Resonant spin amplification in Faraday geometry
P. Schering, E. Evers, V. Nedelea, D. S. Smirnov, E. A. Zhukov, D. R. Yakovlev, M. Bayer, G. S. Uhrig, A. Greilich
RResonant Spin Amplification in Faraday Geometry
P. Schering, E. Evers, V. Nedelea, D. S. Smirnov, E. A. Zhukov,
2, 3
D. R. Yakovlev,
2, 3
M. Bayer,
2, 3
G. S. Uhrig, and A. Greilich Condensed Matter Theory, Technische Universität Dortmund, 44221 Dortmund, Germany Experimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia (Dated: February 5, 2021)We demonstrate the realization of the resonant spin amplification (RSA) effect in Faraday ge-ometry where a magnetic field is applied parallel to the optically induced spin polarization so thatno RSA is expected. However, model considerations predict that it can be realized for a centralspin interacting with a fluctuating spin environment. As a demonstrator, we choose an ensembleof singly-charged (In,Ga)As/GaAs quantum dots, where the resident electron spins interact withthe surrounding nuclear spins. The observation of RSA in Faraday geometry requires intense pumppulses with a high repetition rate and can be enhanced by means of the spin-inertia effect. Poten-tially, it provides the most direct and reliable tool to measure the longitudinal g factor of the chargecarriers. The possibility of using the spin degree of freedom forquantum information [1] continues to drive research onsemiconductor nanostructures [2–5]. The main charac-teristic in this field is defined by the lifetime of the in-formation or the spin coherence time. Complementary,the development of spintronics [6] over two decades gavebirth to a plethora of experimental tools for the investi-gation of the spin dynamics in semiconductor nanostruc-tures. A major part of these methods is based on theinterrelation between the spin of a charge carrier and thepolarization of a photon emitted or absorbed by the semi-conductor structure [7]. The most popular ones are theHanle effect [8] and the time-resolved pump-probe tech-nique, based on the pulsed-laser excitation [9–11], whichcan be extended to detect the spin dynamics on arbitrarylong timescales with femtosecond resolution [12]. Otherpowerful tools are the spin-noise spectroscopy [13, 14]and the spin-inertia technique [15–17].One of the most basic parameters of the spin dynam-ics is the g factor, which is often anisotropic in semicon-ductor nanostructures. Its transverse component can bemeasured very precisely when a magnetic field is appliedin Voigt geometry by means of the resonant spin ampli-fication (RSA) effect [18]. It is based on the pump-probetechnique, where the spin polarization is measured at afixed pump-probe delay as a function of a transverse mag-netic field. Provided the spin relaxation time is longerthan the laser repetition period, the spin polarization isamplified when the Larmor precession period is a multi-ple integer of the laser repetition period [9]. The RSA ef-fect can also be used to evaluate the spin relaxation time,the spread of the transverse g factor, and the strengthof the hyperfine interaction [19]. The RSA method hasbeen successfully applied to a variety of systems, rangingfrom bulk GaAs [18], III-V and II-VI quantum wells andepilayers [20–22], to quantum dots [23, 24].In Faraday geometry where the magnetic field is par-allel to the optical axis, there is no spin precession of the charge carriers on average such that it is difficultto determine their g factor. In this Letter, however,we demonstrate that RSA can emerge in this geometryfor an ensemble of n -doped (In,Ga)As/GaAs quantumdots (QDs), which allows us to measure the longitudinal g factor of the charge carriers with high accuracy. Theeffect is enabled by the hyperfine interaction between theresident electron spins and the fluctuating nuclear spinenvironment [25]. Experimental Details –
The studied sample consistsof 20 layers of (In,Ga)As QDs separated by nm barri-ers of GaAs and grown by molecular beam epitaxy on a(100)-oriented GaAs substrate. A δ doping layer of Sili-con nm above each QD layer provides a single electronper QD on average. Rapid thermal annealing at ◦ Cfor s homogenizes the QD size distribution and shiftsthe average emission energy to . eV. The QD den-sity amounts to cm − .The sample is cooled to . K in a Helium gas atmo-sphere inside a cryostat with a split-coil magnet. A su-perconducting solenoid pair creates an external magneticfield B ext e z in the direction of light incidence, i.e., alongthe optical z axis with an accuracy of 2 degree (Fara-day geometry). The sample is illuminated by laser pulseswith a central optical energy of . eV and a full widthat half maximum of . meV. The pulses have a durationof ps and are emitted with a repetition frequency of GHz. They are split in pump and probe pulses whichare degenerate in photon energy and shifted by . meVto the low energy flank of the QD photoluminescence,see Fig. 1(a). The pump pulses are directed along avariable mechanical delay line. A double modulationscheme reduces the noise arising from separate detectionof the scattered pump and probe light. The helicity ofthe pump is modulated by an electro-optical modulatorat a frequency f m between left- and right-handed circu-lar polarization ranging from . to kHz. The lin-early polarized probe beam is intensity modulated using a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b QDs (a) g , ω N Energy (eV) P ho t o l u m i n e s ce n ce
920 910 900 890
Wavelength (nm)
