Supporting Online Material for

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1 Suorting Online Material for Polariton Suerfluids Reveal Quantum Hydrodynamic Solitons A. Amo,* S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemenager, R. Houdré, E Giacobino, C. Ciuti, A. Bramati* *To whom corresondence should be addressed. alberto.amo@ln.cnrs.fr (A.A.); bramati@sectro.jussieu.fr (A.B.) This PDF file includes: Materials and Methods Figs. S1 to S4 Published 3 June 11, Science 33, 1167 (11) DOI: 1.116/science.137

2 Polariton suerfluids reveal quantum hydrodynamic solitons A. Amo 1,, S. Pigeon 3, D. Sanvitto 4, V. G. Sala 1, R. Hivet 1, I. Carusotto 5, F. Pisanello 1,4, G. Lemenager 1, R. Houdré 6, E Giacobino 1, C. Ciuti 3, A. Bramati 1 1 Laboratoire Kastler Brossel, Université Pierre et Marie Curie-Paris 6, École Normale Suérieure et CNRS, UPMC Case 74, 4 lace Jussieu, 755 Paris, France CNRS-Laboratoire de Photonique et Nanostructures, Route de Nozay, 9146 Marcoussis, France 3 Laboratoire Matériaux et Phénomènes Quantiques, UMR 716, Université Paris Diderot-Paris 7 et CNRS, 7513 Paris, France 4 NNL, Istituto Nanoscienze - CNR, Scuola Sueriore ISUFI, Università del Salento, Via Arnesano, 731 Lecce, Italy 5 INO-CNR BEC Center and Diartimento di Fisica, Università di Trento, via Sommarive 14, I-3813 Povo, Italy 6 Institut de Physique de la Matière Condensée, Faculté des Sciences de Base, bâtiment de Physique, Station 3, EPFL, CH-115 Lausanne, Switzerland Materials and Methods Samle descrition Our samle is a 3λ/ GaAs cavity with three In.5 Ga.95 As quantum wells resulting in a Rabi slitting of 5.1 mev, and a olariton lifetime of about 15 s. The to/bottom distributed Bragg reflectors forming the cavity have 1/4 airs of GaAs/AlGaAs alternating layers with an otical thickness of λ/4, λ being the wavelength of the energy of the confined cavity mode. All our exeriments are erformed at zero exciton-cavity detuning, with a continuous wave single mode laser quasi-resonant with the lower olariton branch. The samle has been grown by molecular beam eitaxy. During the growth of the distributed Bragg reflectors, the slight lattice mismatch between the materials of each layer results in an accumulated stress which relaxes in the form of structural defects. These hotonic defects form a very high otential barrier in the olariton energy landscae. Confocal excitation scheme The data reorted in Fig. 3 have been taken making use of the confocal excitation scheme reresented in Fig. S1. The laser is focalised in an intermediate lane where a mask is laced in order to hide the uer art of the Gaussian sot on that lane. Then, an image of the intermediate lane is done on the samle, roducing a sot with the shae of a half Gaussian (the rofile is deicted in the inset of Fig. S1). Polaritons are resonantly injected in the microcavity with a well defined wavevector, in the region above the red line in the figure. In these conditions, olaritons move out of the excitation sot with a free hase, not imosed by the um beam. This is essential for the observation of hydrodynamic effects involving toological excitations with hase discontinuities. 1

3 Focalisation lens Microcavity Lens Intermediate mask Sot on samle Excitation laser Laser intensity y Fig. S1. Excitation setu used for the exeriments reorted in Fig. 3. The intermediate mask creates a sot on the samle with the shae of a half Gaussian (the inset shows a y cross-section of the sot). Estimation of the sound seed The average sound seeds reorted in the main manuscrit have been obtained from the measured soliton seed v s and hase jum δ, with the use of Eq. 1 ( cos( δ ) = vs cs ). In Figs. 1 and 3 we have estimated the sound seed in the soliton regime (Fig. 1a, 1c and Fig. 3c, 3f) in the region below the otential barrier, where the hydrodynamic effects are observed. We have taken as the soliton seed vs = vflowsinα, where α is the angle of aerture of the soliton air, and v flow is obtained from the injected olariton wavevector and the measured olariton mass via vflow = kh mol. In the case of Fig. 3, the sound seed is estimated from the hase jum at half the total roagation distance in the soliton regime (Fig. 3c, 3f). In order to obtain the sound seed for other two excitation densities (anels a,b,d,e,g,h), we use the measured olariton density relative to the soliton case (c,f) and the sound seed relation s c = h gψ m. Note that the sound seed is roortional to the square root of the density ψ. ol In order to confirm that this relationshi is consistent with our results, we roceed in the same way for the data lotted in fig. 1. In this case we take the sound seed obtained from the hase jum along the right soliton. The sound seed decays as the fluid is further away from the excitation area. The result is shown in black dots in Fig. S. Additionally, we measure the decay of the density on the edges of the soliton along the soliton line. In Fig. S we lot in green triangles the magnitude c s = A I, where I is the emitted intensity (roortional to the olariton density) and A is a fitting constant. The figure shows that the decay of the sound seed obtained from both the hase jum and the measured density follow the same trend.

