SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: 0.038/NPHYS406 Half-solitons in a polariton quantum fluid behave like magnetic monopoles R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese 3, T. Boulier, D. Andreoli, E. Giacobino, J. Bloch 3, A. Bramati *, G. Malpuech, A. Amo 3* Laboratoire Kastler Brossel, Université Pierre et Marie Curie, Ecole Normale Supérieure et CNRS, UPMC case 74, 4 place Jussieu, Paris, France Institut Pascal, PHOTON-N, Clermont Université, University Blaise Pascal, CNRS, 4 avenue des Landais, 6377 Aubière cedex, France 3 Laboratoire de Photonique et Nanostructures, CNRS, Route de Nozay, 9460 Marcoussis, France * address : bramati@spectro.jussieu.fr; alberto.amo@lpn.cnrs.fr. A. COULOMB-LIKE INTERACTION BETWEEN HALF-SOLITONS In order to evaluate the interaction between two half-solitons and its Coulomb-like character, let s first consider the interaction between two point-like electrical charges in the absence of any external field. The solution of Maxwell s equations for a point electric charge in D is a constant electric field pointing in opposite directions on each side of the charge. Such field would produce a force of constant magnitude on a second charge independent of its position, according to the Coulomb law. Another way of tracking this interaction is by looking at the fields generated by the ensemble of the two charges. In D, in the case of two identical charges with opposite signs separated by a distance d, the sum of the fields they generate in the region outside of the charges cancels out exactly and is equal to zero. Between the two charges the fields add and it becomes twice as large as the field created by each individual charge. When the two electric charges have the same sign, the situation is opposite (zero field in between and double amplitude field outside). This is the manifestation of the Coulomb behaviour of point electric charges in D. One would expect magnetic monopoles to interact in a similar way. As explained in the text, the observed oblique half-solitons are D objects replicated in a D plane. In our monopole analogues, the role of the field created by each magnetic charge is played by the pseudospin vector field emanating from the half-soliton. For instance, the wave functions describing two individual half-solitons of opposite charges are: NATURE PHYSICS

2 ( ) ( ) n/tanh x x / ξ n/tanh x x / ξ = = n /, n /. The upper (lower) term indicates the spin-up (-down) component of the spinor wavefunction. In this case, the solitons are present in the spin-up component, located at x and x for and, respectively. This is sketched in the two upper images of Supplementary Fig. a, showing their densities in solid lines. The spin-down component is of constant density, and equal to the spin-up far from x and x. The direction of the pseudospin can be directly extracted from and and it is indicated by the arrows in Supplementary Fig.. Note that the charge of the half-solitons is given by the direction of the divergent pseudospin field S. The spin texture of presents an outgoing divergent spin texture, as shown in the upper image of Supplementary Fig. a. On the other hand, the spin field for is convergent towards the soliton center (middle image). In order to study the interaction between the solitons, we add together (and divide by ) the wavefunctions and : ( ( ) ξ ( ( ) ξ ) ) n/ tanh x x / tanh x x / + = n /. If we concentrate on the spin-up component, this wavefunction gives a finite density between the two half-solitons and zero density far from them, as depicted by the red solid line in the lower panel of Supplementary Fig. a. This profile and the associated spin texture (arrows) coincide exactly with the field expected for two point electric charges that attract each other under Coulomb interaction. Let us note that the attractive/repulsive behaviour of our magnetic charges is opposite to that of electric charges because of the negative effective mass of solitons. The case we just described and depicted in Supplementary Fig. a, takes two half-solitons of opposite charge (given by the sign of the divergence), and the interaction is repulsive, while for electric charges with positive masses it would be attractive. The situation with two half-solitons of same charge is depicted in Supplementary Fig. b, where the resulting interaction is attractive.

3 a b Supplementary Figure : Coulomb-like interaction between two half-solitons. The red arrows show the pseudospin vector field in the (x,z) plane. The solid/red lines stand for the density profiles of the spin component that contains the solitons (σ + in our case), the σ - component remains homogeneous. The black arrows give the propagation direction of each half soliton due to their interaction. a Illustrates the interaction between two solitons of opposite charges that repel each other. b Shows the attractive interaction between two solitons of identical charge. Though the wavefunction profiles we have just described already coincide with those of the field of attracting/repulsing charges, we can also understand the origin of this Coulomb-like force by looking at the energy of the system arising from the interparticle interactions. Polariton-polariton interactions are strongly spin-anisotropic. Polaritons of the same spin (same value of S z, spin-up or spin-down) present strong repulsive interactions (parameterised by the constant α ) which increase the energy of the system ( Δ E ( ) = αn ( ) ). Interactions between opposite spins ( α ) are much weaker and attractive (lowering the energy: Δ E ( ) = α n ( ) n ( ) ). For this reason, a linearly polarised condensate, with equal amount of spin-up and spin-down polaritons, is the lowest energy state of the system (as opposed to a condensate of only spin-up or only spin-down polaritons). In view of this, we can understand that the minimisation of the energy in the case shown in Supplementary Fig. a pushes the half-solitons apart from each other. In this way the system evolves towards a linearly polarized state and reduces its energy. Similarly, in the case depicted in Supplementary Fig. b, the half-solitons attract each other.

