Evolution of the inhomogeneously-broadened spin noise spectrum with ac drive

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by  Z. Yue

Evolution of the inhomogeneously-broadened spin noise spectrum with ac drive
Z. Yue and M. E. Raikh
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
Inthepresenceofrandomhyperfinefields,thenoisespectrum,(cid:104)δs2(cid:105),ofaspinensemblerepresents
ω
a narrow peak centered at ω = 0 and a broad “wing” reflecting the distribution of the hyperfine
fields. In the presence of an ac drive, the dynamics of a single spin acquires additional harmonics
at frequencies determined by both, the drive frequency and the local field. These harmonics are
reflected as additional peaks in the noise spectrum. We study how the ensemble-averaged (cid:104)δs2(cid:105)
ω
evolves with the drive amplitude, ω (in the frequency units). Our main finding is that additional
dr
5 peaks in the spectrum, caused by the drive, remain sharp even when ωdr is much smaller than
1 the typical hyperfine field. The reason is that the drive affects only the spins for which the local
0 Larmour frequency is close to the drive frequency. The shape of the low-frequency “Rabi”-peak in
2 (cid:104)δs2ω(cid:105)isuniversalwithboth, thepositionandthewidth, beingoftheorderofωdr. Whenthedrive
amplitude exceeds the width of the hyperfine field distribution, the noise spectrum transforms into
n
a set of sharp peaks centered at harmonics of the drive frequency.
a
J
PACSnumbers: 85.75.-d,72.25.Rb,78.47.-p
5
1

]
l
al (cid:104)δs2ω(cid:105)=
h  
∞
- π  (cid:90) (cid:104) (cid:105)
s ∆(ω)+ dΩ F(Ω ) ∆(ω Ω )+∆(ω+Ω ) ,
e 6  N N − N N 
m I. INTRODUCTION 0
(1)
.
t
a where
Common experimental techniques for the study of
m
spin dynamics in semiconductors include the polariza- τs
∆(ω)= (2)
- tion of luminescence upon optical spin orientation1 and π(1+ω2τ2)
d s
n the time-resolved Faraday rotation2. Recently, a third
isaLorentzianandτ istheelectronspin-relaxationtime.
o technique, spin noise spectroscopy, had been applied s
Second term in Eq. (1) represents the average over the
c to bulk semiconductors3–5 and various semiconductor
[ structures6–9,seethereviewsRef.10andRef.11forcom- hyperfine fields, ΩN, distributed as20

1 prehensive literature. Within the spin-noise technique, 4 (cid:104) Ω2 (cid:105)
F(Ω )= Ω2 exp N . (3)
v the dynamics of spins manifests itself via random mod- N √πδ3 N − δ2
0 ulation of the refraction indices for the left- and right- e e
6 polarized light. This modulation results in random ro- When the width, δ , of the distribution exceeds τ−1, the
5 tation angle of the plane of polarization of the trans- second term becomees π [F(ω)+F( ω)], i.e. thes shape
3 6 −
mitted light. The power Fourier spectrum of these ran- of the noise spectrum reproduces the distribution of Ω .
0 N
dom rotations is proportional to the spectrum, δs2, of First term in Eq. (1) represents a peak centered at ω =
. ω
1
thespinfluctuations. Originally,thespin-noisemeasure- 0. It originates from the fact that the spin component
0
ments were conducted on atomic vapors12,13. With re- parallel to the hyperfine field does not precess.
5
gard to spin-noise, the principal difference between the Very recently21, in the spin-noise experiment on a va-
1
: vapors and semiconductors is that all spin-related fre- por of 41K alkali atoms, it was found that the ac drive
v
quenciesinvaporarethesame,while,insemiconductors, splits the noise spectrum into a Mollow triplet. This
i
X these frequencies are strongly different for different elec- splitting can be interpreted as a result of modified spin
trons. This is because, without external magnetic field, dynamicsinthepresenceofdrive. Infact,suchevolution
r
a each electron spin precesses around its individual hyper- of the noise spectrum is in accord with the theoretical
fine field created by nuclei which are located within the study of Ref. 22 where the noise of a driven two-level
extent of the electron wave function14,15. Importantly, system was considered. It was demonstrated22 that the
the observation of spin noise for localized electrons and drive-induced additional harmonics in the dynamics of
holes, see e.g. Refs. 5, 7-9, was possible even despite the two-level system manifest themselves as additional
thestronginhomogeneousbroadening. Whentheapplied peaks in the noise spectrum.
magneticfieldismuchsmallerthanthetypicalhyperfine With regard to semiconductors, there is a question:
field, the spin noise spectrum reflects the distribution of whathappenstothespinnoisespectruminthepresence
the hyperfine fields16–19. More precisely, the spectrum of drive when the local hyperfine fields are widely dis-
has the form16 tributed? Since the positions of the drive-related peaks
2
44 G(z)
�s2 H(z)
h !i 33
22
11
00
00..55 11.0.0 11..55 2.2.00 z
! -11
�
0 !RN ⌦dr �!RN ⌦dr ⌦dr +!RN �-22
FIG.1: Acartoonofthenoisespectrumforatypicalrealiza- FIG. 2: (Color online) Dimensionless functions G(z) (pur-
tionofthehyperfinefieldinthepresenceoftheacdrive. Due ple) and H(z) (green) describing the shapes of the peaks at
todrive,azero-frequencypeakdevelopsasatelliteatω=ωN, Rabifrequencyandatdrivingfrequencyintheaveragednoise
R
Eq.(9). Thepeakwhich,intheabsenceofdrive,waslocated spectrum are plotted from Eqs. (30) and (33), respectively.
