Femtosecond electron and spin dynamics probed by nonlinear optics

5 Pages·0.16 MB·English

by  G. P. Zhang

Femtosecond electron and spin dynamics probed by nonlinear optics
G. P. Zhang∗ and W. Hu¨bner
Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
∗ e-mail: [email protected]
(February 1, 2008)
9
A theoretical calculation is performed for the ultrafast spin dynamics in nickel using an exact
9
diagonalizationmethod. Thepresenttheorymainlyfocusesonasituationwheretheintrinsiccharge
9
andspindynamicsisprobedbythenonlinear(magneto-)opticalresponsesonthefemtosecondtime
1
scale, i.e. opticalsecond harmonic generation (SHG)andthenonlinearmagneto-optical Kerreffect
n
(NOLIMOKE). It is found that the ultrafast charge and spin dynamics are observable on the time
a
scale of 10 fs. The charge dynamics proceeds ahead of the spin dynamics, which indicates the
J
existence of a spin memory time. The fast decay results from the loss of coherence in the initial
7
excited state. Both the material specific and experimental parameters affect the dynamics. We
2
findthat theincrease of exchangeinteraction mainly accelerates thespin dynamicsratherthanthe

charge dynamics. A reduction of the hopping integrals, such as present at interfaces, slows down
]
i the spin dynamics significantly. Besides, it is found that a spectrally broad excitation yields the
c
s intrinsicspeedlimitofthecharge(SHG)andspindynamics(NOLIMOKE)whileanarrowerwidth
- prolongsthedynamics. Thismagneticinterfacedynamicsthenshouldbecomeaccessibletostateof
l
r art time resolved nonlinear-optical experiments.
t
m
PACS numbers: 78.47.+p, 78.20.Ls, 75.70.-i
.
t
a
m I. INTRODUCTION effect, resulting from the interplay between band struc-
- ture and electron correlation. In this paper, we focus on
d thenonlinearopticalresponseofspindynamics. Wetake
n Recentlyultrafastspindynamicsinferromagneticmet-
a Ni monolayer as an example.
o als has attracted a great deal of attention due to its
c This paper is arranged as follows. In Section II, we
possible applications, such as the ultrafast magnetic
[
discussour theoreticalschemewhile the mainresultsare
gates. The experimental observation was that, upon

1 the excitation of a femtosecond laser pulse, a sharp given in Section III. Finally, the conclusions are given in
v Section IV.
decrease of the magnetization occurs on a time scale
3
of 100 fs [1–4], which is far beyond the characteristic
0
3 time scale of spin-lattice interaction. Similar results
1 have been independently found in pump and probe lin- II. THEORETICAL SCHEME
0 ear magneto-optics, nonlinear magneto-optics and two-
9
photon-photoemission. However the interpretation of
9 It is well-established that in ferromagnetic transition
this behavior has not been given on the same footing
/
t and remains somewhat speculative. To reconcile these metals, in particularin Ni, the electroncorrelationplays
a
animportantroleeveninthe groundstate andpossesses
m intriguing results from different experimental processes
a significant impact on the excited states, as evidenced
such as linear and nonlinear optics is rather crucial and
-
e.g. by the famous photoemission satellite structure [8].
d isoneofthegoalsofthepresenttheoreticalstudy. More-
n This becomes especially true in the nonlinear optical
over the demagnetization is very similar to the conven-
o process on the ultrafast time scale where highly excited
tional one [2].
