Spin Domains in One-Dimensional Conservative Polariton Solitons

: We report stable orthogonally polarized domains in high-density polariton solitons propagating in a semiconductor microcavity wire. This e ﬀ ect arises from spin-dependent polariton − polariton interactions and pump-induced imbalance of polariton spin populations. The interactions result in an e ﬀ ective magnetic ﬁ eld acting on polariton spin across the soliton pro ﬁ le, leading to the formation of polarization domains. Our experimental ﬁ ndings are in excellent agreement with theoretical modeling taking into account these e ﬀ ects.

T emporal and/or spatial domains of coupled multiple dipoles play a significant role in the properties of various physical systems. Magnetic domains in (anti)ferromagnets have been widely studied to date and are utilized in modern memory devices (hard drives). Domain formation with electric dipoles has also been observed in atomic spinor Bose−Einstein condensates. 1 In optics, polarization domain formation is governed by modulation instability with the resultant separation of adjacent domains by a domain wall, a topological defect closely linked to soliton formation. 2−4 While scalar nonlinear effects related to solitons 5,6 (or optical supercontinuum 7 ) have been studied extensively, little attention has been given to the spatiotemporal evolution of the polarization (or spin) degree of freedom in vectorial optical structures. Only recently have timelocalized polarization rotations or polarization domain walls been reported for light traveling in nonlinear optical fibers 8 and for dissipative solitons in vertical-cavity surface-emitting lasers. 9 In this paper we study spatiotemporal polarization domains in a system of exciton−polaritons in semiconductor microcavities. In such highly nonlinear systems observation of a variety of quantum fluid phenomena, 10,11 Bose−Einstein condensation, 12 Berezinskii−Kostelitz−Thouless phases, superfluidity, and dark and bright solitons 13 have all been reported.
Polaritons are characterized by two possible spin projections on the structure growth axis, which correspond to two opposite circular polarizations. 14 Efficient external control of the polariton spin makes microcavity-based structures promising building blocks for spin-optronic devices, i.e., optical equivalents of spintronic devices. 15 Moreover, due to the exchange terms that dominate the exciton−exciton scattering, 16 polariton− polariton interactions are strongly spin anisotropic: polaritons with the same spin projections strongly repel each other, while polaritons with parallel spins interact more weakly and sometimes even attract each other. 17,18 The interplay between spin dynamics and polariton−polariton interactions leads to a wide variety of nonlinear polarization phenomena in microcavities. This includes spin switching, 19 spin-selective filtering, 20 polarization-dependent stability of dissipative solitons, 21 optical analogues of magnetically ordered states, 22,23 dark half solitons, 24 and spin and half vortices. 25,26 Here we demonstrate formation of polariton spin domains within high-density wavepackets evolving into conservative soliton(s) in a quasi-1D system. Experimentally we resolve the full Stokes polarization vector of the propagating wavepackets under a wide range of excitation powers and develop a theoretical model that we validate through numerical simulations, qualitatively reproducing the experimental results. The experiments were performed in a microcavity wire (MCW): 5 μm wide and 1 mm long mesas etched from planar microcavities. Thanks to the 1D nature of polariton wires and the long polariton lifetimes (∼30 ps), we could reach polariton densities sufficient for formation of solitons. 27 At low excitation powers we observe polarization precession caused by a linear in-plane effective magnetic field inside the sample. At higher excitation powers the high polariton density and an imbalance of polariton spin populations lead to an extra out-of-plane magnetic field. This nonlinear field causes, first, fast polarization oscillations and then the formation of spin domains.

