Twist of generalized skyrmions and spin vortices in a polariton superfluid

Significance

Spin vortices and skyrmions are topological states in exotic phases of matter such as superconductors and superfluids. In these complex states, the spin and orbital angular momentum are both quantized and mixed, resulting in variegated polarization textures. Here we create these composite states in a fluid made of light and matter quasi-particles, polaritons, and demonstrate their reshaping in ultrafast time scales. To describe these dynamics we propose the idea of generalized vortices, a fundamental concept in topology that can be relevant also for applications in the fields of quantum fluids, spintronics, and optical polarization shaping.

Abstract

We study the spin vortices and skyrmions coherently imprinted into an exciton–polariton condensate on a planar semiconductor microcavity. We demonstrate that the presence of a polarization anisotropy can induce a complex dynamics of these structured topologies, leading to the twist of their circuitation on the Poincaré sphere of polarizations. The theoretical description of the results carries the concept of generalized quantum vortices in two-component superfluids, which are conformal with polarization loops around an arbitrary axis in the pseudospin space.

Topological defects represent a wide class of objects relevant to different fields of physics from condensed matter to cosmology. The universality of monopoles, vortices, skyrmions, domain walls, and of their formation processes in different systems, has largely motivated their study in the condensed matter context. In particular, the interplay between the symmetry breaking in phase transitions and the formation of topological defects has been the focus of intensive research in the last century. In high-energy physics, the existence of an isolated point source intrigued a great number of physicists (1). Dirac was “surprised if Nature had made no use of it” and postulated the possibility of the magnetic monopoles linked to the quantization of electric charge (2). However, the elusiveness of their observation in free space has motivated an extensive study of monopole analogues in the form of quasiparticles in many-body systems (3) such as the exotic spin ices (4, 5), liquid crystals (6), exciton–polariton (7), and rubidium Bose–Einstein condensates (BECs) (8, 9), as well as other systems (10, 11). In a 2D multicomponent BEC, an equivalent topological structure to the monopole is given by the hedgehog polarization vortex (12), which together with the hyperspin vortex (13) belongs to the class of spin vortices that when combined lead to a well-defined polarization pattern. Such topological states are characterized by a linear polarization vector which rotates an integer number of times (spin winding number) around a singular central point, in a way analogous to what the magnetization does in the spin vortices of a ferromagnetic spinor BEC (14).

In this work we excite complex vortex states in an exciton–polariton superfluid and study their amplitude, phase, and polarization dynamics. We demonstrate that temporal evolution of such topologies leads, in general, to a twist of the polarization plane in the Poincaré space. The observed features of the vortex behavior are explained within the concept of generalized spin vortices, where the rotation of polarization at large distances occurs around an arbitrary axis on the Poincaré sphere.

Polaritons emerge in planar semiconductor microcavities as eigenmodes of the strong coupling regime between the exciton resonance and the photon cavity mode, combining the properties of light and matter. The photons confer on polaritons a very small effective mass (${10}^{-4}-{10}^{-5}$ of free-electron mass), which, together with nonlinear interactions due to the excitons, leads to effective condensation at a relatively high temperature (up to room temperature for given materials such as ZnO, GaN, or organic dyes). Features related to superfluidity have been observed such as the suppression of scattering from defects (zero viscosity) (15, 16) or persistence of vortex currents (17). Moreover, polaritons with the pseudospin, given by the possibility of polarizing their state, open the opportunity to study condensates with an internal angular momentum degree of freedom, easily detected by optical means thanks to their photonic outcoupling features (18).

In spinor superfluids, the spin degrees of freedom allow for different composite topologies that emerge as the superposition of quantized vortex states. In superfluids with unrestricted geometry, these elementary vortex blocks are half-quantum vortices (HQVs) (3). In exciton–polariton condensates, the HQV is characterized by a $\pi$ phase rotation accompanied by a $\pi$ linear polarization rotation in such an elegant way that the two combine together to ensure the global continuity of the spinor wave function (19–22). For finite-size condensates, where the boundary condition at large distances is not fixed, an HQV can be transformed into a skyrmion. The skyrmions possess a specific circumference of full linear polarization and they are fingerprinted by the inversion of the sign of circular polarization degree when crossing this circumference in the radial direction.

However, the dynamics of the pseudospin vector in semiconductor microcavities is related to the presence of spin-orbital-like coupling, namely, the transverse-electric–transverse-magnetic (TE-TM) splitting of the modes, which is manifested by the optical spin Hall effect (23–25). The TE-TM splitting is often represented by means of an effective magnetic field that produces a precession of the pseudospin vector, which leads to different sectors in circularly polarized states both for real and momentum space, even when starting with a homogeneously polarized field (26, 27).