atω=Ω shiftstoω=Ω −ωN anddevelopstwosatellites
N dr R
at driving frequency and at ω =Ω +ωN. The magnitudes
dr R
ofthesatellitesscalewiththedriveamplitudeasωd2r andωd4r, incorporated into the right-hand side of the equation of
respectively. Both satellites are located to the right from the motion via a term, S. As a result, the constants A
mainpeak,whichcorrespondstodrivingfrequencyexceeding −τs ±
ΩN. andAz inEq. (4)evolvewithtimesimplyas dAdt±+Aτs± =
0 and ddAtz + Aτsz = 0. The fact that the coefficients A±
and A decay as a simple exponent is sufficient16–19 to
depend on the local value of Ω , it might be expected z
N present the noise spectrum as
that they average out. Below we demonstrate that this
is not the case. It appears that the drive-related peaks π
δs2 = ∆(ω) (5)
remainsharpafteraveraging. Thereasonisthatthema- zω 2
jor contribution to the averaged peaks comes from the π(cid:104) (cid:105)
δs2 =δs2 = ∆(ω Ω )+∆(ω+Ω ) . (6)
realizationsofthehyperfinefieldforwhichΩ iscloseto xω yω 4 − N N
N
the drive frequency, Ω . More precisely, the domain of
dr In the presence of the ac drive the equation describing
Ω contributing to δs2 is Ω Ω ω , where ω
N (cid:104) ω(cid:105) | N − dr|∼ dr dr the spin dynamics assumes the form
is the drive amplitude.
IfthefrequencyΩdr exceedsδetherearenorealizations dS (cid:104) (cid:105)
ofthehyperfinefieldinresonancewithdrive. Inthiscase = Ω +2ω cos (Ω t) S, (7)
dt N dr dr ×
the drive has a strong effect on the noise spectrum when
the amplitude ω becomes comparable to Ω . We will where 2ω and Ω are the drive amplitude and fre-
dr dr dr dr
see that δs2 transforms into a sequence of peaks at quency, respectively. We will still assume that the hy-
(cid:104) ω(cid:105)
ω = nΩ . The magnitudes of the peaks behave essen- perfine field is directed along the z-axis. Then the com-
n dr
tially as J2(ω /Ω ), where J (x) is the Bessel function ponent of drive, responsible for the spin precession, is
n dr dr n
of the order n. 2ω , which is the projection of the driving field on the
⊥dr
x-y plane. If the drive amplitude is much smaller than
Ω , the rotating wave approximation applies. Then the
N
II. GENERAL EXPRESSION FOR THE NOISE solutionofEq. (7)iswellknownsincetheclassicalpaper
SPECTRUM WITH DRIVE Ref. 23. We will cast this solution in the form, which, in
the limit ω 0, reduces to Eq. (4). Namely:
dr →
In the absence of spin relaxation and drive, the spin
dynamics is governed by the equation dS = Ω S,
which yields three modes: dt N × S+  α+ei(Ωdr−ωRN)t   β+eiΩdrt 
SS+z=A+eiΩ0Nt+Az01+A− 00 ,(4) SS−z=A+α−eα−zie(−Ωdiωr+RNωtRN)t+Azβ−eβ−ziΩdrt
S− 0 0 e−iΩNt  γ+ei(Ωdr+ωRN)t 
wfiehledreΩS±is=dir√1e2ct(eSdxa±loinSgy)t.heHze-raexwise. Saspsiunmreedlaxtahtaitonthies +A−γ−e−γzi(eΩiωdrRN−tωRN)t, (8)
N
3
whereωN isthefrequencyoftheRabioscillationsdefined �s2
R h !i
as 1.15.5 (a)
ωN =(cid:2)ω2 +(Ω Ω )2(cid:3)1/2. (9) 11.0
R ⊥dr dr− N
The relation between the coefficients for each mode of 0.05.5
precession, say, between α , α , and α , follows from
+ z − 0
Eq. (7) h�s2!i0.0 00..55 11.0 11..55 !/�e22.0
ω2 1.15.5 (b)
α = ⊥dr (10)
+ −2ωN(Ω Ω ωN) 11.0
R dr− N − R
ω
αz = ⊥dr (11) 0.05.5
√2ωN
R
ω2 0
α− = ⊥dr . (12) 0.0 00..55 11.0 11..55 !/�e22.0
−2ωN(Ω Ω +ωN)
R dr− N R
FIG. 3: (Color online) Averaged noise spectra are plotted
The magnitudes of α , α , α are chosen in such a way
+ z − from Eq. (34) for driving frequency, Ω , corresponding to
that the corresponding eigenvector in Eq. (8) is normal- dr
themaximumofthehyperfinefielddistributionandtwoam-
ized. It is easy to see that, as the drive decreases, α+ plitudesofthedrive: 2ωdr =0.15δe (a)and2ωdr =0.4δe (b).
approaches one, while αz and α− vanish. This applies Even for weak drive the drive-induced satellites in the aver-
for Ω > Ω . For the opposite relation, α vanishes aged spectra are well-pronounced. The width of the central
dr N +
upon decreasing drive, while α− approaches one. In a peakatω=Ωdr isτs−1 =10−2δe. Thespectraintheabsence
similar way, for remaining two eigenvectors we have of drive, illustrating the conservation of the net noise power,
are shown with red lines.
