Taming the snake instabilities in a polariton superfluid Ferdinand Claude,1, ∗ Sergei V. Koniakhin,2, 3 Anne Ma^ıtre,1 Simon Pigeon,1 Giovanni Lerario,1 Daniil D. Stupin,3 Quentin Glorieux,1, 4 Elisabeth Giacobino,1 Dmitry Solnyshkov,2, 4 Guillaume Malpuech,2 and Alberto Bramati1, 4 1Laboratoire Kastler Brossel, Sorbonne Universit´e,CNRS, ENS-Universit´ePSL, Coll`egede France, 75005 Paris, France 2Institut Pascal, PHOTON-N2, Universit´eClermont Auvergne, CNRS, SIGMA Clermont, F-63000 Clermont-Ferrand, France 3Alferov University, 8/3 Khlopina street, Saint Petersburg 194021, Russia 4Institut Universitaire de France (IUF), F-75231 Paris, France (Dated: October 27, 2020) The dark solitons observed in a large variety of nonlinear media are unstable against the modu- lational (snake) instabilities and can break in vortex streets. This behavior has been investigated in nonlinear optical crystals and ultracold atomic gases. However, a deep characterization of this phenomenon is still missing. In a resonantly pumped 2D polariton superfluid, we use an all-optical imprinting technique together with the bistability of the polariton system to create dark solitons in confined channels. Due to the snake instabilities, the solitons are unstable and break in arrays of vor- tex streets whose dynamical evolution is frozen by the pump-induced confining potential, allowing their direct observation in our system. A deep quantitative study shows that the vortex street period is proportional to the quantum fluid healing length, in agreement with the theoretical predictions. Finally, the full control achieved on the soliton patterns is exploited to give a proof of principle of an efficient, ultra-fast, analog, all-optical maze solving machine in this photonic platform. I. INTRODUCTION in the last two decades as a very flexible photonic plat- form to study quantitatively topological excitations [12{ Dark solitons are fundamental localized excitations 14]. In these systems, with a resonant excitation, quan- characterized by a density dip on a homogeneous back- tized vortices and oblique dark solitons are generated ground that preserve their shape along the propagation when the flow interacts with a static defect [15, 16]. How- in a nonlinear medium because of the balance between ever in this case the solitons are stable even at subsonic the combined effects of diffraction and non-linearity [1]. flows, due to the intrinsic dissipation of the system [17] They are intrinsically 1-dimensional objects, becoming and their breaking in vortex streets is not observed. In unstable to transverse modulations as soon as they are incoherently pumped polariton condensates, theoretical embedded in a 2 or 3-dimensional environments. This proposals were made to generate dark solitons[18, 19], behavior due to the so called "snake instabilities"[2] has however the very fast relaxation of the solitonic struc- been theoretically predicted and experimentally observed tures is expected to hinder the observation of vortex in optics with self-defocusing nonlinear media [3{5] and street formation. more recently in ultracold bosonic and fermionic gases A scheme which emerged recently is based on resonant [6{8]. In the latter system, the solitons were observed to pumping and the use of the bistability loop of the non- break in sound excitations and vortex rings, which are linear polariton system [20{22] where, for a given pump- 3D dynamically stable structures. ing intensity, the density can be either low or high. When More generally, if not stabilized by a supersonic flow high and low density regions are simultaneously present [9], snake instabilities induce the dark solitons decay in in the system, stationary phase defects of a new type can quantized vortex anti-vortex (VA) pairs leading to the exist in low-density regions [23{28], where the phase is formation of quantum vortex streets, a quantum version not fixed by the resonant excitation, because most parti- of von Karman vortex streets. Interestingly, the creation cles are not directly injected by the laser but diffuse from of VA pairs in a subsonic flow interacting with a defect higher-density regions which represent potential barriers has been reported in time-resolved pulsed experiments in with sharp interfaces. atomic quantum fluids [10]. However, despite several ef- In the resonant configuration, the sharpness of the in- forts devoted to the study of this phenomenon, the quan- terfaces is limited by the healing length ξ of the quantum arXiv:2010.13206v1 [cond-mat.quant-gas] 25 Oct 2020 titative study of the snake instability dynamics remained fluid and not by the thermal diffusion of the exciton reser- elusive so far in equilibrium quantum fluids. voir, as it is the case for non-resonant excitation. The Driven-dissipative quantum fluids, namely cavity control of the spatial distribution of the phase and