Here we are able to initialize the polariton condensate with nontrivial pseudospin patterns. We study the dynamics of exotic topologies such as the lemon and the star skyrmion, the hedgehog, and the hyperspin vortex, in clean regions of the sample, deriving universal observations not linked to specific local disorder/defects pinning (13) or to the effect of sample architecture/confinement (28). The resultant topologies allow us to extend the concept of quantum vortices into a wider class that includes states as generalized skyrmions and spin vortices. These observations subtend the potentialities of resonant excitation of spin and orbital angular momentum states on microcavity (MC) polariton fluids, and of their full control using TE-TM or anisotropy splitting, which is of fundamental importance in the fields of spintronics and polarization shaping.

Setting Up Skyrmions and Spin Vortices

Advanced phase shaping was recently obtained by means of anisotropic and inhomogeneous liquid-crystal devices called $q$-plates (29), which allows an extensive investigation of optical vorticity and of full- and half-quantum vortex dynamics in polariton condensates (22). Phase shaping is applied to an initial Laguerre–Gauss $L{G}_{00}$ laser pulse (4-ps duration and 0.5-nm bandwidth) by sending it across the $q$-plate carrying unitary topological charge. The $L{G}_{00}$ is hence partially or completely transformed into a unitary winding state breaking the chiral symmetry between the two spin populations. Upon proper setting of the incoming/outgoing polarization and tuning of the $q$-plate, we can prepare the specific combination and resultant field pattern (see Supporting Information for experimental details). Indeed, each skyrmion and spin vortex shown here can be thought of as a composite state resulting from the specific superposition of two $LG$ beams with integer phase winding ($L{G}_{0,-1/0/+1}$), one in each of the two spin components.

Projected onto the circular polarization basis, the skyrmions are characterized by the presence of an integer phase winding (orbital angular momentum) in one of the spin components and a zero winding in the opposite one. The resultant vectorial field exhibits an inhomogeneous pattern comprising all polarization states, as is typical of full Poincaré beams. There exists a circle line in real space featuring linear polarization states ($l$-line, at $r=r_l$), which maps to the equatorial loop of the Poincaré sphere (Fig. 1C). The points inside or outside of the circle are associated with either one or the other hemisphere of the sphere. According to the skyrmion definition, the pseudospin vector flips from right-circular at the core to left-circular (or vice versa) at its boundary ($r\approx 2r_l$). Hence, the real-space radius maps to a given meridian on the sphere, and the meridian angle is then associated to the azimuthal real-space angle. The skyrmion polarization field in the region $r<r_l$ covers only one hemisphere of the Poincaré sphere, and it can be mapped to the polarization field of an infinite-size HQV (19). Therefore, similarly to the HQVs (12), the skyrmions can be characterized by two distinct geometries: lemon-like (Fig. 1A) and star-like (Fig. 1B) skyrmions.

Fig. 1.

Generation of vortices. Polarization fields map in real space relating to skyrmion (A and B) and spin (D and E) vortices. The skyrmion covers all of the polarization states (full Poincaré patterns), giving rise to lemon-like (A) and star-like (B) patterns. Hedgehog (D) and hyperspin (E) feature a linear polarization in the whole space, following the radial direction and a hyperbolic pattern, respectively. The circles in the real-space maps (which represent the $l$-line in the case of the skyrmions) correspond to the equatorial line in the polarization spheres, with the associated direction for the different states. The colors in the polarization field maps, blue and red, are associated to either the ${\sigma }_{+}$ ($R$, right) or ${\sigma }_{-}$ ($L$, left) degree of circular polarization, respectively, and black is associated to the linear polarization. (C and F). Conformal mapping onto the Poincaré sphere of the real-space circle line around the vortex cores. Spin vortices follow a double rotation in the Poincaré space, skyrmions a single one. (G–I) Experimental initial Laguerre–Gauss states of the polariton population with (J–L) the associated phase maps, respectively $L{G}_{0-1}$ (G and J), $L{G}_{00}$ (H and K), and $L{G}_{0+1}$ (I and L). (M) The table summarizes all of the possible combinations of $LGs$ in the two cross-spin polarizations of the system leading to the different states of quantum vortices: phase vortices (also known as full-vortices), hedgehog (Hed), hyperspin (Hyp), and lemon and star skyrmions.

