Spontaneous generation, enhanced propagation and optical imprinting of quantized vortices and dark solitons in a polariton superfluid: towards the control of quantum turbulence
Anne Maitre, Ferdinand Claude, Giovani Lerario, Serguei Koniakhin, Simon Pigeon, Dmitry Solnyshkov, Guillaume Malpuech, Quentin Glorieux, Elisabeth Giacobino, Alberto Bramati
SSpontaneous generation, enhanced propagation and optical im-printing of quantized vortices and dark solitons in a polariton su-perfluid: towards the control of quantum turbulence
A. Maˆıtre , F. Claude , G. Lerario , S. Koniakhin , S. Pigeon , D. Solnyshkov , G. Malpuech , Q.Glorieux , E. Giacobino and A. Bramati Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS, ENS-Universit´e PSL, Coll`ege de France - 75005 Paris,France Institut Pascal, PHOTON-N2, Universit´e Clermont Auvergne, CNRS, SIGMA - Clermont, F-63000 Clermont-Ferrand, France
PACS – Optical solitons
PACS – Quantum fluids: general properties
PACS – Vortices and turbulence
Abstract –In resonantly pumped polariton superfluids we recently explored a new regime basedon the bistability of the polariton system to enhance the propagation of polariton fluids up tomacroscopic distances. This technique together with an all-optical imprinting method allowedthe generation and control of various topological excitations such as quantized vortices and darksolitons. The flexibility and scalability of the new experimental scheme opens the way to thesystematic study of quantum turbulence in driven dissipative quantum fluids of light. In thisarticle we review the basic working principles of the bistability enhanced propagation and of theimprinting technique and we discuss the main achieved results as well as the most promising futureresearch directions.
Introduction. –
Exciton-polaritons are half-lighthalf-matter quasi-particles coming from the strong cou-pling between excitons and photons in semiconductor mi-crocavities [1]. They inherit specific properties from theircomponents: a very light mass coming from the photoncomponent and strong mutual interactions from their ex-citonic nature.In the last decade these systems have demonstrated tobe an ideal playground for the study of out-of-equilibriumcondensates and 2D quantum fluid hydrodynamics [2].In particular, the creation of polariton fluids via an alloptical excitation enabling a full control of the speed ofpolariton flows allowed the hydrodynamic generation ofa rich variety of topological excitations in such systems,ranging from quantized vortices to dark solitons, via theinteraction of a supersonic polariton wavepacket with astructural defect [3–9]Pulsed resonant excitation as well as continuous-wave(cw) resonant pumping have been used. However, boththe configurations exhibit the same fundamental limita-tion: the polariton density strongly decays along the prop- agation due to the short polariton lifetime. As a resultthe propagation distances accessible in these early exper-iments are quite short, strongly limiting the study of thedynamics of the topological excitations.In a series of recent articles, devoted to the deep studyof the properties of the bistability exhibited by the polari-ton systems under resonant pumping, we investigated anew configuration and demonstrated that in the bistableregime it is possible to get rid of the polariton densitydecay generating a superfluid flow propagating for macro-scopic distances, typically one order of magnitude longerthan the previous observations.Remarkably, in the bistable regime the topological exci-tations can be generated and their propagation sustainedand strongly enhanced far beyond the polariton free prop-agation length.Moreover, we implemented an all optical imprintingtechnique which allowed us to generate in a fully controlledway dark soliton pairs in different regimes and to studytheir stability against the onset of the snake instabilities.The observation of the breaking of dark solitons in vortexp-1 a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b . Maˆıtre et al. streets illustrates the high potential of this method for thesystematic study of the quantum turbulence in polaritonquantum fluids. Theoretical model. –
