Generation, propagation and control of quantized vortices and dark solitons in polariton superfluids
TTHÈSE DE DOCTORATDE SORBONNE UNIVERSITÉSpécialité : Physique
École doctorale nº564: Physique en Île-de-France réalisée au Laboratoire Kastler Brossel sous la direction de Alberto BRAMATIprésentée par
Anne MAÎTRE pour obtenir le grade de :
DOCTEUR DE SORBONNE UNIVERSITÉ
Sujet de la thèse :
Generation, propagation and control of quantized vortices anddark solitons in polariton superfluids soutenue le 27/11/2020 devant le jury composé de :
Mme Maria CHAMARRO ExaminatriceM. Cristiano CIUTI ExaminateurM. Massimo GIUDICI RapporteurM. Thierry GUILLET RapporteurM. Pavlos SAVVIDIS ExaminateurM. Alberto BRAMATI Directeur de thèseM. Simon PIGEON Invité a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b bstract Exciton-polaritons are quasi-particles arising from the strong coupling regime betweenexcitons and photons. In planar microcavitites, phenomena such as superfluidity or Bose-Einstein condensation can be observed. Those systems have demonstrated to be very effi-cient in the hydrodynamic generation of topological excitations, such as vortex-antivortexpairs or dark solitons. However, the lifetime and motion of those excitations were limitedby the driven dissipative nature of the system.In this thesis, we present a rich variety of results about the generation and control ofsuch topological excitations. Taking advantage of the optical bistability present in oursystem, we were able to greatly enhanced the propagation length of vortices and solitonsgenerated in the wake of a structural defect, revealing in the mean time an unexpectedbinding mechanism of the solitons which propagate parallel. This behaviour was recoveredin a specifically designed experiment, where we artificially imprint dark soliton pairs ondemand on a polariton superfluid. The adaptability of our technique allowed for a detailedstudy of this phenomenon, that we directly connected to the driven-dissipative natureof our system. Finally, confined dark solitons were generated within guided intensitychannels on a static polariton fluid. The absence of flow lead to the development oftransverse snake instabilities of which we studied the interesting properties.1 ésumé
Les polaritons excitoniques sont des quasi-particules créées par le régime de couplage fortentre des excitons et des photons. Dans des microcavités planaires, des phénomènes telsque la superfluidité ou la condensation de Bose-Einstein ont pu être observés. Ces sys-tèmes ont démontrés être particulièrement appropriés pour la génération hydrodynamiqued’excitations topologiques comme des paires de vortex-antivortex ou de solitons sombres.Cependant, le temps de vie et la propagation de ces excitations étaient limités par ladissipation du système.Dans cette thèse, nous présentons une succession de résultats sur la génération et lecontrôle de telles excitations topologiques. En utilisant la bistabilité optique de notre sys-tème, nous avons fortement augmenté la distance de propagation de vortex et de solitonsformés dans le sillage de défauts structurels, ce qui a également révélé un comportementinattendu des solitons liés qui restent parallèles. Ce comportement a été confirmé parl’expérience suivante, où nous avons artificiellement imprimé des paires de solitons som-bres dans des superfluides de polaritons. La flexibilité de notre technique nous a permisd’étudier ce phénomène en détails et de le relier directement à la dissipation du système.Enfin, des solitons sombres ont été produits dans des canaux d’intensités dans un fluidestatique. L’absence de flux a permis le développement d’instabilités transverses ou snakeinstabilities dont nous avons étudié les propriétés.3 emerciements
Avant de rentrer dans le vif du sujet de ma thèse, je voudrais prendre le temps deremercier tous ceux qui ont également participé à ce travail et sans qui mes résultatsn’auraient certainement pas été ce qu’ils sont aujourd’hui.Tout d’abord, j’ai eu la chance d’être encadré par Alberto Bramati, qui a très rapi-dement su m’intégrer à l’équipe et me mettre en confiance sur mon sujet de thèse. Saporte a toujours été ouverte et il a su nous guider sur la bonne voie à chaque étape de nosprojets, prenant toujours en compte nos remarques et nos intuitions. J’aimerais en parti-culier le remercier pour sa patience, sa gentillesse et sa bonne humeur qui ont grandementparticipé à garder ma motivation à son maximum durant ces trois années.J’ai également été énormément conseillé par Élisabeth Giacobino et Quentin Glorieux,qui tout en suivant nos projets d’un peu plus loin, ont toujours été disponibles et ont suprendre le temps de nous aider à résoudre nos problèmes. J’aimerais également remercierparticulièrement Simon Pigeon, qui a travaillé avec nous pendant presque toute la durée dema thèse sur la partie théorique des projets de polaritons, et dont je suis très heureuse qu’ilait accepté de faire partie de mon jury. Il a lui aussi toujours su garder sa porte ouverteet a énormément participé à ma compréhension de la