regulating spin reversal in dipolar systems by the quadratic Zeeman effect
V.I. Yukalov
1, 2 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
A mechanism is advanced suggesting the resolution of the dichotomy of long-lived spin polar-ization storage versus fast spin reversal at the required time. A system of atoms or molecules isconsidered interacting through magnetic dipolar forces. The constituents are assumed to possessinternal structure allowing for the generation of the alternating-current quadratic Zeeman effect,whose characteristics can be efficiently regulated by quasiresonant dressing. The sample is con-nected to an electric circuit producing a feedback field acting on spins. By switching on and off thealternating-current quadratic Zeeman effect it is possible to realize spin reversals with a requireddelay time. The suggested technique of regulated spin reversal can be used in quantum informationprocessing and spintronics.
I. INTRODUCTION
Dipolar interactions are widespread in nature beingtypical of many biological systems [1, 2], polymers [3],magnetic nanomolecules [4–9] and magnetic nanoclusters[10–13]. Many dipolar atoms and molecules can form self-arranged lattices or can be organized in lattice structureswith the help of external fields [14–18]. Dipolar interac-tions are also typical of ensembles of quantum dots [19]and quantum nanowires [20] that possess many proper-ties similar to atoms, because of which they are oftencalled artificial atoms [21].Here we consider lattices formed by constituents pos-sessing magnetic dipolar moments. These constituentsare supposed to enjoy internal structure that can be usedfor inducing the alternating-current quadratic Zeemaneffect by applying quasiresonant linearly polarized lightpopulating internal spin states [22–25]. The alternating-current quadratic Zeeman effect can also be induced byquasiresonant linearly polarized microwave driving fieldpopulating internal hyperfine states [26–28]. It is impor-tant that the optically induced quadratic Zeeman effectcan also be realized with atoms or molecules without hy-perfine structure. Such a quasiresonant driving exertsquadratic Zeeman shift along the field polarization axis.This shift is described by a parameter q Z that does notdepend on a stationary external field. By using eitherpositive or negative detuning, the sign of the parame-ter can be varied. The optically or microwave inducedquadratic Zeeman effect can be easily manipulated andrapidly adjusted, thus providing an efficient tool for reg-ulating the properties of the sample.One of the properties of spin systems, which is ex-tremely important for spintronics, as well as for quan-tum information processing, is the possibility of fast spinreversal. At the same time, this property is in contra-diction with the other important requirement of beingable to keep for long time a fixed spin polarization. Thisis because one can fix spin polarization for sufficientlylong time by choosing materials with a high magnetic anisotropy. However the latter is the major obstacle forrealizing fast spin reversal. The same dilemma of a wellfixed spin polarization versus fast spin reversal, whicharises in spintronic techniques, also exists in quantuminformation processing, where keeping a fixed spin polar-ization is necessary for creating memory devices, whileone needs fast spin reversal for the efficient functioningof such devices. The proposed devices for realizing quan-tum computing are also based on spin systems [29, 30].Generally, spin reversal in magnetic materials can beinduced by inverting a static external magnetic field [31].However this is a rather slow process requiring sufficientlystrong fields. A faster reversal can be realized by apply-ing alternating electromagnetic fields, such as producedby lasers [32, 33].In the present paper, we advance a novel mechanismthat, from one side, allows us to keep for a long time afixed spin polarization, while, from the other side, pro-vides an efficient tool for realizing a fast spin reversal atany time needed. This mechanism suggests a resolutionof the dilemma of the fixed spin polarization versus fastspin reversal. We show that this can be done for dipolarmagnetic systems by employing the alternating-currentquadratic Zeeman effect. II. SCHEME OF SUGGESTED SETUP
The suggested setup is as follows. A magnetic sam-ple is inserted into a magnetic coil with inductance L ,containing n turns and having length l and cross-sectionarea A coil . The coil is a part of an electric circuit alsoincluding capacity C and resistance R . The coil axis isalong the axis x . A constant external magnetic field B isdirected along the axis z . The moving spins of the mag-netic sample induce in the coil electric current j definedby the Kirchhoff equation L djdt + Rj + 1 C Z t j ( t ′ ) dt ′ = − d Φ dt , (1)in which the electromotive force is caused by the mag-netic flux Φ = 4 πc nA coil η f m x formed by the component of the moving magnetic mo-ment of density m x . Here η f = V /V eff is a filling factorbeing the ratio of the sample volume V to the effectivevolume of the coil V eff . The coil inductance is L = 4 π n A coil c l . The circuit natural frequency, circuit damping, and qual-ity factor are ω = 1 √ LC , γ = R L , Q = ωLR . (2)The electric current of the coil produces the magneticfield H = 4 πncl j (3)directed along the coil axis. This field, being induced bymoving spins, acts back on the spins, because of whichit is called the feedback field. The overall scheme of thesuggested setup is shown in Fig. 1. FIG. 1. Scheme of suggested setup, as is explained in thetext.