The standard way to describethe dynamics of a resonantly driven polariton fluid isto use a generalized Gross Pitaevskii equation. In theexciton-photon basis, the system is described by the fol-lowing coupled equations: i ¯ h ddt (cid:18) Ψ γ Ψ X (cid:19) = (cid:18) ¯ hF p (cid:19) + (cid:34) H lin + (cid:18) V γ − i ¯ hγ cav V X − i ¯ hγ X + ¯ hgn (cid:19) (cid:35) (cid:18) Ψ γ Ψ X (cid:19) (1)where the fields Ψ X and Ψ γ , describe the excitons andphotons, respectively.The system losses are represented by the two linewidths γ X and γ cav . The continuous pumping F p applies to thephotonic field while the interactions defined by g occurbetween excitons; n is the excitonic density. The externalpotentials V γ and V X come from the photonic and exci-tonic defects naturally present in the microcavity. H lin isthe linear Hamiltonian: H lin = E X ¯ h R ¯ h R E γ ( − i ∇ r ) (2)The equation (1) can also be written in the polaritonbasis, and in particular if we focus on the lower polaritonbranch, we obtain the driven-dissipative Gross-Pitaevskiiequation for the polariton field: i ¯ h ∂∂t Ψ = (cid:18) − ¯ h m ∗ LP ∇ r + V − i ¯ hγ + ¯ hg LP n (cid:19) Ψ + ¯ hF p (3)with m ∗ LP the effective mass of the lower polariton, V = | X k | V X + | C k | V γ the external potential felt by thelower polaritons and g LP = | X k | g the interaction con-stant between the lower polaritons in the same modes.As we are interested only in the lower polariton branch,to lighten the notation, the indices will be removed: g LP = g and m ∗ LP = m ∗ .This equation is very similar to the standard Gross-Pitaevskii which describes the atomic BECs, except for theloss term − i ¯ hγ and the pump term F p ( r , t ). It is indeedan important difference between our system and cold atomgases: the polariton lifetime τ = ¯ h πγ is of the order ofsome tens of picosecond, compared to seconds for atoms.Hence the fact that the losses need to be continuouslycompensated: our system is out of equilibrium.Let us now focus on the mean-field stationary solutionsof equation (3) in the homogeneous case, i.e. for an ex-ternal potential equal to zero. The system is driven by the pump field F p ( r , t ) = F p ( r ) e i ( k p r − ω p t ) . Therefore, thesolutions can be written as Ψ( r , t ) = Ψ ( r ) e i ( k p r − ω p t ) . Itresults in the mean field stationary equation: (cid:18) − ¯ h k p m ∗ − i ¯ hγ + ¯ hgn ( r ) (cid:19) Ψ ( r ) + ¯ hF p ( r ) = 0 (4)This equation is responsible for the bistability phe-nomenon observed in polariton system and induced bya quasi-resonant pumping. Indeed, the crucial param-eter which determines the bistable behaviour is the de-tuning between the pump and the lower polariton branch∆ E lasLP = ¯ hω p − ¯ hω LP . Multiplying the stationary equa-tion by its complex conjugate, we obtain the equation forthe intensity: I ( r ) = (cid:18) (¯ hγ ) + (cid:16) ∆ E lasLP − ¯ hgn ( r ) (cid:17) (cid:19) n ( r ) (5)with I ( r ) the intracavity intensity. By now this equationcan be derived into: ∂I∂n = 3 g n − E lasLP gn + (¯ hγ ) + ∆ E lasLP (6)the discriminant of which is (2 g ) (∆ E lasLP − hγ ) ).Therefore, the previous equation can have two distinctroots if the detuning satisfies the condition ∆ E lasLP > √ hγ . Fig. 1: Numerical resolution of equation 5 for ¯ hγ = 0 . E lasLP = 1 meV. The green region shows the stablesolutions while the red one indicates the unstable ones [10].The density jumps from one branch to the other at the frontierbetween the green and red regions: we have a hysteresis cycleillustrated by the arrows. In this case, the system presents a bistable behaviour:the polariton density n ( I ) displays a range of intensitiesfor which the system has two possible outputs, as plottedin figure 1. The system is stable on the green parts of thecurve, while the red region shows non-physical solutionsp-2ontrolled Quantum Turbulence in polariton superfluids[10]. The system therefore jumps from one branch to an-other at the limits between the red and green regions. Weobserve a bistability cycle: within the bistable range, theupper branch is only accessible by lowering the pump in-tensity from a higher one, while the lower branch can onlybe reached by increasing the intensity from a lower one,as indicated by the arrows.A very specific behaviour of the system when it is ex-cited in the bistable range was pointed out in [11]: the au-thors suggested to use the optical bistability of an exciton-polariton system to enhance the propagation of the po-lariton superfluid. In particular, the use of two differentbeams, simultaneously exciting the microcavity, can en-able the generation, over a macrocopic scale, of a highdensity, bistable, polariton fluid.Let consider two driving fields, with the same frequency ω p and the same in-plane wavevector k p . The first one,called the seed, is localized in space and has a high in-tensity: I r > I high . It thus produces a nonlinear su-perfluid above the bistability cycle. The second field isthe support, ideally an infinitely extended constant field,stationary in time and homogeneous in space. Its inten-sity is weaker than the intensity of the seed one, and isplaced inside the bistability cycle: I low < I s < I high . Thegoal of this configuration is to enhance the propagationand density of the polariton fluid by combining the prop-erties of both beams. Numerical simulations were doneto understand their combination, presented in figure 2.They have been realized for a cavity without any defects( V = 0) such that: ¯ hω C ( k = ) = 1602 meV, ¯ hω X = 1600meV, ¯ hγ X = ¯ hγ C = 0 .
05 meV, ¯ h Ω R = 2 . hg = 0 .