β =β =α (13)
+ − z
Ω Ω
β = dr− N (14) Suppose that the drive is weak, ω Ω . For a
z ωN ⊥dr (cid:28) N
R typical realization of the hyperfine field the difference
γ+ =α−, γ− =α+, γz =αz. (15) ΩN − Ωdr is much bigger than ω⊥dr. Then the rela-
tive magnitude of the satellites of the zero-frequency
InthepresenceofspinrelaxationthecoefficientsA± and peak is equal to ω2 /2(Ω Ω )2. With regard to
Az inEq. (8)satisfythesameequationasintheabsence the peak at ω = Ω⊥dr, itsNsh−ift ddrue to drive is small,
N
of drive. This allows to establish the form of the noise namely, ω2 /2(Ω Ω ). The magnitudes of these
spectrum, δs2ω = 13(δs2ωz +δs2ω+ +δs2ω−), of the driven satellites e⊥vdorlve diNffe−rentdlry with drive: while the satel-
systeminthesamewayasEq. (5)followedfromEq. (4). lite at ω =Ω grows as ω2 /2(Ω Ω )2, the satellite
One has at ω = Ω +drωR 2Ω ⊥dΩr haNs a−mudrch smaller rela-
D N ≈ dr − N
π(cid:110) tive magnitude ω4 /16(Ω Ω )4. A generic noise
δs2 = β2∆(ω)+γ2∆(ω ωN)+α2∆(ω+ωN) ∼ ⊥dr N − dr
ω 6 z z − R z R spectrum is illustrated in Fig. 1. Overall, the effect of
+β2∆(ω Ω )+β2∆(ω+Ω )+ drive on the noise spectrum for a typical ΩN is weak. In
+ − dr − dr addition,thepositionsofallsatellites,exceptthepeakat
γ+2∆(ω−Ωdr−ωRN)+α−2∆(ω+Ωdr+ωRN)+ ω =Ωdr, depend on ΩN, i.e. these positions are random.
(cid:111) It is not clear whether these satellites manifest them-
α2∆(ω Ω +ωN)+γ2∆(ω+Ω ωN) .(16)
+ − dr R − dr− R selves in the ensemble-averaged noise spectrum. As we
will see in the next Section, the averaging preserves the
It is a direct consequence of normalization of the eigen-
(cid:82) drive-induced peaks in the noise spectrum. The reason
vectors in Eq. (8) that the area dωδs2 does not de-
ω is that the realizations of Ω , which survive the averag-
pend on the drive. Four groups of terms corresponding N
ing, are the those in “resonance” with drive. For such
to the four lines in Eq. (16) can be interpreted as fol-
realizations, with Ω Ω ω , the magnitudes of
lows. The low-frequency peak ∆(ω) in the presence | N − dr| ∼ ⊥dr
∝ the satellites are anomalously big. This compensates for
of drive develops two satellites at ω = ω . From the
± dr small statistical weight of the resonant configurations.
relation β2 +γ2 +α2 = 1, which can be easily checked
z z z
usingEqs. (10),(13),itfollowsthatthenoisepowergets
redistributed between the three peaks. The peak which,
III. ENSEMBLE AVERAGING
in the absence of drive, was located at ω = Ω shifts to
N
the position ω =Ω ωR. It also follows from Eq. (16)
dr− N
that this peak develops two satellites at higher frequen- We will perform the averaging over hyperfine fields in
cies ω = Ω and at ω = Ω +ωR with magnitudes β2 two steps. Firstly, we will average over the magnitudes,
dr dr N +
andγ2,respectively. Again,thenetnoisepowerinthese Ω , with distribution function F(Ω ). and then the As
+ N N
three peaks does not depend on drive. asecondstep,wewillaverageoverdirectionsofΩ with
N
4
respect to the driving field, ω . peak falls off rapidly with ω as the difference ω ω
dr − ⊥dr
increases. From the area conservation, it follows from
Eq. (18) that the area under the peak Eq. (22) should
A. Averaging over the magnitudes of hyperfine be equal to π2ω F(Ω ). On the other hand, from
fields 12 ⊥dr dr
Eq. (22) we see that this area comes from the domain
(ω ω ) ω δ , i.e. the area conservation is
Averaging of the first term in Eq. (16) is straightfor- ens−ured⊥dlroca∼lly.⊥Tdrh(cid:28)is suepports our statement that the
ward, since ∆(ω) does not depend on ΩN. Using the averaged peak remains narrow.
definition Eq. (13), we have Two terms on the second line of Eq. (16) describe the
π β2 ∆(ω)= π(cid:104)1 (cid:90)∞dΩ ω⊥2drF(ΩN) (cid:105)∆(ω). piseankost aatffeωct=ed±bΩydtrh.eSeinmsielmarblteoa∆ve(rωa)gipnega.k,Atvheeriargisnhgapoef
6(cid:104) z(cid:105) 6 − N(Ω Ω )2+ω2 the magnitude is completely analogous to that for ∆(ω)
0 N − dr ⊥dr peak since β2+2α2 =1. Thus, the contribution of these
(17) z z
peaks to the noise spectrum is given by
Taking into account that the typical difference (Ω
N −
Ωwedrg)e∼t δe ismuchbiggerthanthedriveamplitude,ω⊥dr, π2ω F(Ω )∆(ω Ω ), (23)
12 ⊥dr dr ± dr
π(cid:104) (cid:105)
1 πω F(Ω ) ∆(ω). (18) and grows linearly with the drive amplitude.