III. OPERATOR EQUATIONS OF MOTION
We consider a system of constituents (atoms ormolecules) interacting through dipolar forces. The ad-vantage of dealing with such systems is twofold. Fromone side, as is emphasized in the Introduction, systemswith dipolar interactions are widespread in nature, hencethere exists a variety of materials with rather differentproperties. That is, the system parameters can be var-ied in a wide range. From the other side, dipolar in-teractions are much weaker then exchange interactions,because of which the quadratic Zeeman effect can effec-tively influence the properties of the system. While in hard magnetic materials, such as ferromagnets and anti-ferromagnets, the alternating-current Zeeman effect canbe too weak, as compared to the energy of exchange in-teractions, so that the alternating-current Zeeman effectwould not produce the desired regulation of spin dynam-ics.The Hamiltonian of the dipolar lattice system of N sites, each possessing a total spin S and characterized bythe spin operator S j , with j = 1 , , . . . , N , is the sum ofthe Zeeman term ˆ H Z and the part ˆ H D describing dipolarinteractions. Generally, dipolar lattices can also includesingle-site magnetic anisotropy. So that the total Hamil-tonian is the sumˆ H = ˆ H Z + ˆ H D + ˆ H A . (4)The Zeeman Hamiltonian contains a linear Zeemanterm and a quadratic Zeeman term induced by thealternating-current quasiresonant light [22–25]ˆ H Z = − µ S X j B · S j + q Z X j ( S zj ) , (5)where µ S = − g S µ B , with g S being the spin g -factor and µ B , Bohr magneton, while B is an external magneticfield acting on spins. The parameter q Z of the quadraticZeeman effect, induced by a linearly polarized drivingfield coupling internal states, does not depend on the field B . The axis z is assumed to be the polarization axis ofthe driving field. This parameter q Z , for an alternatingfield that is quasiresonant with an internal transition andthat is linearly polarized along the axis z , can be written(see Appendix A) in the form q Z = − ~ Ω res , (6)where Ω is the driving Rabi frequency and ∆ res is thedetuning from an internal transition related to spin orhyperfine structure. The parameter q Z can be tailored athigh resolution and rapidly adjusted. By applying eitherpositive or negative detuning, the sign of this parametercan be made either positive or negative.The dipolar Hamiltonian reads asˆ H D = 12 X i = j X αβ D αβij S αi S βj , (7)where the dipolar tensor D αβij = µ S r ij (cid:16) δ αβ − n αij n βij (cid:17) exp( − κ r ij ) , (8)generally, includes the screening effect, with the screeningparameter κ . The screening of dipolar forces does existin some materials [34–38], while if it is not important, onecan set κ to zero. The following consideration does notdepend on the existence or absence of screening, whichis mentioned here only for generality. Here r ij ≡ | r ij | , n ij ≡ r ij r ij , r ij = r i − r j . The total external magnetic field B includes a constantfield B directed along the z -axis. And the sample isassumed to be placed inside a magnetic coil of an electriccircuit, so that the coil produces a magnetic feedbackfield H directed along the x -axis, B = B e z + H e x . (9)The single-site magnetic anisotropy term can be writ-ten [39] in the formˆ H A = − X j D ( S zj ) . (10)With the use of the ladder operators S ± j = S xj ± iS yj ,the Zeeman term transforms intoˆ H Z = X j (cid:20) − µ S B S zj − µ S H (cid:0) S + j + S − j (cid:1) + q Z (cid:0) S zj (cid:1) (cid:21) . (11)And the dipolar part becomesˆ H D = 12 X i = j (cid:20) a ij (cid:18) S zi S zj − S + i S − j (cid:19) ++ b ij S + i S zj + b ∗ ij S − i S zj + 2 c ij S + i S zj + 2 c ∗ ij S − i S zj (cid:3) , (12)in which the interaction coefficients are a ij ≡ D zzij , b ij ≡ (cid:0) D xxij − D yyij − iD xyij (cid:1) ,c ij ≡ (cid:0) D xzij − iD yzij (cid:1) . (13)Writing down the equations of motion for the spin op-erators, we introduce the notation for the Zeeman fre-quency ω ≡ − µ S B ~ > . (14)Also we define the quantities ξ i ≡ ~ X j (cid:0) a ij S zj + c ∗ ij S − j + c ij S + j (cid:1) (15)and ϕ i ≡ ~ X j (cid:16) a ij S − j − b ij S + j − c ij S zj (cid:17) (16)describing local dipolar fields acting on spins. And weintroduce the effective force f j ≡ − i (cid:18) µ S H ~ + ϕ j (cid:19) . (17) With the above notations, we obtain the spin equationsfor the transverse spin dS − j dt = − i ( ω + ξ j ) S − j + f j S zj −− i ~ ( q Z − D ) (cid:0) S − j S zj + S zj S − j (cid:1) (18)and for the spin z -component, dS zj dt = − (cid:0) f + j S − j + S + j f j (cid:1) . (19)The spin operators in the Heisenberg representationdepend on time t , which is not explicitly shown for thecompactness of notations. At the initial moment oftime, the sample is assumed to be polarized, so that thestatistical average of the spin z -component is nonzero, h S zj (0) i 6 = 0. IV. DIPOLAR SPIN WAVES