01 meV/ µ m . The driving field parameters are δE = ¯ hω p − ¯ hω LP ( k = k p ) = 1 meV and | k p | = (0 . T µ m -1 .The field intensity is shown as a function of space, plot-ted in logarithmic scale. The flow of particles goes fromleft to right. The driving intensity sent in the system ispictured with the red dashed line, while the black solidline shows the steady-state photonic intracavity density.On the upper panel, only the seed is sent on the left,illustrated with the yellow highlighted region ( F r ( x ) (cid:54) = 0; F s = 0). Given the presence of an in-plane wavevector, thepolariton density expands, but it decays exponentially dueto the finite polariton lifetime. Even with the present bestquality samples, the propagation distance in this configu-ration is limited to around 50 microns. Moreover, as thedensity is decreasing all along propagation, all the relatedfluid parameters are also constantly changing.The lower panel shows the behaviour of the system whenthe support, in orange, is added to the previous configu-ration. The support intensity is two order of magnitudelower than the seed one, but infinitely extended in spacefor this simulation. The presence of the support field has astrong impact on the total polariton density: its high levelcreated at the seed location is maintained without any de-cay all over the region where is the support, despite itsmuch weaker intensity. Bistability is essential to explain Fig. 2: The black solid line illustrates the intracavity intensity(logarithmic scale), while the red dashed one is the driving in-tensity sent to the system. The colored regions delimits thedifferent driving fields: the seed in yellow and the support inorange. The polariton flow goes from left to right. On theupper panel, only the seed is sent: it creates a high density po-lariton fluid, decaying exponentially out of the pumping region.The propagation length is limited by the polariton lifetime. Inthe lower panel, the support field is added: even if its densityis two order of magnitude lower than the seed one, the totalpolariton density is maintained at its higher level all along thepresence of the support. From [11] this behaviour. The seed is placed outside the bistabilitycycle, where only one stable state is accessible (see figure3). It therefore ensures the system to be on the upperbranch of the cycle, the nonlinear branch. On the otherhand, the support is chosen to be inside the bistability cy-cle, where two states are reachable. The support itself cannot reach the upper branch and, if alone, would only drivethe system in the linear configuration. Yet, as the seedplaces the system in the nonlinear branch and touches thesupport illuminated region, by extension, the neighbor re-gion jumps also on the upper branch which expands to allthe support area.
Subsonic flow: vortex stream generation. –
Theidea of the seed-support configuration is not only to obtainan extended fluid of polaritons, but also to study its prop-erties, and in particular its ability to generate topologicalexcitations, such as dark solitons and quantized vortex-antivortex pairs. The different hydrodynamic regimes of apolariton fluid and their effects on the generation of topo-logical excitations have been previously studied in the caseof direct injection [12,13] and for a single intense and local-ized pump, placed upstream of a structural defect [6, 14].The presence of the defect creates turbulence along theflow, which evolves differently depending on the ratio be-tween the speed of the fluid and the sound velocity, i.e. the Mach number of the system: M = v f /c s . The speedof the fluid that needs to be taken into account is theone around the defect, therefore always higher than thep-3. Maˆıtre et al. Fig. 3: The seed is chosen to be above the hysteresis cycle.Only one stable state is available at this place, on the up-per nonlinear branch of bistability. The support has muchlower intensity, inside the bistable region. When both fieldsare sent, the seed presence ensures the system to stay on theupper branch, everywhere the support is present. one extracted far from the defect. Indeed, due to the im-penetrable nature of the defect, the particles flowing closeto it are accelerated [15], which induces a phase shift asthe fluid phase and speed are connected. In the case ofa global subsonic flow, but locally supersonic around thedefect, vortex-antivortex pairs emerge in the wake of thedefect. Increasing the Mach number induces an increase inthe emission rate of the vortices, which will finally mergetogether in a pair of dark solitons for a Mach number closeto 1 or higher.To numerically reproduce the effect of a cavity struc-tural defect, a large potential photonic barrier is intro-duced ( V (cid:54) = 0). The seed is placed upstream to it, local-ized and with a fixed intensity above the bistability cycle.The bistable regime ensures the release of the phaseconstraint imposed by a resonant driving. Thus, eventhough the support field possesses a flat phase, vortex-antivortex pairs are generated in the wake of the defect aslong as the support intensity places the fluid in the bistableregime. The seed-support configuration previously de-scribed is used to enhance their propagation length byone order of magnitude.Time resolved numerical simulations have been madein order to precisely locate the vortex-antivotex pairs,however as the experiment is in continuous wave excita-tion, time integrated numerical simulations were also per-formed. Figure 4 shows the comparison between a snap-shot image on the left, and, with the same parameters, animage with 1 ms integration time, on the right.Vortex and antivortex have opposite circulation, andare spotted by red and blue dots, respectively. Vortex-antivortex can stay bounded and propagate along the flowside by side, or annihilate each other and vanish.A time average flow of vortex pairs appears as a thickline of lower density, the dip height of which is proportionalto the vortex density. The phase pattern would be blurriedand a decrease of visibility would appear along the vortex Fig. 4: On the left, the time snapshot shows the position ofvortex and antivortex flowing in the wake of the defect. Onthe right, the time integrated image has blurred the positionsof the vortices and results in a thick shadow along the flow. stream.The experimental setup is displayed in figure 5 [16]. Theinitial laser source is a CW Titanium Sapphire laser. Itis split up a first time into