6 − ⊥dr dr
ThelasttwolinesinEq. (16)describethepeaksinthe
Thus, the reduction of the magnitude of the zero- noise spectrum at frequencies ω = Ω ωN. Firstly,
frequency peak due to drive is linear in drive amplitude we note that the magnitudes of all f±ourdrp±eaksRare equal
andcomesfromthe“resonant”realizationsofthehyper- toeachother. Thisfollowsfromthefactthatthesemag-
fine fields for which ΩN ≈Ωdr. nitudes, γ±2 and α±2 are determined by the values of ΩN
It is less trivial to realize that the peaks at ω = ωN forwhichtheargumentsofthecorrespondingLorentzians
± R
described by the second and third terms in Eq. (16) arezero. Nowtheequalityofallpeakmagnitudesfollows
remainsharpuponaveragingeventhoughtheirpositions from the relation
depend on Ω . The expression for the average of the
N
second term reads γ = α , (24)
+|ΩN=ω−ωRN +|ΩN=ω+ωRN
∞
(cid:90)
π γ2∆(ω ωN) = π ω2 dΩ ∆(ω−ωRN)F(ΩN). whichiseasytocheckusingEqs. (10)and(13). Focusing
6(cid:104) z − R (cid:105) 12 ⊥dr N (ωN)2 onpositiveω,theaveragingoverΩ iseasytoperformby
0 R replacing Lorentzians by correspoNnding δ-functions and
(19)
using the following identities
At this point we make use of the fact that the width
of the Lorentzian, ∆(ω ωN), is much smaller than the (cid:16) (cid:17) (cid:16) (cid:17)
width of the distribution−funRction. Firstly, this allows to δ ω Ω ωN +δ ω Ω +ωN =
− dr− R − dr R
set ωN =ω in the denominator. Secondly, the values Ω
R N (cid:16) (cid:17) ωN
that contribute to the integral are close to 2ωNδ (ω Ω )2 (ωN)2 = R
R − dr − R (cid:112)(ω Ω )2 ω2
(cid:112) − dr − ⊥dr
Ω =Ω ω2 ω2 . (20) (cid:104) (cid:16)(cid:112) (cid:17)
N dr± − ⊥dr δ (ω Ω )2 ω2 (Ω Ω ) +
× − dr − ⊥dr− dr− N
Upon switching to the integration over ωN and taking
R (cid:16)(cid:112) (cid:17)(cid:105)
into account that δ (ω Ω )2 ω2 +(Ω Ω ) . (25)
− dr − ⊥dr dr− N
(cid:113)
dωN (ωN)2 ω2 (cid:112)ω2 ω2 Upon averaging, the last two δ-functions pick the
R = R − ⊥dr = − ⊥dr, (21)
dΩN ωRN ω d(cid:112)istribution F(ΩN) at the values ΩN = Ωdr ±
(ω Ω )2+ω2 . The resulting average shape of the
we find − dr ⊥dr
two peaks at ω =Ω ωN reads
dr± R
π π ω2
γ2∆(ω ωN) = ⊥dr
6(cid:104)(cid:104)(cid:16)z −(cid:112)R (cid:105) 12(cid:17)ω(cid:112)ω2(cid:16)−ω⊥2dr(cid:112) (cid:17)(cid:105) (cid:112)πω⊥4dr
F Ω ω2 ω2 +F Ω + ω2 ω2 . 24ω Ω (ω Ω )2 ω2
× dr− − ⊥dr dr − ⊥dr | − dr| − dr − ⊥dr
(22) (cid:34) F(cid:16)Ω (cid:112)(ω Ω )2 ω2 (cid:17)
dr− − dr − ⊥dr
Sincethedrivingfrequencyismuchbiggerthanthedriv- × (ω Ω +(cid:112)(ω Ω )2 ω2 )2
ing amplitude, both arguments in the distribution func- − dr − dr − ⊥dr
tion can be replaced by Ω . With regard to the fre- F(cid:16)Ω +(cid:112)(ω Ω )2 ω2 (cid:17) (cid:35)
N dr − dr − ⊥dr
quency dependence, Eq. (22) exhibits an integrable di- + (cid:112) . (26)
(ω Ω (ω Ω )2 ω2 )2
vergence near ω ≈ω⊥dr. Most importantly, the averaged − dr− − dr − ⊥dr
5
Note now, that for in the limit ω 0 the above 1/√ω ω and 1/√ω Ω ω anomalies. These
⊥dr → − ⊥dr − dr± ⊥dr
expression reproduces the second term in Eq. (1), i.e. anomalies do not disappear completely but become log-
the background noise spectrum in the absence of drive. arithmical. Both averages can be evaluated analytically.