c states are frequently involved. Therefore, we employ an
: In the SHG experiment [2], the M(T) curve is estab-
v exact-diagonalization method which explicitly avoids a
lished after the electron thermalization is finished while
i perturbative treatment of electron correlation. Within
X theelectronandlatticehavenotreachedacommonequi-
ourscheme,onedoesnotneedtointroduceanydamping
r librium yet, which indicates a purely electronic feature
a term to obtain a correct dephasing time. Our previous
of the ultrafast spin dynamics. Traditionally the spin-
results showedthat the typicaltime scale is basically set
lattice coupling sets the speed limit of the demagneti-
by the dispersion of the bands and the strength of elec-
zation process, typically on 100 ps [5], but here this is
tron correlation. We begin with a generic Hamiltonian
clearly not the case. Our previous theoretical studies
clearly demonstrated that the dephasing of initial states
is the origin of the spin dynamics on the femtosecond H = X Uiσ,jσ′,lσ′′′,kσ′′c†iσc†jσ′ckσ′′clσ′′′
time scale [6,7], as probed by transient reflectivities and i,j,k,l,σ,σ′,σ′′,σ′′′
linearmagneto-optics. We foundthatthe intrinsicspeed + X Eν(K)nνσ(K)+HSO (1)
limit is about 10 fs. This mechanism is a pure quantum
ν,σ,K
1
where Uiσ,jσ′,lσ′′′,kσ′′ is the on-site electron interaction, groundstateisanonzeroCoulombinteractionU andex-
whichcanbedescribedinfullgeneralitybythe threepa- changeinteractionJ. Wecansimplycheckthisbysetting
rameters Coulomb repulsion U, exchange interaction J, bothU andJ tozero. Doingso,wefindthatthe ground
and the exchange anisotropy ∆J [9]. The set of param- state is a singlet,i. e. paramagneticstate, whichcontra-
eters used for Ni is given in Ref. [10]. c† (c ) are the dicts the ferromagnetic nature of nickel. This proves the
iσ iσ
usual creation (annihilation) operators in the orbital i importance of U and J. Once we use the generic sets of
withspinσ (σ =↑,↓). E (K)isthesingle-particleenergy U andJ ofnickel,weobtainatripletasitsgroundstate,
ν
spectrum for band ν of the nickel monolayer. n (K) is from which we can calculate the magnetic moment, 0.88
νσ
the particle number operator in momentum space. HSO µB. This magnetic moment is larger than that in the
is the spin orbit coupling. Since this is a typical many- bulk material, which is consistent with the experimen-
body problem, one cannot solve it without simplifica- tal observation. In this case, also the satellite structure
tion. In order to obtain a tractable model, we first build well-known from photoemission experiments appears in
a two-hole basis set. Within this basis set, for each Ni the spectrum quite naturally.
atom, the dimension of the Hilbert space is 66. The ma-
trix elements of electron correlation for each atom can
In Fig. 1, we firstly show the effect of exchange cou-
be obtained analytically. For each K point, the elec-
pling J on the |χ(2) (ω,t)| and |χ(2)(ω,t)| as a func-
tron correlationis embedded in the crystal field as given xzz zzz
tion of time t. The probe frequency ω is fixed at 2
by the bandstructure. This treatment of correlations is
eV. The initial state is prepared to be 2 eV above the
analogous to a frequency dependent self-energy correc-
ground state with a Gaussian broadening as large as 20
tion. Within this simplification, we are able to exactly
⌈ eV, which opens almost all the possible decay chan-
diagonalize the Hamiltonian for each K point explicitly.
In order to characterize the spin and charge dynam- nels. Such a large distribution width corresponds to a
ics clearly, we calculate both these intrinsic quantities: very large laser spectral width. As previously explained,
S (t) = hΨ(0)|Sˆ |Ψ(t)i and N(t) = hΨ(0)|Nˆ|Ψ(t)i, and thischoiceaimsatrevealingtherealintrinsicspeedlimit
z z
the nonlinear(magneto-)opticalsusceptibilitiesχ(2) and ofthespindynamicsinoursystem,whichisthennotde-
xzz
χ(2). Here Sˆ = 1(nˆ −n ), Nˆ = (nˆ +n ), which are layed by experimental constraints. In Figs. 1(a) and
zzz z 2 ↑ ↓ ↑ ↓ 1(b), the generic parameters of Ni monolayers are used.
directly relatedto the observableNOLIMOKEandSHG
Thereareseveralinterestingfeaturesthatshouldbemen-
yields, respectively. Since |χ(2) (ω,t)| and |Sˆ (t)| mainly
xzz z tioned. One notices that in Fig. 1(a) |χ(2) (ω,t)| first
reflect the spin response while |χ(2)(ω,t)| and |Nˆ(t)| re- xzz
zzz comesupveryquicklyandreachesitsmaximumatabout
flect the charge response, they will be used as indicators
2-3 fs. Then |χ(2) (ω,t)| undergoes a sharply decreas-
to evaluate spin and charge evolutions, respectively. We xzz
find: ing envelope and oscillates with a very short period.