Laser 10 −1 f m (kHz) F a r a d a y E lli p ti c it y ( a r b . un it s ) P pu = 7 mW B ext = 400 mT P pu (mW) / τ * ( m s - ) τ s = 20 µ s τ s = 200 µ s s Figure 1. (a) Photoluminescence of the QD ensemble (gray)at T = 5 . K along with the spectrum of the laser used in thepump-probe measurements (orange). The sketch shows an n -doped QD containing an electron spin (blue) with a g factor,which is interacting with the fluctuating nuclear spin environ-ment characterized by the frequency ω N (red). (b) Faradayellipticity amplitude as a function of the pump modulationfrequency f m for a pump power of P pu = 7 mW measured ata magnetic field of B ext = 400 mT with a pump-probe delayof − ps (blue circles). The black line is the two-componentfit, see main text. The contributions of each component areshown by the solid-orange and dashed-green lines. The in-set shows the power dependence of the two correspondinginverse effective spin lifetimes /τ ∗ s . A linear extrapolationto zero power (black lines) yields τ s = 20 µ s (orange) and τ s = 200 µ s (green). a photo-elastic modulator at a frequency of kHz inseries with a Glan prism. For signal detection, the refer-ence frequency of the lock-in amplifier is running at thedifference frequency. The pump and the probe beams arefocused on the same sample spot, with the pump focusedto a spot diameter of µ m and the probe to a diameterof µ m. We measure the Faraday ellipticity amplitudeof the probe pulses using an optical polarization bridgewhich consists of a lambda-quarter plate, a Wollastonprism, and a balanced photodetector. The Faraday ellip-ticity is proportional to the electron spin polarization ofthe QDs along the optical axis [26]. Results –
Each circularly polarized pump pulse par-tially orients the spins of the resident electrons alongthe optical axis [27]. The spin polarization added byeach pulse in this way depends on its effective pulsearea Θ [28], which is determined by the average pumpbeam power and scales like P pu ∝ Θ when the poweris small [26]. If the added spin polarization exceeds therelaxed polarization along the optical axis until the nextpulse, the spin polarization builds up. This is the case fora magnetic field of B ext = 400 mT using a pump power of P pu = 7 mW, and the origin for the spin polarization inFig. 1(b) displayed by the Faraday ellipticity at − pstime delay between pump and probe pulses studied as afunction of the pump modulation frequency f m . The de- − −
200 0 200 400
Magnetic Field B ext (mT) . . . F a r a d a y E lli p t i c i t y ( a r b . un i t s ) k = +4 +3 +2 +1 − − − − Figure 2. PRCs for different pump powers given next to thecurves at a pump modulation frequency of f m = 10 kHz. Thecolored data is the experimentally measured Faraday elliptic-ity, the smooth black curves are the simulated PRCs. Thepulse area in the simulations (see Supplemental Material) ischosen to fit the experimentally obtained power dependenceof the Faraday ellipticity at large field. The positions of thesmall peaks visible in the PRCs for larger pump powers matchthe k th RSA mode (vertical grey lines) as predicted by thePSC (1), yielding g z = − . ± . for the longitudinal elec-tronic g factor. cay of the ellipticity upon increasing f m stems from thespin-inertia effect [17]. It can be described by the de-pendence E ( f m ) = E / p πf m τ ∗ s ) , where τ ∗ s is theeffective spin lifetime at the corresponding pump power.The extrapolation of τ ∗ s to zero power allows for extract-ing the intrinsic spin relaxation τ s of the electrons [15, 25].The inset in Fig. 1(b) depicts the extracted times for twocomponents present in the spin signal shown as solid-redand dashed-green curves. We relate these to two sub-sets of electrons in the QD ensemble and focus on theregime f m ≥ kHz for which only the shorter living sub-ensemble with τ s = 20 µ s contributes significantly.Without a magnetic field, the spin polarization of theelectrons decays due to the hyperfine interaction with thenuclear spin fluctuations [29]. The application of a longi-tudinal magnetic field suppresses the nuclei-induced spinrelaxation and in turn increases the spin polarization [30].This effect is referred to as polarization recovery. Forthe n -doped QD sample studied in this work, the polar-ization recovery curves (PRCs) have the typical V -likeshape (symmetric around zero field) [16, 17, 25, 30, 31].Figure 2 shows the PRCs for a wide range of pump pow-ers. For small powers, the electron spin polarization isminimal at mT and rises to a saturation level within mT. For zero field, the electron spin is only subjectto the isotropic nuclear fluctuation field characterized bythe frequency ω N / (2 π ) = 140 MHz (equivalent to a fieldof . mT), which is the characteristic frequency of theelectronic Larmor precession in the nuclear fluctuationfield, leading to nuclei-induced spin relaxation; see thesketch in Fig. 1(a). For larger magnetic fields, this lon-gitudinal field begins to dominate the randomly orientednuclear fluctuation field due to the increase of the elec-tronic Zeeman splitting. As a result, the amplitude of thespin precession is reduced and the lifetime of the spin po-larization and therefore the polarization itself increasesuntil saturation is reached [32].For larger pump powers, the spin polarization increasesand the qualitative dependence on the magnetic field isvery similar. But strong pulses suppress the in-plane elec-tron spin components, effectively accelerating the spinrelaxation, which in turn leads to a broadening of thezero field minimum in the PRC [25, 30]. Clearly, this isthe case in Fig. 2: the larger the pump power, the largerthe required magnetic field to reach saturation.The appearance of a modulation in the Faraday ellip-ticity at certain values of the longitudinal magnetic fieldin the case of large pump powers is most striking. Thisis the result of RSA in Faraday geometry enabled by nu-clear fluctuation fields [25]. As demonstrated in Fig. 2,the modulations appear at magnetic fields B ext whichfulfill the phase synchronization condition (PSC) Ω L = | k | ω R , k ∈ Z , (1)for the Larmor frequency Ω L = µ B | g z B ext | ~ − ( µ B isthe Bohr magneton and ~ the reduced Planck constant).These discrete resonance frequencies are given by multi-ples of the laser repetition frequency ω R . We label themby the mode number k . The longitudinal electronic g fac-tor | g z | determines the mode positions. While we cannotextract its sign from this effect, we know that it is neg-ative for similar (In,Ga)As/GaAs QDs [33]. Taking allpeak positions into account, we obtain g z = − . ± . .For comparison, the transverse g factor of the electronsin this sample amounts to g ⊥ = − . ± . [34].The theoretical model is presented in the SupplementalMaterial. It almost perfectly describes the experimentas shown by the black lines in Fig. 