Spin waves are known to exist in ferromagnets andantiferromagnets, where spins interact through exchangeinteractions [40–45]. Here we show that spin waves canalso exist in the systems with pure dipolar interactionsin the presence of quadratic Zeeman effect. These spinwaves are called dipolar, since they arise in a samplewith purely dipolar interactions, without exchange inter-actions.It is necessary to emphasize that the detailed study ofspin waves is not our aim here. But what is important isto show that they do exist. Their existence is importantbecause it is the spin waves that trigger spin motion froma nonequilibrium state.We keep in mind self-organized spin waves caused bydipolar interactions, but not induced by external forces,so that at the initial time, no rotation is imposed on thesystem, h S − j (0) i = h S + j (0) i = 0 , (20)and the feedback field has not yet appeared, that is H =0. Spin waves are small oscillations around the averagespin values, which is described by representing the spinoperators in the form S αj = h S αj i + δS αj . (21)Due to the property of the dipolar tensor, the interactionfunctions (13) satisfy the equality X j a ij = X j b ij = X j c ij = 0 . (22)Therefore, for an ideal lattice, where the statistical aver-age does not depend on the lattice index, the local fields(15) and (16) are actually formed by spin waves, since ξ i = 1 ~ X j (cid:0) a ij δS zj + c ij δS + j + c ∗ ij δS − j (cid:1) ,ϕ i = 1 ~ X j (cid:16) a ij δS zj − b ij δS + j − c ij δS zj (cid:17) . (23)Substituting expression (21) into the equations of mo-tion, it is necessary to be cautious with respect to the lastterm in Eq. (18), taking into account that this term isexactly zero for spin 1 /
2. Then we use the representation[7, 8, 12, 46] S − j S zj + S zj S − j = (cid:18) − S (cid:19) h S zj i S − j (24)that is exact for S = 1 / S → ∞ .Separating in the evolution equations the terms of dif-ferent orders with respect to small spin deviations, inzero order, we have the equations ddt h S − j i = − iω s h S − j i , ddt h S zj i = 0 , (25)where the effective frequency of spin rotation is ω s ≡ ω + (cid:18) − S (cid:19) q Z − D ~ h S zj i . (26)The first equation gives h S − j ( t ) i = h S − j (0) i e − iω s t . In view of the initial condition (20), it follows that h S − j i = 0 , S − j = δS − j . (27)And the second of equations (25) shows that h S zj i = const .To first order with respect to the spin deviations, wefind ddt δS − j = − iω s δS − j − iϕ j h S zj i , ddt δS zj = 0 . (28)Because of the initial condition δS zj (0) = 0, the aboveequations give δS zj ( t ) = 0.Invoking the Fourier transform for the ladder spin op-erators S ± j = X k S ± k exp( ∓ i k · r j )and for the interaction functions a ij and b ij , a ij = 1 N X k a k exp( i k · r ij ) , b ij = 1 N X k b k exp( i k · r ij ) , we reduce the first of equations (28) to the form ddt S − k = − iA k S − k + iB k S + k , (29)in which A k ≡ ω s + a k ~ h S zj i , B k ≡ b k ~ h S zj i . (30)Looking for the solution S − k = u k e − iω k t + v ∗ k e iω k t , (31)we obtain the spectrum of spin waves ω k = q A k − | B k | . (32)Considering the long-wave limit, when k →
0, we keep inmind that the wavelength λ = 2 π/k is much larger thanthe interspin distance but smaller than the sample size.Then the spectrum has the form ω k ≃ | ω s | − h S zj i ~ ω s X h ij i a ij ( k · r ij ) . (33)Here < ij > implies the summation over the nearestneighbors.Generally, the spectrum is well defined when | A k | > | B k | , which yields the stability condition (cid:12)(cid:12)(cid:12) ω s + a k ~ h S zj i (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) b k ~ h S zj i (cid:12)(cid:12)(cid:12)(cid:12) . (34)Explicitly, this condition reads as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2 S − q Z + S h S zj i ~ ω + S a k − (2 S − D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ S | b k | . This means that spin waves exist when the Zeeman fre-quency ω and the parameter q Z of the quadratic Zeemaneffect are sufficiently large, such that condition (34) bevalid. The quadratic Zeeman effect can stabilize dipo-lar spin waves [47]. As is clear, the existence of dipolarinteractions is also crucial.The occurrence of spin waves is very important, sincethey serve as a triggering mechanism initiating spin mo-tion after the system has been prepared in an initialnonequilibrium state [8, 46, 48]. V. AVERAGED EQUATIONS OF MOTION