χ(x2z)z(ω,t)=k,Xl,l′,l′′(pE(kEl′k′l′−′,Et)k−l′p−(Eωk+l′,itη) (pTdlheeteceadydyentpoahm1ai/sciesngoo.ff m|Aχas(x2xz)azimf(oωur,emtm))|,enswettihotilnceehsdd,ino|dwχi(cn2a)ta(erωso,uttnh)d|es1cigo0nmfis--
xzz
− pE(kEl′kl−′,Et)k−l−p(ωE+kl,itη))/(Ekl′′ −Ekl−2ω+i2η) fithees tshpeinsprienlarxealtaixoantitoinmepriosceasbs.ouTth1u0s fws,eweshtiicmhatise tchoant-
×(hkl|Sˆz|kli+hkl′|Sˆz|kl′i+hkl′′|Sˆz|kl′′i−3/2) sistent with our previous results for time-resolved linear
(2) (magneto-)optics based on |χ(1)| and |χ(1)|. For the
xy zz
charge dynamics, we see a different scenario. In Fig.
χ(z2z)z(ω,t)=k,Xl,l′,l′′(Ep(kEl′′kl−′′,Et)k−l′−p(ωEk′+l,itη) s1e(ebs),th|χat(z2z|)zχ((xω2z),zt()ω|,its)|pnloetatrelydsapsenadfsuancstiimonilaorfttimimeetot.reOanche
− pE(kEl′kl−′,Et)k−l−p(ωE+kl,itη))/(Ekl′′−Ekl−2ω+i2η) ictasymoacxciumrsummoarse|χsh(x2az)rzp(ωly,ta)n|ddosetsr,obnugltyt.hAersouubnsdeq5uefns,ttdhee-
(3) value of |χ(2)(ω,t)| is already close to the equilibrium
zzz
where |kl > is the eigenstate with the eigenvalue E ; data. This means that the dephasing already becomes
kl
p(E ,t)=<Ψ(t)|kl>. strong for the charge dynamics before the spin dynam-
kl
ics. If one compares Fig. 1(a) with 1(b), one sees a
clear difference between spin and charge dynamics. Ba-
III. RESULTS sicallythespindynamicslastsabouttwiceaslongasthe
charge dynamics. This has an important consequence
Before we come to our main results, we would like to as it demonstrates the spin memory effect: though the
demonstrate that our Hamiltonian can reasonably de- chargedynamicsfinishes, the spindynamics is stillalive,
scribe some basic experimental results. It has been well- which is crucial for future applications. The main differ-
establishedthataprerequisitetoacquireaferromagnetic ence of the time resolved nonlinear response compared
2
oscillations. For the charge dynamics, the change going
50.0
t)| from J0 to J0/10 is relatively small. This can be seen
ω, (a) fromFig. 1(d)whereweplot|χ(2)(ω,t)|asafunctionof
( 25.0 zzz
z
xz time t. Comparing Figs. 1(b) and 1(d), one finds that
(2)χ the overall variation of |χ(2)(ω,t)| with time is nearly
| 0.0 zzz
identical. This is understandable since the exchange in-

200
)| teractionactsmoredirectlyonthespindegreeoffreedom
t
ω, (b) bychangingthe spindependence ofthe electronicmany-
(z 100 body states microscopically. Consequently, the spin dy-
z
z
2) namics will be affected more strongly than the charge
(χ
| 0 dynamics. However electrons even with different spin-

20.0 orientations play a similar role for the charge dynamics.
)| Thus,thechargedynamicsisbasicallyindependentofthe
t
ω, (c) spinstate. Thatiswhytheexchangeinteractiondoesnot
( 10.0
z
z affect the charge dynamics significantly.