2. Noteworthy, incontrast to Refs. [17, 25, 26], the model accounts for alifetime of the photoexcited trion which is comparable tothe pulse repetition period of ns. The main deviationbetween experiment and theory is found in the regime ofvery small magnetic fields. This deviation has a narrow M -like shape, which is typical for p -doped QD sampleswith a strong field dependent spin generation rate [16,17]. Hence, we attribute the deviation to resident orphotoexcited hole spins with weak hyperfine interaction.In what follows, we describe how the nuclear fluctua-tion fields in QDs enable us to implement RSA in Faradaygeometry. In a single QD, the localized electron spin S precesses in an effective magnetic field being the sumof the external and the Overhauser field (nuclear spinbath) with frequency Ω eff := | Ω L + Ω N | . The Larmorfrequency Ω L points along the external magnetic field.But the Overhauser field Ω N , whose time evolution is Figure 3. Visibility map for the RSA mode | k | = 1 in Faradaygeometry modeled for detection by Faraday ellipticity. Thepump-probe delay is set to zero, the pump modulation fre-quency to kHz. The white cross marks the experimentalconditions for a pump power of . mW ( Θ = 0 . π ) and apulse repetition period of T R = 1 ns [ ω N / (2 π ) = 140 MHz].The sketch illustrates the mechanism leading to RSA in alongitudinal magnetic field Ω L [25]. The transverse compo-nents of the Overhauser field Ω N tilt the effective field fromthe z axis, inducing a precession of the electron spin S withfrequency Ω eff = | Ω L + Ω N | . much slower than the pulse repetition rate [29], has arandom direction, which tilts the effective field from the z axis due to its transverse components. As illustratedby the sketch in Fig. 3, this tilt leads to a precessionmotion which becomes smaller in amplitude for a largermagnetic field. This is the reason why the higher RSAmodes in Fig. 2 are less pronounced. Strong pulses re-sult in a strong generation of spin polarization while alsoaligning the electron spin along the z axis, leading toRSA whenever the PSC Ω eff = | k | ω R for a single QDis met. After averaging over the Overhauser field distri-bution described by Gaussian fluctuations with variance ω / [29, 35], the PSC (1) follows in leading order for Ω L & ω N [25]. Corrections to the resonance frequencyare O ( ω / Ω ) . For Ω L < ω N , the modes appear shifted.We point out that this mechanism is expected to workalso for single QDs because the statistical fluctuationsof the Overhauser field in time can be described by thesame distribution as for the ensemble [29, 35].RSA in Faraday geometry should be seen in a varietyof semiconductor nanostructures. But the ensemble aver-age smears out the RSA modes such that they cannot beobserved unless certain conditions are met. The prereq-uisites are: (i) an efficient hyperfine coupling, (ii) strongpump pulses, and (iii) a laser repetition frequency ω R which on the one hand allows for RSA modes that areseparated enough not to overlap significantly, but onthe other hand fall into magnetic field ranges where thespin polarization is not yet saturated [25]. The condi-tions (i) and (ii) are typically fulfilled for singly-charged n -type (In,Ga)As/GaAs QDs [36]. For p -type QDs withstrongly anisotropic hyperfine interaction [37, 38], the ef-fect is exptected to be much weaker [25]. The condi-tion (iii) is not trivial and potentially leads to a newregime of the spin dynamics. An estimate for its realiza-tion is the condition ω R & √ ω N known from standardRSA [24]. We implemented a laser source with repetitionrate ω R / (2 π ) = 1 GHz to reach this regime.To provide a quantitative basis for the above, we studythe RSA visibility defined as [30] V := E max − E min E max , (2)where E max is the Faraday ellipticity of the first max-imum at the RSA mode | k | = 1 and E min denotes theadjacent minimum for larger magnetic field | B ext | . In or-der to average out statistical fluctuations in both modeland experiment, we fit polynomials to the region aroundthe | k | = 1 mode to determine its visibility. A map of theRSA visibility in dependence of the pulse area Θ (definedin the Supplemental Material) and of the product ω N T R ( T R = 2 π/ω R is the pulse repetition period) is shownin Fig. 3. The visibility is enhanced by an increase ofthe pulse area corresponding to stronger pulses. For atoo large or too small repetition period, the visibility de-creases. Observing RSA in Faraday geometry is easiestin the intermediate regime ω N T R ∼ − . The whitecross on the color map shows that the laser source usedhere is operating at a too small power with a result-ing visibility of merely V ≈ . at a pump power of P pu = 8 . mW. Under optimal conditions, the visibilitycould reach unity. For the commonly used pulse repeti-tion periods T R = 13 . and . ns ( ω N T R ≈ . and . ),no RSA modes are discernible for our QD sample with ω N / (2 π ) = 140 MHz because they overlap.The main obstacle for a larger visibility is the smallpulse area of the applied GHz pulses because a too largepump power would heat the sample too much. Yet, wecan study the power dependence of the experimentallyseen and theoretically modeled visibility in the accessiblerange. Clearly, as demonstrated in Fig. 4(a) by experi-ment and theory, a reduction of the pump power resultsin the disappearance of the RSA modes displayed by avanishing visibility.Remarkably, the RSA visibility can be enhanced byexploiting the spin-inertia effect [15–17, 25] as demon-strated in Fig. 4(b). Due to the periodic modulationof the pump-pulse helicity, the spin-inertia effect leadsto a reduction of the average spin polarization upon in-creasing the modulation frequency f m as demonstratedin Fig. 1(b). The key idea is the following: The ap-plication of a larger frequency results in a decrease ofthe average absolute spin polarization and in turn, eachpulse can better orient the spins along the optical axisbecause a larger number is disordered. In a nutshell,the spin-inertia effect allows us to avoid the influence of P pu (mW) . . . . . R S AV i s i b ili t y V (a) f m = 10 kHz theory k = − k = +1 f m (kHz) . . . . .