4 We can quantify the magnitude of the Coulomb-like force acting on each half-soliton by calculating the gradient of the total energy of the system: Fi = E/ xi, where x i is the position of i th soliton. Assuming that the distance between the solitons is much larger than the healing length x x >> ξ, we obtain a force: F = α n /8, which does not depend on x i, and where we neglect the weak interaction between the opposite spins ( α = 0 ). The total polariton density is given by n. In order to write this force as a product of a charge and a field F = qs, we use the equivalent of Maxwell s equation div S = ρ / α, where α plays the role of the dielectric permittivity ε 0, and ρ is the charge density. Thus, in our case the field of a single charge is given by S = n/4, and the divergence theorem gives the charge of the topological defect: q= α n/. We obtain the repulsion of two charges with opposite signs since the mass of half-solitons is negative. We have chosen here to consider the solitons within the same spin component (spin-up), however, the same result would be obtained for solitons lying in different components. The sign of the charge will be given by the direction of the divergent pseudospin field S, not the polarisation at the core. While the half-soliton configurations described in this section and shown in Supplementary Fig. are not those reported in our experiments, in which the effect of the external magnetic field is dominant, we have shown here that half-solitons behave like point-like charges subject to a Coulomb-like force when interacting between each other. This behaviour reinforces the interpretation of half-solitons in terms of magnetic monopole analogues. B.- CASE OF CIRCULARLY POLARISED PUMP The generation of oblique half-solitons requires exciting the system with linearly polarised light 6. As a proof of this statement, we show in this section that pumping with a circularly polarised beam, as performed in Ref. 3, leads to integer-like solitons of a scalar condensate which, contrary to half-solitons, are not affected by the effective magnetic field.

5 Supplementary Figure shows the density, phase and pseudospin of a polariton fluid created with right circularly polarised (σ + ) light upstream from the defect. The corresponding initial pseudospin vector S therefore points in the z direction (along the growth axis of the sample, north pole of the Bloch sphere Fig. a). The effective magnetic field Ω induced by the TE-TM splitting lies on the plane of the microcavity (red arrow in Fig. b). The pseudospin and the magnetic field are then perpendicular to each other and this should lead to a precession of S, even upstream from the defect. However, the density imbalance between σ + and σ populations introduces an additionally energy splitting between these two components arising from the spin anisotropy of the polariton-polariton interaction. Polaritons with the same spin interact via the parameter α (see below), while polaritons of opposite spin present a much weaker interaction: α 0.α. This spin splitting gives rise to an extra effective magnetic field along the z axis (parallel to the injected pseudospin): Ω z = S3( α α )/ uz. With a large spin up population, this effective field is much higher than Ω, and almost no pseudospin precession takes place. Such a behaviour is clearly visible in the Supplementary Fig. c, where the degree of circular polarisation remains close to and almost unperturbed close to the barrier. The strength of Ω z is however reduced far away from the defect where the density of polaritons decays due to their finite lifetime which allows the conversion from spin up to spin down polaritons (yellow regions). Therefore, under strong σ + pumping, the σ population remains weak everywhere, being completely determined by the transfer of particles from the other component. This is seen in Supplementary Figs. a and b, where one can observe integer (scalar) solitons in the σ + component, and their copy (but with a smaller density) in the other component. These integer solitons are not affected by the in-plane field Ω and no acceleration is observed in that case. The phase slices along the white dotted line shown in Supplementary Fig. e undergo a phase jump of approximately 0.7 π in the σ + component (as well as in the σ, which just follows the dominant component) across each soliton, as expected for integer grey solitons. Finally, the rotation of the linear polarisation angle is a specific signature of half-solitons, and it is strongly suppressed in this configuration as one can see in Supplementary Fig. c,

6 representing the pseudospin, and in Supplementary Fig. f, which shows a slice of the polarisation angle along the white dotted line. Supplementary Figure : Generation of dark solitons using a σ + polarised pump. a Intensity of the light emitted by the microcavity in the σ + (left panel) and σ - (right panel) polarisation using a logarithmic scale. b Normalized fringes obtained from the interferences between the real space emission of the microcavity and a reference beam of constant phase. c Pseudospin map of the emission. The colour map shows the degree of circular polarisation and the arrows direction stand for the projection of the pseudospin on the linear polarisation plane (equator of the Bloch sphere). d Density profiles extracted from a and along the white dotted line. e Corresponding phase slices extracted from b. f Linear polarisation angle along the same dotted line. C.- NUMERICAL SIMULATIONS We have performed numerical calculations reproducing the experimental results shown in Figs. to 4 in the main text. The result is shown in Supplementary Fig. 3. To accurately model the polariton fluid, we need to take into account its driven-dissipative character, that is, we should include the continuous pumping of polaritons in the excitation area and the losses