x
2)
(χ 0.0
| Next, we wish to gain some physical insights into the

60.0 effects of the experimental constraints, such as the spec-
)|
t tral width of the excited state distribution, on the spin
ω, (d)
( 30.0 dynamics. In Fig. 2, we perform a detailed compari-
z
z
z sonbetweenthoseexperimentalobservablesandintrinsic
2)
(χ | 0.0 quantities(i.e.,|Sˆz(t)|,|Nˆ(t)|). Theinitialexcitedstate
0 5 10 15 20 isprepared2eVabovethegroundstatewithaGaussian

t (fs) broadeningofnowonly0.2eV,whichsimulatesanarrow
0.13 8.00
FIG. 1. The femtosecond time evolution of the nonlin- (a) (c)
e|pinχaag(zrr2zea)zmds(ωaw2g,iendtV)ee|ta,oas-rboe2ops0vpteeiecVctath.lieIvnaegnl(rydao.)uHonapendtrdiesct(atabhlt)ee,reaiwnsspiiettohitanolasfeegGsx,ecani|uteχesr(xdsi2ciz)aszpnt(aaωbtr,eartmo)isa|edptaeernnreds-- Im(S(t))z −00..0103 (2)ωχ(,t)||xzz0246....00000000
ofnickelisused;in(c)and(d),theexchangeinteraction J is −0.13 0.00 0.13 0 20 40 60 80
Re(S(t)) Time(fs)
reducedtoJ /10 whiletherest oftheparametersiskeptun- z
0
changed. Thespindynamicsisdelayedwithrespecttocharge
dynamics,whichindicatestheexistenceofaspinmemoryef-
fect. Additional‘bunching’occurs,whichhasnotbeenfound 1.00 20.0
flaotretshweiltihneaarlaprguemrpp-eprrioobde. responses of Ni. |χ(x2z)z(ω,t)| oscil- N(t)) 0.00 (b) ω,t)| 1105..00 (d)
Im( 2)(zzz 5.0
to the linear one, which is particularly evident for the (χ
−1.00 | 0.0
magnetic dynamics, consists in an additional ‘bunching’ −1.0 0.0 1.0 0 20 40 60 80
Re(N(t)) Time(fs)
of the structures resulting from the simultaneous pres-
ence of ω and 2ω resonances in Eqs. (2) and (3).
FIG.2. Femtosecond time evolution of the intrinsic spin
In order to get a handle on the microscopic origin of
(a)andcharge(b)dynamics,whichisnotdirectlyobservable,
the observed magnetic dynamics, we try to investigate
incomparisonwiththemagneto-optical(c)andoptical(d)re-
the effect of the on-site exchange coupling J. We re-
sponse functions. |Sˆz(t)| and |Nˆ(t)| , (a) and (b) show the
duceJ toJ /10. Thecorrespondingtime-dependencesof
0 clearly differentbehaviorbetween spin andcharge dynamics.
|χ(x2z)z(ω,t)| and |χ(z2z)z(ω,t)| are shown in Figs. 1(c) and Here the initial state is also prepared to be 2 eV above the
1(d), respectively. It can be seen that |χ(2) (ω,t)| first groundstate, butthespectralwidth is0.2eV.Theotherpa-
xzz
comes up within 2 fs. After that a recurrence appears rametersaretakenasthegenericparametersofNi. Thetime
interval is [0,40 fs] for (a) and (b). The calculated intrinsic
witharatherlargeamplitude. ComparedwithFig. 1(a),
((a), (b)) and experimentally accessible ((c), (d)) quantities
|χ(2) (ω,t)|oscillateswithalonger‘bunching’periodand
xzz showa strongdependenceon thelaser spectral width (corre-
the loss of coherenceis weaker. We estimate that the re-
spondingtotheinitial excitedstate width). Comparing with
laxationtimeisabout10fsbuttheperiodisnearlytwice Fig. 1,wenoticethatforthesamesetofgenericparameters,
as long as that in Fig. 1(a). This demonstrates that the thedecrease of thelaser spectral width yields a slower decay
decrease of exchange interaction prolongs the period of of the spin and charge response.
3
laser spectral width, in contrast to the previously used 80.0
width of 20 eV. This allows us to see the effect of the
60.0 (a)
laser spectral width clearly. All the parameters for the |
)
t
Hamiltonian are the generic set of parameters of Ni. In ω, 40.0
Figs. 2(a)and2(b),weshowtheresultsupto40fs. The (
z
z
abscissa and ordinate in Figs. 2(a) and 2(b) denote the 2) x 20.0
realand imaginary parts of the intrinsic spin and charge (χ
|
dynamics, S (t) and N(t), respectively. Note that these 0.0
z
quantitiescannotdirectlybeobservedandareonlytheo-
reticallyaccessible. Thearrowsrefertothetimedirection 200
and the centers are the final positions of S (t) and N(t).