Let us consider the temporal behavior of the averagedquantities, the transverse spin polarization function u ≡ SN N X j =1 h S − j i , (35)coherence intensity w ≡ SN ( N − N X i = j h S + i S − j i , (36)and the longitudinal spin polarization s ≡ SN N X j =1 h S zj i . (37)Notice that if one resorts to the standard mean-fieldapproximation, then the averages of the local fields (15)and (16), because of property (22), become zero, h ξ j i = h ϕ j i = 0 . Thus the influence of the dipolar interactions would belost. However these interactions are principally impor-tant, since they are necessary for the existence of spinwaves triggering the initial spin motion.To take the dipolar interactions into account, we em-ploy a more refined stochastic mean-field approximation [8, 46, 49]. In the process of averaging over the spin vari-ables, we set the notation * N N X j =1 ξ j S αj + = ξ S N N X j =1 h S αj i , * N N X j =1 ϕ j S αj + = ϕ S N N X j =1 h S αj i , (38)where ξ S and ϕ S are treated as stochastic variables re-lated to local spin-wave fluctuations.Realizing statistical averaging over the spin variables,we use the mean-field approximation for the spin corre-lation functions h S αi S βj i = h S αi ih S βj i ( i = j ) (39)corresponding to spins at different lattice sites. And forthe single-site term, we employ the decoupling followingfrom Eq. (20), h S αj S βj + S βj S αj i = (cid:18) − S (cid:19) h S αj ih S βj i , (40)which is exact for S = 1 / S → ∞ .The stochastic local fields ξ S and ϕ S are defined asrandom variables satisfying the stochastic averaging con-ditions hh ξ S ( t ) ii = hh ϕ S ( t ) ii = 0 , hh ξ S ( t ) ξ S ( t ′ ) ii = 2 γ δ ( t − t ′ ) , hh ξ S ( t ) ϕ S ( t ′ ) ii = hh ϕ S ( t ) ϕ S ( t ′ ) ii = 0 , hh ϕ ∗ S ( t ) ϕ S ( t ′ ) ii = 2 γ δ ( t − t ′ ) , (41)in which γ is the relaxation rate caused by fluctuatingspins interacting through dipolar forces. To evaluate thevalue of γ , we may notice that, in view of Eqs. (41), therate γ can be represented as γ = (cid:12)(cid:12)(cid:12)(cid:12) Z ∞ hh ξ S ( t ) ξ S (0) ii dt (cid:12)(cid:12)(cid:12)(cid:12) . (42)The fluctuating field ξ S ( t ) behaves according to the law ξ S ( t ) ∝ γ exp {− i ( ω s − iγ ) t } , where ω s is the effective spin rotation frequency (26) and γ = 1 ~ ρµ S S (43)is the dipolar transverse attenuation rate, in which ρ ≡ N/V is average spin density, with V being the samplevolume. The effective spin-rotation frequency (26), thatreads as ω s = ω + (2 S − q Z − D ~ s , (44)can be represented as ω s = ω (1 + As ) , (45)where the dimensionless parameter A ≡ (2 S − q Z − D ~ ω (46)plays the role of an effective magnetic anisotropy renor-malized by quadratic Zeeman effect.From the integral (42), we find γ ∼ = γ p ω s + γ . (47)The effective force (17), under averaging over spins,becomes f = − i (cid:18) µ S H ~ + ϕ S (cid:19) . (48)In the equations of motion, we take into account the ex-istence of the transverse spin attenuation rate γ and thelongitudinal attenuation rate γ .Finally, averaging Eqs. (18) and (19), we derive theequations for the transverse polarization function dudt = − i ( ω s + ξ S − iγ ) u + f s , (49)coherence intensity dwdt = − γ w + ( u ∗ f + f ∗ u ) s , (50)and the longitudinal spin polarization dsdt = −
12 ( u ∗ f + f ∗ u ) − γ ( s − s ∞ ) , (51)where s ∞ is an equilibrium (or stationary) spin polariza-tion. VI. FEEDBACK MAGNETIC FIELD
According to the setup mentioned in Sec. II, the sam-ple is inserted into a coil of an electric circuit. Therefore,moving spins induce electric current in the coil, which isdescribed by the Kirchhoff equation. In turn, this cur-rent creates a feedback magnetic field inside the effectivecoil volume V eff . Such a coupling with a resonance elec-tric circuit induces in the system the so-called radiationdamping [50–54]. The feedback magnetic field satisfiesthe equation [7, 8, 46, 48] dHdt + 2 γH + ω Z t H ( t ′ ) dt ′ = − πη f dm x dt (52)following form the Kirchhoff equation. Here γ is the cir-cuit ringing rate, ω is the circuit natural frequency, and η f is the filling factor η f = V /V eff . The electromotiveforce is created by the motion of spins forming the mag-netic moment with the effective density m x = µ S V N X j =1 h S xj i . Equation (52) can be rewritten [7, 8, 46, 48] as theintegral equation H = − π Z t G ( t − t ′ ) ˙ m x ( t ′ ) dt ′ , (53)in which ˙ m x = N V eff µ S S ddt ( u ∗ + u ) , the transfer function is G ( t ) = (cid:20) cos( ω eff t ) − γω eff sin( ω eff t ) (cid:21) e − γt , with the effective frequency ω eff ≡ p ω − γ . The electric circuit can be tuned close to the Zeemanfrequency ω , so that the detuning be small, (cid:12)(cid:12)(cid:12)(cid:12) ∆ ω (cid:12)(cid:12)(cid:12)(cid:12) ≪ ≡ ω − ω ) . (54)And, as usual, all attenuations are supposed to be small,such that γω ≪ , γ ω ≪ , γ ω ≪ , γ ω ≪ . (55)The coupling between the magnetic coil of the electriccircuit and the sample is characterized by the couplingrate γ ≡ π ~ ρη f µ S S = πη f γ , (56) which is close to γ , if the volumes of the sample and coilare close to each other. Solving Eq. (53) by an iterativeprocedure, to first order with respect to the coupling rate,we find µ S H ~ = i ( uX − X ∗ u ∗ ) , (57)where the coupling function is X = γ ω s (cid:20) − exp {− i ( ω − ω s ) t − γt } γ + i ( ω − ω s ) ++ 1 − exp {− i ( ω + ω s ) t − γt } γ − i ( ω + ω s ) (cid:21) . (58)When ω s >