Formally this follows from the fact that either the first The averaging of Eq. (22) yields
denominators in Eq. (26) (for ω < Ω ) or the second
odredneormtionaitsoorla(tfeorthωe<driΩvedr-r)eblaetceodmpeesaskmdsraflrlo,m∝Eωq⊥4.dr.(2I6n) π6F(Ωdr) G(cid:16)ωωdr(cid:17), (29)
one has to subtract
wherethedimensionlessfunctionG(z)describingtheav-
π (cid:16) ω Ω (cid:112) (cid:17) eraged shape is given by
F Ω + − dr (ω Ω )2 ω2 (27)
6 dr ω Ω − dr − ⊥dr
| − dr| 1(cid:90)π sin3θ (cid:40)z2+1ln√z+1 1, z <1,
ftrioonmoEfqω.. (O26n).thTehoitshseurbhtarnacdt,eadftteerrmtheissuabstmraocottiohnf,uEnqc-. G(z)= 20 dθ z(cid:112)z2−sin2θ = z224+zz1lnzz−+111−z−2 −12,2 z >1.
(26) would describe two narrow peaks at ω+Ω ω . (30)
dr± ⊥dr
This again allows to set Ω = Ω in the argument of The divergence near z =1 should be cut off at (1 z)
N dr − ∼
distribution function. It is convenient to cast the final 1/ω τ . The large-z behavior of Eq. (30) is G(z)
dr s ≈
result in the form 2/3z2.
The result of averaging of Eq. (28) can be presented
(cid:112)
1π2(cid:104)(cid:112)(ω |ωΩ−Ω)2dr| ω2 + (ω−ωΩdrΩ)2−ω⊥2dr−2(cid:105)F(Ωdr). in the form similar to Eq. (30)
− dr − ⊥dr | − dr| (28) π F(Ω )H(cid:16)|ω−Ωdr|(cid:17), (31)
It is worth noting that the peaks described by Eq. (26), 12 dr ω
dr
having the same width ω , are “shaper” than the
peak Eq. (22) at the R∼abi⊥fdrrequency. They decay as where the function H(z) is defined as
ω4 /(ω Ω )4, while the peak at the Rabi frequency
de⊥cdarys as−ω2dr/ω2. 1(cid:90)π 2zsinθ sin3θ
⊥dr H(z)= dθ (cid:112) − 2. (32)
2 z2 sin2θ −
0 −
B. Averaging over orientations of hyperfine fields
Similarly to Eq. (30), the integral can be evaluated ana-
lytically yielding
The shape of the peaks in the noise spectrum derived
abovedependsonω⊥dr,theprojectionofthedrivingfield (cid:40)3z2−1ln√z+1 3, z <1,
on the plane normal to the local hyperfine field. If the H(z)= 2z 1−z2 − 2 (33)
anglebetweenΩ andω isθ,thenω =ω sinθ. To 3z2−1lnz+1 3, z >1.
N dr ⊥dr dr 4z z−1 − 2
find the ensemble-averaged shape of the noise spectrum
withdriveonehastoaverageoverθallfourcontributions The large-z behavior of the combination in the square
Eqs. (18), (22),(23), and (28) as 1(cid:82)πdθsinθ (......). brackets is 1/z4. Dimensionless functions G(z) and
2 0 ∝
Averaging of Eqs. (18), (23) simply reduces to H(z) are plotted in Fig. 2.
the replacement of ω by πω without affecting the Combining all the above, the final result for the
⊥dr 4 dr
Lorentzian shapes of the narrow peaks. The prime ef- ensemble-averaged noise spectrum of the driven system
fect of averaging of Eqs. (22) and (28) is the rounding of can be cast in the form
(cid:40) (cid:41)
π(cid:104) π2ω (cid:105) π π (cid:16) ω (cid:17) π2 1 (cid:16) ω Ω (cid:17)
δs2 = 1 drF(Ω ) ∆(ω)+ F(ω)+ F(Ω ) G + ω ∆(ω Ω )+ H | ± dr| . (34)
(cid:104) ω(cid:105) 6 − 4 dr 6 6 dr ±ω 8 dr ± dr 2 ω
dr dr
Itisnaturalthatthedrive-relatedcontributionsarepro- peak due to drive. The area under all three peaks in the
portional to the density, F(Ω ), of the resonant real- secondlineofEq. (34)is ω . Inthisregard,theweak-
dr ∼ dr
izations of hyperfine fields. The net effect of drive on nessofdrivemeansthattheportionofthenoisespectrum
the noise spectrum is maximal if Ω is chosen near the affectedbydriveisrelativelysmall. However,withinthis
dr
maximum of the distribution F(Ω ). Then, at frequen- portion, the spectrum is fully dominated by drive, since,
N
cies ω ω , the background noise is determined by for the resonant realizations, the drive changes the spin
∼ dr
F(ω) ω2 and is much weaker than the low-frequency dynamics completely. Formally, it is the consequence of
∝
6
realizations, the spin dynamics will be affected by drive
0.06h�s2!i (a) 0.06h�s2!i (b) when the amplitude, ω , becomes comparable to Ω .
