Spin noise spectroscopy of a single-quantum-well microcavity
S.V. Poltavtsev, I.I. Ryzhov, M.M. Glazov, G.G. Kozlov, V.S. Zapasskii, A.V. Kavokin, P.G. Lagoudakis, D.S. Smirnov, E.L. Ivchenko
SSpin noise spectroscopy of a single-quantum-well microcavity
S.V. Poltavtsev, I.I. Ryzhov, M.M. Glazov,
1, 2
G.G. Kozlov, V.S. Zapasskii, A.V. Kavokin,
1, 3
P.G. Lagoudakis, D.S. Smirnov, and E.L. Ivchenko Spin Optics Laboratory, St.-Petersburg State University,1 Ul’anovskaya, Peterhof, St.-Petersburg 198504, Russia Ioffe Physical-Technical Institute of the RAS, 26 Polytekhnicheskaya, St.-Petersburg 194021, Russia School of Physics and Astronomy, University of Southampton, SO17 1 BJ, Southampton, UK
We report on the first experimental observation of spin noise in a single semiconductor quan-tum well embedded into a microcavity. The great cavity-enhanced sensitivity to fluctuations ofoptical anisotropy has allowed us to measure the Kerr rotation and ellipticity noise spectra in thestrong coupling regime. The spin noise spectra clearly show two resonant features: a conventionalmagneto-resonant component shifting towards higher frequencies with magnetic field and an un-usual “nonmagnetic” component centered at zero frequency and getting suppressed with increasingmagnetic field. We attribute the first of them to the Larmor precession of free electron spins, whilethe second one being presumably due to hyperfine electron-nuclei spin interactions.
Introduction.
In the present-day physics of semicon-ductor nanostructures, a considerable interest is shownfor the fundamental spin-related properties which arealso promising in applications. Among optical methodsof spin dynamics studies, an important place is givento the Faraday-rotation-based spin noise spectroscopy(SNS) which became well-known and popular during thelast several years [1]. The advantages of SNS are primar-ily owed to its nonperturbative nature because probingthe sample response by a weak laser beam in the regionof transparency does not lead to any real electronic tran-sitions. Extreme smallness of the magnetization fluctua-tions detected with the SNS technique calls for the high-est polarimetric sensitivity which is achieved by usingvarious electronic or optical means. A real breakthroughoccurred when the fast-Fourier-transform (FFT) spec-trum analyzers were applied in electronics of the SNStechnique [2]. The most straightforward optical way toenhance the polarimetric sensitivity implies increasing in-tensity of the probe light beam and, simultaneously, leav-ing the input power of photodetector on the admissiblelevel. This can be implemented either by using high-extinction polarization geometries [3] or by placing thesample inside a high-Q optical cavity [4]. In both cases,the light power density on the sample can be increasedby a few orders of magnitude, with the light power onthe photodetector and, therefore, the photocurrent shotnoise remaining on the same low level.For low-dimensional semiconductor structures (quan-tum wells, wires and dots) the problem of polarimetricsensitivity is especially topical. In Ref. [5], in order to in-crease the signal, the spin noise spectra of n -doped GaAsquantum wells were studied in the samples containing tenidentical quantum wells (QWs). The measurement of thespin noise spectrum of a layer of InAs/GaAs quantumdots (QDs) in a high-finesse microcavity allowed Dah-bashi et al. [6] to perform unique investigation of spindynamics of a single heavy hole localized in a selectedQD. We are not aware of any experimental study of spin
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Wavelength, nm R e fl ec ti on X lh X hh X − ( a )( c ) ( b ) AlGaAsλ-cavity GaAs QW DBRDBR tunable Ti:Sa cw-laser 650nm diode laserbalancedphotodetectorFFT spectrum analyzercryostat with sample
Figure 1: (Color online) (a) Schematic of the sample. (b)Refection spectra measured at different points of the sample,i.e. at different detunings. Arrows mark the positions ofnegative trion ( X − ), heavy hole ( X hh ) and light hole ( X lh )resonances. Different curves are shifted along vertical axisfor clarity. Dotted lines are guides for eye and demonstrateanticrossings. (c) Schematic of the experimental setup. noise in a single quantum well.In this paper, we report on the first observation of spinnoise in a single GaAs QW embedded inside a high-finessemicrocavity operating in the strong coupling regime. Adramatic increase of the sensitivity has made it possibleto observe, in addition to the Kerr rotation fluctuations,the noise of ellipticity, the effect reported previously foratomic gases only [7]. We demonstrate also that an in-crease of the probe beam intensity from weak to moder-ate values significantly perturbs the spin system in themicrocavity making it possible to study the spin noise insteady nonequilibrium states as well [8–10]. Experiment.