théorie en répondant patiemmentà mes questions plus ou moins pertinentes.I would also probably not have been able to achieve this work if it was not for GiovanniLerario. During his time as a post-doc with us, he taught me how to build and run anoptical experiment, how to get information on your system from tiny details you wouldn’thave notice, but also useful Italian words, quite interesting psychedelic music and manymysterious chess moves. I am very glad to have had the opportunity to work with himand to rely on his knowledge to make me understand the bigger picture, even after he leftthe group and until the very last night before my defense.J’ai eu aussi l’occasion d’interagir régulièrement avec nos collaborateurs de l’InstitutPascal à Clermont-Ferrand, et en particulier avec Sergei Koniakhin, qui a réalisé sa thèsethéorique simultanément à la mienne, a réalisé beaucoup de simulations de nos travaux etest même venu passer quelques jours en salle de manip pour découvrir avec nous la partieexpérimentale de nos recherches. Je laisse maintenant les expériences de polaritons entrede bonnes mains, celles de Ferdinand, qui a également beaucoup participé à ce travail,5aujourd’hui accompagné de Maxime et dont je suis sûre qu’ils obtiendront encore de beauxrésultats. J’ai également une pensée pour tous les membres du groupe qui se sont succédésdurant ma thèse, et qui ont tous contribué à la bonne ambiance qui y régnait : Maxime,Tom, Popi, Stefano, Lorenzo, Chengjie, Rajiv, Murad, Huiqin, Thomas, Marianna, Weiet Tanguy.Une thèse expérimentale dépend aussi beaucoup de l’environnement dans lequel elle estréalisée, et nous sommes très bien entourés au laboratoire Kastler-Brossel. Je voudraisdonc remercier tous ceux qui contribuent à nos recherches d’un peu plus loin, au serviceadministratif, à l’atelier mécanique ou à l’atelier d’électronique. Enfin, je pense quema motivation a aussi été renforcée par les amitiés que j’ai pu créer au-delà de notreéquipe, et c’est pourquoi j’ai également une pensée pour Tom, Tiphaine, Yohann, Ferhat,Michaël, Arthur, Paul, Thomas, Adrien, Jérémy, Félix, et tous ceux que j’ai croisé unpeu moins régulièrement, pour ces longues discussions animées autour d’un café ou dequelques bières.Enfin, je sais que ma thèse a parfois occupé mes pensées en dehors des murs du labora-toire, et j’ai été très heureuse de la patience et de la très grande curiosité de mes procheset amis quant à mes recherches (pas encore de sabre laser, ce sera pour la prochaine fois).Je tiens donc à les remercier tous pour leur soutien, que ce soit les anciens mini ou leshabitués du BDF qui ont suivi ça très régulièrement, mais aussi les petits merdeux, lescrazy Wednesdays ou encore Solène, Manu et Jeanne, avec qui les emplois du temps sontplus difficiles à accorder mais les retrouvailles toujours aussi intenses.Bien évidemment, je terminerai par remercier ma famille, mes grandes sœurs, mon pèreainsi que mes adorables neveux qui ont chacun à leur manière activement participé à marédaction confinée. ontents
Introduction 91 Exciton-polaritons quantum fluid of light 13
CONTENTS
Conclusion 143Communications 145Bibliography 147 ntroduction
The concepts of light and matter are fascinating topics whose description stronglyevolved over history. Light in particular remained an enigma for physicists for centuries,before the debates over its wave or particle nature were definitely settled by Louis deBroglie in 1923, who theorized the light matter-wave duality [1]. It became the startingpoint of quantum mechanics, which the basis were developed in the following years [2].This new mechanics allows both light and matter to have similar behaviour. In partic-ular, the light quanta or photons follow a bosonic statistic, identical to the one describingatoms with integer spins. They can have therefore identical properties, like the ability tocondense in a single state, leading to a macroscopic quantum state where all particles areindistinguishable, called a Bose-Einstein condensate.An important difference still remains between light and matter in the fact that pho-tons do not interact in vacuum. However, photons can interact with themselves whenpropagating in a non linear medium. Such effects have been observed with the discoveryof the laser: a sufficiently high intensity beam passing through a material medium canperturb it; and this perturbation also induces modifications on the light field. In particu-lar, nonlinear effects such as self focusing and self defocusing can be observed , which areequivalent respectively to an effective photon-photon attractive or repulsive interaction.This interactions between light and matter can be strongly enhanced by correctly adapt-ing the environment. One possibility is for instance to confine an electromagnetic mode inorder to couple it to the energy transition of the material emitter. This can be achieved byplacing the emitter inside an optical cavity, whose resonance corresponds to the transitionfrequency [3]. If the system losses are low enough, a weak coupling is established whenthe confined light changes the emission properties of the transition, known as the Purcelleffect [4]. The coupling can even be enhanced to the point where the rate of coherentexchange of energy between light and matter is higher than their decay rates [5]. Thesystem enters then the strong coupling regime where the initial states splits into two: thischaracteristic energy anticrossing is proportional to the coupling strength and is calledthe Rabi splitting.Such optical cavities can be developed in many geometries, although we have focusedour work on a particular type of material known as semiconductors. Those insulators90
