Spin waves in semiconductor microcavities
SSpin waves in semiconductor microcavities
M.M. Glazov
1, 2 and A.V. Kavokin
2, 3, 4 Ioffe Institute, 26 Polytechnicheskaya, St.-Petersburg 194021, Russia Spin Optics Laboratory, St. Petersburg State University,1 Ul’anovskaya, Peterhof, St. Petersburg 198504, Russia School of Physics and Astronomy, University of Southampton, SO17 1NJ Southampton, United Kingdom Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia (Dated: May 8, 2019)We show theoretically that a weakly interacting gas of spin-polarized exciton-polaritons in asemiconductor microcavity supports propagation of spin waves. The spin waves are characterised bya parabolic dispersion at small wavevectors which is governed by the polariton-polariton interactionconstant. Due to spin-anisotropy of polariton-polariton interactions the dispersion of spin wavesdepends on the orientation of the total polariton spin. For the same reason, the frequency ofhomogeneous spin precession/polariton spin resonance depends on their polarization degree.
PACS numbers: 72.25.Rb, 75.30.Ds, 72.70.+m, 71.36.+c
Introduction . Spin waves are weakly damped harmonicoscillations of spin polarization. Predicted to appear inFermi liquids over 50 years ago [1] and discovered in theend of 1960s in metals [2] are among the most fascinatingmanifestations of collective effects in interacting system.Later it was understood theoretically that the spin wavescan exist in a non-degenerate electron gas [3] as well asin atomic gases [4, 5], and the spin waves were indeedobserved in a number of interacting gases such as Hydro-gen and Helium [6–8], see Ref. [9] for review. Spin waveswere also observed in atomic Bose gases, namely, in Rbvapors at temperatures of about 850 nK which exceededthe Bose-Einstein condensation temperature [10], see alsoRefs. [11–13] where this experiment was interpreted.Recently, semiconductor microcavities with quantumwells sandwiched between highly reflective mirrors haveattracted a lot of interest in the solid state and photonicscommunities [14]. In these artificial structures the strongcoupling is achieved between excitons, being material ex-citations in quantum wells, and photons confined betweenthe mirrors [15]. Resulting mixed light-matter particles,exciton-polaritons, demonstrate the Bose-Einstein statis-tics and may condense at critical temperatures rangingfrom tens Kelvin [16] till several hundreds Kelvin [17, 18],which exceeds by many orders of magnitude the Bose-Einstein condensation temperature in atomic gases. Hightransition temperatures and the strong coupling withlight makes semiconductor microcavities perfectly suitedfor benchtop studies of collective effects of Bosons.In typical GaAs based microcavities, exciton-polaritons may have two spin projections onto the struc-ture growth axis, ±
1, corresponding to right- and left-circular polarizations of photons (and spin moment ofexcitons) forming polaritons. Owing to the compositenature of exciton-polaritons, the interactions betweenthem are strongly spin-dependent [14, 15, 19]. A numberof prominent spin-related phenomena both in interact-ing and in noninteracting polariton systems have alreadybeen predicted and observed in the microcavities, suchas, e.g., polarization multistability [20, 21] and optical spin Hall effect [22, 23], see Refs. [14, 15, 19] for reviews.Here we predict the existence of weakly-damped spinwaves for a non-degenerate or weakly degenerate polari-ton gas in a microcavity with embedded quantum wells.We show that the system sustains the spin wave solu-tions, where the spin of polaritons S is harmonic functionof the coordinate, r , and time, t , S ∝ exp (i qr − i ωt ),and calculate their dispersion, ω ≡ ω ( q ). The stability ofspin waves is analyzed. The