12 (b) P pu = 8 . Figure 4. RSA visibility V as a function of the (a) pumppower P pu and (b) pump modulation frequency f m for thefirst RSA mode | k | = 1 . The experimental data is plottedas blue squares ( k = − ) and orange circles ( k = 1 ), thetheoretical data in black. The error bars represent the root-mean-square error of the polynomial fits around the respectiveRSA mode. spin saturation, which is detrimental to RSA in Faradaygeometry. Indeed, Fig. 4(b) shows that an increase ofthe modulation frequency increases the visibility signifi-cantly in agreement with the theoretical prediction. Forinstance, we find a visibility of V = 0 . ± . for the k = +1 mode using f m = 40 kHz. The experimental lim-itation is the deteriorated signal-to-noise ratio for largerfrequencies.In the Supplemental Material, we provide complemen-tary results which demonstrate RSA in Faraday geometryfor another QD ensemble with a weaker nuclear fluctu-ation field. There, we measure the Faraday rotation in-stead of the Faraday ellipticity, but we emphasize thateffect is easier to detect by measuring the ellipticity. Conclusion –
We demonstrated that the detrimen-tal nuclear spin fluctuations in QDs can be exploited forone’s own advantage: they enable RSA in Faraday ge-ometry. The positions of the RSA modes directly yieldthe longitudinal g factor of the charge carriers, whichwe determine very precisely to be g z = − . ± . forthe studied n -doped (In,Ga)As/GaAs QDs. This methodsolves the long-standing problem of measuring the longi-tudinal g factor of the charge carriers in weak magneticfields [39] and puts the characterization of the spin dy-namics in a longitudinal magnetic field on equal foot-ing with the case of a transverse field. The theoreticalanalysis paves the way to achieve a better visibility ofthe effect: use of strong pump pulses combined with ahigh laser repetition rate. A significant enhancement isachieved by exploiting the spin-inertia effect. We believethat this technique will also be useful for the investiga-tion of other semiconductor nanostructures, e.g., quan-tum wells.We thank M. M. Glazov for helpful discussions and ac-knowledge the supply of the QD samples by D. Reuterand A. D. Wieck. P.S. and G.S.U. gratefully acknowledgethe resources provided by the Gauss Centre for Super-computing e. V. on the supercomputer HAWK at High-Performance Computing Center Stuttgart and by the TUDortmund University on the HPC cluster LiDO3, par-tially funded by the German Research Foundation (DFG)in project 271512359. D.S.S. gratefully acknowledgesthe RF President Grant No. MK-5158.2021.1.2 and theFoundation for the Advancement of Theoretical Physicsand Mathematics “BASIS”. This work has been supportedby the DFG in the frame of the International Collabora-tive Research Centre TRR 160 (Projects A1, A4, A7) andby the Russian Foundation for Basic Research (GrantsNo. 19-52-12038, 20-32-70048). [1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010).[2] D. Loss and D. P. DiVincenzo, Quantum computationwith quantum dots, Phys. Rev. A , 120 (1998).[3] D. A. Gangloff, G. Éthier-Majcher, C. Lang, E. V. Den-ning, J. H. Bodey, D. M. Jackson, E. Clarke, M. Hugues,C. Le Gall, and M. Atatüre, Quantum interface of anelectron and a nuclear ensemble, Science , 62 (2019).[4] E. V. Denning, D. A. Gangloff, M. Atatüre, J. Mørk,and C. Le Gall, Collective Quantum Memory Activatedby a Driven Central Spin, Phys. Rev. Lett. , 140502(2019).[5] E. A. Chekhovich, S. F. C. da Silva, and A. Rastelli, Nu-clear spin quantum register in an optically active semi-conductor quantum dot, Nature Nanotechnology , 999(2020).[6] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu,B. Diény, P. Pirro, and B. Hillebrands, Review on spin-tronics: Principles and device applications, J. Magn.Magn. Mater. , 166711 (2020).[7] F. Meier and B. P. Zakharchenya, eds., Optical Orienta-tion (North Holland, Amsterdam, 1984).[8] W. Hanle, Über magnetische beeinflussung der polarisa-tion der resonanzfluoreszenz, Zeitschrift für Physik ,93 (1924).[9] D. D. Awschalom, D. Loss, and N. Samarth, Semicon-ductor Spintronics and Quantum Computation (Springer,Berlin, 2002).[10] M. I. Dyakonov, ed.,
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2, 3
D. R. Yakovlev,
2, 3
M. Bayer,
2, 3
G. S. Uhrig, and A. Greilich Condensed Matter Theory, Technische Universit¨at Dortmund, 44221 Dortmund, Germany Experimentelle Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia (Dated: February 5, 2021)
This Supplemental Material is divided in two parts. InSec. I, we provide all the details of the model used to an-alyze the performed experiments which reveal resonantspin amplification (RSA) in Faraday geometry. Comple-mentary results, which demonstrate RSA measured inthe Faraday rotation for another quantum dot (QD) en-semble, are presented in Sec. II. We will refer to this QDensemble as ‘sample 2’ and to the one studied in the maintext as ‘sample 1’.
I. THEORETICAL MODEL
The applied theoretical model is a combination of themodel for the spin dynamics used in Ref. [1] with theformalism of Ref. [2] which allows for a detailed descrip-tion of the pumping and probing of spins in QD ensem-bles. As a novel ingredient, we account for trion stateswith a lifetime comparable to the pulse repetition pe-riod. The layout of this section is as follows. First, weintroduce the equations of motion describing the spin dy-namics for a single QD which is singly charged by anelectron. Then, the pulse model to describe the opticalspin orientation induced by a pump pulse is introduced.In order to model the experimental setup, we repeat theformalism of Ref. [2] which allows us to calculate the ex-perimentally probed Faraday ellipticity and rotation fora single QD. Subsequently, averaging over the inhomoge-neous ensemble of QDs takes place. Finally, we have toaccount for the experimental detail that an accumulatedsignal is measured and the helicity of the pump pulses ismodulated.
A. Statement of the Theoretical Problem
We consider an ensemble of (In,Ga)As/GaAs QDssingly charged by electrons ( n doped) in a pump-probetype experiment as described in the main text and inSec. II. A magnetic field is applied along the axis of lightpropagation ( z axis, Faraday geometry). The periodicpump pulses excite singlet trion states leading to the spinorientation of the resident electrons according to the op-tical selection rules [3]. The spin polarization along theaxis of light propagation is probed by weak linearly po-larized pulses measuring the Faraday ellipticity or rota-tion [2, 4]. The difference between these two quantitiesis that they probe the polarization of different subsets of the QD ensemble, depending on the detuning betweenthe pump and probe pulses [2]. In our case, the pumpand probe pulses have the same photon energy.The localized spins in the QDs are excited by very longtrains of pump pulses following one another with repe-tition period T R . Each pulse is circularly polarized andthe helicity of the pulses is alternated with modulationfrequency f m . The spin polarization is probed by weakprobe pulses with the same repetition period. Experi-mentally, they arrive shortly before the next pump pulsewith a delay of −
60 ps (main text) or −
50 ps (Sec. II).