dr dr
0.05 0.05 Suppose that ω is directed along x, while the projec-
0.04 0.04 tionsofΩ areedqrualtoΩx,Ωy,andΩz. Withoutdrive,
0.03 0.03 N N N N
the two frequencies of the spin dynamics are ω = 0 and
0.02 0.02
(cid:104) (cid:105)1/2
0.01 !/�e 0.01 !/�e ω =ΩN = (ΩxN)2+(ΩyN)2+(ΩzN)2 . Theprimeeffect
0.060h�s2!i 5 10 15 20(c) 0.06h0�s2!i 5 10 15 20(d) of a fast drive is that the frequency ΩN transforms into
0.05 0.05 (cid:34) (cid:35)1/2
0.04 0.04 λ = (Ωx)2+(cid:110)(Ωy)2+(Ωz)2(cid:111)J2(cid:16)ωdr(cid:17) , (35)
0.03 0.03 N N N N 0 Ω
dr
0.02 0.02
0.01 0.01
!/�e !/�e where J0(x) is the zero-order Bessel function. Besides,
0 5 10 15 20 0 5 10 15 20 both frequencies ω = 0 and ω = λ acquire satellites at
N
ω =nΩ andω =λ +nΩ . ThederivationofEq. (35)
dr N dr
is given in the Appendix. According to Eq. (35), as the
FIG. 4: (Color online) Evolution of the averaged noise spec- (cid:16) (cid:17)
tra in the case of a fast drive, Ωdr =6.7δe, with drive ampli- drive amplitude increases, J0 Ωωddrr falls off, so that the
tude, ωdr. The values of the amplitudes are ωdr =0.4Ωdr(a), spin precesses only in the y-z plane.
ω = 0.8Ω (b), ω = 1.2Ω (c), and ω = 1.6Ω (d).
dr dr dr dr dr dr The derivation of the noise spectrum in the case of a
Narrow peaks have the width τ−1 = 0.0067δ . Satellites of
s e fast drive is in line with procedure employed in Sect II.
thezero-frequencypeakatω=nΩ graduallydevelopupon
(cid:16)dr (cid:17) In the Appendix, along with deriving Eq. (35), we find
increasing ω . Since the value J ωdr is close to 1 for all
dr 0 Ωdr the solutions of the equations of motion of a driven spin
ωdr thelow-frequencypartsofthespectrahasthesameshape corresponding to the frequencies ω =0 and ω =λN and
as in the absence of drive.
present these solutions in the form of combination of the
normalized eigenvectors similar to Eq. (8)
0.020h�s2!i (a) 0.020h�s2!i (b) S+ µ+eiλNt+iΦ(t)  η+eiΦ(t) 
0.015 0.015 Sx=B+ µxeiλNt +Bx ηx 
0.010 0.010 S− µ−eiλNt−iΦ(t) η−e−iΦ(t)
0.005 0.005 ν+e−iλNt+iΦ(t)
!/�e !/�e +B− νxe−iλNt , (36)
0.020h0�s2!i 10 20 30 (c4)0 0.020h0�s2!i 10 20 30 4(0d) ν−e−iλNt−iΦ(t)
0.015 0.015 where S± = √1 (Sy iSz), the oscillating phase Φ(t) is
2 ±
0.010 0.010 defined as
0.005 0.005 (cid:90) t ω
!/�e !/�e Φ(t)=ωdr dt(cid:48)cos(Ωdrt(cid:48))= Ωdr sin(Ωdrt), (37)
0 10 20 30 40 0 10 20 30 40 0 dr
and the components of the eigenvectors are related as
FIG. 5: (Color online) Same as Fig. 4 for stronger drive am- follows
plitudesωdr =2.5Ωdr(a),ωdr =3.5Ωdr (b),ωdr =4.5Ωdr (c), Ωx
and ω = 5.6Ω (d). The peaks at ω = nΩ have gaus- η = N, (38)
dr dr dr x
λ
sian shape. Their amplitudes evolve with drive in oscillating N
fashion.
(cid:104) (cid:105)1
Eq. (9), that for (Ω Ω ) (cid:46) ω all the frequencies µ =ν = iη =iη = (ΩyN)2+(ΩzN)2 2J (cid:16)ωdr(cid:17),
N − dr dr x x − + − − √2λ 0 Ω
ωN are close to ω . The evolution of the noise spectrum N dr
R dr (39)
with drive amplitude is illustrated in Fig. 3.
(cid:104) (cid:105)
µ = ν = i (ΩyN)2+(ΩzN)2 J2(cid:16)ωdr(cid:17), (40)
IV. FAST DRIVE + − − 2 λ (λ Ωx) 0 Ω
N N − N dr
If the drive frequency is much bigger than the width (cid:104) (cid:105)
of distribution of the hyperfine fields, there are no real- µ = ν = i (ΩyN)2+(ΩzN)2 J2(cid:16)ωdr(cid:17). (41)
izationsof Ω which areinresonance withdrive. Forall − − + 2 λ (λ +Ωx) 0 Ω
N N N N dr
7
The components S of the eigenvectors are simple expo- At zero drive the ensemble averaging over Ω resulted
x N
nents. Thenthecontributionδs2 tothenoisespectrum in the noise spectrum given by F(ω), Eq. (3). By con-
xω
follows directly from Eq. (36) trast, from Eq. (45) we see that, for a strong drive, the
π(cid:104) (cid:105) ensemble averaging of each term yields the distribution
δs2xω = 2 |ηx|2∆(ω)+|µx|2∆(ω−λN)+|νx|2∆(ω+λN) . function of ΩxN, i.e. the shapes of the satellites are gaus-
(42) sian. Theoverallnoisespectruminthepresenceofafast
Concerning the contributions δs2 and δs2 , they origi- drive is illustrated in Figs. 4, 5.