experimental manifestationsof spin waves in the photoluminescence spectroscopy andspin noise studies are discussed. Due to the strong light-matter interaction in microcavities, which allows one toobserve directly the spin states of quasi-particles, micro-cavities may become one of the most suitable systems forspin waves experimental studies. Model . We consider non-degenerate or weakly de-generate polariton gas at a temperature higher thanthe Berezinskii-Kosterlitz-Thouless transition tempera-ture with weak interactions, in this case the single-particle spin density matrix is parametrized as ˆ ρ k =( N k /
2) ˆ I + S k · ˆ σ , where N k is the occupancy of the or-bital state with the wavevector k , S k ≡ S k ( r ) is thecoordinate r -dependent spin distribution function, ˆ I andˆ σ are 2 × S k change in time as a result ofthe particle propagation with the group velocity v k , spinprecession in the effective field Ω (eff) k , as well as gener-ation and scattering processes described by the collisionintegral Q { S k } [24]: ∂ S k ∂t + v k · ∂ S k ∂ r + S k × Ω (eff) k = Q { S k } , (1)Here Ω (eff) k = α (cid:88) k (cid:48) S k (cid:48) ,z e z + Ω L , (2)the constant α describes interaction of polaritons withparallel spins. We recall that the polariton-polariton a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r interactions are strongly spin-anisotropic and neglectweak interaction of particles with opposite signs of cir-cular polarization [25–28]. The effective field Ω L = (cid:126) − gµ B B z e z + Ω a e x + Ω ( k ), where e i are unit vectorsof Cartesian axes, i = x, y, z ; is interaction-independent,generally, it is contributed by an external magnetic field B , g is the exciton-polariton g -factor [29], the splittingof linearly polarized polariton states due to the structureanisotropy, Ω a (the anisotropy field is parallel to x -axis)and TE-TM splitting of the cavity modes, Ω ( k ). Weassume that the structure anisotropy is strong enough,Ω a (cid:29) Ω( k ) for the relevant wavevectors range and ne-glect TE-TM splitting [30]. Under our assumptions theeffective field is independent on k and acts similarlyto the real magnetic field. Depending on the ratio of gµ B B z / (cid:126) and Ω a this field can be arbitrarily oriented, seeFig. 1(a). The collision integral in the right hand side ofEq. (1) accounts for the polariton generation, scatteringand decay processes, Q { S k } = − S k τ + g k + (cid:88) k (cid:48) [ W kk (cid:48) S k (cid:48) − W k (cid:48) k S k ] . (3)Here τ is the lifetime of polaritons, g k is the polari-ton generation rate accounting for the in-coming flow ofquasiparticles from the reservoir, and W kk (cid:48) is the scatter-ing rate from the state k to the state k (cid:48) which accountsfor both elastic and inelastic scattering processes. Due tothe bosonic nature of exciton-polaritons and polariton-polariton interactions g k and W kk (cid:48) depend, generally,on the occupancies and spin polarizations in the states k , k (cid:48) [24, 25, 33]. The dynamics of polaritons can bedescribed by Eq. (1) which is valid provided that therenormalization of spectrum due to polariton-polaritoninteractions is negligible, otherwise the excitations spec-trum should be found from the spin-dependent Gross-Pitaevskii equation [34–36].Under the steady-state excitation, the quasi-equilibrium distribution S (0) k of exciton-polaritonsis formed whose shape is determined by the generation,thermalization and interactions. This function satisfiesEq. (1) with derivatives ∂/∂t , ∂/∂ r being equal tozero. For simplicity we assume that the pumping isisotropic, hence S (0) k depends only on the absolute valueof the polariton wavevector k = | k | . Its specific formis determined by the pumping conditions. In whatfollows