B. Equations of Motion for the Spin Dynamics of aSingle Quantum Dot
We apply the same phenomenological model for thespin dynamics as used in Ref. [1]. Here, we consideronly electrons as resident charge carriers ( n doped); seeRef. [1] for the model describing p -doped QDs. Our mainfocus is on the electron spin dynamics, but the photoex-cited trion determines the spin generation rate, which isessential for the physics [5, 6].The dynamics between consecutive pump pulses of asingle electron spin S in a QD can be described by theequation of motion ddt S = ( Ω N + Ω L ) × S − S τ s + J z τ e z , (S1)where Ω N is the frequency of the spin precession in therandom nuclear field (Overhauser field) due to the hyper-fine interaction (HI) and Ω L = Ω L e z = g z µ B B ext ~ − e z is the Larmor frequency due to the external longitudinalmagnetic field B ext ( g z is the longitudinal electronic g fac-tor, µ B the Bohr magneton, ~ the reduced Planck con-stant). Furthermore, the phenomenological term − S /τ s describes the spin relaxation unrelated to the hyperfineinteraction with the nuclear spins in the QD. The com-ponent J z is the z projection of the trion pseudospin J and τ = 0 . − . Itconsists of two electrons in a spin singlet and a heavyhole with unpaired spin, which can be represented by an a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b effective pseudospin J [4]. Its dynamics between con-secutive pump pulses is described similarly to Eq. (S1)by ddt J = (cid:0) Ω TN + Ω TL (cid:1) × J − J τ Ts − J τ . (S2)The label ‘T’ refers to the parameters of the heavy holespin in the trion. The nonradiative trion recombination,which does not contribute to the spin polarization in theground state and which is unrelated to the hyperfine in-teraction, is accounted for by the relaxation time τ Ts .Combined with Eq. (S1) and according to the opticalselection rules, only the z component of the trion pseu-dospin J is transferred back to the ground state S on thetimescale τ due to the radiative trion decay.Due to the large number of O (10 ) nuclear spins ineach QD coupled to the electron spin via the hyperfineinteraction, the Overhauser field can be treated as a clas-sical fluctuation field [10, 11]. According to the cen-tral limit theorem, its probability distribution is Gaus-sian [10, 12, 13], p ( Ω N ) = λ ( √ πω N ) exp (cid:18) − λ (Ω x N ) + (Ω y N ) ω − (Ω z N ) ω (cid:19) , (S3)where ω / λ parameterizes the po-tential degree of anisotropy of the hyperfine interaction.For electrons the hyperfine interaction is isotropic, i.e., λ = 1. Generally, the frequency ω N represents the char-acteristic fluctuation strength of the nuclear spin bathseen by the electron spin. In contrast, for heavy holesthe hyperfine interaction is strongly anisotropic with λ > T ,x N = χ λλ T Ω x N , Ω T ,y N = χ λλ T Ω y N , (S4a)Ω T ,z N = χ Ω z N , (S4b)where χ = ω TN /ω N describes the relative strength of thehyperfine interaction for the trion state and λ T quan-tifies its anisotropy. Physically, the averaging over thedistribution (S3) models two cases: (i) the average overa homogeneous QD ensemble and (ii) the time averagein an experiment where the signal is probed over a timemuch longer than the typical correlation time of the nu-clei (about 200 ns for sample 2 [6]).For n -doped QDs, it is typically sufficient to neglectthe precession-term in Eq. (S2). The reason is that inthis case, the trion pseudospin stems from the heavy holewith weak and anisotropic hyperfine interaction, i.e., itscontribution to the generation rate of spin polarizationis negligible [5]. The ensemble of (In,Ga)As/GaAs QDsstudied in Sec. II (sample 2) is well characterized [6]; thisis not the case for sample 1 studied in the main text sothat more parameters need to be fitted. Hence, we ne-glect the precession-term in Eq. (S2) in the analysis of Table I. System parameters and their physical meaning usedin the model calculations for the two different n -doped(In,Ga)As/GaAs QD ensembles studied in the main text andin Sec. II, respectively. The parameters used in Sec. II arebased on a previous sample characterization [6]. Parameter Value Physical meaningSample 1 (main text) ω N / (2 π ) 140 MHz electron HI strength τ s µ s electron spin relaxation time τ Ts . µ s hole spin relaxation time g z − .
69 longitudinal electron g factor Sample 2 (Sec. II) ω N / (2 π ) 70 MHz electron HI strength ω TN / (2 π ) 16 MHz hole HI strength λ T τ s . µ s electron spin relaxation time τ Ts . µ s hole spin relaxation time g z − .
64 longitudinal electron g factor g T z − .
45 longitudinal hole g factor Common parameters τ . λ sample 1 but include it for sample 2. Despite the bet-ter characterization of sample 2, the results for sample 1shown in the main text are more suitable to demonstratethe effect of RSA in Faraday geometry due to a strongerhyperfine interaction leading to more visible RSA modes.In our model calculations presented in Sec. II, we ap-ply the system parameters listed in Table I. The two fitparameters varied to reproduce the experimental resultsare the longitudinal electronic g factor, which is deter-mined by the positions of the experimentally observedRSA modes, and the trion spin relaxation time τ Ts (esti-mated in Ref. [6] to be τ Ts < g factor is also determined by the posi-tions of the RSA modes. The parameter ω N is estimatedbased on an extrapolation of the polarization recoverycurve (PRC) to zero pump power; the spin relaxationtime τ s is determined from spin-inertia measurements [1].The applied parameters are also summarized in Table I.We point out that the simulated PRCs are fairly sensi-tive to the choice of τ Ts and of the effective pulse area Θ(defined in the following), and both quantities can onlybe estimated. It is possible that other combinations existwhich yield a similarly good agreement between simula-tion and experiment. C. Spin Polarization Induced by Optical TrionExcitation