the main beam and the ref-erence beam, later used to realize interferograms and getinformation on the phase. The main beam is split againto generate the seed and the support beams. The seed isthen focused on the sample into a spot of 30 microns di-ameter, for an intensity I r = 10 . . On the otherhand, the support is shaped by two cylindrical lenses inan elliptical spot of 400 microns length, elongated in the y direction with an intensity of 5.8 W/mm . The inset offigure 5 gives a representation of the relative position ofthe beams; the seed is not centered on the support so thatthe topological excitations can be studied on the flat partof the Gaussian support beam.The real space detection gives access to the polaritondensity in the plane of the cavity. The reference beaminteracts with the signal of the cavity and the resultinginterferogram is used to reconstruct the phase map. Allthe other parameters of the excitation are coming fromthe momentum space data, acquired via a spectrometer.The sample under investigation is a GaAs/AlGaAs mi-crocavity with 21/24 (front/back) layers of DBR andIn . Ga . As quantum wells at each of the three antin-odes of the confined electromagnetic field [17]. A smallwedge is inserted between the Bragg mirrors during thefabrication process allowing to precisely select the cavity-exciton detuning. The half Rabi splitting is 2.55 meV. Thepolariton mass, extracted from the dispersion, is 7 · − free electron mass. The polariton lifetime is about 14 ps.The experiments are performed in a liquid helium cryostatat cryogenic temperature in transmission configuration.The results when both seed and support are sent to-gether are shown in Figure 6. As predicted, a shadowappears in the wake of the defect, which corresponds ex-p-4ontrolled Quantum Turbulence in polariton superfluids Fig. 5: Experimental setup. The CW Ti:Sa laser beam is splitinto three using half waveplates (HWP) and polarizing beam-splitter (PBS). A reference beam, in light red, is set aside forinterferogram on the detection part; the seed beam in yellowis focused on the sample to a spot of 30 microns diameter; thesupport in orange is extended and elongated in the verticaldirection by cylindrical lenses (CL) and sent into the micro-cavity. The seed and the support, not centered to one another(see inset), share the same wavevector. The detection is donein real space, from which we can get information on the den-sity and phase maps, and also in momentum space through thespectrometer. actly to the time integrated image numerically calculatedand presented in figure 4. The pairs of vortex-antivortexare generated around the defect and follow the flow, lead-ing to a decrease of density along their propagation pathon the time integrated image.
Fig. 6: Vortex stream generation. a. Time integrated densitymap of a flow of vortex pairs generated in the wake of a defect.As the vortices move along the flow, the time integrated imageresults in a blurried density dip in the wake of the defect. b.Interferogram of the previous image, showing phase irregulari-ties along the vortices propagation. c. Visibility map extractedfrom the interferogram, displaying a lower visibility along thevortex stream.
To confirm that this shadow is indeed due to the pres-ence of vortices, an interferogram has been realized as wellas a visibility map, displayed on figures 6b. and c. As thevortices are moving along the flow, one cannot expect toobserve forks, the typical signature of vortices, due to thelow time resolution of the detection (the integration timeis typically of few milliseconds, while the flow speed is about 1 µ m/ps). However, the presence of the vorticesis visible on the phase map as a blur of the fringes alongtheir path.In order to further insure that the vortex generationis responsible for the shadow, a visibility map has beenextracted from the interferogram and shown in figure 6c.If the irregularities of the phase pattern are indeed a phaseblur due to the flow of vortices, the fringes visibility shoulddecrease. This is exactly what is observed: the shadowand the phase variations coincide with a dip in visibility,confirming its attribution to the vortex stream.Another interesting feature visible in this figure is thatat the top end of the fluid, the stream separates into twothinner and darker lines. This is due to the fact thatin this region, vortices merge together and become greysolitons. Indeed, the vortex core size is controlled by thehydrodynamic properties of the system, and in particularits healing length ξ , defined as [2]: ξ = ¯ h √ m ∗ ¯ hgn with m ∗ the effective mass of the polariton and the prod-uct gn the polariton interaction energy. On the top partof figure 6 a. the density of the non linear fluid on the up-per bistability branch is slowly decreasing. This densitydecrease has a direct impact on the healing length whichinversely increases, leading to vortex cores bigger and big-ger: eventually vortices merge together into a grey soliton.A precise phase jump is indeed visible at the correspondingposition of the interferogram. It can also be explained interms of Mach number, as the decrease of density inducesa decrease of the sound speed and therefore an increase ofthe Mach number: the flow becomes supersonic there. Supersonic flow: parallel dark soliton pair gen-eration. –
In order to study the generation of solitons,the supersonic configuration is investigated. According toprevious studies in polaritons superfluid [6], high Machnumbers allow for the generation of dark solitons. How-ever, the system configuration, without the support beam,did not allow for long propagation: the quasi-resonantpump beam was sent upstream of the defect and the soli-tons could only be observed for short distances (around30 micrometers), limited by the exponential decay of thepolariton density due to the finite polariton lifetime.The goal here is to use the configuration of seed-supportexcitation to generate dark solitons propagating over longdistances and to enable the study of their hydrodynamicbehaviour.The setup of this experiment is similar to the previousone for the vortex generation, with two co-propagatingbeams sent to the cavity: the seed, localized and intense,and the support, extended and with an intensity withinthe bistability cycle.The position of the seed and the support beams on thesample are shown in figure 7. The seed, shifted comparedto the support center, is placed upstream of the consideredp-5. Maˆıtre et al.