■ RESULTS
We performed our experiments on a 3λ/2 microcavity composed of three embedded InGaAs quantum wells (10 nm thick, 4% indium) and GaAs/AlGaAs (85% Al) distributed Bragg mirrors with 26 (23) repeats on the bottom (top) mirror. The Rabi splitting and the polariton lifetime are ∼4.12 meV and ∼30 ps in this sample, which was previously described in refs 27 and 28. The detuning between the exciton and the photon modes is ca. −2 meV. The top mirror was partially etched (down to the last few layers of the top distributed Bragg reflection (DBR)) defining 1000 μm long mesas (MCWs) of different widths. In all our measurements we used the same 5 μm wide MCW as in ref 27.
We excited the sample using a TE-polarized pulsed (∼5 ps full width at half-maxima, fwhm) laser quasi-resonant with the lower polariton branch corresponding to the ground MCW mode. The angle of incidence was k x ≃ 2.2 μm −1 , at which the effective polariton mass is negative, favoring formation of solitons. 27 We employed transmission geometry, where we applied the excitation beam on one side of the cavity (bottom) and collected emission on the opposite side (top) to avoid the reflected pump beam saturating our detectors. We then changed the laser power and measured the full Stokes vector of the emitted light as a function of time and propagation coordinate, x, for each excitation power. In the experiment we separately detected emission intensity in six polarizations: horizontal (I h ), vertical (I v ), diagonal (I d ), antidiagonal (I ad ), and right-hand (I σ + ) and left-hand (I σ − ) circularly polarized. In the circular polarization basis of the cavity light {ψ + ; ψ − }, the total intensity is written S 0 = |ψ + | 2 + |ψ − | 2 and the Stokes vector (normalized to unity) S* =(S 1 *, S 2 *, S 3 *) is given by At the lowest excitation power, P =1 3μW, polariton nonlinearity is very weak and the wavepacket propagates in the linear regime, experiencing dispersive spreading ( Figure 4 in the SI). At higher power, P =8 7μW, a soliton is formed, characterized by nonspreading propagation 27,29 as shown in Figure 2a. Both circular and linear polarization degrees experience oscillations between negative and positive values with time as shown in the S 1 * and S 3 * components (Figure 2b,c) and the S 2 * component ( Figure 3 of the SI). We note that even though the pump beam is TE-polarized, the polarization of the emission at time t = 0 has some diagonal and circular components, likely due to birefringence in the substrate and the influence of the edges of the MC wire on the polarization of the pump field before it couples to the polariton field inside the wire.
The polarization beats observed in Figure 2b,c correspond to motion of the Stokes vector S* (which is also referred to as the polariton pseudospin or spin) around the unit Poincarésphere, as shown in Figure 2d. where we plot the trajectory that the tip of the normalized Stokes vector follows as the soliton propagates. The direction of the Stokes vector is constructed by measuring the values of the Stokes components at different times at the spatial points of the soliton profile shown by the green lines in Figure 2a−c. The Stokes vector clearly precesses around the sphere as the soliton propagates. We note that at the lowest excitation power (P =1 3μW) the polariton wavepacket does not propagate long distances due to fast spreading, and the onset of the polarization beats and Stokes vector precession are barely visible (see Figure 3 of the SI).
Such motion of the Stokes vector around the Poincarésphere can be described mathematically as a precession of the polariton pseudospin around a time-varying effective magnetic field. 30 Physically, this effective magnetic field arises from three mechanisms: TE−TM (V−H in the laboratory frame) splitting of polaritons propagating with nonzero momenta; 24,31 strain, electronic anisotropy, or anisotropy related to crystallographic lattices 32,33 inducing splitting between D and AD polarized components; and polariton−polariton interactions. The last mechanism is weak at small powers (P =8 7μW), but starts playing a critical role at higher polariton densities (P > 0.95 mW), as we discuss below in the modeling section. The  Figure 3d) with a period of T ≃ 10 ps, corresponding to an out-of-plane increased effective magnetic field due to the spin-dependent polariton nonlinearity which induces an energy splitting ΔE eff ≃ 130 μeV. As the polariton density and hence nonlinearity are reduced at times t > 50 ps the precession occurs at slower speed with a period of ∼100−125 ps. In this time range the Stokes vectors taken at positions of trace 1 and trace 2 precess in the south and north Poincaréhemispheres, respectively, remaining almost orthogonal.
At the highest pump power, P = 3.1 mW (see Figure 4), a further onset of cascaded polariton−polariton scattering leads to the occupation of states on the lower polariton branch at energies below the pump energy in a process resembling the optical continuum generation as observed and discussed in ref 27 for the same MC wire polariton system. In such a process polariton relaxation results in a significant occupation of the states described by positive effective mass, 27 this results in both coexisting solitons and a spreading dispersive polariton wavepacket with time. At such high polariton density the nonlinearity results in more complex spatiotemporal polarization dynamics, which is shown in Figure 4. The domains with negative and positive circular polarization degree remain for the first ∼50−70 ps as seen for the S 3 * component in Figure 4c. However, at later times when the nonlinearity becomes weaker, the wavepacket breaks into a set of domains with different polarizations, the Stokes vectors of which evolve in time in a complex manner as shown in Figure 4d and e. This temporal evolution originates from the time-and space-dependent out-ofplane effective magnetic field due to the spin-dependent polariton nonlinearity.

■ MODEL
In order to describe the polarization dynamics of the nonlinear polariton wavepacket and understand the physical mechanisms responsible for the observed behavior, we work with the coupled spinor macroscopic cavity photon field ψ =( ψ + , ψ − ) T and exciton wave function χ =( χ + , χ − ) T written in the circular polarization basis. 34 The TE−TM splitting of the photon modes is described by the operator Σ k,± = β(k x ± ik y ) 2 written in reciprocal space 35,36 and characterized by a splitting constant β.
Propagation distance of polaritons, as well as the spatial scales of the observed effects, significantly exceed the width of the channel. This allows us to take advantage of the 1D nature of the microwire in order to simplify the formalism. Let us first consider an infinite rectangular potential well width w y where we assume the photon wave function to be in the form Excitons possess a much larger effective mass than cavity photons, and their dispersion can be safely approximated as flat compared to the photon dispersion. Here, m is the effective mass of cavity photons; Δ is the exciton-photon detuning; Γ is cavity photon decay rates corresponding to photons leaking from the cavity; Γ χ is the exciton decay rate corresponding to nonradiative dephasing processes; Σ x = −β(∂ x 2 + π 2 /w y 2 ) is the real-space TE− TM operator along the microwire; δ describes a fixed splitting in the diagonal polarizations that arises from the optical and electronic anisotropy in a microcavity; 32,33 and ℏΩ is the Rabi splitting giving rise to the exciton−polariton eigenmodes of the system. The parameters α 1 and α 2 are the interaction constants in the triplet configuration (parallel spins) and in the singlet configuration (opposite spins), respectively. The last term in eq 4 describes the resonant optical pumping of the lower branch polaritons where E ± describes the pump pulse amplitude and phase; w x and w t are the spatial and temporal pulse width, respectively; ℏω p is the pump energy; and k p is the pulse wavevector along the wire.
It is instructive to describe the polarization effects in the microwire in terms of an effective magnetic field which acts on the Stokes vector. 37 The magnetic field vector can be written, in units of energy, as Ω =(Ω x , Ω y , Ω z ) where Ω x = β(k x 2 − π 2 /w y 2 ), Ω y = δ, and Ω z =( α 1 − α 2 )(|χ + | 2 − |χ − | 2 ). The last term is a consequence of the nonlinear interactions between polaritons