yω zω
natefromtheS andS componentsoftheeigenvectors,
+ −
which are not simple exponents. This gives rise to the
satellitesspacedbynΩ inthenoisespectrum. Relative V. DISCUSSION
dr
magnitudes of the satellites are found from the Fourier
expansion
Our main result is Eq. (34) for the averaged noise
(cid:88) (cid:16)ω (cid:17) • spectrum. This result was obtained within the ro-
exp(iΦ(t))= J dr exp(inΩ t). (43)
n Ω dr tating wave approximation and applies for large
n dr
enough drive amplitudes ω τ 1. Fig. 3 il-
Since δs2 and δs2 give equal contributions to the lustrates the evolution of thdressp(cid:29)ectrum with ω .
yω zω dr
ensemble-averaged spectrum, it is convenient to average As ω increases, the magnitude of a central peak
dr
the combination δs2 +δs2 . For this combination the at ω = Ω grows linearly with drive, while the
yω zω dr
resultforagivenhyperfinefieldassumesacompactform satellites at ω Ω = ω broaden linearly with
| ± dr| dr
drive. Central peak and satellites merge at weak
π(cid:104) (cid:88) (cid:16)ω (cid:17)
δs2yω+δs2zω = 2 (|η+|2+|η−|2) Jn2 Ωdr ∆(ω−nΩdr) drive ωdrτs ∼1. For smaller ωdr the effect of drive
n dr on the spin dynamics is weak even for “resonant”
+(µ 2+ µ 2)(cid:88)J2(cid:16)ωdr(cid:17)∆(ω λ nΩ )+ hyperfine field configurations and can be treated
| +| | −| n Ω − N − dr perturbatively. The effect of drive amounts to re-
n dr
(|ν+|2+|ν−|)2(cid:88)Jn2(cid:16)Ωωdr(cid:17)∆(ω+λN +nΩdr)(cid:105). (44) ptalalcpeemake.ntTωhderrbeylaωtid2vreτscoinrrtehcetioanmtpolittuhdeeboafcktghreocuennd-
n dr value of δs2 due to drive is ω2 τ2 1. Despite
(cid:104) ω(cid:105) dr s (cid:28)
From Eqs. (42) and (44) we can trace the evolution of beingsmall,theeffectofdrivecanbedistinguished
the averaged noise spectrum upon increasing the drive in the derivative with respect to ω. Indeed, the
amplitude. Firstly, in the weak-drive limit, ω Ω , derivative of the background can be estimated as
when the magnitudes of the satellites are negldigrib(cid:28)le, advr- δ1eF(Ωdr), while the estimate for the derivative of
eraging of Eqs. (42), (44) reproduces the result Eq. (1). the central peak is ω2 τ3F(Ω ). Thus the drive
(cid:16) (cid:17) dr s dr
Indeed, when J ωdr 1, the frequency λ returns to dominates the derivative in the domain
0 Ωdr ≈ N
Ω . Themagnitudes, η 2 and(η 2+ η 2),ofazero- 1
farneNdqu[(eΩncyy)2p+ea(kΩszin)2E]/qΩs|2.x,(|4a2s)inantdh|e(4a+4b|)sebne|cceo−mo|fed(rΩivxNe.)2S/iΩm2N- ωdrτs (cid:29) (δeτs)1/2. (46)
N N N
siluamrlye,stihtsezfearcot-dthriavtetvhaelumeafgonlliotwudsefroofmωg=eneλrNalpreealaktiaosn- Largetypicalvalueofthehyperfinefield,δe (cid:29)τs−1,
which is presumed, allows to distinguish the effect
µ 2+ µ 2+ µ 2 =1.
x + − of drive even when it is weak.
| | | | | |
As the drive amplitude increases, the magnitude of a
ω = 0 peak first decreases but eventually returns to its
AsinRefs. 16-19,weassumedthatspin-relaxation
(cid:16) (cid:17)
zero-drive value. Indeed, in the limit J ωdr 0, we • time, τ , resulting from random short-time corre-
0 Ωdr → s
have η 2 1, while η and η vanish. This suggests lated fields different from hyperfine field, is the
x + −
| | ≈ same for all elements of the ensemble. ac-driven
that the ω = nΩ satellites of a zero-frequency peak
dr system is stationary but not equilibrium. Still we
develop at ω Ω , but disappear in the strong-drive
dr ∼ dr calculated the noise spectrum from eigenmodes.
limit. By contrast, the satellites at ω = λ + nΩ
± N dr Justification for doing this is that the temperature
persistinthestrong-drivelimit. Inthislimitµ vanished,
x
and thus we have µ 2+ µ 2 1. This suggests that is much higher than all the frequencies involved.
+ −
| | | | ≈ Underthiscondition,alltheeigenmodesareequally
all the noise power in ω = λ peak at zero drive gets
N represented in the spin dynamics18.
redistributedbetweenthesatellitesatstrongdrive. Also,
in the limit of strong drive, we have λ Ωx , so that
N ≈ | N| In a recent paper Ref. 24 a direct measurement of
Eq. (44) assumes the form • the spin-relaxation rate, τ−1, was reported. Such
s
δs2 +δs2 = 1 (cid:88)J2(cid:16)ωdr(cid:17) τs . ameasurementbecamepossibleduetoimplement-
yω zω 2 n Ω 1+(ω+Ωx +nΩ )2τ2 ing of the spin noise correlation techniques, which
n(cid:54)=0 dr N dr s involves two laser beams and allows to probe only
(45)
specificconfigurationsofthehyperfinefield. Inthis
8
regard, the effect of the ac drive is prominent be- Asanextstep, weaverageEqs. (53)overthetimeinter-
(cid:16) (cid:17)
cause it also results from specific “resonant” con- val π , π . The justification for this step is that,
figurations. −Ωdr Ωdr
since Ω Ω , the spin projections do not change sig-
dr (cid:29) N
In experiments on different semiconductor nificantlyduringthisinterval. Thusonecanaverageonly
• structures4,5,7,8 the measured width of the noise cos(cid:0)Φ(t)+Ωxt(cid:1) and sin(cid:0)Φ(t)+Ωxt(cid:1)
N N
spectra ranged from 2MHz to 50MHz. Appli-
∼ ∼
cation of the ac drive with comparable frequency
does not constitute a problem, see e.g. Ref. 25.