In a spin vortex, the two spin components feature counterrotating phase windings (Fig. 1 D and E) (13). The central phase singularities in the two spin populations convert into a polarization singularity at the core. There are two principal types of polarization vortices: the hedgehog with a purely radial direction of polarization (Fig. 1D) and the hyperspin vortex, characterized by a hyperbolic polarization pattern (Fig. 1E). Upon changing the phase delay between the two spins, the hedgehog can transform into an azimuthal polarization pattern, whereas the hyperspin undergoes a texture rotation. These vortex states can be described by an equivalent form when conformally mapping them to points on the Poincaré sphere: The circulation along every circle centered at the vortex core in the real space can be associated with a closed double loop lying in the equatorial plane (Fig. 1F) of the pseudospin space. To make the classification clearer, in the third and fourth rows of Fig. 1 we show examples of the density and phase profiles of the fundamental building blocks of all type of vortices in circular polarization basis: clockwise $L{G}_{0-1}$ (Fig. 1, G and J), zero-winding $L{G}_{00}$ (Fig. 1, H and K), and counterclockwise $L{G}_{0+1}$ (Fig. 1, I and L) states. Different combinations of the three $LG\text{s}$ in the two circular polarizations ${\sigma }_{+}$ and ${\sigma }_{-}$ (the two opposite pseudospins) leading to skyrmion, spin and phase vortices are shown in Fig. 1M.

These photonic states are set as the initial conditions of the polaritonic population dynamics by resonant excitation on the MC sample, at the energy of the lower polariton branch (LPB). On the detection side, we extract both the instantaneous local density and the phase of the polariton emission during its time evolution, by means of an interferometric setup performing real-time digital Fourier transform and off-axis selection (30–32). Using polarization filtering in detection, we can project the single components in each of the three polarization bases: the linear horizontal-vertical ($H$-$V$), diagonal-antidiagonal ($D$-$A$), and circular right-left ($R$-$L$). From these measurements it is possible to retrieve the full map of the different degrees of polarization and Stokes parameters $S_{\mathrm{1,2,3}}$, respectively, which are used to plot the resultant polarization vector field, and to associate every point in real space to the pseudospin space (33) (Supporting Information).

Twist of the Vortex Polarization Field

The variation of the $S_{\mathrm{1,2,3}}$ parameters in real space for the hyperspin polarization vortex is presented in the first row of Fig. 2 A–C. These plots relate to the emission from the polariton condensate at the initial time (after the laser pulse has arrived), and the degrees of polarizations are clearly mapped also in the regions of weak or null intensity, such as in the center of a vortex. The twofold symmetry ($C_2$ symmetry) of the $S_1$ (Fig. 2A) and $S_2$ (Fig. 2B) results in a petal shape of the polarization distribution, whereas $S_3$ is ∼0 over all space, as expected. The presence of a very small component in the $S_3$ can be attributed to the $q$-plate device, which cannot be simultaneously tuned at all of the wavelengths composing the pulsed beam. However, the circular degree of polarization is far weaker than the linear components of the $S_1$ and $S_2$ parameters.

Fig. 2.

Stokes maps. (A–C) Polarization maps of the polariton hyperspin vortex at the initial time, given as a projection onto the three Stokes bases, horizontal-vertical ($S_1$, A) diagonal-antidiagonal ($S_2$, B), and circular right-left ($S_3$, C). (D) Time evolution of the circular degree of polarization, plotted as $S_3$ angular profiles around the vortex core at different times. (E–G) $S_1$, $S_2$, and $S_3$ maps of the lemon-like skyrmion at initial time and (H) associated time evolution of the spin degree profile $S_3$. All of the profiles have been taken along the black/white circle plotted in the real-space maps, which represent the $l$-line of the skyrmion.

The most pronounced effect appears in the time evolution of the spin degree: The $S_3$ component increases during the system evolution, becoming almost as large as the linear degree of polarization. This effect is clearly visible in Fig. 2D, where we show the azimuthal profile of $S_3$ (taken along the gray circle in Fig. 2C) whose sinusoidal modulation increases with time. Similar dynamics are observed also when starting with a skyrmion state. Fig. 2 E–G shows, for the skyrmion configuration, the associated $S_{\mathrm{1,2,3}}$ maps with a lowered symmetry in the $S_1$ and $S_2$ linear degrees and a concentric distribution for $S_3$, which changes from $-1$ to $+1$ from the center outward. We plot the azimuthal profile of $S_3$ along the $l$-line of purely linear polarizations (gray circle in Fig. 2G), at different times, in Fig. 2H (see also Movie S1). Although the circular degree of polarization is approximately zero at the initial time, during time evolution an increasing imbalance of right and left spin polarizations develops. Also, in this skyrmion case, the profile assumes a sinusoidal modulation growing in amplitude, and rising even larger than in the case of the polarization vortices. We checked that this effect is not due to a real-space movement of the whole topological state with respect to the initial circle. Indeed, the phase singularities (vortex cores), which can be tracked for each spin component possessing a nonzero phase winding (either two phase singularities for the polarization vortices or one for the skyrmions), remain quite stable during the whole dynamics with just a few micrometers displacement even after 45 ps (see also Movie S2).