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Article giving rise to effective Zeeman splitting when the spin populations are imbalanced.
For modeling, we take the following values of the parameters: β =12μeV μm 2 ; δ l =20μeV; α 1 =2μeV μm; α 2 = −0.1α 1 , Γ =1/ 30 ps −1 , Γ χ =2Γ; m =5× 10 −5 m 0 , where m 0 is the free electron mass; Δ = −2 meV; ℏΩ = 4.12 meV; ℏω p = −1.3 meV; k p = 2.2 μm −1 ; w x and w t were set to have fwhms of 20 μm and 5 ps, respectively. The initial polarization of the resonant laser is chosen to fit the experimental results: E + = E 0 and E − = 2.65e iϕ E 0 where ϕ = 1.9 and E 0 denotes the overall excitation amplitude of the beam.  (Figure 2b,c). We note that in the weak pulse regime the effect of TE−TM splitting does not account for the appearance of oscillations in the linear polarization S 1 * component since the effective magnetic field is oriented in the x-direction. The oscillations of S 1 * emerge when the additional splitting δ between the diagonal polarizations is accounted for in the microwire system.  Figure 5e,h, again as in the experiment (see Figure 3b). The reason for this effect is that the Ω z component decreases as the polariton intensity decays, becoming small or comparable to the in-plane magnetic field (Ω x , Ω y ) around ∼50 ps. Consequently, the rotation of the S 1 * Stokes vector component halts and long streaks are formed in the microwire.
At the low excitation power in Figure 5a−c the effective, interaction-induced, z-magnetic field has its maximum absolute value of Ω z ≈ 0.01 meV at around 5 ps after the pulse. The field is small and the polarization precession is mainly governed by splitting between linearly polarized components. Increasing the excitation power, the Ω z component becomes significant, leading to fast polarization changes.

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Article Cherenkov radiation is also observed before the pulse separation (Figure 5d We finally point out that although the polariton nonlinearity plays a crucial role in the spin domain formation, the formation of solitons in the system is only weakly influenced by the presence of effective magnetic fields from various polarization splitting mechanisms. We find that approximately the same spatial patterns of solitons appear in the intensity of scalar wave functions and the total intensity of the spinor wave functions considered here (see Section B in the SI). Thus, the initial formation of the solitons at the pump spot (e.g., the pattern of total intensity summed over all polarizations, which leads to nondispersive propagation) is dominantly a scalar effect. Separately, the nonlinear pseudomagnetic field leads to the formation of the polarization domain pattern on top of the solitonic total-intensity pattern.

■ CONCLUSION
We have observed nonlinear spin dynamics of polariton wavepackets in a microcavity wire. At low excitation just above the threshold of soliton formation the polarization precession during the propagation is caused by an effective magnetic field due to splittings between linearly polarized polariton components. With increasing polariton density an extra pump-induced out of plane magnetic field due to spin imbalance in the initial polariton wavepacket results in formation of spatially separated polarization domains. While soliton formation is largely a scalar effect (not dependent on polarization), it likely helps to preserve spin domains over the propagation distance due to the nonspreading nature of solitons; hence polarization domains experience little dispersion.
Polariton wires considered in this work can represent building blocks for future polaritonic devices, such as polariton-based spin transistors 38,39 or soliton-based logic gates. 39 In the context of the recent demonstration of bright temporal conservative solitons in such a system 27 polarization domains may be used for the nonbinary information encoding and transfer. This observation also opens a possibility for further fundamental studies of the system, such as description of domain wall formation mechanisms. It would be also interesting to investigate polarization dynamics of bright solitons in high velocity thick waveguide (>1 μm) systems, 40 where the splitting between linearly polarized components is comparable to polariton blue-shifts. Recently, waveguide polaritons have been also reported in a screening-resistant GaN system, 41 which may potentially work at 300 K, paving the way toward polaritonic device applications.

* S Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.8b01410.
(1) Numerical simulations demonstrating more clearly the effects of the polariton−polariton interaction induced out-of-plane effective magnetic field on the wavepacket; (2) comparison of scalar and spinor polariton theory on soliton formation; (3) higher resolution experimental images (PDF)

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