CONTENTS become conductive when electrically or optically excited. Electrical excitation have beendeeply studied and developed in the electronic domain; but what interests us in this workis the optical excitation and the fast response and strong control it offers. In particular,the development of optically controlled semiconductor heterostructures opened the wayto many possibilities of light-matter interaction.In the end of the twentieth century, new techniques of heterostructures growth reachedthe nanometer scale of layers thickness. This allowed to design optical microcavities,in which were embedded planar quantum wells. It is in such a structure that ClaudeWeisbuch observed for the first time in 1992 the strong coupling between light and matterin semiconductors [6]. This configuration, previously theorized by John J. Hopfield in1958 [7], results in the appearance of bosonic quasiparticles known as exciton polaritons ,made of a superposition of cavity photons on one hand and electron-hole pairs on theother hand.Polaritons are of particular interest as they inherit properties from both their con-stituents. Their photonic part gives them a low effective mass, while the excitonic oneensures strong interactions and non-linearity. They have been intensively studied in thepast few decades, so that their essential properties are well known, such as their dispersioncurve [8], lifetime [9] or relaxation [10]. All those properties allow these systems to exhibitinteresting phenomena, like Bose Einstein condensation at high temperature [11, 12], fourwave mixing [13, 14] or squeezing [15, 16].Excitons polaritons also appear as ideal candidates to study the hydrodynamic prop-erties of quantum fluids. Indeed, many parameters are easily accessible and controlledin those systems, such as momentum, density and phase which have already lead to theobservation of superfluidity [17–20] and of topological excitations [21–24].Those last ones in particular are the guiding line of this thesis work. The differentexperiments indeed followed each others in the study of those phenomena, and moreimportantly on their sustainability and control. This thesis is therefore organized asfollowed.The first chapter focuses on the description of our system. We first study indepen-dently its two components: the light part with the photons confined within a planarmicrocavity, and the matter part represented by the quantum well excitons. We thencombined these two together by embedding a quantum well inside a microcavity, whichallows to reach the strong coupling between the two oscillators, and therefore to obtainthe polariton quasiparticles.We focus then on the two main ways of exciting such a system, namely resonantly ornon-resonantly, and describe how they influence the polaritons properties and the fluidbehaviour.The second chapter details the experimental tools available in the lab that we usedto implement our work. The first part lists the devices used in the excitation part in
ONTENTS
Chapter 3 begins by reporting a theoretical proposal [25] which is the starting point ofthe experimental work of this thesis. It suggests to use the property of optical bistability,present in our system due to a strong non linearity, to get rid of the phase constraintusually imposed by the pump in a quasi-resonant excitation scheme. To exploit thisphenomenon, a novel implementation is suggested, using two excitation beams and leadingto a large fluid area within the bistability cycle.This idea was put in application experimentally to generate and sustain topologicalexcitations, such as vortex-antivortex pairs or dark solitons [26, 27], depending on the hy-drodynamic conditions of the system. They form spontaneously in the wake of a structuraldefect hit by a nonlinear flow of polaritons. A comparable configuration had previouslybeen implemented [22], but the solitons were generated within the decaying tale of theexcited polaritons, resulting in a short propagation distance through an intensity decay-ing fluid. Our configuration not only greatly enhances the propagation distance, but alsohighlights an unexpected behaviour of the dark soliton pairs, which under the influenceof the driving field align to each other and propagate parallel. This phenomenon is intotal opposition to all former results in atomic quantum fluids as well as in polariton sys-tems. Indeed, the only observation of an attractive behaviour between dark solitons hasbeen realized in