The same effects are observed when starting with a hedgehog vortex and with a star skyrmion. In particular, in Fig. 3 we plot the full polarization vectors in real space, retrieved from the $S_{\mathrm{1,2,3}}$ maps. The first row shows the polarization vector for the hedgehog at three different time frames. At the initial time, Fig. 3A, the field pattern follows the classic hedgehog structure schematically introduced in Fig. 1D. The colors used here refer directly to the degree of spin polarization $S_3$. In addition, in Fig. 3D we show the double loop of the pseudospin along the Poincaré sphere. The subsequent dynamics are presented as vector maps in Fig. 3 B and C (see also Movie S3). On the Poincaré sphere, Fig. 3 E and F, we observe a clear twist of the plane containing the double loop away from the equatorial plane, where this effect grows in time (see also Movie S4). The twist angle $\beta$ is directly linked to the maximum degree of circular polarization assumed by the polariton population, as $\beta =\max \text{arcsin}{S}_3$. By sinusoidal fitting of the azimuthal profiles of $S_{\mathrm{1,2,3}}$, we retrieve the trajectory and twist angle $\beta$ of the double loop.

Fig. 3.

Polarization textures. (A–C) Experimental polarization textures of the hedgehog vortex at 20, 45, and 54 ps and (D–F) conformal mapping onto the Poincaré sphere of the real-space profile (solid blue circle line in the both kinds of maps). (G–I) Polarization texture of the star-like skyrmion at 20, 45, and 54 ps and (J–L) associated plotting in the Poincaré space. The solid blue circle represents the initial boundary line between ${\sigma }_{+}$ and ${\sigma }_{-}$ spin domains ($l$-line), and its evolution at later time is reported as a dashed black line. (M) Twisting dynamics, represented by the angle $\beta$ between the plane containing the single and double loops around the polarization sphere and the equatorial plane as a function of time. The time behavior of the total population (gray shaded area) shows that the twisting dynamics are basically independent on the polariton density (straight dashed lines are just a guide for the eye).

Analogous effects are observed for the star skyrmion, whose vector textures are shown in the third row. The experimental map at the initial time, Fig. 3G, is very close to the sketch pattern of Fig. 1B. The polarization reshaping at later times, as shown in Fig. 3 H and I, results in an apparent spin transport with respect to the inner circle, initially containing prevalent positive spin (${\sigma }_{+}$), both outward (from the top-left area) and inward (to the bottom-right part). Overall, by considering the sign of the spin, the entering negative currents (${\sigma }_{-}$) contribute to the net outgoing positive spin flux. We should emphasize that the spin transport is decoupled from the mass transport and from the phase singularity movement. Yet again, it is possible to clearly follow the dynamics on the Poincaré sphere, as in Fig. 3 J–L, where the single loop initially on the equator undergoes a large twist also for this case. A similar reshaping effect has been observed before for the spontaneous hyperbolic spin vortex generated in nonresonant quasi-cw condition by Manni et al. (13). However, in that case the vortex was instead pinned by a defect and the causes for twisting were ascribed to the interplay between the disorder potential and the finite-$k$ TE-TM splitting, relevant due to radial flows of polaritons.

Theory Models and Discussion

To understand the physical origins of our observations we perform numerical modeling of the system’s dynamics using two-component open-dissipative Gross–Pitaevskii equations, which describe the MC photon field and the quantum well (QW) exciton field coupled to each other. The excitonic coupling between differently polarized populations is usually represented by the interspin nonlinearity term (34), although here we are interested in the linear regime. Fundamental to the present work are the terms directly acting on the photonic fields (see Supporting Information for the model details), as discussed in the following. The photonic coupling between different polarizations is given by the finite-$k$ TE-TM splitting term $\chi$ and the $k$-independent anisotropy splitting ${\chi }_0$. The former appears due to the difference of transverse-electric and transverse-magnetic masses of MC modes (35), and the $k$-independent splitting ${\chi }_0$ between linearly polarized modes can be present in some samples due to strain effects (36, 37) and heavy–light hole mixing (38) on the QW interfaces. In such cases two linearly polarized waves with specific polarization directions (say, $x$ and $y$) are subject to a slightly different energy shift, regardless of the direction and of their wavevector. Hence, also the $k=0$ state can be subject to a dephasing between the two linear components, and as a result there could be a precession of an initial polarization state (different from the linear $x$ and $y$ ones) at each point in space.