The sample under study represents a20 nm GaAs QW with AlAs barriers grown along z k [001] axis, placed into the λ -cavity formed by two dis-tributed Bragg mirrors (DBR) comprised of 25 and 15pairs of AlAs/AlGaAs layers. Two additional narrow 2.6-nm QWs were grown on both sides of the central wellwhich enabled us to use photodoping by means of theabove-barrier illumination. The sample had a gradient of a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n thickness that made it possible to vary the detuning bymoving the light-spot on the sample. In more detail, thestructure is described in [11]. The schematic of the sam-ple and its reflection spectra are presented in Fig. 1(a)and Fig. 1(b), respectively. The reflection spectra underour experimental conditions of cw excitation were some-what smoother than those presented in Ref. [11], but stillallowed one to trace anticrossings of the cavity mode withmaterial excitations of the QW, namely, the negativelycharged trion ( X − ), heavy- ( X hh ) and light-hole ( X lh )excitons. The observation of trion resonance means elec-tron density n e not to be higher than ∼ × cm − , seeRef. [11] for details. The sample was placed at a temper-ature of about 6 K into a small transverse magnetic field B = 0 . . . mT (Voigt geometry), and the fluctuationsof the polarization plane rotation were detected in thereflection geometry (Kerr rotation noise) using a stan-dard setup with a balanced photoreceiver (bandwidth 200MHz) and an FFT spectrum analyzer, see Ref. [1] for de-tails. The signal of ellipticity noise was also measured byplacing a properly oriented quarter-wave plate in front ofthe balanced detector. The probe light from a tunablecw Ti:sapphire laser was tightly focused on the sample(diameter of the spot ∼ µ m) and tuned to the cavityresonance at the chosen point of the sample. In somecases, the probed area of the sample was additionally il-luminated by a laser diode with shorter wavelength ∼ nm and power density about mW / cm . Schematic ofthe experimental set-up is shown in Fig. 1(c).Under above experimental conditions, in most cases,the Kerr rotation and ellipticity noise were comparable toor even exceeded the shot noise level, so that the noise sig-nals could be easily detected. At the same time, these sig-nals were spatially inhomogeneous with a typical lengthscale of about µ m. Specifically, depending on partic-ular area of the sample the spin noise signals could be alsoobserved in the absence, rather than only in the persence,of the additional short-wavelength illumination. In thiscommunication, we restrict ourselves to systematic re-sults obtained in our studies of spin noise (SN) spectraof the system at the negative photon-exciton detuning,with the cavity mode lying below the exciton and trionresonances. The dependence of the signals on the mag-netic field and probe power was similar in different spotsof the sample. Experimental results.
Figure 2 demonstrates Kerr rota-tion noise (a) and ellipticity noise (b) at negative detun-ing from the X hh resonance δ ≈ − . meV. At the chosenpoint, the noise was observed only under additional illu-mination. The noise spectrum has been found to containgenerally two resonant features with essentially differentsensitivity to the applied magnetic field. The frequencyof one of them, as expected for a spin resonance, linearlyvaried with the field (this component is termed “mag-netic” hereafter), while the other peak centered at zerofrequency did not exhibit any shift with the applied field S i gn a l - t o - s ho t - no i s e r a ti o B = 0 mTB = 6.5 mTB = 13.0 mTB = 19.4 mTB = 25.9 mTB = 32.4 mT S i gn a l - t o - s ho t - no i s e r a ti o Ellipticity noise λ = 814.16 nmT = 3.6KP Probe = 1mWKerr-rotation noise ( a ) ( b ) K R po w e r , × − r a d / H z E lli p ti c it y po w e r , × − r a d / H z Figure 2: (Color online) Kerr-rotation (a) and ellipticity (b)noise spectra measured at fixed probe power and differentmagnetic fields indicated at the legend. Shot noise is sub-stracted from the data and the signals are normalized tothe shot noise level (see [19] for details of the normaliza-tion). Right-hand axis shows the Kerr rotation noise powerin rad /Hz. Frequency, MHz S p i n N o i s e P o w e r Probe power:
B = 29 mT
Figure 3: (Color online) Spin noise power density extractedfrom total Kerr-rotation noise by normalizing signal-to-noiseratio by the probe power measured at B = 29 mT and T =3 . K. (“nonmagnetic” component). As seen from Fig. 2, the“nonmagnetic” feature decreases in amplitude with in-creasing magnetic field. Moreover, the amplitudes andwidths of both components depend strongly on the probebeam intensity, as illustrated in Fig. 3. Particularly, withthe decrease of the probe intensity, both “magnetic” and“nonmagnetic” resonances narrow down and the relativemagnitude of the “magnetic” resonance increases makingit possible to observe the field-dependent component ofspin noise in the pure form.Figure 4(a) presents the Kerr rotation noise spectraat different transverse magnetic fields measured with-out above-barrier illumination at the sample point wherethe magnetic component is most pronounced. A field-induced shift of the “magnetic” component correspondedto the effecive g -factor equal to | g | ≈ . , which corre-lates with the electron g -factor value in the 20 nm GaAsQW [12]. The shape of this resonance can be well approx-imated by a field-independent Lorentzian with a FWHMof 60 MHz corresponding to the dephasing time of ns. Frequency, MHz S i gn a l - t o - s ho t - no i s e r a ti o B=29mTB=9.5mT 0 50 100 150 20000.20.40.60.8