It will require adding a coil to the conventional cos(cid:0)Φ(t)+Ωxt(cid:1) =J (cid:16)ωdr(cid:17)cosΩxt, (54)
setup4,5,7,8. (cid:104) N (cid:105) 0 Ωdr N
sin(cid:0)Φ(t)+Ωxt(cid:1) =J (cid:16)ωdr(cid:17)sinΩxt. (55)
Due to isotropy of the hyperfine fields the noise (cid:104) N (cid:105) 0 Ω N
• dr
spectrum calculated above does not depend on the
direction of the ac magnetic field. This is the case
when the electron g-factor is isotropic. In experi-
ment Ref. 7 it was established that the g-factor is It is also convenient to switch in the averaged equations
strongly anisotropic27. This conclusion was drawn to S+(cid:48) = √1 (Sy(cid:48) +Sz(cid:48)) and S−(cid:48) = √1 (Sy(cid:48) Sz(cid:48)). Then
2 2 −
from the analysis of the shift of the noise spec- we get
trummaximumwithexternalmagneticfield. With
anisotropic g-factor, drive-induced features of the
noise spectrum will depend on the direction of ω .
dr
tWivheilteotthheepaonsiistoiotnroopfy,atpheeaskepatarωat=ionΩdorfitshiensseantesli-- ∂∂Stx(cid:48) =−√12ΩzNJ0(cid:16)Ωωdr(cid:17)(cid:104)S+(cid:48)eiΩxNt+S−(cid:48)e−iΩxNt(cid:105),
dr
ltihteesbwigigllerbeg-bviaglgueer. for the drive polarization along ∂S∂t+(cid:48) = √12ΩzNJ0(cid:16)Ωωdr(cid:17)Sx(cid:48)e−iΩxNt,
dr
VI. APPENDIX ∂S∂t−(cid:48) = √12ΩzNJ0(cid:16)Ωωddrr(cid:17)Sx(cid:48)eiΩxNt. (56)
WithoutlossofgeneralitywecansetΩy =0. Westart
N
from the equations of motion for the spin projections
We see that the dynamics after averaging is slow, which
justifiestheaveragingperformed,seeRef. 26forrigorous
∂S
x = ΩzS , (47) justification. One can also see that Eqs. (56) have the
∂t − N y
form of equations of motion in a constant magnetic field
∂∂Sty =−(ωdrcosΩdrt+ΩxN)Sz+ΩzNSx, (48) with x- and z-components being ΩxN and ΩzNJ0(cid:16)Ωωddrr(cid:17),
∂S respectively. Finite Ωy is naturally included as a y-
z =(ω cosΩ t+Ωx)S . (49) N
∂t dr dr N y component. This immediately leads us to Eq. (35) of
the main text. Three eigenvectors correspond to rota-
To take advantage of the fact that the drive is fast it is tions with frequencies ω = λ , ω = 0, and ω = λ .
convenient26 to switch to the new variables To return to the lab frame onNe has to multiply S+−(cid:48) bNy
Sx(cid:48) =Sx, (50) edxopes(inΦo(tt)ch+aniΩgexNtt)haenrdelSat−i(cid:48)onbybeextwpe(e−niΦth(et)c−omiΩpxNont)e.nTtshoisf
Sy(cid:48) =Sycos(cid:0)Φ(t)+ΩxNt(cid:1)+Szsin(cid:0)Φ(t)+ΩxNt(cid:1), (51) the eigenvectors which have the form Eq. (36).
Sz(cid:48) =−Sysin(cid:0)Φ(t)+ΩxNt(cid:1)+Szcos(cid:0)Φ(t)+ΩxNt(cid:1), (52)
wherethephaseΦ(t)isdefinedbyEq. (37). Thephysical
meaning of the above transformation is moving into the
rotating frame in which the ac field is canceled. The
equations of motion for the new variables read
∂Sx(cid:48) =ΩzS sin(cid:0)Φ(t)+Ωxt(cid:1) ΩzS cos(cid:0)Φ(t)+Ωxt(cid:1), VII. ACKNOWLEDGEMENTS
∂t N z N − N y N
∂Sy(cid:48) =ΩzS cos(cid:0)Φ(t)+Ωxt(cid:1),
∂t N x N
We are grateful to C. Boehme, Y. Li, and E. G.
∂∂Stz(cid:48) =−ΩzNSx(cid:48)sin(cid:0)Φ(t)+ΩxNt(cid:1). (53) MsuipsphochrteendkobyfoNrSiFnstihgrhotufuglhdMisRcuSsEsiConDs.MRT-h1i1s21w2o5r2k. was
9
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