To reproduce the polarization twisting observed in our experiment we perform different sets of simulations. In the first set we take the $k$-independent anisotropy splitting ${\chi }_0$ to be zero, and in the second set we assume it to be in the ${\chi }_0=0.01\hspace{0.17em}\text{meV}-0.04\hspace{0.17em}\text{meV}$ range, estimated on the basis of polarization-resolved photoluminescence. We see no twist effect in the dynamics when ${\chi }_0=0$ and instead see a significant twist, comparable to experimental, in the simulations with ${\chi }_0$ inside the said range (Fig. 4). The initial $S_{\mathrm{1,2,3}}$ Stokes maps for the spin vortices are reported in Fig. 4 A–C, respectively, for the case of a hedgehog state. Here the initial degree of circular polarization is homogeneously null ($S_3$ map, Fig. 4C). The evolution of $S_{\mathrm{1,2,3}}$ at later times ($t=67\hspace{0.17em}\text{ps}$), presented in the second row of Fig. 4 D–F, shows emergence of a strong circular polarization under the action of a ${\chi }_0\equiv {\chi }_{02}=0.02\hspace{0.17em}\text{meV}$. The $S_3$ map (Fig. 4F) exhibits a symmetric division in four quadrants aligned as those of the $S_2$ parameter (Fig. 4E). $S_2$ is maintaining the same orientation as in the initial state but is decreasing in its intensity, whereas $S_1$ is essentially unmodified in both orientations and intensity. This effect is indeed observed only in the presence of a $k=0$ $xy$ anisotropy, which in the simulations has the specific orientation along the $x$ and $y$ axis and thus is not affecting the $S_1$ pattern. On the Poincaré sphere, the space circulation of the polarization vortices around the cores at the initial time can be mapped to a double rotation on the sphere lying in the equatorial plane as in Fig. 4M. The effect of the dynamical polarization reshaping is equivalent to a twist of the geodesics around the $S_1$ axis toward the circular poles, which grows in time.

Fig. 4.

Theoretical analysis. (A–C) Polarization maps of the hedgehog vortex from numerical simulations at the early stage and (D–F) final time ($t=67\hspace{0.17em}\text{ps}$). (G–I) $S_1$, $S_2$ and $S_3$ maps of the star-like skyrmion at initial and (J–L) final time ($t=67\hspace{0.17em}\text{ps}$). (M and N) Twisting of the associated double-loop (M) and single-loop (N) trajectory in the Poincaré sphere, retrieved along the black circle in the real-space maps (which is the initial $l$-line of the skyrmion). In M the loops are traced at different times ($t=6\hspace{0.17em}\text{ps}$, 34 ps, and $67\hspace{0.17em}\text{ps}$) and constant anisotropy (${\chi }_0\equiv {\chi }_{02}$), whereas in N the loops are traced at fixed time ($t=55\hspace{0.17em}\text{ps}$) and different ${\chi }_0$ values (${\chi }_{01}=0.01\hspace{0.17em}\text{meV}$, ${\chi }_{02}=0.02\hspace{0.17em}\text{meV}$, ${\chi }_{03}=0.03\hspace{0.17em}\text{meV}$, ${\chi }_{04}=0.04\hspace{0.17em}\text{meV}$). (O) Time evolution of the twist angle $\beta$ for the same anisotropy values used in N. The twist rate is constant in time and linear on the ${\chi }_0$ value and independent of the topology state [that is, each $\beta (t)$ twist curve in O is the same for all of the four states, hedgehog and hyperspin and lemon and star skyrmion].

Similar effects are observed when starting with a skyrmion. As an example in Fig. 4 G–I we show the star-like states, with their associated two-sector symmetry in the linear polarizations. Here the degree of circular polarization is not zero at the initial state due to the skyrmion structure, which translates to a vortex in one circular polarization and the Gaussian state in the other. The polarization evolves in time in a similar way to what we have seen for the skyrmions in the experiment and is caused by the mechanism associated with the ${\chi }_0$ splitting, as described earlier. Fig. 4 J–L shows the Stokes maps obtained at later time ($t=67\hspace{0.17em}\text{ps}$) again under the action of a ${\chi }_0\equiv {\chi }_{02}$ anisotropy value. We also examine the polarization profile along a circle in real space taken along the so-called $l$-line, marked on the maps as a solid black circle. This is conformal to a single loop around the equator of the Poincaré sphere as shown in Fig. 4N. Here we report the loops at fixed time ($t=55\hspace{0.17em}\text{ps}$) and for different increasing ${\chi }_0$ values (${\chi }_{01}=0.01\hspace{0.17em}\text{meV},{\chi }_{02}=0.02\hspace{0.17em}\text{meV},{\chi }_{03}=0.03\hspace{0.17em}\text{meV},{\chi }_{04}=0.04\hspace{0.17em}\text{meV}$). The polarization reshaping with its associated Stokes twist is once again happening along the $S_1$ axis and it is proportional to the ${\chi }_0$ strength (see also Movie S5). The initially circular symmetry of the spin degree in real space evolves as well, as seen in Fig. 4L. It assumes a distribution that is somehow complementary to that of the $S_2$ one.