Next, we turn to the description of the action of acircularly polarized pump pulse on the localized electronspin in a single QD. In the experiments, a laser with apulse repetition period of T R = 1 ns is used. In this case,the optically excited trion state is still slightly populatedbefore the arrival of the next pulse due to its comparablelifetime τ = 0 . n G ( t ) = n Ga + n Ta (cid:20) − exp (cid:18) − tτ (cid:19)(cid:21) , (S5a) n T ( t ) = n Ta exp (cid:18) − tτ (cid:19) . (S5b)Here, n Ga and n Ta denote the occupation numbers of theground and trion state immediately after (label ‘a’) apump pulse. Before the arrival of the very first pulse,we start from the initial conditions S b = 0, J b = 0, n Gb = 1, and n Tb = 0 in our model calculations (the la-bel ‘b’ denotes the value before a pump pulse), i.e., thereis no finite polarization and all electron spins are in theirground state. The initial condition represents a com-pletely disordered state.The pulses in the experiment have a duration of 2 ps,which is much shorter than all other timescales of thesystem. Hence, we can treat them as instantaneouspulses. Under this condition and following the formal-ism of Ref. [2], the spin components of the electron spinbefore ( S b , J b ) and after ( S a , J a ) a pump pulse can berelated to each other by S x a = Q cos(Φ) S x b + P Q sin(Φ) S y b , (S6a) S y a = Q cos(Φ) S y b − P Q sin(Φ) S x b , (S6b) S z a = −P − Q n Gb − n Tb ) + 1 + Q S z b + 1 − Q J z b , (S6c)and similary for the trion pseudospin by J x a = Q cos(Φ) J x b − P Q sin(Φ) J y b , (S7a) J y a = Q cos(Φ) J y b + P Q sin(Φ) J x b , (S7b) J z a = P − Q n Gb − n Tb ) + Q + 12 J z b + 1 − Q S z b . (S7c)These relations depend on the occupation numbers ofthe ground and trion state, which are also affected bythe pump pulse. Their values before ( n Gb , n Tb ) and af-ter ( n Ga , n Ta ) the pulse are related by n Ga = 1 + Q n Gb + 1 − Q n Tb − P (1 − Q )( S z b − J z b ) , (S8a) n Ta = 1 + Q n Tb + 1 − Q n Gb + P (1 − Q )( S z b − J z b ) . (S8b) Here, 0 ≤ Q ≤ Q is related to the pumpingefficiency. For Q = 0, which is the most efficient case,each pulse completely aligns the electron spin S along the z axis because the transverse components are set to zero.The parameter Φ describes the spin rotation induced bydetuned pulses, and P = ± Q and Φ depend on the detuning of the pump pulse fromthe trion transition energy. This is discussed in the fol-lowing subsection. Note that one easily retains the pulserelations of Ref. [2] by inserting n Gb = 1 and n Tb = 0.This simplification would be valid for τ (cid:28) T R , e.g., forthe commonly used pulse repetition period T R = 13 . T R = 1 ns. D. Inhomogeneous Ensemble of Quantum Dots
All QDs in a real ensemble are slightly different, e.g.,in size, shape, composition, and strain. In particular,due to the inhomogeneous broadening of the trion tran-sition, each QD of the ensemble has a slightly differenttrion transition energy E T . We model this situation byassuming that the transition energies are normally dis-tributed according to p ( E T ) = 1 q πσ T exp (cid:18) − ( E T − µ E T ) σ T (cid:19) . (S9)This situation implies that for a fixed energy of the pumppulses all the individual QDs are pumped with differentefficiencies due to detuning between laser and trion tran-sition energy. Eventually, this can be described by anassociated pair { Q, Φ } of the pulse parameters for eachQD. Similarly for the probe pulses, different QDs have adifferent contribution to the measured Faraday elliptic-ity and rotation. Tuning the pump and probe energiesallows for an analysis of different subsets of the QD en-semble. For precise details, we refer the interested readerto Ref. [2] where this formalism is established.
1. Pulse Parameters for a Single Quantum Dot
The energies of the pump and probe pulse are denotedas E pu and E pr , respectively. In the experiments, a de-generate pump-probe setup is used, i.e., E pu = E pr . Thefinite spectral width of the pulses is accounted for by theinverse pulse duration τ − , i.e., we assume that the pulsesare Fourier-transform limited. For simplicity, we modelthe pulse shape as Rosen-Zener pulses [16] for which onecan derive analytical expressions for the pulse parame-ters Q and Φ [2]. The amplitude of the electric field ofsuch pulses with duration τ p has the form f ( t ) = µ sech (cid:18) π tτ p (cid:19) , (S10)where µ is a measure for the electric field strength. In theexperiments, the actual pulse shape is Gaussian and thepulses are not perfectly Fourier-transform limited. Weuse τ p = 1 . Q = s − sin (Θ / ( πy ) , (S11a)Φ = arg " Γ (cid:0) − iy (cid:1) Γ (cid:0) − Θ2 π − iy (cid:1) Γ (cid:0) + Θ2 π − iy (cid:1) , (S11b)where Γ( z ) is the Gamma function,Θ = 2 Z ∞−∞ f ( t ) dt = 2 µτ p (S12)is the effective pulse area, and y = ( E pu − E T ) τ p π ~ (S13)is the dimensionless detuning of the pump pulses.For small pump powers P pu , the pulse area scales likeΘ ∝ p P pu [2]. We use this scaling to obtain a first esti-mate of the pulse area in the experiments. For the exper-iments described in the main text, the best fit is achievedusing Θ = 0 . π for a pump power of P pu = 8 . ≈ . π for a pump power of P pu = 5 mW. The pulse areas forthe other pump powers are chosen such that the resultingspin polarization for large magnetic fields fits the pumppower dependence observed in experiment.
2. Probing the Faraday Ellipticity and Faraday Rotation
The Faraday ellipticity and rotation, which are propor-tional to the spin polarization, can be probed by weaklinearly polarized pulses in the experiments. Dependingon the energy E pr of the probe pulse, different subsets ofthe QD ensemble are probed, i.e., the contribution of asingle QD to the probed signal depends on the detuningof the probe pulse from the trion transition energy E t .In the experiments, the energy of pump and probe aredegenerate, i.e., E pu = E pr .For a single QD, the Faraday ellipticity E and rota-tion R are proportional to the spin polarization J z − S z weighted by an additional function which depends on theprobe detuning and pulse duration [2], E ∝ ( J z − S z ) Re G ( E pr − E T , τ p ) , (S14a) R ∝ ( J z − S z ) Im G ( E pr − E T , τ p ) , (S14b) with G ( E pr − E T , τ p ) = τ π ζ (cid:18) , − i ( E pr − E T ) τ p π ~ (cid:19) , (S15)where ζ ( z ) is the Hurwitz Zeta function. With respect tothe detuning E pr − E T , the real part Re G ( E pr − E T , τ p )is symmetric around zero while the imaginary partIm G ( E pr − E T , τ p ) is antisymmetric. The prefactors inEq. (S14) are identical but sample dependent. The exactprefactors do not matter for our considerations becausewe scale the simulated polarization recovery curves to theexperimental data by a global factor.