0, the first, quasiresonant, term in the cou-pling function prevails over the second, since (cid:18) ω − | ω s | ω + | ω s | (cid:19) < . By the same reason, the second term is larger than thefirst, if ω s <
0. Both these cases can be summarized inthe expression X ∼ = γ ω s − exp( − i ∆ s t − γt ) γ + i ∆ s sign ω s == γγ ω s γ + ∆ (cid:8) − (cos(∆ s t ) − δ s sin(∆ s t )) e − γt −− i (cid:2) δ s − (sin(∆ s t ) + δ s cos(∆ s t )) e − γt (cid:3)(cid:9) , (59)where ∆ s ≡ ω − | ω s | = ω − ω | As | ,δ s ≡ ∆ s γ sign ω s . (60) VII. REGULATING SPIN REVERSAL
Substituting the feedback field H into Eq. (49) givesthe equation dudt = − i Ω u − iξ S u − iϕ S s − X ∗ u ∗ s , (61)where Ω = ω s − i ( γ − Xs ) . From Eqs. (49) to (51) it follows that the functionalvariable u can be classified as fast, while the variables w and s as slow. This allows us to employ the scaleseparation approach [8, 46, 49] that is a variant of theaveraging techniques. To this end, we solve equation (61)for the fast variable u treating there the slow variables w and s as quasi-integrals of motion, which yields u = u exp (cid:26) − i Ω t − i Z t ξ S ( t ′ ) dt ′ (cid:27) −− is Z t ϕ S ( t ′ ) exp (cid:26) − i Ω( t − t ′ ) − i Z tt ′ ξ S ( t ′′ ) dt ′′ (cid:27) dt ′ . (62)The nonresonant counter-rotating term of order γ /ω isomitted here. Then we substitute the feedback field H and the fast variable u into equations (50) and (51) forthe slow variables w and s and average these equationsover time and over the stochastic variables ξ S and φ S .This results in the equations for the guiding centers dwdt = 2 γ w ( αs −
1) + 2 γ s (63)and dsdt = − αγ w − γ s − γ ( s − s ∞ ) , (64)with the coupling function α ≡ Re Xγ = gγ γ + ∆ s (1 + As ) ×× (cid:8) − [cos(∆ s t ) − δ s sin(∆ s t )] e − γt (cid:9) , (65)in which g ≡ γ ω γγ (66)is the dimensionless coupling parameter characterizingthe coupling between the sample and the electric circuit.Analyzing equations (63) and (64), we take into ac-count that the dipolar relaxation rate γ is smaller thentransverse attenuation rate γ , and the longitudinal at-tenuation rate γ is usually much smaller than γ . Mea-suring time in units of 1 /γ , we come to the equations dwdt = 2 w ( αs −
1) + 2 γ γ s ,dsdt = − αw − γ γ s . (67)Assume that the system is polarized at the initial time,but no coherence from external sources is imposed, sothat the initial conditions are w (0) = 0 , s (0) = s . (68)The external magnetic field B at the initial time is di-rected along the z axis, so that the system is in a nonequi-librium state. The regulation of spin dynamics is based on thepossibility of varying in time the parameter q z of thealternating-current quadratic Zeeman effect. The valueof this parameter can be varied in a rather wide range.For example, dipolar lattices, organized by means oflaser beams [14–18] have the mean interatomic distance a ∼ (10 − − − ) cm, hence the average density ρ ∼ (10 − ) cm − . For the magnetic moments µ S ∼ µ B , the dipolar transverse rate (42) is γ ∼ (10 − )s − . And the value | q Z | / ~ can reach 10 s − , as can beinferred from Refs. [22–25].There may happen two situations.(i) First, if the dipolar system has no single-siteanisotropy, then one can create a nonzero parameter q Z for the required time, say between zero and τ , duringwhich the initial spin polarization is preserved due to thenonzero value of parameter (46) that equals A = (2 S − q Z ~ ω (0 ≤ t < τ ) . After this, one switches off the quadratic Zeeman effectsending q Z to zero, hence making zero the parameter A .This corresponds to the temporal behavior A ( t ) = (cid:26) A , ≤ t < τ , t ≥ τ . (69)(ii) A similar procedure can be realized when thesingle-site anisotropy parameter D is not zero. Then onecan either keep q Z zero, if the value of D is sufficientfor freezing the initial spin direction, or create a nega-tive value of q Z for increasing the effective anisotropy tothe needed magnitude. After the required time τ , oneshould switch on the quadratic Zeeman effect so that tocompensate the value of D , thus sending A to zero.Numerical solutions to Eqs. (67) are presented in Fig.2, where we set γ/γ = 10, ω = ω = 1000 γ , and g = 100. For the delay time, we take τ = 0 . /γ , whichcan be about 10 − − τ , during which the spin polar-ization s practically does not change, can reach 100 /γ ,which amounts to 0 . −