7 arising from the polaritons escaping out of the cavity due to their finite lifetime. In order to do so we have used a set of modified Schrödinger and Gross-Pitaevskii equations for the spin- rt, rt, respectively, coupled dependent photonic mean field φ ± ( ) and the excitonic one χ ± ( ) via the light-matter interaction given by a Rabi splitting Ω R = 5.meV: φ i ± i = Δ φ± +Ω Rχ± + φ± + β φ + ± φ± t mφ x y τ φ i D i Pe ( ) χ± i i = Δ χ± +Ω Rφ± + α χ± + α χ χ± χ± t m τ χ χ ( k P r ωpt) f ( r) This set of equations is derived by minimizing the following Hamiltonian and adding pumping and dissipative terms treated via a dissipative formalism: ΩR * * α 4 α H = σ + Dφ± ( χφ σ σ χφ σ σ ) χσ χσ χ σ S = { φ, χ} m Ω σ= { +, } σ = { +, } The very last term is the magnetic energy of the condensate where S is the normalized pseudospin vector defined by: Here S = s + s + s 0 3 * ( φφ + ) * ( φφ + ) S S 0 R S = S = I S 0 S 3 S0 ( φ+ φ ) s j are the components of the pseudospin (without normalisation). effective field induced by the photonic TE-TM splitting given by: ( θ ) ( θ ) cos Ω = β ( kx + ky) sin 0 Ω is the where θ is the polar angle. In the equations () and () we have used the transformations k i ; τ φ = 0 ps and τ χ = 400 ps are the photonic and excitonic lifetimes f r and P ± are respectively the spatial extension and amplitudes of the cw x, y x, y respectively; ( ) pump of frequency ω P = ELPB( kp ) simulation) and wave vector (we pump exactly on the bare polariton branch in this k P =.3μm, acting on each spin component σ ± =±. The constant α = Ea S (where E = 0 mev is the exciton binding energy, 3 b B b ab = 0 μm is

8 its Bohr radius and S is the normalization area) describes the intra-component excitonexciton repulsion while α = 0.α stands for the weak attractive inter-component interaction. D( r ) is a potential barrier that accounts for the photonic defect in the TE TM microcavity. The constant β = ( mφ mφ ) is the strength of the effective magnetic field induced by the k -dependent photonic energy splitting between TE and TM eigenmodes 36. We note that the excitonic TE-TM splitting is orders of magnitude smaller and TE TM is therefore neglected. Here m = 0.95m and φ φ m TM 5 φ = 5 0 m0 are the masses of the transverse electric and transverse magnetic cavity eigenmodes, and m 0 is the free electron TE TM TE TM mass. The photonic effective mass is taken as mφ ( mφ mφ ) ( mφ mφ ) = +. Separation of oblique half-solitons with opposite charges Simulations based on the solutions of the above equations are shown in Supplementary Fig. 3. To illustrate the clearest way the action of the effective magnetic field on the trajectory of the half-solitons, and then to evidence their monopole behaviour, we have deliberately chosen a defect with a small radius (around µm) to avoid the nucleation of multiplets of oblique solitons. Supplementary Fig. 3 reproduces all the panels of the Fig., and shows a remarkable agreement with the experiment. In the simulations we are obviously able to analyse the flow at larger distances than in the experiment, where the separation is larger. Downstream and close to the obstacle, we see the formation of a pair of integer solitons. Each one of them can be considered as the composition of two half-solitons with equal contributions in the σ + and σ components. As we get farther from the obstacle, Ω accelerates each of the half-solitons in a direction depending on its charge. The action of this field results in the separation of the initially integer solitons into two sets of half-solitons, and it is the smoking gun of the monopole behaviour of half-solitons. This is clearly visible in panels I and II of Supplementary Fig. 3a, showing the emission from both circular components, as well as panels I and II of Supplementary Fig. 3b, showing the phase shifts undergone by the condensate.