Frequency, MHz S p i n N o i s e po w e r , a . u . Simulation λ =814.10 nm,P Probe =2mW, T=6K ( a ) ( b ) Experiment K R po w e r , × − r a d / H z Figure 4: (Color online) (a) Measured Kerr-rotation noisespectra for the magnetic field varied from . to mT inequal steps. Parameters of the experiment are given in thepanel. (b) Calculated spin noise power spectra for g = − . , τ s ≈ ns, δ e ≈ . × s − ( ≈ MHz). The spread ofelectron g -factor values is taken into account, see [13, 19] fordetails. Wavelength, nm S N po w e r , a . u . KerrEllipt. ×10 00.20.40.60.81 R e fl ec ti on Wavelength, nm S N po w e r , a . u . KerrEllipt.×10 00.20.40.60.81 R e fl ec ti on SimulationExperiment ( a ) ( b ) P Probe =1mWT = 3.6K F
Detect =35MHz
Figure 5: (Color online) (a) Reflectivity (top blue curve,right axis), Kerr rotation (solid/dark red) and ellipticity(dashed/red) optical spectra. (b) Results of calculation afterEq. (1) for ~ ω c = 1523 . meV (wavelength 813.8 nm), ~κ =0 . meV, κ = 0 . meV, taking into account only X − reso-nance with ~ ω X − = 1526 . meV (812 nm), ~ g X − = 0 . meV, ~ γ X − = 0 . meV, and taking into account the inhomogeneousbroadening . meV of the trion resonance [19]. The narrow peak at zero frequency can be attributedto the hyperfine interaction with lattice nuclei [13]. Itswidth of about 15 MHz corresponds to the spin relaxationtime τ s = 25 ns. Overall, such a behavior of the experi-mental data is well reproduced theoretically, as shown inFig. 4(b), see below for details. Discussion.
The noise of Kerr rotation and ellipticityis caused by the fluctuations of reflection coefficients r ± of the microcavity for right ( + ) and left ( − ) circularlypolarized components of the probe beam. If the probefrequency ω is close to the cavity resonance frequency ω c , the reflection coefficients can be presented as [14, 15] r ± = − κ ω − ω c + i κ + κ + P j g j, ± ω − ω j, ± +i γ j, ± . (1)Here, κ and κ are the photon escape rates through themirrors (light is incident on the mirror characterized by κ ), j enumerates resonances in the active layer, namely, X − trion and X hh , X lh excitons, ω j, ± are the correspond-ing resonance frequencies, g j, ± and γ j, ± are the couplingconstants and damping rates, respectively. In general,the differences ω j, + − ω j, − , g j, + − g j, − , γ j, + − γ j, − are pro- portional to the z -component of magnetization in the sys-tem, making the instant values r + and r − different [16].As follows from Eq. (1), the reflection coefficient as afunction of the probe frequency has dips at the resonantfrequencies of mixed modes, or polaritons [Fig. 1(b)]. Weassume that the main contribution to the Kerr rotationand ellipticity fluctuations results from the spin noise ofresident electrons and, thus, take into account only trionresonance. In this case, the fluctuations of the trion oscil-lator strength cause the fluctuating splitting of polaritonresonance for σ + and σ − polarizations. As a result theellipticity noise ∝ | r + | − | r − | should reveal two peaksat the slopes of the resonance and vanish in its center,where | r + | = | r − | , while Kerr rotation noise governedby the phase of the reflection coefficient should be peakedat resonance center. This is demonstrated in Fig. 5 wherethe experimental data and calculated optical spectra areshown in panels (a) and (b), respectively, see [19] for de-tails.Strong sensitivity of the spin noise spectra on theprobe intensity, particularly, the effects of probe lighton amplitudes and widths of the “magnetic” and “non-magnetic” components in the spin noise spectra, clearlydemonstrates that in the strongly-coupled quantum mi-crocavity even a moderate probe perturbs the system.Such a nonequilibrium system calls for special theoreti-cal treatment. The unambiguous presence of the “mag-netic” component demonstrates that the noise of Kerr ro-tation and ellipticity can be attributed to the spin fluc-tuations of resident electrons, which can be present inthe structure due to unintentional doping and/or above-barrier illumination. For relatively low electron densities, n e ∼ cm − , the carriers are localized at QW imper-fections and their spins are affected by both the externalmagnetic field B and the nuclear field fluctuations. Inthe strong couling regime, the probe beam, even detunedfrom material resonances, generates exciton-polaritonsand trion-polaritons in the structure. Here we considerthe simplest model which takes into account (i) the pre-cession of a localized electron spin in the nuclear fieldfluctuation with the frequency Ω N which is randomlydistributed as F ( Ω N ) = ( √ πδ e ) exp ( − Ω N /δ e ) with δ e being the nuclear spin fluctuation [13], (ii) the effect ofexternal magnetic field B with the Larmor frequency Ω B = gµ B B/ ~ , and (iii) probe-induced coupling of elec-trons and trions neglecting a contribution from excitons.The coupled dynamics of electron and trion spins is de-scribed by [16, 17] d S d t = ( Ω N + Ω B ) × S − S τ s − G S + S T e z τ T , (2a) d S T d t = − S T τ T + GS z . (2b)Here S is the electron spin pseudovector with the com-ponents S x , S y and S z , S T is the trion pseudospin, S T = Frequency, MHz ( δ s Z ) ω , a . u . Frequency, MHz ( δ s Z ) ω , a . u . B = 9.5 mTB = 14.5 mTB = 19.5 mTB = 24.5 mTB = 29.5 mT