two important limiting cases are addressed:(i) quasi-resonant pumping which creates monoener-getic polaritons (so-called excitation of elastic circle),such a situation can be experimentally realized if thepump energy is slightly above the inflection point onthe dispersion Fig. 1(b), so that the phonon-assistedrelaxation towards the ground state is suppresseddue to polaritons strong energy dispersion (bottleneckeffect), and (ii) nonresonant pumping which createsthermalized distribution of particles. In the case ofquasi-resonant excitation of polaritons with the sameenergy ε by a polarized light, the spin distribution (a)(c) (b) Figure 1: (a) Schematics of the microcavity structure and ex-ternal fields acting on the polariton pseudospin. (b) Sketchof polariton dispersion (surface) and elastic circle (red cir-cle). The distribution function of polariton spin in the caseof linearly polarized excitation of the states on elastic circleis illustrated by arrows. (c) Schematic illustration of the spinwave propagating along x -axis at B z = 0; red and blue arrowsdemonstrate δs z and δs y components. function S (0) k ∝ δ ( ε − ε ) e i , where i = x or y for thelinearly polarized excitation and i = z for the circularlypolarized one, ε ≡ ε k is the polariton dispersion, seescheme in Fig. 1(b), while for thermalized polaritons(at the temperature T higher than the degeneracytemperature), S (0) k ∝ S exp ( − ε/T ) with the prefactor S dependent on the effective field Ω L . In order toanalyze the spin excitations, the total spin distributionfunction is presented as a sum of its quasi-equilibriumpart S (0) k and the fluctuating correction δ s k (cid:28) S (0) k .A standard linearization of Eq. (1) and substitution of δ s k = exp (i qr − i ωt ) s k with q being the wavevectorand ω being the frequency of the fluctuation yields,cf. [1, 9]: (cid:2) τ − c − i ω + i( q · v k ) (cid:3) s k + α S (0) k × e z (cid:88) k (cid:48) s k (cid:48) ,z + s k × (cid:32) Ω L + α e z (cid:88) k (cid:48) S (0) k (cid:48) ,z (cid:33) = − s k − ¯ s k τ , (4)where we introduced the lifetime of the fluctuation τ c and the isotropization time τ . Bar simbolizes averagingover possible orientations of k . In the simplest approx-imation, polaritons are assumed to be supplied directlyby the polarized pump or from the incoherent but spin-polarized reservoir, the polariton-polariton scattering isneglected as well as inelastic processes, and the elas-tic scattering is assumed to be isotropic, in which case W k , k (cid:48) = W ( ε k ) δ ( ε k − ε k (cid:48) ), τ − = (cid:80) k (cid:48) W ( ε k ) δ ( ε k − ε k (cid:48) ).The lifetime of a fluctuation is governed by an interplayof polariton decay processes accounted for by the life-time τ in our formalism, and by the bosonic stimulationeffect, which increases the lifetime of the fluctuations, τ c = τ (1 + N k ) [24]. Equation (4) determines the dy-namics of spin fluctuations in the system. Its eigenmodesrepresent the spin waves in the interacting polariton en-semble. Results . The solution of Eq. (4) can be expressed bydecomposing the function s k in the angular harmonics ofthe polariton wavevector k as s k = (cid:80) m exp (i mϕ ) s m ( ε ),with ϕ being the azimuthal angle of k , and reducingEq. (4) to a system of equations for the energy depen-dent functions s m ( ε ). The condition of compatibility forthis system of equations yields dispersions of the waves.Below we analyze the spectrum of excitations and eigen-modes for different particular cases. Homogeneous excitations . We start the analysis fromthe homogeneous case, q = 0. In this case the angu-lar harmonics exp (i mϕ ) s m ( ε ) are the eigensolutions ofEq. (4). For all m (cid:54) = 0 one eigenmode corresponds tothe damped solution with s m parallel to the