Fig. 7: The support beam is elongated along the vertical direc-tion and its intensity is within the bistability cycle. The flowis from bottom to top, and a structural defect is placed in thecenter of the support. Upstream to it is sent the seed, localizedand with an intensity above the bistability cycle. The distri-bution of the total intensity along the vertical axis is displayedon the right as well as the upper and lower bistability limits. structural defect - the flow is from bottom to top. Thecurve on the right shows the distribution of the intensity inthe cavity along the vertical axis. The region close to theseed is above the upper bistability limit, while the mainpart of the support is inside the cycle. The combinationof the two beams ensures the fluid to be bistable and onthe upper branch of the cycle.To generate dark solitons in the wake of a defect, super-sonic conditions need to be created. The in-plane wavevector is chosen to be high ( k = 1 . µ m -1 ) in order to en-sure a high velocity of the fluid: in this case, v f = 1 . µ m/ps. The speed of sound is extracted through theenergy renormalization and is measured to be c s = 0 . µ m/ps: the supersonic conditions are reached. Fig. 8: Spontaneous generation of a pair of parallel dark soli-tons. Intensity (left) and interferogram (right) of a dark solitonpair spontaneously generated in the wake of a structural de-fect. The solitons propagate after a short distance align andpropagate parallel for over a hundred microns.
The figure 8 presents the observation of a spontaneousgeneration of dark solitons in the wake of a defect: inten-sity on the left and interferogram on the right. The firstconclusion that can be done from those results is that thepropagation length is indeed greatly enhanced. The scalebar illustrates 20 microns: the solitons are sustained formore than hundred microns, one order of magnitude morethan previously reported.Moreover the solitons have a surprising behaviour. Theyare generated at the same position, close to the defect,and propagate away from each other for a few microns;eventually, they reach an equilibrium separation distanceof about 8 microns, align and stay parallel as long as theyare sustained. The observation of such a bound state ofdark solitons is quite unexpected as solitons have repulsiveinteractions and usually constantly repel each other [18].The observed solitons are fully dark: the intensity dipgoes to zero and the phase jump across the solitons isnearly π all along the propagation, confirming their trans-verse velocity is close to zero.To fully understand the phenomenon taking place inthe system, numerical simulations based on the coupledequations of the excitons and cavity photons fields, ψ X and ψ γ were realized to reproduce the solitons behaviour.The presence of a structural defect is modeled in theequations, by the term V ( r ), representing a 10 meV po-tential barrier, with a Gaussian shape of 10 µ m width. Thespatial profiles of the seed and support are also accuratelymodeled through the pumping term. Fig. 9: Simulation of the spontaneous generation of dark paral-lel solitons. Using the driven-dissipative Gross-Pitaevskii equa-tion, intensity (left) and interferogram (right) simulation of adark soliton pairs spontaneously generated in the wake of adefect.
The results of the numerical simulations are presentedin figure 9. They show an excellent agreement with theexperimental data of figure 8. The left image is the po-lariton density map, switched on to the upper branch ofthe bistability cycle by combination of both the localizedintense seed and the bistable extended support. The in-terference pattern on the right exhibits a clear phase jumpall along the propagation, confirming the solitonic naturep-6ontrolled Quantum Turbulence in polariton superfluidsof the intensity pattern. As expected, the solitons are sus-tained for a macroscopic distance and stay parallel to oneanother during their propagation.
All-optical imprinting of dark soliton moleculesin a polariton superfluid. –
We described in the pre-vious section how to use the bistable behaviour of thepolariton system to enhance the propagation length ofa polariton superfluid, simultaneously getting rid of thephase constraint of the pump. This allowed us to observethe spontaneous generation of quantized vortices and darksolitons and their propagation for over a hundred microns.However, their generation was not controlled, since it de-pends on parameters out of reach; in particular, the nec-essary presence of a structural defect to induce the turbu-lence leading to the topological excitations [16, 19].The goal in this section is to show how to overcomethis limitation and to be able to generate solitons on de-mand. It is realized by imprinting a phase modulationon the system, leading to the formation of dark solitonsthat can evolve freely on the nonlinear fluid. Such con-trolled impression of solitons provides many tools for thedetailed study of their hydrodynamic behaviour, as themain parameters of the soliton pattern, such as their shapeand position, can be tuned at will. It results once againin the unexpected binding mechanism between the im-printed solitons, leading to the propagation of a dark soli-ton molecule. [20].The main difficulty of this experiment is to combine tworegions of the systems which exhibit a different behaviour:namely a region whose phase is imposed by the pump anda free propagation area, where despite to the fact that thephase of the pump beam is flat, yet the system can developand sustain topological excitations.