Probe intensity dependence Magnetic field dependence ( a ) ( b ) Figure 6: (Color online) Calculated spin noise spectra. (a)Different curves correspond to different trion generation rates G = 0 . . . × s − (in equal steps), the magnetic field isfixed, B = 24 mT. (b) Different curves correspond to differentmagnetic fields B = 9 . . . . . mT (in equal steps), genera-tion rate is fixed, G = 4 × s − . Other parameters are asfollows: τ T = 11 ps, τ T = 9 . ps, δ e = 2 . × s − , spreadof g -factor values is disregarded. ( T + − T − ) / , with T ± being the occupation numbers ofheavy hole trions with the spin / and − / , respec-tively, e z is the unit vector along the growth axis z , τ s isthe electron spin relaxation time and, for simplicity, theeffects of spin relaxation anisotropy related to crystallo-graphic orientation of the quantum well are disregardedfor simplicity [18], τ T is the lifetime of the trion, τ T isthe spin lifetime of the trion given by τ Ts τ T / ( τ Ts + τ T ) with τ Ts being the trion spin relaxation time, and G isthe trion generation rate. The latter includes the for-mation of trions both directly by the probe absorptionand via the capture of excitons by resident electrons, itis proportional to the probe intensity and increases withdecreasing absolute value of the detuning | δ | . The spinprecession in the trion is neglected. We stress that forthe linearly polarized probe, the trions thus created con-tain electrons with any spin orientation, but, accordingto the optical selection rules, the electron, returned afterthe trion recombination, has a spin S = S T e z .Figure 4(b) demonstrates the calculation of the elec-tron spin noise spectra, ( δS z ) ω , by using Eqs. (2) in thelimit of low probe intensity, G → . The parameters ofcalculation are given in the caption. The model repro-duces the main features of measured spin noise spectra,Fig. 4(a): the narrow peak at ω = 0 , which vanishes withthe increase of the field, and the “magnetic” peak.An increase in probe intensity and, hence, the triongeneration rate G drastically changes the spin noise spec-tra, as shown in Fig. 6. In qualitative agreement with ex-perimental data presented in Fig. 3, the magnetic peakin the spin noise spectrum decreases, while the peak at ω = 0 becomes broader and relatively more pronounced.Such a behavior can be qualitatively understood bear-ing in mind that the probe-induced coupling of the elec-tron with trion leads to the anisotropic spin relaxationof the electron. Indeed, at τ Ts (cid:29) τ T , the electron spin z component is conserved, while its in-plane components S x and S y vanish after the trion decay. As a result,the electron effective spin relaxation rate γ eff increases with the field giving rise to the broadening of the spinnoise spectrum [5]. Calculation shows that for Gτ T (cid:28) , Ω B < Gτ T / (2 τ T ) and in the absence of random nuclearfields [19] γ eff = 1¯ T − G s τ T τ T − B G , (3)where ¯ T − = 1 /τ s + G [1 − τ T / (2 τ T )] . It follows then thatfor small enough magnetic fields the spin noise spectrumis centered at ω = 0 and its width increases quadraticallywith the magnetic field. For Ω B > Gτ T / (2 τ T ) , the “mag-netic” component in the spin noise spectrum appears.The simulation after Eqs. (2) presented in Figs. 6(a) and6(b) reproduces well not only the magnetic field depen-dence of the spin noise spectrum, shown in Fig. 2 andmeasured at the moderate probe intensity, but also thedependence of the spin noise spectrum on the probe in-tensity, Fig. 3.A detailed fitting of experimental data by the devel-oped model needs allowance for other possible sources ofthe “nonmagnetic” component of the spin noise spectrum,e.g., spin fluctuations of holes (in the generated trionsor captured in the sample as a result of above-barrierillumination), spin noise of excitons [20] and exciton-polaritons [9], spin noise of electrons and holes trappedin narrow quantum wells or at the localization centersin the barriers. Additionally, “nonmagnetic” componentof the ellipticity noise can result from fluctuations of theoff-diagonal component of the background dielectric sus-ceptibility tensor, Re { ε xy } , caused, e.g., by the phonons.To elucidate the contributions of particular mechanisms,the application of magnetic field in the Faraday geom-etry which enhances hyperfine-interaction-induced zero-frequency peak could be useful [6, 13, 21, 22]. All theseeffects are, however, beyond the scope of present workand deserve further studies. Conclusion.