total field Ω (tot) = Ω L + α S ,z e z , S = (cid:80) k (cid:48) S k (cid:48) , whose dampingrate is ν = τ − c + τ − , and two other eigenmodes precess-ing in the plane perpendicular to Ω (tot) with frequenciesΩ (tot) and the damping rate ν .The harmonic with m = 0 is isotropic in k -space,its eigenfrequency corresponds to the spin resonance fre-quency. We introduce ˜ s = (cid:80) k s ( ε ) and perform thesummation of Eq. (4) over k which yields( τ − c − i ω )˜ s − α ˜ s ,z e z × S + α ˜ s × e z S ,z + ˜ s × Ω L = 0 . (5)We recall that in the case of spin-anisotropic interactionsthe Larmor theorem [37] in not applicable, and the homo-geneous spin excitation frequency can be renormalized bythe interactions. To illustrate it we consider the depen-dence of the spin resonance frequency on the orientationof effective Larmor field Ω L and spin polarization S (0).The orientation of Ω L in ( xz )-plane can be varied bychanging the external magnetic field contributing to Ω L,z or mechanical strain contributing to Ω
L,x = Ω a . We as-sume efficient thermalization in the spin space, S (cid:107) Ω L ,introduce the angle θ between S and z -axis and presentthe complex eigenfrequencies of Eq. (5) in a form [28] ω = − i τ c , (6) ω ± = − i τ c ± (cid:113) Ω L + α S cos θ + α Ω L S (3 cos θ − . For instance, if Ω L and S are parallel to z -axis, θ =0, the frequencies of the precessing modes are ±| Ω L,z + α S ,z | and the damping rate is 1 /τ c .Real part of ω + and imaginary parts of ω ± are shownin Fig. 2 as a function of angle between the field and the z -axis θ . Depending on the sign α S Ω L the frequencycan increase or decrease with an increase of θ . Note,that for large enough α S , hence ω ± become imaginaryas shown by red/solid curve in Fig. 2. Moreover, theimaginary part of one of the frequencies can be positivewhich manifests the instability of the system, see inset -- Figure 2: Eigenfrequencies of homogeneous precessing modeswith m = 0 as a function of angle θ calculated after Eq. (6).Main panel shows real part of the frequencies and insetshows imaginary parts. The parameters of calculation are: α S Ω L = 0 . . L τ c = 10. in Fig. 2. To analyze it in more detail we put θ = π/ B z = 0, S and Ω L are in the structure plane). In thiscase ω ± = − i /τ c ± (cid:112) Ω L − α S Ω L . The system becomesunstable for α S Ω L > Ω L + 1 τ c , (7)and small fluctuations of s y and s z grow exponen-tially. This is because the anisotropic interactions be-tween polaritons favor in-plane orientation of the pseu-dospin [19, 20]. The instability of small spin fluctuationscan result in the nonlinear oscillations of spin polariza-tion [31] similar to those discussed in Refs. [1, 39] or inchanges in the polarisation of the ground state accompa-nied by change of orientation of S (cf. [40]) where thecondition (13) no longer holds. Spin waves . Let us consider the spatially inhomoge-neous solutions of Eq. (4) which describe the propagationof spin fluctuations and spin waves. To be specific, weconsider the case where S k and Ω L are parallel to x -axis,and to simplify the treatment we assume τ (cid:29) τ c [28].Moreover, we assume that the system is stable at q = 0,i.e. the condition (13) is not fulfilled. We seek the solu-tion, which corresponds to the precessing mode at q = 0,where s k ,x = 0. From Eq. (4) we arrive to the set oflinear homogeneous integral equations for s k ,y and s k ,z ,whose self-consistency requirement yields (cid:88) k α Ω L τ c S (0) k [1 − i ωτ c + i( qv k ) τ c ] + (Ω L τ c ) = 1 . (8)This equation describes the dispersion of spin waves. Ithas a more complex form compared with the dispersionequation for the spin waves in the systems with spin-isotropic interactions [1, 3, 9].To solve Eq. (8) one has to specify the function