Fig. 10: Beam intensity and bistability. Left, spatial distribu-tion of the excitation beam. The white dashed line illustratesthe position of I high , upper intensity threshold of the bistabilitycycle, as shown in the intensity profile on the center image. Onthe right, the theoretical bistability cycle with the definition ofI high and I low
Once again, it is achieved by using the properties of the optical bistability. Indeed, the input intensities abovethe bistability cycle impose their phase to the fluid, whilethis constraint is released for the input intensities withinthe bistability cycle. The two situations are obtained byexploiting the gaussian shape of the excitation beam, asillustrated in figure 10. The spatial distribution of thebeam is plotted on the left, where the white dashed lineindicates the position of the high intensity threshold ofthe bistability I high (see right picture). All the area in-side this circle is above the bistability cycle, as shown inthe profile in the center. Therefore, it fixes the phase ofthe fluid and corresponds to the effective impression re-gion. Outside of this circle, most of the beam intensitiesare within the bistability cycle: the phase is not imposedanymore and the system is able to sustain the free prop-agation of topological excitations. The beam is elongatedin the y direction in order to flatten its profile and extendthe bistable region where the solitons free propagation willbe studied.As the solitons induce a phase jump on the system, theirimplementation can be done by modeling the phase of theexcitation. To do so, we use a Spatial Light Modulator, aliquid-crystal based device that can shape the wavefrontof an incident light beam.The phase modulation induced by dark solitons is aphase jump of π : the phase profile corresponding to apair of dark solitons is thus an elongated region π -shiftedcompared to the background beam.The figure 11 shows a typical phase pattern generatedby the SLM on the beam wavefront: it results, as desired,in a rectangular phase jump of π which consequently in-duces an intensity dip. The dashed line delimits the regionof the beam which is above the bistability cycle. Fig. 11: SLM profile. Typical π -shifted rectangular shape gen-erated by the SLM on the excitation beam. Its position andshape are easily tunable. The dashed line delimits the regionof the beam which is above the bistability cycle. The experimental setup is sketched in figure 12. Thelaser source is a Titanium Sapphire, and its spot is elon-gated in the y direction by two cylindrical lenses (CL).The beam is then split in two by a polarizing beam split-ter (PBS) preceded by a half-wave plate (HWP), whichallows for a precise control of the power sent in each arm.The phase front of the main beam is shaped by the SLM,then is filtered by the slit to smooth the phase jump in thedirection of the flow. This beam is sent collimated to thep-7. Maˆıtre et al. cavity, so that the phase jumps are well defined onto thesample. The inset illustrates the excitation beam configu-ration on the sample: the solitonic pattern is placed in thecenter, where the intensity is above the bistability limit,delimited by the white dashed line. The beam enters thecavity with an appropriate in-plane wave vector that givesan upward flow to the polaritons. The black rectangle isthe detection field of view: it is shifted on top of the il-luminated region to observe the solitons free propagationthrough the bistable area. Fig. 12: Experimental setup. The excitation beam is designedby the SLM and filtered through the slit. It is collimated on thesample, imaging the SLM plan. The inset illustrates the beamconfiguration: the solitonic pattern is in the center, where theintensity is above the bistability limit, located at the whitedashed line. The detection field of view is delimited by theblack rectangle, shifted from the center in order to focus on thebistable region and the solitons free propagation. The detectionis done in real and momentum space, so that the experimentalconditions can be associated with the corresponding intensityand phase maps.
As usual, the detection is done in both real and momen-tum space. The real space gives the intensity map of thecavity plan, as well as information on the phase patternthrough the interference with the reference beam previ-ously separated from the laser beam. The experimentalconditions of the system are extracted from the momen-tum space images.The essential role of the bistability in the propagationof the solitons is explained in figure 13. The figure a. re-minds the S shape of the bistability curve and the threeassociated intensity regions: below the cycle, the low den-sity region in grey, denoted as LD; above the cycle, thehigh density region highlighted in yellow and denoted asHD; and the bistable cycle left blank.On b. is plotted the phase pattern designed by the SLM,with the same field of view as the detection: we can seethat the phase modulation is only present on the bottompart of the images. Figures c. and d. are realized in theexact same conditions except for the total intensity of theexcitation. In c., the total laser power is high, which putsalmost all the illuminated area above the bistability cycle:the yellow HD region covers the major part of the picture.
Fig. 13: Impression of dark parallel solitons. a. Theoreticalbistability profile and the three associated intensity ranges. b.SLM phase pattern with the same field of view as the detection:the phase modulation are only sent on the bottom part of theimages. c. High power density and phase maps. Almost all theilluminated area is in the HD regime: the phase is fixed by thedriving field and replicates the pattern designed by the SLM. d.Same configuration as c. at lower power. The bistable regionhas extended toward the beam center, and reached the toppart of the solitons. The dark solitons propagate through thebistable area, until the low density region where the nonlinearinteractions are too low to sustain them.