The electron spin noise in a single QW mi-crocavity operating in the strong coupling regime is ob-served via the Kerr-rotation and ellipticity fluctuations.The spin noise spectrum contains both a “magnetic” com-ponent, with its maximum located at the frequency ofLarmor precession of the electron spin around the ex-ternal magnetic field, and a “nonmagnetic” one centeredat zero frequency. The magnitudes and widths of thesecomponents strongly depend on the probe intensity. Theexperimental findings are described in the framework ofproposed model which takes into account the spin preces-sion of resident electrons in the external magnetic fieldand the field of nuclear fluctuations as well as the effectof trion generation by the probe beam.
Acknowledgments.
The financial support from theRussian Ministry of Education and Science (ContractNo. 11.G34.31.0067 with SPbSU and leading scientistA. V. Kavokin), Dynasty Foundation, RFBR, and EUprojects POLAPHEN and SPANGL4Q is acknowledged.The work was fullfilled using the equipment of SPbSUresource center “Nanophotonics”. [1] V. S. Zapasskii, Adv. Opt. Photon. , 131 (2013).[2] M. Romer, J. Hubner, and M. Oestreich, Review of Sci-entific Instruments , 103903 (2007).[3] P. Glasenapp, A. Greilich, I. I. Ryzhov, V. S. Zapasskii,D. R. Yakovlev, G. G. Kozlov, and M. Bayer, Phys. Rev.B , 165314 (2013).[4] A. V. Kavokin, M. R. Vladimirova, M. A. Kaliteevski,O. Lyngnes, J. D. Berger, H. M. Gibbs, and G. Khitrova,Phys. Rev. B , 1087 (1997).[5] G. M. M¨uller, M. R¨omer, D. Schuh, W. Wegscheider,J. H¨ubner, and M. Oestreich, Phys. Rev. Lett. ,206601 (2008).[6] R. Dahbashi, J. H¨ubner, F. Berski, K. Pierz, andM. Oestreich, ArXiv e-prints (2013), 1306.3183.[7] T. Mitsui, Phys. Rev. Lett. , 5292 (2000).[8] E. L. Ivchenko, Sov. Phys. Solid State , 998 (1974) [Fiz.Tverd. Tela , 1489 (1974)].[9] M. M. Glazov, M. A. Semina, E. Y. Sherman, and A. V.Kavokin, Phys. Rev. B , 041309 (2013).[10] F. Li, Y. V. Pershin, V. A. Slipko, and N. A. Sinitsyn,Phys. Rev. Lett. , 067201 (2013).[11] R. Rapaport, E. Cohen, A. Ron, E. Linder, and L. N. Pfeiffer, Phys. Rev. B , 235310 (2001).[12] I. A. Yugova, A. Greilich, D. R. Yakovlev, A. A. Kiselev,M. Bayer, V. V. Petrov, Y. K. Dolgikh, D. Reuter, andA. D. Wieck, Phys. Rev. B , 245302 (2007).[13] M. M. Glazov and E. L. Ivchenko, Phys. Rev. B ,115308 (2012).[14] D. F. Walls and G. J. Milburn, Quantum optics (Springer, 2008), 2nd ed.[15] C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, andJ. G. Rarity, Phys. Rev. B , 085307 (2008).[16] M. M. Glazov, Physics of the Solid State , 1 (2012).[17] G. V. Astakhov, M. M. Glazov, D. R. Yakovlev,E. A. Zhukov, W. Ossau, L. W. Molenkamp, andM. Bayer, Semiconductor Science and Technology ,114001 (2008).[18] N.S. Averkiev, L.E. Golub, Phys. Rev. B , 15582(1999).[19] See supplemental materials for details.[20] D.S. Smirnov, M.M. Glazov, E.L. Ivchenko, to be pub-lished.[21] The effect of magnetic field in the Faraday geometry onspin noise spectra was addressed theoretically in Ref. [13]and experimentally in Refs. [6, 22]. Such an experimentalgeometry is not possible in our setup.[22] Yan Li, N. Sinitsyn, D. L. Smith, D. Reuter, A. D. Wieck,D. R. Yakovlev, M. Bayer, and S. A. Crooker, Phys. Rev.Lett. , 186603 (2012). upplemental Material to“Spin noise spectroscopy of a single-quantum-well microcavity” S.V. Poltavtsev, I.I. Ryzhov, M.M. Glazov,
1, 2
G.G. Kozlov, V.S. Zapasskii, A.V. Kavokin,
1, 3
P.G. Lagoudakis, D.S. Smirnov, and E.L. Ivchenko Spin Optics Laboratory, St.-Petersburg State University,1 Ul’anovskaya, Peterhof, St.-Petersburg 198504, Russia Ioffe Physical-Technical Institute of the RAS, 26 Polytekhnicheskaya, St.-Petersburg 194021, Russia School of Physics and Astronomy, University of Southampton, SO17 1 BJ, Southampton, UK
S1. NONEQUILIBRIUM SPIN NOISE THEORY