S (0) k whose form is determined by the excitation conditions. It Figure 3: Dispersion of spin waves in the case of resonant exci-tation calculated after Eq. (9). The parameters of calculationare indicated at each curve. is instructive to consider the case of resonant excitationof polaritons at the elastic circle where a monoenergeticdistribution of particles is generated. For the isotropicdispersion the product qv k = qv cos ϕ , v = (cid:126) − dε k /dk is the polariton velocity on the elastic circle. Here thesummation over k reduces to the averaging over the az-imuthal angle ϕ and Eq. (8) takes the form1 (cid:113) ˜ ω − ( qv ) − (cid:113) ˜ ω − − ( qv ) = 2 α S , (9)where ˜ ω ± = ω ± Ω L − i /τ c . Equation (9) determines thedispersion of the spin waves. For q = 0 it passes to ω ± inEq. (6). For small qv (cid:28) | α S | and | α S | (cid:28) | Ω L | thedispersion of the spin waves reads ω ( q ) = Ω L − α S / − ( qv ) / ( α S ) − i /τ c , (10)where we took the solution which passes to ω + in Eq. (6).As follows from Eq. (10) the dispersion is parabolic forsmall qv and the “effective mass” is proportional to α .Similarly to the previously studied electronic and atomicsystems [1, 3, 9, 41] the dispersion of spin wave resultsfrom an interplay of the gradient, ∝ qv k , and interac-tion, ∝ α , terms in the kinetic equation (4). Indeed,owing to the gradient contribution, the spin density in k -space acquires ∝ ( qv ) correction, yielding gain or lossof energy depending on the sign of α S . The spin wavefrequency increases with the increase of the wavevectorfor α S < α S >
0. The behavior of ω ( q ) is illustrated in Fig. 3. Noteworthy, for the solutionswith α S > , the real part of the frequency vanishes atsome q , in which case the solutions of Eq. (10) may be-come unstable. For arbitrary direction of Ω L and S thedispersions of waves has a similar form, but its parame-ters depend on the orientation of the magnetic field andthe total spin due to anisotropy of polariton-polaritoninteractionsIt is instructive to compare the parabolic dispersion ofspin waves in a weakly interacting polariton gas with the dispersion of excitations of an interacting polariton con-densate [34–36]. In the case of a condensate of polaritons(in the absence of TE-TM splitting) the dispersion of ex-citations is linear. By contrast, the dispersion of spinwaves for non-condensed polaritons is parabolic at smallwavevectors.In the case of a non-resonant excitation where a con-tinuous distribution S (0) k is formed, the analysis of thedispersion of spin waves is more complex, particularlybecause the additional channel of damping caused by thespatial dispersion appears [3, 9], but the basic physics re-mains the same. To illustrate this we consider the case ofa thermalized non-degenerate gas where S (0) k is describedby the Boltzmann function characterised by an effectivetemperature T . The evaluation of the sum in Eq. (8)under the assumptions | α S | (cid:28) | Ω L | , | α S | τ c (cid:29) qv T (cid:28) | α S | , where v T = (cid:112) k B T /m is the thermal ve-locity with m being the polariton effective mass and k B being the Boltzmann constant, and the solution of theresulting equation yields the dispersion in the form ω ( q ) = Ω L − α S / − qv T ) α S − i γ L , (11)where the Landau damping can be estimated as γ L = (cid:114) π α S ) qv T exp (cid:18) − ( α S ) qv T ) − (cid:19) , (12)and it is exponentially small for small wavevectors, inagreement with Refs. [3, 9]. The allowance for Landaudamping in Eq. (11) is correct only if the damping islarge enough compared with 1 /τ c but small compared to | α S | . Conclusions . To conclude, we predicted the existenceof exciton-polariton spin waves in semiconductor micro-cavities with embedded quantum wells. The dispersionand damping of spin waves were calculated in two impor-tant particular cases: (i) resonant excitation of a quasi-monoenergetic distribution of polaritons at an elastic cir-cle and (ii) nonresonant excitation, where the Boltzmanndistribution of quasi-particles is formed. In the state-of-the-art microcavities the polariton polarisation splittingsinduced by the cavity anisotropy, (cid:126) Ω L , and interaction-induced effective field (cid:126) α S are of the order of 100 µ eV,usually [42–44]. For the polariton lifetime (cid:38)