10 s. The polarization reversalis very fast, being approximately equal τ p ≈ /gs γ ,which makes 10 − − − s. The polarization reversal isaccompanied by a coherent pulse, shown in Fig.2b andcorresponding to spin superradiance [55]. In that way,we achieve the desired goal, being able to keep for longtime a fixed longitudinal spin polarization, while quicklyreversing it as soon as we need.Moreover, it is straightforward to repeat the spin re-versal several times by inverting the external magneticfield B during the stage of frozen spin, which impliesthe change of ω s by − ω s . This procedure, illustrated inFig. 3, goes as follows. The value of A is kept nonzeroduring the time interval 0 ≤ t < τ . At the moment oftime τ , by regulating the quadratic Zeeman effect, thevalue of A is sent to zero. Thus the first reversal occurs,as in Fig. 2. The value of A is kept zero till some time t -1-0.500.51 s(t) (a) t w(t) (b) FIG. 2. Longitudinal spin polarization (a) and coherence intensity (b) as functions of time measured in units of 1 /γ . A singlespin reversal for the parameters γ/γ = 10, ω = ω = 1000 γ , A = 1, and τ = 0 . /γ . t -1-0.500.51 s(t) (b) t w(t) (b) FIG. 3. Sequence of longitudinal spin reversals (a) realized by inverting the external magnetic field at the moments of thedimensionless time t n = ( n +1) τ , measured in units of 1 /γ , and the related sequence of superradiant bursts (b). The parametersare as in Fig. 2. t . At this time, the external field B is inverted and A is set to a nonzero value, which is kept nonzero till thetime t + τ . At the moment of time t + τ the param-eter A is switched off, which results in the second spinreversal. And then the process is repeated as many timesas necessary. The values t n and τ n can be varied, thusrealizing the required sequence of spin reversals. VIII. POSSIBILITY OF EXPERIMENTALIMPLEMENTATION
Choosing appropriate materials for the physical imple-mentation of possible experiments, the main point is toselect such atoms with internal spin structure that al-low for an efficient variation by means of the alternating-current Zeeman effect of the dimensionless anisotropy pa-rameter A in Eq. (46) between small values close to zeroand the values of order of unity or higher. A collection of such atoms can be arranged in a lattice either in aself-organized way or by means of external fields. Also,the atoms can be incorporated into a solid-state matrixas a kind of admixture.One way is to deal with atomic systems without mag-netic anisotropy. For example, one can take the atomsof Cr that has the effective spin S = 3 and magneticmoment 6 µ B . The nucleus of this atom has zero spin,because of which the atom does not possess hyperfinestructure, but the alternating Zeeman effect can be in-duced by a quasiresonant light field [22, 23]. Since theatomic system does not have magnetic anisotropy, thestabilization of an initial nonequilibrium state has to bedone by the alternating-current Zeeman effect followingthe procedure explained above in paragraph (i). Thealternating-current Zeeman parameter q Z and the Zee-man frequency ω should be taken such that the param-eter A could reach at least unity.The other way is to take a system possessing magneticanisotropy which could be compensated for the requiredtime by switching on the alternating-current Zeeman ef-fect to provoke the reversal of the magnetization. Con-sequently, one should follow the way described in para-graph (ii). This mechanism sounds more promising forapplications in view of the smaller energy consumption.The solid-state materials, commonly employed in spin-tronic devices [56, 57] in the majority of cases correspondto ferromagnetic or antiferromagnetic systems, whosespins interact through exchange interactions. If we addto Hamiltonian (4) the exchange spin termˆ H exc = − X i = j (cid:2) J ij (cid:0) S xi S xj + S yi S yj (cid:1) + I ij S zi S zj (cid:3) , the overall procedure of solving the equations remains thesame. The main difference is that the effective anisotropyparameter (46) now becomes A = 1 ~ ω [(2 S − q Z − D ) − S ∆ J ] , including the exchange anisotropy∆ J ≡ N X i = j ( I ij − J ij ) . In many cases, the latter gives ∆ J/ ~ ∼ s − . Such ahigh value, to our understanding, cannot be compensatedby the alternating-current Zeeman effect.More promising could be the collections of atoms ab-sorbed on the