Signature of the microcavity exciton-polariton relaxation mechanism in the polarization of emitted light
Georgios Roumpos, Chih-Wei Lai, T. C. H. Liew, Yuri G. Rubo, A. V. Kavokin, Yoshihisa Yamamoto
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Signature of the microcavity exciton polariton relaxation mechanism in thepolarization of emitted light
Georgios Roumpos, ∗ Chih-Wei Lai,
1, 2
T. C. H. Liew, YuriG. Rubo,
4, 5
A. V. Kavokin,
4, 6 and Yoshihisa Yamamoto
1, 2 E. L. Ginzton Laboratory, Stanford University, Stanford, CA, 94305, USA National Institute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan Centre for Quantum Technologies, National University of Singapore, Singapore 117543 School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK Centro de Investigaci´on en Energ´ıa, Universidad Nacional Aut´onoma de M´exico, Temixco, Morelos, 62580, Mexico Marie-Curie Chair of Excellence “Polariton devices”,University of Rome II, 1, via della Ricerca Scientifica, Rome, 00133, Italy (Dated: November 29, 2018)We have performed real and momentum space spin-dependent spectroscopy of spontaneouslyformed exciton polariton condensates for a non-resonant pumping scheme. Under linearly polarizedpump, our results can be understood in terms of spin-dependent Boltzmann equations in a two-statemodel. This suggests that relaxation into the ground state occurs after multiple phonon scatteringevents and only one polariton-polariton scattering. For the circular pumping case, in which onlyexcitons of one spin are injected, a bottleneck effect is observed, implying inefficient relaxation.
PACS numbers: 78.67.De, 03.75.Nt, 78.70.-g
I. INTRODUCTION
Bose-Einstein condensation (BEC) is an active field ofresearch, especially after its realization in dilute alkaligases . Microcavity exciton polaritons , compositequasi-particles consisting of quantum well (QW) excitonand microcavity photon components, have been proposedas candidates for BEC . Due to their low mass, the crit-ical temperature for BEC is expected to be high, evenup to room temperature . The confinement in two di-mensions, along with the dual exciton-photon characterof polaritons, enables interesting optical studies. Indeed,several characteristic signatures of dynamical condensa-tion have been reported in recent years .However, the lifetime of polaritons is short, on theorder of 10 psec in our GaAs-based sample when con-densation is observed, so the system is inherently dy-namical. In previous studies, the final energy distribu-tion of polaritons was compared to the Bose-Einsteindistribution for steady-state or time-resolved data.These results are explained by modeling the relaxationmechanism in terms of polariton-acoustic phonon andpolariton-polariton scattering . However, takinginto account the polariton spin degree of freedom intro-duces further complications, due to the interplay betweenenergy and spin relaxation .Here, we report the insights we gained on the re-laxation mechanism, based on polarization-dependentstudies of exciton polariton condensation under non-resonant incoherent pumping. For linearly polarizedpump, the condensate emission develops both non-zerolinear and circular polarization. We observed rotationof the linear polarization axis by ∼ ◦ between thepump and condensate. The exact rotation angle is corre-lated with the handedness of the observed circular po- larization. These signatures are similar to the obser-vations of a parametric oscillator experiment , whichwere interpreted in terms of spin-asymmetric polariton-polariton interaction . We use a two-state modelemploying the spin-dependent Boltzmann equations tounderstand our experimental results. The agreementwe obtain reveals the similarities of the non-resonantpumping scheme to parametric oscillator (magic angle)geometries . In the former case, it is believed thatpolaritons suffer multiple scatterings with phonons andother polaritons before reaching the k x ∼ . Further, the observed spectra undercircular pumping, show a bottleneck effect. This suggeststhat polaritons cannot efficiently relax into the groundstate when only one spin species is present. A similarsuppression of the scattering rate was observed in para-metric amplification experiments .In Section II we describe our experimental setup. Ourmeasurements of the Stokes vector and the correspondingtheoretical model are presented in Sections III and IVrespectively. Section V covers the relaxation bottleneckunder circularly-polarized pumping. Our conclusions aredrawn in Section VI. In the Appendix, we write downthe equations used in our theoretical model. II. EXPERIMENTAL SETUP