Conclusions and Perspectives

In summary, the dephasing of $x$ and $y$ linear polarization components leads to a transformation of the diagonal-antidiagonal degree of polarization into a circular spin degree, for both spin vortices and skyrmions. By comparing with the experiments, we deduce that the axis of the $k=0$ splitting anisotropy in our experimental configurations is oriented along the diagonal and antidiagonal directions. The twist speed induced by the ${\chi }_0$ term in the simulations is the same for all four considered states. The perfectly linear trend in time (starting from $t=8\hspace{0.17em}\text{ps}$, that is, when the fluid is left free to evolve after the arrival of the exciting pulse) shown in Fig. 4O demonstrates the effect to be independent of the instantaneous density of polaritons, which decay according to the ${\tau }_{\text{LP}}=10\hspace{0.17em}\text{ps}$. However, the strength of the twist is directly proportional to the anisotropy value, as demonstrated by looking at the slopes of the $\beta$ curves in Fig. 4O corresponding to different ${\chi }_0$. We can evaluate a theoretical twist speed of $\sim 40°{(\text{ps}\cdot \text{meV})}^{-1}$. Noticeably here (for the skyrmions) we also observe an interesting evolution of the $S_1$ pattern in real space (Fig. 4J). There is a sort of rotation of the sectors with some features of spiraling. This additional effect, that is, the rotations of the $S_{\mathrm{1,2,3}}$ sectors in real space, is instead associated to the action of the finite-$k$ TE-TM splitting term $\chi$ in our model (27). We would like to stress that our simulations clearly confirm that it is the $xy$ anisotropy that is the cause for the polarization twisting of vortex states. However, the disorder potential term, produced by the inhomogeneities inside the cavity mirror, was not needed to reproduce the observed dynamics.

From a theoretical point of view, the reshaping of the polarization field, and more specifically the Stokes twist, can be a convenient way to define the generalized quantum vortex, where the angle $\beta$ measures the inclination between the plane of polarization rotation and the equatorial plane in the Poincaré sphere. We note that the concept of generalized quantum vortex can be used to describe the new type of half-quantum circulation recently found in a macroscopic ring by Liu et al. (39) under nonresonant pulsed pumping. Namely, this vortex corresponds to the polarization rotation around a tilted axis on the pseudospin sphere. We can hence define the generalized skyrmion as a full Poincaré topology whose real-space circuitations are conformal to a family of single-loop curves around an arbitrary axis on the pseudospin sphere (see Movie S6). The same concept can apply to the spin vortices, whose generalized version maps to a double loop along an arbitrary great circle of the Poincaré sphere.

Methods Summary

The experimental polariton device is a typical photonic MC embedding QWs kept at cryogenic temperature. A picosecond laser pulse tuned on the LPB energy works as the excitation and reference beams. Optical vortices and their composition are obtained by means of a liquid crystal $q$-plate device, waveplates, and polarizers. Space-temporal dynamics are retrieved upon implementing the off-axis digital holography technique on a custom interferometric setup. The modeling of the system is based on coupled two-component open-dissipative Gross–Pitaevskii equations for the MC photons and QW excitons. Dynamical simulations of the equations are implemented on the XMDS2 software framework (40). For experimental and theoretical details see refs. 22, 30, and 33 and Supporting Information.

SI Text

Experimental Methods.

The experimental polariton device is an AlGaAs 2$\lambda$ MC with three 8-nm In_0.04Ga_0.96As QWs. All of the experiments shown here are performed at a temperature of 10 K in a region of the sample clean from defects. The excitation beam is a 4.0-ps Gaussian laser pulse with a repetition rate of 80 MHz selectively tuned on the LPB energy. Its intensity is adjusted so as to keep the resonantly excited fluid in a linear regime during the whole dynamics. To obtain the four different initial topological patterns (as reported in the table of Fig. 1M) we used a combination of impinging polarization, electrical tuning of the $q$-plate, and waveplates as described below. In the case of the spin vortices the pulse is linearly polarized and the tuning of the $q$-plate is complete (100%). This allows us to directly obtain a hedgehog pattern at the exit. Upon insertion of a half-wave plate (HWP) after the $q$-plate, we locally rotate the linear vectors of such a pattern, obtaining the hyperspin topology. In the case of the skyrmion, we send the pulse with a circular polarization onto the $q$-plate, which is now partially tuned (50%). This results in an outcoming lemon skyrmion, which can be rotated by means of an HWP into its conjugated state, the star skyrmion.