10 ps thespin waves can be readily detectable even at the rela-tively weak pump. The spin waves can be excited, e.g., intwo-beam photoluminescence experiments where the cw beam creates the desired steady distribution of polaritonswith a given spin polarization S (0) k and the probe beaminjects a small non-equilibrium portion of polaritons withthe spin polarization different from S (0) k . The time-resolved micro-photoluminescence spectroscopy as usede.g. in Refs. [45, 46] would be a suitable tool for detec-tion of the spin waves. Another possibility to observe thespin waves is to use the spin noise spectroscopy [47, 48]and measure temporal and spatial correlations of spinfluctuations in the presence of the pump only [31]. Acknowledgements . We are grateful to V.A. Zyuzinfor discussions. This work was supported by RFBR, RFPresident grant MD-5726.2015.2, Dynasty Foundation, Russian Ministry of Education and Science (Contract No.11.G34.31.0067 with SPbSU and leading scientist A. V.Kavokin), and SPbSU grant 11.38.277.2014. [1] V. P. Silin. JETP , 870 (1959).[2] S. Schultz, G. Dunifer. Phys. Rev. Lett. , 283 (1967).[3] A. G. Aronov. JETP , 301 (1977).[4] E. P. Bashkin. JETP Lett. , 8 (1981).[5] C. Lhuillier, F. Lalo¨e. J. Phys. France , 197 (1982); ibid , 225 (1982).[6] B. R. Johnson, J. S. Denker, N. Bigelow, L. P. L´evy, J. H.Freed, D. M. Lee. Phys. Rev. Lett. , 1508 (1984).[7] P. Nacher, G. Tastevin, M. Leduc, S. Crampton, F. Lalo¨e.Journal de Physique Lettres , 441 (1984).[8] W. J. Gully, W. J. Mullin. Phys. Rev. Lett. , 1810(1984).[9] E. P. Bashkin. Sov. Phys. Usp. , 238 (1986).[10] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, E. A.Cornell. Phys. Rev. Lett. , 070403 (2002).[11] M. O. Oktel, L. S. Levitov. Phys. Rev. Lett. , 230403(2002).[12] J. N. Fuchs, D. M. Gangardt, F. Lalo¨e. Phys. Rev. Lett. , 230404 (2002).[13] J. E. Williams, T. Nikuni, C. W. Clark. Phys. Rev. Lett. , 230405 (2002).[14] V. Timofeev, D. Sanvitto (eds.). Exciton Polaritons inMicrocavities (Springer, 2012).[15] A. Kavokin, J. Baumberg, G. Malpuech, F. Laussy.
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The linear analysis of kinetic equation (5) in the maintext shows that the polariton spin system becomes un-stable at θ = π/ S (cid:107) Ω L is in the structure plane)provided that α S Ω L > Ω L + 1 τ c , (13)see Eq. (7) of the main text. In the linear regime the ex-citations are grow exponentially (in the unstable regime)with the increment λ = (cid:113) α S Ω L − Ω L − /τ c (14)In order to analyze the instability in more detail we con-sider the simplest possible case where polariton genera-tion and dissipation are absent and the spin dynamics isdescribed by the following equation ∂ S ∂t + S × Ω (eff) = 0 , (15)with Ω (eff) k = Ω L + α (cid:88) k (cid:48) S k (cid:48) ,z e z , (16)see Eqs. (1) and (2) of the main text. - -- (a) (b) Figure 4: Panels (a) and (b) show temporal dynamics ofthe total spin components nonlinear Eqs. (15), (16) for ho-mogeneous case for Ω L (cid:107) S || x (at t = 0) and small δs y (0) = S /
10. (a) corresponds to stable and (b) to un-stable regimes. Spin components are marked at each curve,thin solid line shows exponentially growing solution calculatedafter Eq. (14).