3. Ensemble Average
In order to correctly account for the ensemble of QDs,two averaging processes must take place. First, we haveto average over the fluctuations of the nuclear spin bathdescribed by the distribution (S3). Second, we have toaverage over the distribution of the trion transition en-ergies (S9). For this purpose, we calculate the ensembleaverage over 2 . × (sample 1) or 10 (sample 2) inde-pendent trajectories starting from random initial condi-tions sampled from these two distributions. Numerically,this is extremely expensive and requires a massive paral-lelization. The parameters E T , σ E T , E pu , E pr , and τ p areobtained by fitting the assumed shapes to the measuredphotoluminescence spectra.As mentioned before, the Faraday ellipticity and rota-tion reveal the spin dynamics of different subsets of theQD ensemble [2]. In the experiments presented in Sec. II,a degenerate pump-probe setup is used with the pulse en-ergy being shifted by 2 . . Q is enhanced (given the pulse area Θ is identical).For the degenerate pump-probe setup, the Faraday el-lipticity yields a better RSA visibility than the Faradayrotation. It is the subject of future research to determineif a controlled pump-probe detuning [17] can be used toenhance the visibility. E. Accumulated Signal in the Modulated PulseScheme
In the experiments, the helicity of the pump pulsesis modulated with the frequency f m to enhance RSAand prevent the buildup of dynamic nuclear polarization.This modulation scheme leads to the formation of alter-nating spin polarization, which is zero on average [5].For this reason, the probed signal must also be modu-lated. The actually measured Faraday ellipticity E orrotation R is given by [5, 18] E ( f m ) = 1 n p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n p X k =1 E ( kT R + τ d ) e i πf m ( kT R + τ d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (S16a) R ( f m ) = 1 n p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n p X k =1 R ( kT R + τ d ) e i πf m ( kT R + τ d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (S16b)These expressions represent the accumulation of theprobed signal for the ensemble of QDs (ensemble averagedenoted by the overline), modulated with frequency f m and averaged over the number of applied pulses n p (cid:29) τ d = −
60 ps(sample 1) or −
50 ps (sample 2).In our numerical simulations, it is not feasible to cal-culate the spin dynamics for more than a few modulationperiods 1 /f m while in the experiments, the pumping isperformed over many modulation periods. However, forsmall enough modulation frequencies as applied in ex-periment it is sufficient to simulate only two modulationperiods and then to sum over the last period. The firstperiod is neglected because it shows a slight transient be-havior due to the simulations starting from equilibriumconditions. II. DEMONSTRATION OF RESONANT SPINAMPLIFICATION IN FARADAY GEOMETRYFOR ANOTHER QUANTUM DOT ENSEMBLE
In this section, we complement the results of the maintext by studying another (In,Ga)As/GaAs QD ensem-ble (sample 2) whose parameters are well known [6].Resonant spin amplification in Faraday geometry is alsodemonstrated for this sample.
A. Experimental Details
The sample 2 consists of 20 layers of (In,Ga)As QDsseparated by 60 nm barriers of GaAs and grown by molec-ular beam epitaxy on a (100)-oriented GaAs substrate.A δ doping layer of Silicon 16 nm above each QD layerprovides a single electron per QD on average. Rapidthermal annealing at 945 ◦ C for 30 s homogenizes the QDsize distribution and shifts the average emission energyto 1 . cm − . Pump-Probe Delay (ns) F a r a d a y R o t a ti on ( a r b . un it s ) B ext = 200 mT1.38 1.39 1.4 Energy (eV) P ho t o l u m i n e s ce n ce (b) QD ensemble PumpProbe T = 5.3 K (a) P pu = 4 mW Supplementary Figure 1. (a) Photoluminescence of the QDensemble (sample 2, gray) at T = 5 . B ext = 200 mT. The orange arrow marks the delay ( −
50 ps)at which the polarization recovery curves are taken.
The setup is very similar to the experimental detailsprovided in the main text; the differences are given inthe following. The sample is illuminated with a centraloptical energy of 1 . . . µ m and theprobe to a diameter of 45 µ m. The Faraday rotation am-plitude of the probe pulses is measured using an opticalpolarization bridge which consists of a Wollaston prismand a balanced photodetector.We point out that our setup differs from the earlierexperiments of Ref. [19] on CdTe quantum wells wherethe external magnetic field itself is tilted from the opticalaxis such that different components of the g factor tensordetermine the electronic Larmor frequency; in our casethe external field points purely along the optical axis withan accuracy of 2 degree (Faraday geometry). B. Results
As discussed in the main text, each circularly polarizedpump pulse partially aligns the spin polarization of theresident electrons along the optical axis [20]. The spinpolarization added by every pulse in this way depends −
100 0 100 200 300
Magnetic Field B ext (mT) . . . F a r a d a y R o t a t i o n ( a r b . un i t s ) k = +1 k = − k = − Supplementary Figure 2. PRCs for different pump powersgiven next to the curves. The colored data is the experimen-tally measured Faraday rotation, the black curves are the sim-ulated ones. The simulated PRCs are smoothed by a movingaverage over a range of 7 mT, the pulse area in the simulationsis chosen to fit the experimentally obtained power dependenceof the saturation amplitude A , see Suppl. Fig. 3(a). The po-sitions of the small peaks around B ext = −
112 mT, 112 mT,and 224 mT match the RSA modes (vertical grey lines) pre-dicted by the PSC (S18) for k = +1, −
1, and −
2, yielding g z = − . ± .