9 If we concentrate on the left part of the flow (left side of the yellow line) we see that the soliton present in the σ component [S, visible in the panels a-ii and b-ii], initially overlaps with the σ + soliton, and it is bent towards the right at large downstream distances, becoming almost parallel to the flow direction. This displacement stabilizes the soliton, making it deeper. On the other hand, the soliton present in the σ + component has an opposite magnetic charge and is accelerated towards the left [panels 3a-I and 3b-I], with an orientation that tends to set the soliton trajectory perpendicular to the polariton flow. The σ + half-soliton thus gains speed with respect to the flow. As the soliton depth ns n is related to its speed via the expression ( ) s s s v = c n n, where c s is the sound speed of the fluid, the σ + halfsoliton soliton becomes shallower (as evidenced by the reduced phase shift visible in panel b- I) and, eventually, disappears. We clearly observe that the trajectories of the half-solitons are not linear but curved, with a quadratic dependence with the distance from the defect, which is a clear signature of their acceleration. The spatial separation between deep and shallow solitons is clearly visible in panel 3c showing a density profile along the white dotted line 5 µm away from the defect. The acceleration of the oblique half-solitons with opposite polarisation under the effect of the TE-TM effective magnetic field reveals the monopole-like behaviour of these objects. The sign of the magnetic charge associated to the half-solitons is given by the pseudospin texture 7. As obtained in the experiment and as originally predicted in Ref. 6, each oblique half-soliton also appears as a domain wall with respect to linear polarisations, which is most clearly observed on the diagonal (D)/anti-diagonal (A) polarisation basis defined by: ( ) ( ) ( π ) ( π ) D cos π / 4 sin π / 4 x = A sin / 4 cos / 4 y x + = y i i The corresponding photonic emission intensities I j = are shown in the panels 3a-III and j 3a-IV and the respective phases in the panels 3b-III and 3b-IV. The domain wall behaviour is underlined in the density profiles of the panel 3d.

10 Finally, the half-integer character of the oblique solitons is demonstrated in the panels 3f and 3g, where the calculated phase profiles along the white dotted line are displayed. As one can see, the phase jumps through the deepest half-solitons on the diagonal linear basis (panel 3g) are one half of the one obtained on the circular polarisation basis (panel 3f). As well, the polarisation changes from diagonal to anti-diagonal, which means a rotation of its direction of π across the deep solitons as it can be seen in the panels c and h. Supplementary Figure 3. Numerical simulations showing the monopole-like behaviour of the oblique half-solitons (linearly polarised pump). a Calculated emission in the circular (I-II) and diagonal (III-IV) polarisations. b Corresponding interference patterns showing the local phase shifts of the fluid. c Calculated pseudospin extracted from the linear and circular degree of polarisation of the emission: the black arrows stand for the in plane projection of the pseudospin S =(S,S ), [equator of the Bloch sphere] showing the inversion of the linear polarisation across the solitons (also visible in h showing a profile of the linear polarisation direction), and the colormap depicts the S 3 component evidencing the strong local degree of circular polarisation along the deep half-solitons. c and e Profile of the σ + and σ - densities and their related phase, respectively, along the dotted line in a and b. d and e Density and phase profiles of the emission in diagonal polarisations.

11 Supplementary videos In order to further clarify the monopole behaviour of the half-soliton in the presence of the effective magnetic field, we have performed a simulation summarized in Supplementary Video, which shows the calculated polariton density. The parameters are the same as those corresponding to Supplementary Fig. 3. In the simulation, the excitation source is switched on at t=0. We have set the value of the effective field Ω to zero for the first 00 ps, when a steady state of the flow is attained. We can see the formation of integer oblique dark solitons (or, equivalently, two superimposed oblique half-solitons). At t=00ps we start to ramp up linearly in time the effective field Ω. At that moment, the half-solitons start to split and to bend. If we concentrate on the σ + panel, we see that the left half-soliton is expelled outwards becoming shallower and hardly visible at some point. The right half-soliton is bent inwards more and more parallel to the fluid, to a limit where it starts to exhibit snake instability at higher values of Ω.( t > 350 ps). The role of the left and right soliton is exchanged in the σ emission. We note that thanks to the continuous injection of new particles in the system, we are allowed to perform simulations of this kind, namely changing parameters continuously. Supplementary video shows the corresponding normalized pseudospin vector S zooming on the left half-soliton: the colormap shows the S 3 component (the degree of circular T S = S, S. Initially the polarisation) and the black arrows the in-plane vector field ( x y) solitons show no circular polarisation. As the magnetic field is switched on and the halfsolitons start to separate showing a strong positive and negative values of S z (the red and blue regions corresponding to circularly polarised emission). Simultaneously, the in-plane vector field shows a shift when traversing this half-solitons attaining a value of π, which corresponds to a rotation of π / of the angle of linear polarisation. This shift defines a domain wall between diagonal and anti-diagonal polarisations. Supplementary references 3 Panzarini, G., Andreani, L. C., Armitage et al., Exciton-light coupling in single and coupled semiconductor microcavities: Polariton dispersion and polarization splitting. Phys. Rev. B 59, (999).