The sample is the same as in Ref. 27. It consistsof an AlAs λ cavity sandwiched between two distributedBragg reflector (DBR) mirrors. The upper and lowermirrors are made of 16 and 20 pairs respectively of AlAsand Ga . Al . As. 3 stacks of 4 GaAs QW’s are grownat the central three antinodes of the cavity. The spec-troscopy setup is described in Ref. 28, and it allows usto perform near field (NF - real space) and far field (FF- momentum space) imaging and spectroscopy. That is,we can measure energy-resolved luminescence as a func-tion of position or of in-plane momentum. The mea-surements reported here are taken from a spot on thesample with photon-exciton detuning δ = +6 meV , whilethe Rabi splitting is 2 ~ Ω Rabi = 14 meV . The sampleis kept at a temperature of 7 − K on the cold fingerof a He flow cryostat. The system is pumped with amode-locked Ti-Sapphire laser of 2psec pulse width and76MHz repetition rate focused on an ellipse of diameters50 µm and 30 µm . For FF data, luminescence is collectedthrough an aperture at the first image plane correspond-ing to a circular area of 30 µm diameter on the sample.The pumping laser is incident at an angle of 55 ◦ (Fig. 1inset, corresponding wavenumber k y = − µm − ), at theexciton resonance wavelength. The setup employs liquidcrystal polarization components as shown in Fig. 1(a).We can pump with linear polarization of varying angle θ p , as well as general elliptical polarization. The detec-tion can be performed for linear polarization of arbitraryangle θ d , or for right- and left-circular polarization.Using the transfer matrix method for exciton inho-mogeneous broadening of 3 meV , as measured at the farblue detuned regime, we estimate that the absorbed laserpower for TM ( θ p = 90 ◦ ) and TE ( θ p = 0 ◦ ) pumping is ∼
4% and ∼ .
9% respectively of the incident power. Weassume that the absorption efficiency is independent ofpower. In the rest of the paper, the various pump polar-ization states refer to the actually absorbed light insidethe cavity, taking into account the calculated differentialabsorption of TM and TE pumping.A ground state ( k x,y = 0) linear polarization splittingof ∼ µeV , similar to earlier studies , is measuredfor low excitation power and the current sample orien-tation, (Fig. 1(b-c)) possibly due to crystal asymmetryor strain. The observed superimposed linear polarizationsplitting for k x = 0 is in quantitative agreement with atransfer matrix calculation (Fig. 1(b) inset). III. STOKES VECTOR MEASUREMENT
The polarization state of light is characterized by thefollowing three parameters (normalized with respect tothe total power), which are equivalent to the Stokes pa-rameters as originally defined : S = I ◦ − I ◦ I ◦ + I ◦ , S = I ◦ − I − ◦ I ◦ + I − ◦ , S = I L − I R I L + I R . (1) I ◦ , I ◦ , I ◦ , and I − ◦ are the intensities of the linearlypolarized components at θ d = 0 ◦ , 90 ◦ , 45 ◦ , and − ◦ re-spectively. I L and I R are the intensities of the left- andright-circularly polarized components respectively. Fromthe above parameters, we can calculate the degree of lin-ear polarization (DOLP) and the angle of the major lin- s a m p l e xy z QWP1HWP1 θ LP1VR1 45 o pump VR245 o QWP290 o VR345 o LP2 detect s a m p l e xy zpump θ p detect θ d (a) −3 −2 −1 0 1 2 31610161116121613 k x ( µ m −1 ) E ( m e V ) TMTE −2 0 2 00.10.2
TM−TE (b)
E (meV) i n t e n s i t y ( a r b i t r a r y ) (c) FIG. 1: (color online) (a) The polarization measurementsetup. Vectors label the fast or polarization axes of the op-tical components. The laser pump is initially horizontallypolarized ( θ p = 90 o ), and is incident at an angle of 55 o withrespect to the growth direction z . Luminescence is collectedalong the z-axis. The first variable retarder (VR1) and linearpolarizer (LP1) work as a variable attenuator. By rotatinga half waveplate (HWP1), and by using a removable quarterwaveplate (QWP1), we can implement various polarizationstates for the pump. The second variable retarder (VR2) isused as a zero, half, or quarter waveplate. The combinationof a quarter waveplate (QWP2), variable retarder (VR3) andlinear polarizer (LP2) is used for detection of a particular lin-ear polarization state, depending on the retardance of VR3.Inset: Definition of angles θ p and θ d corresponding to the po-larization axes of the pump and detection respectively. (b)Measured dispersion curves for TM ( θ d = 0 ◦ - blue dots) andTE ( θ d = 90 ◦ - red squares) luminescence for low excitationdensity (60 µm − per pulse per QW). The plotted points arethe first moments of measured spectra for every k x . A smallground state splitting is visible. k x = 3 µm − corresponds to21 ◦ in air. Inset: The measured TM-TE splitting (black dots)and the theoretical prediction for our sample parameters witha superimposed ground state splitting of 50 µeV (red line). (c)Measured spectra for k x = 0 (points) fitted with Lorentzians(lines). ear polarization axis ψ DOLP = q S + S , ψ = 12 arctan (cid:18) S S (cid:19) . (2) I n t e n s i t y ( a r b i t r a r y un i t s ) θ d =0 ° θ d =90 ° −0.3−0.10.10.30.5 S , S , S ( no r m a li z e d ) S S S −1−0.8−0.6−0.4−0.200.20.4 density n ( µ m −2 ) S , S , S ( no r m a li z e d ) S S S D O L P θ p =0 o θ p =19 o θ p =45 o θ p =90 o ψ − θ p ( d e g r ees ) −0.8−0.6−0.4−0.200.20.40.60.8 density n ( µ m −2 ) D O C P ( S ) (a)(b)(c) (d)(e)(f) FIG. 2: (color online) Measurement of Stokes parameters(markers) compared with the theoretical model (solid lines).(a) Horizontal pumping ( θ p = 90 ◦ ). Collected luminescencefor θ d = 0 ◦ (red squares) and θ d = 90 ◦ (blue circles) vs. in-jected particle density in µm − per pulse per QW. A clearthreshold is observed at 5 × µm − . (b-c) Degree of po-larization measurement for (b) θ p = 90 ◦ linear pumping and(c) left circularly polarized pumping. Blue circles: S , greendiamonds: S , red squares: S defined in eq. (1). (d-f) Cal-culated polarization parameters from the measurement of theStokes parameters for linear pumping (eqs. (1-2)). (d) DOLP.(e) Angle for major axis of linear polarization ψ relative to θ p (f) Degree of circular polarization ( S ). We record the far field spectra for varying pumpingpower and polarization angles θ p and θ d , and sum theintensities inside the area | k x | < . µm − (correspond-ing to 4 ◦ ). The observed normalized intensities I θ d areonly weakly dependent on the choice of this area, andare shown in Fig. 2. In Fig. 2(a) we plot the measuredluminescence intensity for linearly polarized light along θ d = 0 ◦ and θ d = 90 ◦ as a function of pumping powerin units of the generated polariton density per pulse perQW. The pump is horizontally polarized ( θ p = 90 ◦ ). Thedata show a non-linear increase above a threshold densityof ∼ µm − , which marks the onset of condensation.By measuring all six intensities required by equation (1),we calculate the three Stokes parameters. The resultsfor this pump polarization ( θ p = 90 ◦ ) are plotted in Fig.2(b) along with the theoretical curves, to be discussed inthe next section.For circularly polarized pump (Fig. 2(c)) and well S S S (a) n A B (b)
FIG. 3: (color online) (a) The path followed by the polariza-tion vector for increasing excitation power ( θ p = 90 ◦ linearlypolarized pumping). The projections on the three normalplanes are shown with colored dots, the color scale corre-sponding to the injected polariton density n in µm − perpulse per quantum well (QW), as shown in the colorbar. (b)Schematic of the proposed relaxation mechanism: the opti-cally excited polariton A, first looses energy by phonon scat-tering, then scatters with one other polariton (B) and popu-lates the ground state. above threshold, the signal is perfectly circularly polar-ized, up to − . ∼ ps ), which is shorter than the spin relax-ation time. The negative sign of S means that the an-gular momentum of the emitted photons along the z-axisis the same as that of the optically injected exciton po-laritons, since we pump and detect from the same side ofthe sample (Fig. 1(a)).We next focus on linearly polarized pumping and varythe direction of linear polarization for the pump ( θ p ).Above threshold, a non-zero degree of linear polarizationdevelops (Fig. 2(d)), while the polarization direction isrotated by ∼ ◦ compared to the pump (Fig. 2(e)).Also, a circularly polarized component emerges, with S changing sign for varying θ p (Fig. 2(f)). The sign changeis correlated with the deviation of ψ − θ p from 90 ◦ . Thepath followed by the polarization vector for increasingpower and θ p = 90 ◦ linearly polarized pumping is plottedin Fig. 3(a). IV. THEORETICAL MODEL
To interpret these results we have used a simplifiedmodel based on the spin-dependent Boltzmann equationsfor polaritons in microcavities . Our model is basedupon two states, representing the condensate and reser-voir, each characterized by a 2 × α (positive), is believed to bemuch greater in magnitude than that in antiparallelconfiguration, α (negative). Therefore, calculating thetransition rates we keep only terms ∝ α and the inter-ference terms ∝ α α . We assume the reservoir is quicklypopulated by the pump from fast polariton-phonon relax-ation. Then we consider the polariton-polariton scatter-ing processes, which populate the condensate (Fig. 3(b)).The spin-anisotropy of the polariton-polariton inter-actions gives