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# Twist of generalized skyrmions and spin vortices in a polariton superfluid

Edited by G. G. Lonzarich, University of Cambridge, Cambridge, United Kingdom, and accepted by Editorial Board Member Zachary Fisk November 11, 2016 (received for review June 23, 2016)

## 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 (

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

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 *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

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 (*C*). 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 (*A*) and star-like (Fig. 1*B*) 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. 1*D*) and the hyperspin vortex, characterized by a hyperbolic polarization pattern (Fig. 1*E*). 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. 1*F*) 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 *G* and *J*), zero-winding *H* and *K*), and counterclockwise *I* and *L*) states. Different combinations of the three *M*.

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 (*Supporting Information*).

## Twist of the Vortex Polarization Field

The variation of the *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 (*A*) and *B*) results in a petal shape of the polarization distribution, whereas

The most pronounced effect appears in the time evolution of the spin degree: The *D*, where we show the azimuthal profile of *C*) 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 *G*), at different times, in Fig. 2*H* (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 *A*, the field pattern follows the classic hedgehog structure schematically introduced in Fig. 1*D*. The colors used here refer directly to the degree of spin polarization *D* 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

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. 3*G*, is very close to the sketch pattern of Fig. 1*B*. 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 (*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-

## 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-

To reproduce the polarization twisting observed in our experiment we perform different sets of simulations. In the first set we take the *A–C*, respectively, for the case of a hedgehog state. Here the initial degree of circular polarization is homogeneously null (*C*). The evolution of *D–F*, shows emergence of a strong circular polarization under the action of a *F*) exhibits a symmetric division in four quadrants aligned as those of the *E*). *M*. The effect of the dynamical polarization reshaping is equivalent to a twist of the geodesics around the

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 *J–L* shows the Stokes maps obtained at later time (*N*. Here we report the loops at fixed time (*L*. It assumes a distribution that is somehow complementary to that of the

## Conclusions and Perspectives

In summary, the dephasing of *O* demonstrates the effect to be independent of the instantaneous density of polaritons, which decay according to the *O* corresponding to different *J*). There is a sort of rotation of the sectors with some features of spiraling. This additional effect, that is, the rotations of the

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

## 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 *Supporting Information*.

## SI Text

### Experimental Methods.

The experimental polariton device is an AlGaAs 2_{0.04}Ga_{0.96}As 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. 1*M*) we used a combination of impinging polarization, electrical tuning of the

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 (

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

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

The photonic linear coupling between different polarizations is given by the finite-

The disorder potential term

### 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 ^{2}. 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.

## Acknowledgments

We thank R. Houdré for the microcavity sample and L. Marrucci and B. Piccirillo for the *Engineering and Physical Sciences Research Council* Grants EP/I028900/2 and EP/K003623/2 (to M.H.S.), and Consejo Nacional de Ciencia y Tecnología Grant 251808 (to Y.G.R.).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: lorenzo.dominici{at}gmail.com or m.szymanska{at}ucl.ac.uk.

Author contributions: L.D., M.H.S., and D.S. designed research; S.D., L.D., G.D., D.B., M.D.G., G.G., and D.S. performed research; A.B. provided expertise with the microcavity sample; S.D., L.D., G.D., Y.G.R., and M.H.S. analyzed data; and S.D., L.D., G.D., A.B., Y.G.R., M.H.S., and D.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.G.G.L. is a Guest Editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1610123114/-/DCSupplemental.

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