In the high density region the properties of the fluid arefixed by the pump: this area is therefore a replica of thedriving pump field. Indeed, the solitons are imprinted onlyin the bottom part of the picture, while on top, the phaseand the intensity of the pump beam are flat.Figures d. are obtained from the c. ones by gradu-ally decreasing the input intensity. The bistable regionexpands toward the center of the beam, and eventuallyreaches the top part of the imprinted solitons. The solitonsthen propagate through the bistable region, even thoughthe region between the solitons is out of phase with thedriving field. Indeed, the dark solitons in d. are clearlyvisible within the bistable region, inducing a phase jumpof π all along their propagation. The propagation is sus-tained as long as the system is in the bistable regime. Asthe illuminated region is finite, the solitons will reach itsborder: in the low density region, the nonlinear interactionare too low to sustain dark solitons.To find the good configuration for the solitons to propa-gate through the bistable region, several parameters needto be finely tuned. In particular, the total intensity ofthe pump has an important impact on the soliton propa-gation, as they need a bistable fluid to be sustained andpropagate.To qualitatively study the influence of the pump inten-sity, several images are recorded for different values of thetotal pump intensity. The results are presented in figure14. The SLM phase pattern is presented on a., again withthe same field of view as the detection images, and theexperimental images are shown in b. The top line showsthe density maps and the bottom one the interferograms.The input power is gradually decreased from picture (i) to(vi). The flow is from bottom to top, and the yellow col-ored regions indicates the area above the bistability cycle.In picture (i), the total power is high: the fluid is abovep-8ontrolled Quantum Turbulence in polariton superfluids Fig. 14: Scan of the input power. a. Phase pattern designedby the SLM with the same field of view as the detection. b.Density (top) and phase (bottom) maps for different valuesof the total input intensity. On (i), the power is maximumand the phase is fixed everywhere. The intensity graduallydecreases and with it the size of the HD region in yellow: thesolitons propagate and open, until the bistable part reachesthe imprinted solitons in (vi) where they align and propagateparallel. the bistability cycle on the whole picture. Its phase istherefore fixed, consequently the solitons are imprintedonly in the bottom part of the image, where the phasefront of the beam is shaped by the SLM. From picture (ii)to (v), as the power decreases, the bistable region expands.The solitons propagate further and further but the systemstill can not perfectly sustain them: they are grey as theirphase jump is lower than π and they open and vanishalong the flow. The phase maps confirm as well that thephase modulation induced by the solitons vanishes withthem.Finally, in picture (vi), the bistable area joins the toppart of the imprinted solitons. They are then able toalign to each other, and to remain dark and parallel allalong their propagation. In this case, their phase jumpis π and stays constant. They are sustained through thewhole bistable region, and vanish only at its edge, wherethe system jumps to the low density regime.This set of measurement clearly confirms the necessityto be in the bistable regime (namely inside the bistabilitycycle) to achieve the free propagation of dark solitons ina resonantly pumped polariton fluid.The imprinting method provides a large flexibility onthe shape of the imprinted phase pattern: the SLM pat-tern is easily tuned and several other configurations canbe implemented, in order to study the solitons behaviour.In particular the imprinting technique is fully scalable andallows generating stable pattern of multiple solitons, whichopens the way of the study of soliton lattices in polaritonsuperfluids.As an example of the scalability of such method, we Fig. 15: Four solitons imprinting. a. SLM phase pattern corre-sponding to the detection region. b. Density map of the fluid.The two solitons pairs are imprinted on the yellow region andpropagate through the bistable white one. c. Interferogram ofthe fluid. The phase jump propagates with the solitons. show in figure 15 a double pair of solitons. On the left,figure a. is a scheme of the SLM phase pattern. It rep-resents the top part of the beam, coinciding with the cor-responding detection pictures, plotted on figure b. andc. and showing the density map and the interferogram,respectively.The double pair of solitons is realized by sending tworectangular shapes in phase opposition with the back-ground thanks to the SLM. The imprinted phase patternis designed so that each of the four solitons is equidistantfrom its neighbor (yellow part of figure 15.b.). Duringtheir free propagation in the bistable region of the fluid(grey part of figure 15.b.), they get closer to their respec-tive pair, so that the area in phase with the driving ex-pands while the one in phase opposition is reduced.
Snake instabilities, soliton breaking and vortexstreets. –
In the experiments described in the previoussection we decided to generate and study stable solitonsand for this purpose we deliberately chose to work in asupersonic regime with high speed polariton flows [21].However, the flexibility of the imprinting technique andthe full control of the fluid velocity that can be achieved,easily allow one to imprint dark solitons on polariton flu-ids with deeply subsonic velocities. In these conditions,it is well known that dark solitons are unstable againstthe snake instabilities and break into quantum vortex-antivortex pairs, a behaviour which is a quantum analog ofthe classical von K´arm´an vortex street [22]. Therefore theimprinting technique can be exploited for the systematicstudy of quantum turbulence.As an example of such possibilities, in this section weinvestigate a new configuration to generate solitonic pat-tern within a static polariton fluid. It uses a transverseconfinement within an intensity channel to create a pairof dark solitons, which decays into vortex streets due tothe disorder of the system. We observe the soliton snakeinstability leading to the formation of symmetric arraysof vortex streets, which are frozen by the pump-inducedp-9. Maˆıtre et al. confining potential allowing their direct observation and aquantitative study of the onset of the instabilities.The setup we used for this experiments is quite simi-lar to the one of the soliton impression, presented in theprevious section. Two main differences have yet to be no-ticed. First of all, to facilitate the development of theinstabilities, the experiment needs a static fluid: the ex-citation is therefore sent at normal incidence, and the in-plane wavevector is zero. Then, the 1D elongated channelswhich confine the dark solitons are created by shaping theintensity of the excitation beam, while its phase is notmodulated anymore.Indeed, although the Spatial Light Modulator (SLM)usually shapes the phase front of the incoming light beam,we can also use it as a intensity modulator, and thus de-sign at will an intensity pattern on the beam. To do so,a grating with a controlled contrast is imprinted in a spe-cific region of the pumping beam by the SLM. When thefull range of the SLM grey scale is used for the gratingimpression, all the light is diffracted to the grating firstorder. However, by using a fraction of the scale, only aportion of light is diffracted, resulting in a darker regiondug inside the non diffracted zero order beam. This way,by tuning the grating contrast of the SLM screen and fil-tering out the first order in the Fourier plane, we can shapea beam with a designed intensity pattern. Again, as it iscontrolled via a computer, the shape is easily tunable.The detection is realized in both real and momentumspace. The excitation conditions are extracted from themomentum images, while the real space gives us the den-sity map as well as the phase from interferences with areference beam.We consider a channel with both the ends closed byhigh-density walls (Figure 16). The channel width of 23 µ m is chosen in order to generate a single dark soliton pair,which evolves toward a stationary frozen vortex street dueto the snake instability.In figure 16, the panel (a) shows the measured inten-sity and phase distributions in the channel for increas-ing ratios of the intensity inside and outside the channel,called S and P respectively. The panels (b) shows thecorresponding numerical simulations obtained by solvingthe system of coupled equations. A symmetric array ofvortex-antivortex (VA) pairs, similar to a Von-Karmannvortex street is visible for S/P = 0 . .