surface of graphene [58–61]. Such adatomsusually also interact through exchange forces, but the re-lated magnetic anisotropy can be smaller than in hardmagnetic materials.There exists a large class of magnetic molecules [4–8, 62–67] interacting through dipolar forces, possessingvarious spins, between 1 / T = τ exp (cid:18) U eff k B T (cid:19) , in which U eff = | D | S is the effective energy bar-rier. Clearly, at sufficiently low temperatures, lowerthan a blocking temperature T B , a molecule can be ina metastable state for rather long time. For instance,the molecule, labeled as Mn , having the spin S = 10,is characterized by the blocking temperature T B = 3K,below which it has the metastable state lifetime of order10 s and longer. But the magnetic anisotropy of thismolecule is too high, with D/ ~ ∼ s − .Fortunately, there are so many various magneticmolecules that it is possible to find among them themolecules with much lower magnetic anisotropy. For ex-ample, the molecule, labeled as Mn , has the magnetic anisotropy parameter D/ ~ = 7 × s − . At the sametime, this molecule possesses a very large spin S = 83 / U eff / ~ ∼ s − . The re-lated blocking temperature, for which U eff is much largerthan k B T , is T B ∼ . A contains the ratio D/ ~ ω .Therefore the parameter D can be suppressed by increas-ing the external magnetic field B , that is, by increasing ω .In order to find out an explicit expression for the re-versal time, during which the average spin of the systemreverses from its initial value s to the value about − s ,let us consider more in detail the situation, when the ef-fective anisotropy parameter A is of the order of one orlarger till some time τ , after which this parameter A isswitched off or suppressed.Thus, at the beginning | A | & t < τ ) . (70)To simplify the following formulas, we take into accountthe inequalities γ ≪ γ ≪ γ . Under condition (70), we have γ = γ ω | As | . The coupling of the sample with the resonant circuit isweak, since α ∼ γ γγ ω | As | ≪ t < τ ) . We assume that at the initial time no coherent pulses acton the sample, so that w = 0. Then Eqs. (67), with thecondition γ t ≪
1, result in the solution that at time τ gives w ( τ ) ≃ γ γ s , s ( τ ) ≃ s . (71)At time τ the parameter A is assumed to be sup-pressed, so that | A | ≪ t ≥ τ ) . (72)In the case of the resonance, when ω = ω , we have ω s ≃ ω , hence ∆ s ≃
0. For the time t > τ , when γτ ≫
1, the coupling with the resonator becomes strong,such that α ≃ g . The ratio γ γ = γ ω ≪ t ≥ τ ) , being small, allows us to neglect the term γ /γ in Eqs.(67). This results in the equations dwdt = 2 w ( gs − , dsdt = − gw . (73)0These equations enjoy the exact solution w = (cid:18) γ p gγ (cid:19) sech (cid:18) t − t τ p (cid:19) ,s = − γ p gγ tanh (cid:18) t − t τ p (cid:19) + 1 g , (74)in which we return to the time measured in time units.Here γ p ≡ /τ p and t are the integration constants de-fined by sewing this solution with the values (71) at thetime t = τ . Then, assuming a strong resonator-samplecoupling, such that gs ≫
1, we find γ p = γ gs (cid:18) γ γ (cid:19) ,t = τ + τ p (cid:18) γ γ (cid:19) . (75)The time τ p ≡ /γ p describes the width of the coherencepulse w and also it shows the time during which the spinpolarization s reverses form the initial value s to thefinal value − γ p gγ + 1 g ∼ = − s . That is, τ p is the reversal time, for which we have τ p = γγ ω s . (76)In this way, the reversal time depends on the resonatordamping γ that can be varied, the coupling rate γ that,according to Eq. (56), is close to γ , the Zeeman fre-quency ω , and the initial spin polarization s . For anexternal magnetic field B ∼ µ S ∼ µ B , wehave ω ∼ s − . Choosing s = 1 and γ ∼ γ , we getthe reversal time τ ∼ − s. IX. CONCLUSION
We have suggested a novel mechanism of regulatingspin reversal in a system of atoms or molecules possess-ing internal spin states. The mechanism is based on theuse of the alternating-current quadratic Zeeman effectoccurring when applying quasiresonant linearly polarizedlight populating internal spin states. This quasiresonantdriving exerts quadratic Zeeman shift along the field po-larization axis. The optically induced quadratic Zeemaneffect can be easily manipulated and rapidly adjusted.The appearance of the quadratic Zeeman shift is equiva-lent to the induction of an effective anisotropy that can beeasily varied. Therefore, it is possible to solve the prob-lem of creating a device that could keep spin polarizationfor long time, but quickly reversing this polarization atthe required moments of time. The process can be re-peated many times, producing a sequence of polarizationreversals with desired intervals of time.