4 These results justify our method to obtain the sound seed in the suerfluid and vortex emission regimes (Fig. 3a,b) from the measured sound seed in the soliton regime (Fig. 3c, obtained from the hase jum) and the relative olariton density. 6 5 From hase jum From density sound seed (μm/s) Δy (distance from defect; μm) Fig. S. Sound seed estimation for the data of Fig. 1. Black dots show the sound seed obtained from the hase jum and Eq. 1 along the soliton trajectory. Green triangles show the fit from the measured square root of the emitted intensity (roortional to the olariton density). The dashed line shows the fluid seed. Degree of first order coherence The degree of first order coherence, g (1), is defined as: ψ ( r, ) (, ) (1) 1 t ψ r t+ τ g ( r1, r, t, τ ) =. ψ r, t ψ r, t+ τ ( ) ( ) 1 In our cw exeriments in stationary conditions, g (1) is indeendent of t. In order to measure g (1) (τ=), we direct the emitted light from the olariton condensate, which contains all the coherence information of the wavefunction, into a modified Mach- Zehnder interferometer. The interference image is obtained from the comosition of the real sace emitted field with coordinate r 1, and a reference beam issued from the enlarging of a small area of the emission with a fixed osition r with a well defined satial hase. By varying the length of the reference beam arm by u to two wavelengths around zero delay, we measure the visibility of the fringes of the interferometric image, giving direct access to the time averaged real sace degree of ψ r 1,t with resect to a coherent coherence of the condensate wavefunction ( ) reference ψ ( r,t). Gross-Piteavskii equation Figure shows simulations based on the solution of a generalized non-equilibrium Gross-Pitaevskii equation describing the olariton condensate subject to interarticle interactions. In the basis of the confined exciton and hoton wavefunctions it has the form: 3

5 where,,, where, x is a two-dimensional satial vector, ψ X( C) is the exciton (cavity hoton) wavefunction, F, k and ω are, resectively, the amlitude, momentum and energy of the um field. The k-deendent energy of the excitons (cavity hotons) is described by ω X ( C ), γ X( C) is the decay rate of the excitons (cavity hotons), with a value of 16 s, ΩR is the vacuum Rabi slitting between the olariton modes (5.1 mev), V( C x ) is the hotonic otential barrier, g the exciton-exciton interaction constant, taken to be.1 mev μm. x indicates the osition of the centre of the Gaussian sot on the samle, while δ X is its radial width. In the simulations shown Fn fig., k =.73 μm -1 and the um energy is detuned from the lower olariton branch at that k by. mev. The defect is simulated as a rectangle of 5x3 μm and a height of 8 mev. 4

6 Suorting Figure 3 A High ower B +1 Δy (μm) C Low ower D +1 Δy (μm) Δx (μm) - Δx (μm) Fig. S3. (A) Real sace emission showing oblique dark solitons at high excitation density (85 mw), and (B) the corresonding interference attern showing the hase sli along the soliton trajectory (reroduced from Fig. 1A and C). (C and D) Real sace emission and corresonding interference attern at low excitation density (1 mw). In this case, olariton-olariton interactions are negligible and solitons are neither formed nor sustained in the fluid. The arabolic wave atterns observed in (C) arise from the interference between the injected olaritons and those elastically scattered by the defect. The interference attern (D) does not show hase jums as those associated to solitons. 5

7 Suorting Figure 4 Polariton density (arb. units) π Δφ = 95. π Relative hase 1 Δx (μm) Fig. S4. Intensity (blue line) and hase rofile (red dots), along the dashed line indicated in Fig. S3, showing a hase jum of the condensate wavefuntion of almost π across the the soliton. 6