The nonequilibrium system of resident electrons andphotoexcited X − trions is considered. We assume thatthe excitation is unpolarized or linearly polarized, andit does not serve as a source of spin fluctuations. Dur-ing photoexcitation the resident electrons are capturedto trions and the trion recombination serves as a sourceof electrons. In the absence of magnetic field the rateequations for the number of spin-up/spin-down electrons, N ± / , and corresponding heavy-hole trions, T ± , read [1] d N / d t = − N / − N − / τ s − GN / + T + τ T , (S1a) d N − / d t = − N − / − N / τ s − GN − / + T − τ T , (S1b) d T + d t = − T + − T − τ Ts + GN / − T + τ T , (S1c) d T − d t = − T − − T + τ Ts + GN − / − T − τ T . (S1d)We denote the total number of electrons in the systemas N e = N / + N − / , the total number of trions in thesystem as T = T + + T − and the number of resident elec-trons in the absence of pumping as N . Since a residentelectron can remain resident or can be captured to trionone has N e + T = N . (S2)The solution of the set (S1) and (S2) in the steady stategives N e = N Gτ T , T = N Gτ T Gτ T , (S3a) N / − N − / = T + − T − = 0 . (S3b)In the presence of transverse magnetic field, B k x ,Eqs. (S3) holds, while spin fluctuations obey the followingset of equations [1, 2]: d δS z d t = Ω B δS y − δS z τ s − GδS z + δS T τ T , (S4a) d δS y d t = − Ω B δS z − δS y τ s − GδS y , (S4b) d δS T d t = − δS T τ T + GδS z . (S4c)Here we denote as δ S = ( δS x , δS y , δS z ) the electronspin pseudovector fluctuation, δS T is the trion pseu-dospin fluctuation, δS T = ( δT + − δT − ) / , and τ T = τ Ts τ T / ( τ Ts + τ T ) . The hole-in-trion spin precession isneglected. Set of Eqs. (S4) is equivalent to Eqs. (2) ofthe main text in the absence of nuclear fields. The inclu-sion of spin precession in the total field being the sum ofexternal field and the field of nuclear spin fluctuation isstraightforward [3].The correlation functions of spin fluctuations are in-troduced as follows: C αβ ( t ) = h δS α ( t ) δS β (0) i , α, β = z, y, T, (S5a) C (+) αβ ; ω = Z + ∞ e i ωt C αβ ( t )d t, (S5b) C αβ ; ω = Z + ∞−∞ e i ωt C αβ ( t )d t. (S5c)The same-time correlators read C zz (0) = C yy (0) = N e N Gτ T ) , (S6a) C T T (0) = T N Gτ T Gτ T ) , (S6b) C zT (0) = C T z (0) = 0 . (S6c)Taking into account the fact, that the correlation func-tions of fluctuations obey the same kinetic equationsas fluctuations and making the Fourier transform ofEqs. (S4) we immediately obtain the solutions for C (+) αβ ; ω [4]. Similar procedure allows us to calculate cor-relation functions in the presence of nuclear fields Ω N ,in which case the correlation functions should be av-eraged with the appropriate distribution of the nuclearspin precession frequencies [3]. The results of simulationin Fig. 4(b) of the main text include also the effect ofspread of the electron g -factor values [3]. For the calcu-lation presented in Fig. 6 the spread of electron g -factorhas been disregarded, but the value of the nuclear fieldfluctuation has been somewhat increased to obtain thesimilar width of the magnetic peak.For the sake of example, let us analyze in detail thecontribution ( δS z ) ω = C zz ( t ) caused by electron spinfluctuations. We focus on the typical case, where τ s (cid:29) τ T , τ T . First, we consider the case of zero field, Ω B = 0 .In the limit of vanishing trion generation rate G , we havestandard result ( δS z ) ω = N τ s ω τ s . (S7)If Gτ T (cid:28) (but Gτ s can be on the order of 1) one has ( δS z ) ω = N τ s ω τ s , (S8)where τ s = τ s / [1 + Gτ s (1 − τ T /τ T )] < τ s . Hence, thespin noise spectrum somewhat broadens with an increaseof the trion generation rate due to the depolarization ofthe electron after the trion recombination: Factor − τ T /τ T = τ T / ( τ T + τ Ts ) is important if hole spin relaxationin the trion is fast enough.Now we address the case where a transverse magneticfield is applied, Ω B τ s ∼ . It is noteworthy that, ow-ing to the coupling with trion, the electron in-plane spincomponent S y decays with the rate /τ s + G , while S z component relaxes with smaller rate /τ s . Hence, the sit-uation of the anisotropic spin relaxation