APPENDIX A. ALTERNATING-CURRENTZEEMAN EFFECT
The physics of the alternating current quadratic Zee-man effect [22, 70–73] is similar to the alternating currentStark effect [74–77]. Let us consider a system of atomsenumerated by j = 1 , , . . . , N . Atoms are assumed tobe identical, each possessing energy levels labeled by anindex n , with the energies E n = ~ ω n and level widths γ n .In the ground state, a j -th atom has the energy E a andspin S j . Atoms are subject to an alternating externalfield that can be written as B alt = 12 (cid:0) h e − iεt + h ∗ e iεt (cid:1) , where ε is the field frequency. This field interacts withthe atomic magnetic moment of each atom M j = µ S S j . The interaction energy of the field with a j -th atom, tofirst order, is zero on average, since the term − B alt · M j ,being averaged over time, is zero. To second order ofperturbation theory, the interaction energy is∆ E j = − ~ X n Re (cid:20) |h n | h · M j | a i| ω na − ε − iγ na ++ |h a | h · M j | n i| ω na + ε + iγ na (cid:21) , with the transition frequencies and transition widths ω na ≡ ω n − ω a , γ na ≡
12 ( γ n + γ a ) . The summation goes over all level indices, except n = a .Let the alternating field be linearly polarized along theaxis z , so that h = h e z . Then, defining the Rabi fre-quency Ω ≡ | µ S h | ~ , we have ∆ E j = − ~ Ω X n |h a | S zj | n i| ×× (cid:20) ω na − ε ( ω na − ε ) + γ na + ω na + ε ( ω na + ε ) + γ na (cid:21) . The alternating field is tuned close to one of the tran-sition frequencies, corresponding to some fixed n = b , sothat the quasiresonance condition be valid (cid:12)(cid:12)(cid:12)(cid:12) ∆ res ω ba (cid:12)(cid:12)(cid:12)(cid:12) ≪ res ≡ ω ba − ε ) . X n |h a | S zj | n i| = X n h a | S zj | n ih n | S zj | a i == h a | ( S zj ) | a i , we come to the expression∆ E j ∼ = − ~ Ω ∆ res res + γ ba ) h a | ( S zj ) | a i . The Hamiltonian of the effect for a j -th atom is de-fined as the operator whose quantum-mechanical averageyields the additional energy∆ E j ≡ h a | ∆ ˆ H j | a i . This results in the Hamiltonian∆ ˆ H j = − ~ Ω ∆ res res + γ ba ) ( S zj ) . Respectively, the corresponding Hamiltonian term for thewhole collection of N atoms isˆ H QZ = X j ∆ ˆ H j . As is evident, it would not be reasonable to take theexact resonance condition ∆ res = 0, since then the in-teraction energy tends to zero. Therefore on takes ε not too close to the transition frequency, in the sense thatthe off-resonance condition be true, (cid:12)(cid:12)(cid:12)(cid:12) ∆ res γ ba (cid:12)(cid:12)(cid:12)(cid:12) ≫ . Under this condition, the Hamiltonian term becomes∆ ˆ H j = − ~ Ω res ( S zj ) . Finally, summing over all atoms in the system, we getthe interaction term corresponding to the alternating-current quadratic Zeeman effectˆ H QZ = q Z X j ( S zj ) , with the parameter q Z defined in Eq. (6).In order to exhibit the alternating-current Zeeman ef-fect, an atom, or molecule, needs to possess an internalspin structure. If the nucleus of an atom has a nonzerospin, then there exists hyperfine structure. And even ifthere is no the latter, when the nuclear spin is zero, therealways exists the spin structure of energy levels, as soonas an atom contains electrons [22–28, 70–73]. Since allatoms have electrons, their energy levels depend on thepresence of external magnetic fields, including alternatingfields. Therefore, the alternating-current Zeeman effectoccurs for practically all atoms and molecules [78–80]. [1] L.F. Cameretti, Modeling of Thermodynamic Propertiesin Biological Solutions (Cuviller, G¨ottingen, 2009).[2] T.A. Waigh,
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