4, respec-tively. A soliton pair is indeed unstable in this regime[23] against modulational ”snake” instability. It thereforebreaks into the observed VA chain. In free space, thesechains would dynamically evolve [24] to eventually disap-pear. Remarkably, the presence of the confining poten-tial allows to freeze the snake structures at a given stageof their evolution and to easily observe them in a steadystate CW experiment.The particle density in the channel is associated withthe healing length of the fluid ξ which sets the dark solitonwidth and the vortex core size.It also naturally sets the spatial period at which the Fig. 16: Panel (a): experimental fluid density and phase mapsfor a channel with length and width respectively l = 150 µ m, L= 23 µ m; the ratio S/P increases from left to right. Panel (b):corresponding numerical simulations. Panel (c): Evolution ofthe number of VA pairs N top in the channel for different valuesof 1 /ξ . The black dots are the experimental data. The yellowcurve gives the simulations results under the same experimentalconditions whereas the blue curve shows the vortex density foran infinite channel. instability develops along the main channel axis [23] andtherefore the number of VA pairs which appear in a chan-nel of a finite length L . The panel (c) of figure 16 shows thenumber of pairs experimentally observed versus 1 /ξ and inred, the theoretical value obtained by numerically solvingthe system of the coupled equations, which are in excel-lent agreement. They both show a step-like increase dueto the quantization imposed by the finite channel length.The blue line shows the expected number of vortices perlength L in an infinite channel which is proportional to ∼ /ξ . Conclusions. –
In this article we have reviewed ourrecent results on the polariton optical bistability and itsrelated properties.By implementing the theoretical proposal [11] wedemonstrated the possibility to use bistability exhibitedby the polariton system under resonant pumping togetherwith a pump-support configuration to sustain topologicalexcitations such as quantized vortices and dark solitons[16, 19] over hundreds of microns, greatly enhancing theirp-10ontrolled Quantum Turbulence in polariton superfluidspropagation length compared to the previous observations[6].This experiment also revealed a very unexpected be-haviour of the dark solitons: the presence of the drivingfield imposed by the bistable pump compensates the darksolitons repulsion as usually observed in an undriven sys-tem [6, 25]. In this new configuration, dark solitons alignto each other and propagate parallel. Moreover, to achievea full control on the formation of the topological excita-tions we developed a new all optical imprinting methodby accurately shaping the excitation beam with a SpatialLight Modulator, and we managed to generate on demanddark solitons on a polariton fluid [20]. Once again, due tothe presence of the driving field, we observed this bind-ing between the solitons, propagating parallel as a darksoliton molecule.The flexibility of the imprinting method opens the wayto deeper studies of quantum turbulence phenomena. Inparticular we generated solitonic structures in guided low-density channel in a static polariton fluid [23] and observedtheir breaking into vortex streets due to transverse snakeinstabilities [26].The imprinting technique could give significant contri-butions to elucidate open questions as the formation ofturbulent cascades in the presence of dissipation at alllength scales [27], as in polaritons, and the validity of theentropy arguments that explain the formation of inversecascades in equilibrium two-dimensional systems [28–30]. ∗ ∗ ∗
This work has received funding from the French ANRgrants (”C-FLigHT” 138678 and ”Quantum Fluids ofLight”, ANR-16-CE30-0021), from the ANR program ”In-vestissements d’Avenir” through the IDEX-ISITE ini-tiative 16-IDEX-0001 (CAP 20-25), from the EuropeanUnion Horizon 2020 research and innovation programmeunder grant agreement No 820392 (PhoQuS). QG, AB,and DS thank the Institut Universitaire de France (IUF)for support. SVK and DDS acknowledge the support fromthe Ministry of Education and Science of the Russian Fed-eration (0791-2020-0006).
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