is realised here(similar to considered in Ref. [5–8]). Making use of thesolution for the electron spin dynamics of Ref. [7] andintroducing the average relaxation rate T = 1 τ s + G (cid:18) − τ T τ T (cid:19) , (S9)and effective precession frequency ˜Ω = s Ω B − G τ T τ T , (S10)we obtain for the electron spin noise power spectrum ( δS z ) ω = N ¯ T τ s ω − ˜Ω) ¯ T ][1 + ( ω + ˜Ω) ¯ T ] × (cid:26) τ s τ s (cid:2) ω τ s + (1 + Gτ s ) (cid:3) + (1 + Gτ s )Ω B τ s (cid:27) (S11)In small magnetic fields, where Ω B < G/ the effectiveprecession frequency is imaginary, giving rise to the ex-ponential decay of spin despite the presence of magnetic field. The spin decoherence rate (determining the spinnoise spectrum width) is given by γ eff = 1¯ T − i ˜Ω = 1¯ T − G s τ T τ T − B G , (S12)in agreement with Eq. (3) of the main text. For strongenough magnetic field, Ω B > G/ , the effective spinprecession frequency ˜Ω becomes real, giving rise to themagnetic peak in the spin noise power spectrum. For Ω B (cid:29) G one has ˜Ω ≈ Ω B . S2. PROBE FREQUENCY DEPENDENCE OFSPIN SIGNALS
In order to describe the optical spectra of Kerr andellipticity noise signals, it is important to include inho-mogeneous broadening of the trion resonance frequencyin Eq. (1) of the main text. The reflection coefficient hasthe form r ± = (S13) − κ ω − ω c + i κ + κ R dω F δω ( ω ) g ± ω − ω + i γ . where g ± are the coupling constants for right and left cir-cular polarizations differing due to the fluctuating num-bers of spin-up/spin-down electrons, γ is the dampingrate. We have assumed that ω is normally distributedwith the variance defined by parameter δω and the corre-sponding probability density function F δω . In derivationof Eq. (S13), we made use of the fact that the main con-tribution to the spin-Kerr and ellipticity signals causedby the trion resonance are given by spin-induced modu-lation of trion oscillator strength [2]. Here, for simplicity,we used the notations ω , g and γ instead of ω X − , g X − and γ X − used in the main text.The fluctuating difference of g ± causes Kerr rotation ofthe incident light polarization plane by angle θ K , whichis defined by [9, 10] sin(2 θ K ) = 2 Im r + r ∗− | r + | + | r − | . (S14)The ellipticity angle, θ E , can be defined in a similar way sin(2 θ E ) = | r + | − | r − | | r + | + | r − | . (S15)Taking into account that reflection coefficients are nearlyequal to each other, r + ≈ r − ≡ r , and the spin signalsare very small, θ K , θ E (cid:28) , one can find that θ K = Im r ∗ r | r | dg, θ E = Re r ∗ r | r | dg, (S16)
811 812 813 814 81500.20.40.60.81
Wavelength, nm S N po w e r , a . u . KerrEllipticity ×10 00.20.40.60.81 R e fl ec ti on Figure 1: Reflectivity (dark blue), Kerr rotation (brown)and ellipticity (red) optical spectra. Lines present theoreticalcalculations of θ K and θ E after Eqs. (S16), points are theexperimental data. The parameters of the calculation are asfollows ~ ω c = 1523 . meV (wavelength 813.8 nm), ~κ =0 . meV, ~κ = 0 . meV, ~ ω = 1526 . meV (812 nm), ~ g = 0 . meV, ~ γ = 0 . meV, and inhomogeneous broadeningof the trion resonance is . meV. where r = d r/ d g , dg = g + − g − .Qualitative behavior of the Kerr rotation noise and el-lipticity noise as functions of the probe frequency couldbe understood as follows. The momentary fluctuationof resident electron spin results in the splitting of eachpolariton state into σ + and σ − . As a result, the re-flection coefficients r + and r − have features at differentfrequencies, ω ± = ω pol ± CδS z , where ω pol is the fre-quency the given polariton state and C is a coefficient.Thus, the dips in | r + | and | r − | are shifted with respectto each other, resulting in vanishing ellipticity exactly at ω = ω pol . The Kerr rotation is controlled by the phaseof the reflection coefficient and has a maximum at ω pol .The frequency dependencies of the reflectivity, Kerr andellipticity noises calculated after Eqs. (S13),(S16) are pre-sented in Fig. 5(b) of the main text in the