rise to two important effects. First, a90 ◦ rotation of the linear polarization appears upon onepolariton-polariton scattering, which has been evidencedin parametric oscillator experiments in magic angle aswell as degenerate configurations . This is because ofthe difference between the scattering matrix elements oflinearly polarized polaritons h φ, φ | V | φ, φ i = 12 ( α + α ) , (3) h φ + 90 ◦ , φ + 90 ◦ | V | φ, φ i = 12 ( α − α ) . (4) V is the polariton-polariton interaction operator, and | φ i is the linear superposition √ (cid:0) |↑i + e iφ |↓i (cid:1) of spin-up and spin-down polaritons. We note that if multi-ple polariton-polariton scattering events are involved, theinitial polarization information should be lost.Second, if there is an imbalance of the populations inthe two spin components (in either the condensate or thereservoir) then a self-induced Larmor precession of thecondensate and reservoir Stokes vector occurs. This isbecause of the difference in the polariton-polariton in-teraction energy between the different spin components.This precession becomes faster by increasing the polari-ton population. Therefore, at high pumping rates, thedegree of linear polarization of the luminescence decaysin our time-integrated data (Fig. 2(d)).Other polarization sensitivity derives from an assumedenergy splitting between states linearly polarized at 19 ◦ and 109 ◦ , as is evidenced from Fig. 1(c) and from thelack of circularly polarized component in the lumines-cence for excitation with θ p = 19 ◦ . This splitting causesa rotation of the Stokes vector if the reservoir state isnot an eigenstate with linear polarization of 19 ◦ or 109 ◦ ,which results in non-zero S (Fig. 2(f)). The condensateStokes parameters are time integrated and normalized bythe time integrated condensate population for compari-son to the experimental results.The results of our model are represented by solidlines in Fig. 2. We assumed a condensate lifetime of2 ps , reservoir lifetime of 100 ps , pulse duration of 2 ps , α /α = − . µeV forboth the condensate and reservoir. The final equationsand the value we used for α are provided in the Ap-pendix. The main features of our experimental resultsare explained within this model. V. RELAXATION BOTTLENECK UNDERCIRCULARLY-POLARIZED PUMPING
In Fig. 4 we compare the FF and NF spectra for twopumping schemes, namely linear ( θ p = 90 ◦ , Fig. 4(a-b)) and left-circular (Fig. 4(c-d)) polarizations. Underlinear pumping, we observe that the linewidth narrowsat threshold, and luminescence is concentrated around k x = 0 and x = 0. For higher excitation power, the × µ m −2 × µ m −2 µ m −2 k x ( µ m −1 ) w ave l e ng t h ( n m ) µ m −2 −2 −1 0 1 2768768.5769769.5770770.5 (a) x ( µ m) −20 0 20 (b) × µ m −2 × µ m −2 µ m −2 k x ( µ m −1 ) µ m −2 −2 −1 0 1 2 (c) x ( µ m) −20 0 20 00.20.40.60.81 (d) FIG. 4: (color online) Far-field (FF) ( k x in µm − vs. wave-length in nm ) and near-field (NF) ( x in µm vs. wavelength in nm ) spectra for various injected particle densities (in µm − per pulse per QW). (a) FF, θ p = 90 ◦ , θ d = 0 ◦ . (b) NF, samepumping-detection scheme. (c) FF, left circular pump, rightcircular detection. Note that the projection of angular mo-mentum along the z-axis (Fig. 1(a)) has the same sign forboth pump and detected photons. (d) NF, same pumping-detection scheme. momentum and position distributions broaden and thecondensate energy blue-shifts. Under circular pumpingand at just above threshold, relaxation bottleneck is ob-served in momentum space at k x ∼ ± . µm − ( ± ◦ in air), while in real space luminescence is concentratedat the center of the excitation spot. This implies thatrelaxation into the zero momentum region is only effi-cient when both spin species are present. For higher ex-citation power, luminescence is mainly observed around k x = 0 and x = 0, similar to the linear pumping case.This result is consistent with previous parametric am-plification experiments , where a suppression of thescattering rate towards the zero-momentum region wasobserved when only one spin species was present.Polariton condensation is a competition between relax-ation and decay from the cavity. Our data suggest thatrelaxation is more efficient in the linearly polarized pump µ m −2 µ m −2 µ m −2 µ m −2 µ m −2 µ m −2 µ m −2 k y ( µ m −1 ) k x ( µ m − ) −2 −1 0 1 2−2−1012 µ m −2 FIG. 5: (color online) Momentum space images for left-circularly polarized pumping and right-circularly polarizeddetection (same scheme as in Fig. 4(c-d)). For increasingpumping power, a ring pattern develops and the images losereflection symmetry. The cyan crosses mark the origin in eachfigure. The