On the detection side, to obtain polarization-resolved imaging, a waveplate and a linear polarizer are inserted before the charge-coupled device. Upon using a HWP before the polarizer it is possible to resolve every direction of the linear polarization ($H$, $V$, $D$, and $A$), and by replacing the HWP with a quarter-wave plate it is possible to map the circular polarizations ($R$ and $L$). In this way we perform six dynamical sequences for each initial topology, from which it is possible to extract each independent degree of polarization. The three Stokes parameters are effectively derived as $S_1=\frac{I_H-I_V}{I_H+I_V}$, $S_2=\frac{I_D-I_A}{I_D+I_A}$, and $S_3=\frac{I_R-I_L}{I_R+I_L}$, where the intensities are a function of both time and space [e.g., $I_H(x,y,t)$]. We checked that the total intensity in each of the three bases is the same at each point in space and time, $(I_H+I_V)(x,y,t)=(I_D+I_A)(x,y,t)=(I_R+I_L)(x,y,t)$. In other words, we checked that there is no significant depolarization and the six measurements are consistent with each other.

To obtain the time dynamics, the emission profiles are made to interfere with a delayed expanded reference beam carrying homogeneous density and phase profiles. Such a technique is known as off-axis digital holography and relies on the use of fast Fourier transform (FFT) to filter only the information associated with the simultaneity between the emission and the delayed reference pulse. In this way it is possible to study the dynamics of the polariton fluid, by obtaining the 2D real-space snapshots of both the emission amplitude and phase, at a given time frame set by the delay. Each final snapshot results from thousands of repeated events, whose stability is based on the repeatability of the dynamics (with respect to the physics of the polaritons) and on the acquisition speed of each single interferogram (with respect to the experimental setup). Despite the fact that here we mostly used intensity features in each of the six pseudospin vectors, to study the polarization degree distribution and evolution it is also possible to look at the phase maps to devise the phase singularities at the cores of the vortex states (which here we did to check their stability in time). Additional details on the technique and the sample can be found in refs. 22, 30, and 33.

Theory Models.

To understand the physical origins of our observations, we perform numerical modeling of the system’s dynamics using two-component open-dissipative Gross–Pitaevskii equations, which describe the MC photon field ${\varphi }_{\pm }$ and the QW exciton field ${\psi }_{\pm }$ coupled to each other:

$$\begin{array}{c}i\mathrm{\hslash }\frac{\partial {\varphi }_{\pm }}{\partial t}=\left(-\frac{{\mathrm{\hslash }}^2}{2m_{\varphi }}{\nabla }^2-i\frac{\mathrm{\hslash }}{2{\tau }_{\varphi }}\right){\varphi }_{\pm }+\frac{\mathrm{\hslash }{\mathrm{\Omega }}_{\text{R}}}{2}{\psi }_{\pm }+\chi {\left(\frac{\partial }{\partial x}\mp i\frac{\partial }{\partial y}\right)}^2{\varphi }_{\mp }+\frac{1}{2}{\chi }_0{\varphi }_{\mp }+D{\varphi }_{\pm }+F_{\pm }\\ i\mathrm{\hslash }\frac{\partial {\psi }_{\pm }}{\partial t}=\left(-\frac{{\mathrm{\hslash }}^2}{2m_{\psi }}{\nabla }^2-i\frac{\mathrm{\hslash }}{2{\tau }_{\psi }}\right){\psi }_{\pm }+\frac{\mathrm{\hslash }{\mathrm{\Omega }}_{\text{R}}}{2}{\varphi }_{\pm }+{\alpha }_1{|{\psi }_{\pm }|}^2{\psi }_{\pm }+{\alpha }_2{|{\psi }_{\mp }|}^2{\psi }_{\pm }.\end{array}$$ [S1]