z
Onenoticesthatthespindynamicsneedsaboutsixcycles 150 (b)
|
)
toreachitsfinalvaluewhileforthechargedynamicsonly t
,
three cycles are needed. This demonstrates again that ω 100
(
z
the spin dynamics is delayed with respect to the charge z
dynamics. Wefindthatcomparingthoseintrinsicquanti- (2)χz 50
tiesonecanseeacleardifferencebetweenspinandcharge |
0
responses, which is not blurred by the details of the ex- 0 20 40 60 80
perimental conditions. Nevertheless, from our results,
TIME (fs)
one can still identify the differences alsoin the nonlinear
optical and magneto-optical responses. We show the re-
sults in Figs. 2(c) and 2(d). In Fig. 2(c), |χ(x2z)z(ω,t)| is FIG. 3. The femtosecond time evolution of the nonlin-
plottedasafunctionoftimeupto80fs. Onemaynotice ear magneto-optical and optical responses, |χ(x2z)z(ω,t)| and
that a decay occurs within 30-40 fs, which is consistent |χ(z2z)z(ω,t)|,respectively. Thesameinitialconditionhasbeen
withourpreviouslinearresultsobtainedfrom|χ(1)| and used as in Fig. 1 except for the hopping integrals which are
xy
|χ(1)| [6]. SHG probes more bands, which causes more taken to be one-tenth of the original ones. It turns out that
zz
thechangeofbandstructurehasasignificantimpactonspin
‘beating’ and ‘bunching’. This might then result in an
and charge dynamics. It slows down the dynamics by modi-
effectivelyslowerresponsethanthatseeninlinearpump- fying theenvelopes of theresponse functions |χ(x2z)z(ω,t)|and
probeexperiments,inadditiontotheslowingdownwhich |χ(z2z)z(ω,t)|.
results from the narrower bandwidth at interfaces. For
|χ(2)(ω,t)| we present the results in Fig. 2(d). Compar-
zzz
(2)
ing with |χ(2) (ω,t)| in Fig. 2(c), |χ(2)(ω,t)| drops even Fig. 3. Onefindsthatthechangeofboth|χxzz(ω,t)|and
xzz zzz
more sharply. One can see a clear drop at 18 fs, which |χ(z2z)z(ω,t)| with time is very different from the previous
sets its relaxation time for charge dynamics. A delay ones. From Fig. 3(a), one notices that |χ(x2z)z(ω,t)| in-
about 10 fs between spin and charge responses is found creases sharply within 5 fs. The strong oscillation lasts
for this specific set of parameters. Comparing Figs. 1(a) about 20 fs, where no clear decay can be seen. The out-
and 2(c), one can immediately see that a narrow initial lineof|χ(2) (ω,t)|rangingfrom0fsto20fsformsabroad
xzz
state distribution width prolongs the relaxation process. peak. After 20 fs, dephasing occurs, but the envelope of
In Fig. 1, one knows that the relaxation process basi- |χ(2) (ω,t)| decays only slowly. Comparing Figs. 1(a)
xzz
callyfinisheswithin10fs,butheretherelaxationtime is and 1(b) with Figs. 3(a) and 3(b), respectively, one sees
around 30-40 fs. that the reduction of the hopping integrals slows down
both the spin and charge dynamics considerably. The
Finally,asourpreviousstudiesalreadyshowed[6,7]for envelope of |χ(2) (ω,t)| now decays on the time scale of
xzz
the linear pump-probe calculations, the band structure
30-40 fs. The appreciable fast oscillation in the long-
will also influence the relaxationprocess. Its effect is ac-
time tail survives beyond 80 fs. Comparing Figs. 3(a)
tuallyverysignificant. Ourresultson|χ(x1y)| and|χ(z1z)| al- and 3(b) with Figs. 2(c) and 2(d), respectively, indi-
ready showed that the hopping integrals can modify the cates that the elementary oscillations are more rapid.