pump is incident at ( k x , k y ) = (0 , − µm − . density n ( µ m −2 ) I n t e n s i t y ( a r b i t r a r y un i t s ) o o +45−45RL (b) µ m −2 k x ( µ m −1 ) w ave l e ng t h ( n m ) −2 −1 0 1 2768.5769769.5770 µ m −2 × µ m −2 × µ m −2 (c) −3 −2 −1 0 1 2 310 k x ( µ m −1 ) I n t e n s i t y ( a r b i t r a r y un i t s ) (a) FIG. 6: (color online) (a) The momentum space distributionalong the x-axis for various polariton densities n (in µm − perpulse per QW). The pump is left-circularly polarized, and thedetection right-circularly polarized (same scheme as in Figs.4(c-d) and 5). At n ∼ µm − two peaks appear around k x = ± . µm − , which move towards k x = 0 µm − for in-creasing n . Eventually, a central peak appears and domi-nates the luminescence. (b) Luminescence inside the area | k x | < . µm − for the six different polarization states ofeq. 1 as a function of polariton density under left-circularlypolarized pumping. A stimulation threshold is observed at n ∼ µm − . (c) Far-field (FF) spectra for left-circularlypolarized detection (represented by magenta stars in (b)) forvarious pumping powers. A broad distribution following thelower polariton dispersion is always observed. case, whereas decay is more efficient in the circularly po-larized pump case. On the other hand, our simple two-state model treats the relaxation rate as a free param-eter. Derivation of this rate involves a full many-bodycalculation, where all states in momentum space need tobe considered. A more sophisticated model is thereforeneeded to understand the results of this section. w ave l e ng t h ( n m ) density n ( µ m −2 ) (a) (b) FIG. 7: (color online) (a) The measured spectra near zeromomentum ( | k x | < . µm − ) for linearly polarized pump-ing ( θ p = 90 ◦ , θ d = 0 ◦ ) as a function of polariton density.(b) Same spectra for left-circularly polarized pump and right-circularly polarized detection. The inefficient cooling for the circular pumping case isfurther evidenced in the FF images presented in Fig. 5 forvarious pumping powers. Above threshold, they do notpossess the k y ↔ − k y reflection symmetry. The laserpump is incident at ( k x , k y ) = (0 , − µm − , so the po-lariton distribution is shifted towards the source. On thecontrary, under linearly polarized pumping the momen-tum space distribution is always spherically symmetric.Detailed data of the momentum space distribution alongthe x − axis for increasing pumping power are shown inFig. 6(a). The cross-circularly polarized component ismuch weaker above a threshold pumping power, as shownin Fig. 6(b)), and does not condense (Fig. 6(c)).Fig. 7(a) shows the measured spectra near zero mo-mentum ( | k x | < . µm − ) for linearly polarized pump-ing ( θ p = 90 ◦ , θ d = 0 ◦ ). We observe a linewidth decreaseand blue shift just above threshold. We note that theobserved energy shift shows an almost logarithmic in-crease as a function of pumping power, similar to Ref.35. From a polariton-polariton interaction point of view,a linear increase would be expected. Fig. 7(b) showsthe same spectra for left-circularly polarized pump andright-circularly polarized detection. We observe a simi-lar blue shift, but no linewidth narrowing. The reasonfor the different spectral linewidths is not well under-stood. It might indicate that the temporal coherence isnot necessarily enhanced with increasing accumulation ofpolaritons near the zero in-plane momentum. VI. CONCLUSIONS
In conclusion, we studied polarization-dependent lumi-nescence from an exciton polariton system as a functionof pump power and polarization in a non-resonant pump-ing geometry. Spin-dependent polariton-polariton inter-action manifests itself in the rotation of the linear polar-ization axis by ∼ ◦ under linearly polarized pumping.This can be understood in terms of a two-state model,suggesting that polaritons populate the condensate af-ter multiple phonon scatterings and only one polariton-polariton scattering. In addition, when only one spinspecies is injected, we observed a relaxation bottleneck.This phenomenon is typically attributed to inefficient re-laxation, leading to photon leakage from the cavity beforepolaritons reach the zero-momentum region. Full deter-mination of the polarization of polariton condensates re-veals that the spin degree of freedom plays an importantrole in understanding the relaxation mechanism of mi-crocavity exciton polaritons. Acknowledgments
G.R. acknowledges support from JST/SORST andSpecial Coordination Funds for Promoting Science andTechnology. T.C.H.L., Y.G.R., and A.V.K. would like tothank E.P.S.R.C. for financial support. A.V.K. thanksIvan Shelykh for useful comments.