Here the upper lines of both equations represent analogous terms for the two fields, which are the kinetic energy, the decay time, and the Rabi coupling strength between photons and excitons, respectively. In practical terms, excitons have an effective mass $m_{\psi }$ of four to five orders of magnitude greater than that of the MC photons $m_{\varphi }$, resulting in their kinetic energy’s being negligible. The exciton and photon lifetimes are ${\tau }_{\psi }=$ 1,000 ps and ${\tau }_{\varphi }=5\text{ps}$, respectively, giving the lower polariton lifetime of ${\tau }_{\text{LP}}\approx 10\text{ps}$ at zero detuning and $k=0$. The Rabi coupling is $\mathrm{\hslash }{\mathrm{\Omega }}_{\text{R}}=5.3\text{meV}$. Selective excitation of the LPB can be obtained by using picosecond pulses with less than $1\text{meV}$ energy width, tuned on the lower polariton mode as in the experiments. The bottom lines in both equations represent the specific terms acting on the two fields. The exciton–exciton interaction strengths used in the simulations are ${\alpha }_1=+2\mathrm{\mu }\text{eV}\cdot \mathrm{\mu }{\text{m}}^2$ for the intraspin nonlinearities and ${\alpha }_2=-0.2\mathrm{\mu }\text{eV}\cdot \mathrm{\mu }{\text{m}}^2$ for the interspin ones (34). However, in the present work we are interested in the linear regime and specifically in the terms directly acting on the photonic fields, as discussed in the following.

The photonic linear coupling between different polarizations is given by the finite-$k$ TE-TM splitting term $\chi$ and the $k$-independent anisotropy splitting ${\chi }_0$. The former appears due to the difference of transverse-electric and transverse-magnetic masses of MC modes (35) as $\chi =\frac{{\mathrm{\hslash }}^2}{4}(\frac{1}{m_{\varphi }^{\text{TE}}}-\frac{1}{m_{\varphi }^{\text{TM}}})$, where the two effective masses’ imbalance is assumed $m_{\varphi }^{\text{TE}}/m_{\varphi }^{\text{TM}}=0.95$ in our case. The $k$-independent splitting ${\chi }_0$ between linearly polarized modes, which is due to strain effects (36, 37) and heavy–light hole mixing (38) on the QW interfaces, results in a different energy shift between the relevant linear polarized modes, and in the accumulation of a relative phase. In our simulations, we assumed $xy$ directions for the anisotropy axis and used four different ${\chi }_0$ values (${\chi }_{01}=0.01\text{meV},{\chi }_{02}=0.02\text{meV},{\chi }_{03}=0.03\text{meV},{\chi }_{04}=0.04\text{meV}$).

The disorder potential term $D(x,y)$, produced by the inhomogeneities inside the cavity mirror, which is reported here only for completeness, was not used and not needed to reproduce the observed dynamics. Finally, the initial laser pulse is described as a pulsed Laguerre–Gauss $F_{\pm }$:

$$F_{\pm }(𝐫)=f_{\pm }{r}^{|l_{\pm }|}{e}^{-\frac{1}{2}\frac{r^2}{{\sigma }_r^2}}{e}^{i{l}_{\pm }\theta }{e}^{-\frac{1}{2}\frac{{\left(t-t_{\text{0}}\right)}^2}{{\sigma }_t^2}}{e}^{i({𝐤}_p\cdot 𝐫-{\omega }_{p}t)}$$

with a winding number of the vortex state in the ${\sigma }_{\pm }$ component represented by $l_{\pm }$, and a strength $f$ that reproduces the total number of output photons. The parameters of the initial state are chosen to reproduce the experimental specifics (${\sigma }_{\text{r}}$ and ${\sigma }_t$ resulting in space and time FWHM equal to 30 μm and $4\text{ps}$, respectively). The initial state is centered on the LPB mode at 836 nm and at ${𝐤}_p=0$.

Computational Methods.

The dynamics of Eq. S1 is simulated using the XMDS2 software framework (40). We used an adaptive step-size algorithm based on a fourth- and fifth-order “embedded Runge–Kutta” (ARK45) method with periodic boundary conditions. This algorithm was also tested against eighth and ninth order (ARK89) of embedded Runge–Kutta method. The periodic boundary condition is an artifact of using FFT to efficiently switch between the real space to compute the potential energy and the momentum space to evaluate the kinetic energy. This method ensures very fast computation of each time step. To ensure that all flux leaving the system is not coming back from the other side due to the periodic boundary conditions, we implemented additional circular/ring absorbing boundary conditions, with the depth and the width carefully adjusted to the geometry of current experiments. We solve the equations on a 2D finite grid of $N\times N=$ 1,024 × 1,024 points and lattice spacing $l=$ 0.54 μm in a box of $L\times L=556\times 556$ μm². The large size of the simulation box ensures that polariton density drops practically to zero at the boundary. However, in all of the maps we plot the physically relevant central region only, where the density of polaritons is still significant.