in- exciton-polaritons [11] (polaritons), which are 2D pho- tensity of the pump with spatial light modulators (SLM) tonic modes interacting via their excitonic parts, emerged allows to realize various confining potentials, such as 1D channels, 0D traps [29], or circuits made by the combi- nation of both. In this work we use an all optical imprinting technique ∗ [email protected] together with the polariton bistability to create dark soli- 2 tons confined inside 1D elongated channels. We observe staying on the upper branch with a large density of po- the soliton snake instability leading to the formation of laritons in the system. symmetric arrays of vortex streets, which are frozen by Ti:Sa the pump-induced confining potential allowing their di- PBS HWP rect observation and a quantitative study of the onset of the instabilities. top-hat Moreover, by exploiting the full optical control and the flexibility offered by our photonic platform we demon- strate its applied potential by implementing an all opti- slit cal, programmable maze solving machine. e SLM c n A maze of arbitrary shape is optically imprinted as a e r sample e complex set of 1D channels with dead ends and only an objective f e entrance and an exit. We observe that in a certain range BS r camera of parameters the vortex streets disappear from the dead real space ends and remain only along the path connecting the en- BS trance and exit, then "solving" the maze within a pi- cosecond timescale. This ultra fast all-optical maze solv- k-space ing machine represents an important step for the large spectrometer interdisciplinary field of analog graph solving algorithms + camera [30{36], opening a new field of applications for the quan- Sample cross-section Bistability loop GaAs/AlAs tum fluids of light. Bragg mirrors II. EXPERIMENTAL SETUP density INPUT OUTPUT In0.04Ga0.96As substrate excitation intensity Our sample is a planar 2λ GaAs high-finesse semi- (a)quantum wells (b) conductor microcavity made of 21 top and 24 bottom GaAs/AlAs Bragg mirror layer pairs, with one embed- SLM phase pattern Pumping scheme ded In0:04Ga0:96As quantum well (QW) at each of the π three antinodes of the confined electromagnetic field (fig- sample ure 1(a)). QW excitons strongly couple to the first con- 0 fined radiative mode of the microcavity to form polariton Intensity profile quasi-particles. Polaritons are massive (mass m) bosons L2 slit m with repulsive interactions (interaction constant g). SLM µ 0 0 The Rabi splitting and the polariton lifetime were mea- 2 (c) L1 sured to be respectively around ~Ω = 5:1 meV and M1 τpol = 15 ps in experiments conducted in an open-loop flow helium cryostat at a temperature of 5 K. The energy FIG. 1. Experimental setup. Panel (a) shows a cross-section detuning between the photon and exciton modes can be skecth of the microcavity. Panel (b) is a sketch of the bistabil- tuned around 0 meV by changing the working point on ity loop obtained with a quasi-resonant excitation. Panel (c) the microcavity, due to the presence of a small wedge gives in detail the shaping method to dig a rectangular verti- between the cavity mirrors. cal channel in the center of the driving field. In the scheme A cw single mode Ti-Sapphire laser injects photons at of the experimental set-up the red beam after the SLM cor- normal incidence with respect to the microcavity plane, responds to the laser intensity diffracted by the grating; the non-diffracted purple beam corresponds to the zero order of so that the fluid has no imposed flow velocity in the trans- the grating, which is cut by a slit in the Fourier plane to verse plane. It has a small positive energy above the obtain the intensity profile shown in the input plane of the resonant states of the lower polariton branch, typically cavity. The laser spot is flat and square due to the use of a blue-detuned by ~! = 0:1 meV. For homogeneous pho- top-hat lens. The real space detection arm gives access to the tonic systems with repulsive (defocusing) interactions, density and phase maps of the fluid, while the k-space one this type of pumping is associated with a hysteresis bista- to the energy-momentum distribution of the fluid by using a bility cycle, as shown in figure 1(b). Starting from an spectrometer. empty system, the rise of the pumping intensity takes place with a weak absorption up to the turning point, The confinement potential is obtained by shaping the where the system jumps on the higher bistability branch. intensity of the driving field using a spatial light modula- The interaction energy between the created polaritons is tor (SLM). The latter displays a blazed diffraction grat- there slightly larger than the laser detuning, which gives ing, which deviates a part of the laser intensity in a given a good estimate of the height of the induced confining po- off-axis direction with respect to the main optical path.