01 for the longitudinal electronic g factor. on its effective pulse area Θ [8], scaling like P pu ∝ Θ for small powers [2]; see also Sec. I and Fig. 3(d). Fora train of pump pulses, the spin polarization builds upwhen the relaxed polarization along the optical axis be-tween two pulses is smaller than the polarization addedby a pump pulse. This is the case for an external fieldof B ext = 200 mT along the optical axis using a pumppower of P pu = 4 mW and it is the origin of the offsetvisible in the time-resolved Faraday rotation shown inSuppl. Fig. 1(b). The initial signal variations after eachpump pulse are related to the optically excited hole intrion which recombines on the timescale of 0 . z axis. Any external field com-ponent perpendicular to the optical axis would lead to aprecession of the electron spin about this component.The polarization recovery curves (PRCs) studied inthis section are measured by setting the delay betweensubsequent pump and probe pulses to −
50 ps, i.e., theFaraday rotation is measured right before the arrival ofthe next pump pulse after a delay of 950 ps as indicatedby the orange arrow in Suppl. Fig. 1(b). The longitudi-nal field is varied from −
150 to 300 mT with a rate of45 mT min − .Supplementary Figure 2 shows the PRCs for a widerange of pump powers. They have a V -like typical for n -doped QDs [5, 6, 21]. For small pump powers, theelectron spin polarization is minimal at 0 mT and risesto a saturation level within 30 mT. Without a magnetic . . . A ( a r b . un i t s ) (a) theoryexperiment ∆ B ( m T ) (b) . . . . − C (c) Pump Power P pu (mW) . . . Θ / π (d) fit Θ ∝ p P pu fit without P pu = 3 , , Supplementary Figure 3. Characterization of the PRCs forexperiment and theory. The saturation level A , the width ∆ B of the zero field minimum, the ratio 1 − C which describes thedepth of the zero field minimum relative to A , and the pulsearea Θ are shown versus the pump power P pu . The pulsearea is set in the model such that it fits the saturation levelextracted from the experiment; the parameters A , ∆ B , and C are determined by fitting (S17) to the PRCs. The horizontaldashed line in (c) marks the expected ratio 1 − C = 1 / ∝ p P pu . field, the electron spin is only subject to the isotropicnuclear fluctuation field described by the characteristicfrequency ω N , leading to nuclei-induced spin relaxation.For rather small fields, the electronic Zeeman effect doesnot dominate over the hyperfine interaction between theelectron spin and nuclear spins. For larger fields, the ex-ternal field dominates the nuclear fluctuation field andeventually, no nuclei-induced spin relaxation takes place.This results in a plateau region for the spin polariza-tion [22].We characterize the measured and simulated PRCs byapplying the fit function [21] R ( B ext ) = A (cid:20) − C B ext − B ) / ∆ B (cid:21) . (S17)The parameter A defines the saturation level for largemagnetic fields, 1 − C is the ratio of spin polarizationsfor zero and large field, ∆ B characterizes the width of thePRC, and B accounts for a potential shift. The widthof the PRC for small pump power is determined by thenuclear fluctuation field and can be extracted from thedata by extrapolating the width to zero pump power, seeSuppl. Fig. 3(b). It amounts to an effective Overhauserfield characterized by ω N / (2 π ) = 70 MHz (equivalent toa field of 7 . A asa function of the pump power P pu . For strong pulsescorresponding to a larger pump power, a larger magneticfield is required to suppress the nuclei-induced spin relax-ation [1, 23], leading to an increased width of the PRCas shown in Suppl. Fig. 3(b). The ratio 1 − C deviatesfrom the standard value 1 / B ext which fulfillthe phase synchronization condition (PSC) [1]Ω L = | k | ω R , k ∈ Z , (S18)for the Larmor frequency Ω L = µ B | g z B ext | ~ − ( µ B isthe Bohr magneton and ~ the reduced Planck constant).These discrete resonance frequencies are given by multi-ples of the laser repetition frequency ω R . We label themby the mode number k . The longitudinal electronic g fac-tor | g z | is the relevant system parameter which fixes themode positions. By determining the maxima of thesemodulations, we obtain | g z | = 0 . ± .
01. From earlierexperiments we know that in (In,Ga)As/GaAs QDs the g factor has a negative sign [24], i.e., g z = − . ± . τ Ts = 0 . µ s, andthe longitudinal electronic g factor g z = − .
64. The re-maining parameters are taken from the earlier samplecharacterization presented in Ref. [6]; they are listed inTable I.Analogously to the main text, we benchmark our lasersource by studying the RSA visibility [23] V := R max − R min R max , (S19)where R max is the Faraday rotation of the first maxi-mum at the RSA mode | k | = 1 and R min denotes theadjacent minimum at a larger field | B ext | . Note thatthe Faraday rotation is more difficult to calculate nu-merically than the Faraday ellipticity. The reason is asign problem arising in the applied Monte Carlo sam-pling which results in much larger statistical fluctuationsthan for the Faraday ellipticity studied in the main text.Since these fluctuations hinder the reliable detection ofphysical minima and maxima in the simulated PRCs, weset V = 0 whenever V < .
02. A map of the RSA visibil-ity in dependence of the pulse area Θ and of the productof repetition period T R = 2 π/ω R and nuclear fluctua-tion field ω N / (2 π ) = 70 MHz is shown in Suppl. Fig. 4. ω N T R . . . . . . P u l s e A r e a Θ / π . . . . . . R S AV i s i b ili t y V Supplementary Figure 4. Visibility map of the RSA modesin Faraday geometry modeled for detection by Faraday ro-tation. The pump-probe delay is set to zero. The whitecross marks the experimental conditions for a pump power of5 mW (Θ ≈ . π ) and a pulse repetition period of T R = 1 ns[ ω N / (2 π ) = 70 MHz]. The visibility becomes larger upon increasing the pulsearea, which corresponds to a larger pump power. Fora too large or too small repetition period, the visibil-ity vanishes. Observing RSA in Faraday geometry whenmeasuring the Faraday rotation appears to be easiest inthe intermediate regime ω N T R ∼
1. As evident from thewhite cross on the color map, the laser source used inthe experiment does not operate at optimal conditions.For a pump power of P pu , the visibility amounts to only V ≈ .
02. Under better conditions, the visibility couldreach up to V ≈ .
5. No RSA modes are visible for thecommonly used repetition periods T R = 13 . . ω N T R ≈ . .
9) in theory nor in experiment (notshown here).The visibility for this QD sample with ω N / (2 π ) = 70 MHz could be improved by increas-ing the pump power and by using a slightly largerrepetition period of about 2 − g factor. In contrast, forsample 1 analyzed in the main text [ ω N / (2 π ) = 140 MHz]the choice T R = 1 ns is perfectly suited to reveal theRSA modes because the nuclear spin fluctuationscharacterized by ω N are larger by a factor of 2. Thelarger value also results in broader PRCs such thatRSA modes are also visible at larger magnetic fields.Our theoretical modeling reveals that measuring theFaraday ellipticity instead of the Faraday rotation fora degenerate pump-probe setup yields a larger RSAvisibility. Hence, we opted for it in the experimentspresented in the main text. Another possibility toenhance the visibility is to exploit the spin-inertia effect;see the main text for details.Generally, the RSA visibility strongly depends on theaverage spin polarization, which in turn depends on thespin relaxation time τ Ts of the hole in the intermediatetrion, with a smaller time corresponding to a larger po-larization [5]. As discussed in the main text, approaching the saturation limit of the spin polarization is detrimen-tal for RSA in Faraday geometry, and hence, the RSAvisibility is fairly sensitive to the choice of τ Ts . [1] P. Schering, G. S. Uhrig, and D. S. Smirnov, Spin iner-tia and polarization recovery in quantum dots: Role ofpumping strength and resonant spin amplification, Phys.Rev. Research , 033189 (2019).[2] I. A. Yugova, M. M. Glazov, E. L. Ivchenko, and A. L.Efros, Pump-probe Faraday rotation and ellipticity in anensemble of singly charged quantum dots, Phys. Rev. B , 104436 (2009).[3] E. L. 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