Examples of temporal dynamics of the total polaritonpseudospin in the stable and unstable regimes calculatednumerically after Eq. (15) are shown in Fig. 4, panels (a)and (b), respectively. It is seen that the instability ofsmall spin fluctuations can result in the nonlinear oscil-lations of spin polarization similar to those discussed inRefs. [2 ? ] [Fig. 4(b)]. The detailed analysis of the finalstate of the system and its stability with allowance fordissipative processes is beyond the scope of the presentpaper. B. Spin precession caused by TE-TM splitting
Here we demonstrate that the TE-TM splitting of thepolariton states can also result in the weakly dampedspin resonance and diffusive spin modes. We considersymmetric cavity with the TE-TM splitting Ω k in theform Ω k = Ω( k )[cos ϕ, sin ϕ, , (17)where ϕ is the angle between k and x -axis in the struc-ture plane, Ω( k ) is the amplitude of the TE-TM splitting.For simplicity we consider monoenergetic polaritons withthe energy ε ≡ ε ( k ), introduce Ω = Ω( k ), and as-sume that both Ω τ c , Ω τ (cid:29)
1. We focus on dynamicsof spin z component. For q = 0 we obtain (note that S z is isotropic function of ϕ , while S x and S y are stronglyanisotropic and their angular averages are 0): (cid:18) − i ω + 1 τ c (cid:19) S k,z + S k ,x Ω ,y − S k ,y Ω ,x = 0 , (18a) (cid:18) − i ω + 1 τ c + 1 τ (cid:19) S k ,x − S k,z Ω ,y = 0 , (18b) (cid:18) − i ω + 1 τ c + 1 τ (cid:19) S k ,y + S k,z Ω ,x = 0 . (18c)Solution of Eqs. (18) yields the frequency of homogeneousspin z component oscillations ω = Ω (cid:112) − [2Ω τ ] − − i τ c − i2 τ . (19)In the relevant limiting case Ω( k ) τ (cid:29) z compo-nent demonstrates oscillations with the frequency Ω( k )similarly to the oscillations of spin in high-mobility elec-tron gas [3–5]. Note that for Ω( k ) τ (cid:28) ω = − iΩ τ − i /τ c .The spectrum of inhomogeneous spin excitations canbe conveniently found in the limit of qv τ (cid:28)
1, in whichcase the gradient term ∝ ( qv k ), Eqs. (1) and (4) of themain text, can be taken into account by perturbationtheory. After some algebra we obtain for monoenergeticparticles in the limit of Ω τ c , Ω τ (cid:29) qv τ (cid:28) ω ( q ) = Ω − i τ c − i2 τ − i( qv ) τ. (20)Unlike the spin waves predicted in the main text, Eq.(10), here the spatial inhomogeneity results in the diffu-sive damping of the spin precession mode. The analysisof the interplay between interactions and TE-TM split-ting is beyond the present paper. C. Spatial correlations of polariton spins
Spin waves describe the time-space correlations of po-lariton spins and can be addressed in the two-beam spinnoise spectroscopy technique (for reviews on the spin
Figure 5: Correlator of polariton spins C zz = (cid:104) δS z ( r, t ) δS z (0 , (cid:105) calculated after Eq. (23). The cal-culation parameters are as follows: Ω = 1, τ c = 10, A (cid:48) = 0 . A (cid:48)(cid:48) = 0 .
01 (coordinates and time are given indimensionless units). noise spectroscopy see, e.g., Refs. [6, 7], temporal andspatial fluctuations of spin density were studied for elec-trons for the first time in Refs. [8, 9], respectively). The correlation function of polariton spin fluctuations can beexpressed via the spin waves spectrum ω ( q ) as (cid:104) δS z ( r , t ) δS z (0 , (cid:105) ∝ (cid:90) d ω π (cid:88) q exp (i qr − i ωt ) ω − ω ( q ) . (21)In the long-wavelength limit [see Eqs. (10) and (11) ofthe main text] ω ( q ) = Ω − A q − i /τ c , (22)where Ω and A are the parameters depending on theeffective field, S and interactions. Note that diffusionof particles yields imaginary part A (cid:48)(cid:48) of A = A (cid:48) + i A (cid:48)(cid:48) inaddition to its real part A (cid:48) determined by interactions.Making use of Eq. (22) we arrive from Eq. (21) to (cid:104) δS z ( r, t ) δS z (0 , (cid:105) = Im exp (cid:104) i (cid:16) r A t − Ω t (cid:17)(cid:105) π A t e − t/τ c . (23)The spin correlation function is presented in Fig. 5.The oscillations of the correlation function as a functionof coordinate and time are clearly seen demonstratingwave-like propagation of spin excitations. [1] D. Read, T. C. H. Liew, Y. G. Rubo, A. V. Ka-vokin. Stochastic polarization formation in exciton-polariton Bose-Einstein condensates . Phys. Rev. B ,195309 (2009).[2] E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi,P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Kavokin,P. G. Lagoudakis. Nonlinear Optical Spin Hall Effect andLong-Range Spin Transport in Polariton Lasers . Phys.Rev. Lett. , 036404 (2012).[3] V. N. Gridnev.
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