vicinity of thecavity mode and in Fig. 1 for the wide range of wave-lengths. We stress that for large negative detunings, theinclusion of inhomogeneous broadening of the material(in our case, trion) resonance results in vanishing noisein the vicinity of the trion resonance, while the fluctu-ations in the vicinity of the cavity mode frequency areobservable. The improvement of agreement between theexperiment and theory could be reached if one takes intoaccount additional resonances as well as background ab-sorption in the wells and barriers. Such a considerationis beyond the scope of the present work.Finally, we note that in the presence of photocreatedtrions, the Kerr (or ellipticity) effect is caused by both electron and trion spins [2], hence, ϑ = αS z + βS T , (S17)where α and β are coefficients. The correlation functionof fluctuations is then given by h δϑ ( t ) δϑ (0) i = α h δS z ( t ) δS z (0) i + β h δS T ( t ) δS T (0) i + αβ [ h δS z ( t ) δS T (0) i + h δS T ( t ) δS z (0) i ] . (S18)In addition to the spin noise of electrons and trions thisexpression contains cross-correlations. However, our es-timations show that, for the parameters used to calculateFig. 4 and Fig. 6 of the main text, the trion and cross-correlation contributions to the observed noise spectraare negligible because the condition Gτ T (cid:28) holds. S3. NORMALIZATION OF SPIN NOISE SIGNALS
The measured signal at the balanced detector (inVolts) caused by the Kerr rotation or ellipticity is givenby B = AϑP, (S19)where ϑ is the Kerr rotation angle, θ K , or ellipticity angle, θ E [see Eq. (S16)], P is the probe power, and A is thecoefficient dependent on the detector parameters. Theautocorrelation function of signal at a detector is givenby hB ( t ) B (0) i = A P h ϑ ( t ) ϑ (0) i , and the noise power density (in W/Hz) is given by S ω = 1 R Z d t e i ωt hB ( t ) B (0) i = A P R ( ϑ ) ω , (S20)where R = 50 Ohm is the input resistance and ( ϑ ) ω isthe rotation noise spectrum (in rad /Hz). The rotationnoise power density is quadratic in the probe power P .The shot noise power density scales linearly with theprobe power: S ω = BP, (S21)with the coefficient B . Hence, at a given probe powerthe shot noise is equivalent to the rotation noise with thedensity ( ϑ ) ω = BRA P . (S22)In our setup, A = 2 . × Volt/(rad · W), B = 1 . × − Hz − .The spin signals presented in Fig. 2(a) and 4(a) of themain text is normalized as follows:Signal-to-shot-noise ratio = SN ( B ) − SN (0 . T ) SN (0 . T ) − EN , (S23)where SN ( B ) = S ω + S ω + EN is the noise measured atthe magnetic field B , EN is the noise measured in theabsence of probe, i.e. electronic noise.In Fig. 3 of the main text, in order to elucidate theeffect of probe on the spin noise, we additionally dividedsignal-to-shot-noise ratio by the probe power P . In thiscase, the presented data are proportional to S ω /P or ( ϑ ) ω with a probe power independent coefficient, seeEq. (S20). [1] G. V. Astakhov, M. M. Glazov, D. R. Yakovlev, E. A.Zhukov, W. Ossau, L. W. Molenkamp and M. Bayer,Semiconductor Science and Technology , 114001(2008).[2] E. A. Zhukov, D. R. Yakovlev, M. Bayer, M. M. Glazov,E. L. Ivchenko, G. Karczewski, T. Wojtowicz and J. Kos-sut, Phys. Rev. B , 205310 (2007). [3] M. M. Glazov and E. L. Ivchenko, Phys. Rev. B ,115308 (2012).[4] L. Landau and E. Lifshitz, Physical Kinetics (Butterworth-Heinemann, Oxford, 1981).[5] S. D¨ohrmann, D. Hagele, J. Rudolph, M. Bichler,D. Schuh and M. Oestreich, Phys. Rev. Lett. , 147405(2004).[6] G. M. M¨uller, M. R¨omer, D. Schuh, W. Wegscheider,J. H¨ubner and M. Oestreich, Phys. Rev. Lett. ,206601 (2008).[7] M. M. Glazov and E. L. Ivchenko, Semiconductors ,951 (2008).[8] V. Kalevich, B. Zakharchenya, K. Kavokin, A. Petrov,P. Jeune, X. Marie, D. Robart, T. Amand, J. Barrauand M. Brousseau, Physics of the Solid State , 681(1997).[9] I. A. Yugova, M. M. Glazov, E. L. Ivchenko, and Al. L.Efros, Phys. Rev. B , 104436 (2009).[10] M. M. Glazov, Phys. Solid State , 1 (2012) [Fiz. Tverd.Tela54