relaxation process strongly. Analogously, this will be re- Thus from the comparison of all the three figures, we
(2)
flectedinthe nonlinearopticalresponses|χxzz(ω,t)|and conclude that the reduction of the excited states dis-
|χ(2)(ω,t)|. Inordertoinvestigatetheeffectoftheband tribution width affects the elementary oscillation more
zzz
structure, we reduce the hopping integrals to one-tenth strongly than the envelope of the spin and charge re-
of the originalnickelhopping integrals while keeping the sponses while decreasing the hopping integrals does the
rest of parameters unchanged. Here, the initial excited opposite and mainly slows down the response envelope
stateisalsoprepared2eVabovethegroundstatewitha of |χ(2) (ω,t)| and |χ(2)(ω,t)| . It is remarkable that in
xzz zzz
Gaussian broadening of 20 eV. The results are shown in allcasesthe spinmemory effectpersists. Inaddition, we
4
wouldliketopointoutthatthechangeofbandstructure slowsdownthe dynamics. This is understandable as, for
imposed by spin-orbit coupling as well affects the spin asmallerspectralwidth, the number ofpopulatedstates
dynamics observed in nonlinear magneto-optical experi- is smaller. The dissipation becomes weaker and eventu-
ments. For linear time-resolved magneto-optical experi- ally the spin dynamics is prolongedsignificantly. Finally
ments,thishasbeendemonstratedtheoretically(seeRef. we studied the effect of the band structure on the spin
[6,7]). Wefoundthatspin-orbitcouplinghadtoberaised dynamics. We found that the reduction of the hopping
to an unphysical large value of 1 eV (the generic value integrals slows down both the spin and charge dynam-
of spin-orbit coupling in nickel is 0.07 eV) in order to be ics by modifying the envelope of the response functions
relevant for the spin dynamics. Thus we do not expect |χ(2) (ω,t)| and |χ(2)(ω,t)| . This is different from the
xzz zzz
anymajorinfluencefromspin-orbitcouplingonthe10fs effect of a narrower distribution width, where one basi-
spin dynamics of transition metals in the valence band. cally prolongs the oscillation periods. It is important to
note that materials with a small hopping integral corre-
spondtonanostructurematerials,clusters,quantumwell
IV. CONCLUSIONS states, magnetic insulators (such as oxides), defects, and
impurities. Thus it is expected that with the presence
of those nanostructurematerials, the observeddynamics
In conclusion, we performed an exact-diagonalization
will be significantly slower. Naturally, this is very mean-
scheme to study the spin dynamics in ferromagnetic
ingful for the design of materials for ultrafast magnetic
nickelonthefemtosecondtimescale. Thisstudyisexclu-
devices. Finally and mostly importantly, the processes
sivelydevotedtothenonlinearopticalresponses. Atfirst,
whichweinvestigatedallexhibitthespinmemoryeffect,
wecheckedthatourHamiltoniancangiveacorrectferro-
a crucial property for future applications.
magnetic ground state and a correct order of magnitude
ofthemagneticmoment,whichislargerthanitsvaluein
bulk materials, as is physically expected. For a generic
setofNiparameters,the intrinsicspeedlimit ofspindy-
namics is about 10 fs, which is on a similar time scale as
revealed by |χ(1)| although additional ‘bunching’ struc-
xy
ture appears for the nonlinear time-resolved magneto- [1] E.Beaurepaire,J.-C.Merle,A.Daunois,andJ.Y.Bigot,
optical response. Our theory clearly yields the memory Phys. Rev.Lett. 76, 4250 (1996).
effect of the spin dynamics also in nonlinear optics. It is [2] J. Hohlfeld, E. Matthias, R. Knorren, and K. H. Benne-
mann, Phys. Rev. Lett. 78, 4861 (1997); ibid. 79, 960
found that the spin dynamics is delayed with respect to
(1997) (erratum).
the charge dynamics. The spin dynamics survives even
[3] M. Aeschlimann, M. Bauer, S. Pawlik, W. Weber, R.
when the charge dynamics ceases. This is very impor-
Burgermeister, D. Oberli, and H. C. Siegmann, Phys.
tant for future applications, such as ultrafast magnetic Rev. Lett.79, 5158 (1997).
gates. We also examined the effect of exchange inter-
[4] A.Scholl,L.Baumgarten,andW.Eberhardt,Phys.Rev.
action. It is found that a decrease of the exchange in- Lett. 79, 5146 (1997).
teraction prolongs the relaxation process. Especially we [5] A.Vaterlaus,T. Beutler,andF.Meier, Phys.Rev.Lett.
noticedthatforasmallerexchangeinteractionJ thespin 67, 3314 (1991).
response ‘bunches’ with longer periods. This again con- [6] W. Hu¨bner and G. P. Zhang, Phys. Rev. B 58, R5920
firms our earlier results that the observedspin dynamics (1998).
results from the spin dependent (J-dependent) dephas- [7] W. Hu¨bner and G. P. Zhang, J. Magn. Magn. Mater.
189, 101 (1998).
ing in the excitedmany-body states. In thatsense, SHG
[8] P. Fulde, Electron Correlation in Molecules and Solids,
and NOLIMOKE reveal the same fundamental physics
3rd edit. (Springer, Heidelberg, 1995); J. Wahle, N.
as time-resolved linear (magneto)-optics. In order to re-
Blu¨mer, J. Schling, K. Held, and D. Vollhardt, cond-
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mat/9711242.
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5