APPENDIX
Here we present the equations used in the theoreti-cal model of Section IV. The approach we have takenis based on the spin-dependent Boltzmann equations forexciton-polaritons in microcavities of Ref. 16. We haveconsidered two states, reservoir and condensate, eachcharacterised by a 2 × (cid:20) R ↑ ( R x − iR y )( R x + iR y ) R ↓ (cid:21) , (cid:20) N ↑ ( S x − iS y )( S x + iS y ) N ↓ (cid:21) . (A.1)Here R ↑ and R ↓ are the reservoir populations for spin-upand spin-down polaritons, R x and R y are the pseudospincomponents that characterize the linear polarization de-gree measured in the horizontal-vertical and diagonal ba-sis, respectively. The circularly polarized component R z of reservoir pseudospin is R z = ( R ↑ − R ↓ ) /
2. The cor-responding numbers for the condensate are given by N ↑ , N ↓ , S x , S y , and S z = ( N ↑ − N ↓ ) / P ↑ , P ↓ , P x , and P y describe the pump. For example, for TE pumping( θ p = 0 ◦ ), we have P ↑ = P ↓ = P x . The full rate equa-tions we used are as follows,d N ↑ d t = − Γ N ↑ + ( ω x S y − ω y S x ) + W R ↑ ( N ↑ + 1) , (A.2)d N ↓ d t = − Γ N ↓ − ( ω x S y − ω y S x ) + W R ↓ ( N ↓ + 1) , (A.3)d S x d t = − Γ S x + ω y S z − ( α − α ) ~ ( S z + R z ) S y + W (cid:0) R ↑ + R ↓ (cid:1) S x + W α α ( R ↑ + R ↓ ) ( N ↑ + N ↓ + 2) R x , (A.4) d S y d t = − Γ S y − ω x S z + ( α − α ) ~ ( S z + R z ) S x + W (cid:0) R ↑ + R ↓ (cid:1) S y + W α α ( R ↑ + R ↓ ) ( N ↑ + N ↓ + 2) R y , (A.5)d R ↑ d t = − γR ↑ + (Ω x R y − Ω y R x ) − W R ↑ ( N ↑ + 1)+ P ↑ , (A.6)d R ↓ d t = − γR ↓ − (Ω x R y − Ω y R x ) − W R ↓ ( N ↓ + 1)+ P ↓ , (A.7)d R x d t = − γR x + Ω y R z − ( α − α ) ~ ( S z + R z ) R y − W N ↑ + 1) R ↑ + ( N ↓ + 1) R ↓ ] R x + P x , (A.8)d R y d t = − γR y − Ω x R z + ( α − α ) ~ ( S z + R z ) R x − W N ↑ + 1) R ↑ + ( N ↓ + 1) R ↓ ] R y + P y . (A.9)Here ω x,y and Ω x,y are the Larmor frequencies corre-sponding to the effective magnetic field due to the polar-ization splitting. ω x,y refer to the condensate and Ω x,y refer to the reservoir. Γ and γ are the decay rates forthe condensate and reservoir, respectively. As discussedin Section IV, we use the values ω x = Ω x = 50 µ eV ~ cos (2 × ◦ ) ,ω y = Ω y = 50 µ eV ~ sin (2 × ◦ ) , Γ = 0 . − , γ = 0 .
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