a thermo-optic medium with non-local interactions, whereas polaritoninteraction are based on the exciton exchange interaction and are fully local.Those surprising results still relies on the spontaneous generations of such turbulence,under specific conditions around a structural defect on which we do not have any con-trol. The next step of our work, presented in chapter 4 , was therefore to get rid theseconstraints and to artificially imprint dark solitons, to have full control over them andto lead deeper studies. A smart design of the excitation beam allowed us to artificiallycreate dark solitons within a polariton flow, and to let them propagate freely through abistable fluid.Once again, the soliton propagate parallel and experience a binding mechanism, lead-ing to the formation of a dark soliton molecule [28]. The scalability of our new techniqueallowed us to further investigate this unexpected behaviour. In particular, we observedthat the solitons always reach the same equilibrium separation distance, no matter theinitial conditions. We managed to understand this phenomenon as a consequence of thedriven-dissipative nature of our system. The implementation of this new method opensthe way to a fine control and manipulation of collective excitations, which could lead tothe generation of multiple soliton pattern or a quantitative study of quantum turbulencephenomena.We followed this direction in the fifth chapter , where we once again artificially createsolitons in our system, but within a static fluid. They are this time generated inside a2
CONTENTS low density channel, stabilized by the pressure from the high density walls [29]. Thisconfiguration allows the observation of transverse instabilities called
Snake instabilities ,inhibited by the fast flow in the previous implementations. Moreover, the behaviour ofthose solitons strongly depends on the ratio of intensities between the high and low densityregions, and on the connection of the channel to low density fluid surrounding the exciteddomain. A careful implementation can therefore leads to interesting phenomena such asthe resolution of a maze pattern. hapter 1
Exciton-polaritons quantum fluidof light
Exciton-polaritons are quasiparticles arising from the strong coupling between light andmatter. They are indeed a superposition of cavity photons and quantum well excitons.In order to discover all the tools of our system, we start this chapter by independentlydescribing the two elements of the polaritons. We will focus on particularities of photonwithin a planar microcavity, then study the consequences of bidimensional confinement onsemiconductor excitons. We will then be able to strongly couple those elements togetherand describe this new system of excitons-polaritons.On the second part of this chapter, we will extend our field of view to the ensembleof the polariton fluid: we will see how the excitation configuration influences the systemproperties, and describe the main regimes of such a fluid.134
CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
Polaritons are made of light and matter strongly coupled. To figure out how thosequasi-particles arise, it is essential to study their components independently and howtheir environment influences their behaviour. Then, by combining them together, we willbe able to understand their interactions and how to describe our new system.
This section focuses on the photonic part of the polaritons. As we ultimately want toachieve a strong coupling with some matter particle, the electromagnetic field needs tobe enhanced, which is done by confining it within a microcavity. i Optical microcavityMicrocavity properties
A cavity is an optical resonator where a standing wave is am-plified. The light is confined within the cavity through a particular mirrors arrangementand follows a well defined path: interferences take place and lead to the amplification ofsome specific modes.A microcavity is a particular case of optical cavity, which dimension is of the order ofthe light wavelength.Two parameters are mainly used to describe the optical cavity properties: the qualityfactor and the finesse. The quality factor or Q-factor characterizes the frequency widthof the field enhancement. It is defined as the ratio between the resonant frequency ω cav and the full width at half maximum (FWHM) of the cavity mode δω cav : Q = ω cav δω cav (1.1)The Q-factor describes the rate of the energy decay, which can be due to losses ofthe mirrors, absorption, or scattering on the cavity imperfections: Q − is the fractionof energy lost in one round-trip of the cavity [30]. Consequently, the photon populationdecays exponentially and its lifetime can be defined as τ = Qω cav (1.2)The finesse connects the FWHM of the cavity mode to the free spectral range (FSR), i.e. the frequency separation between two consecutive longitudinal resonant modes. Itcharacterizes the spectral resolution of the cavity. It can also be defined using the totalreflectivity R of the cavity: F = ∆ ω cav δω cav = π √ R − R (1.3)The frequency separation is connected to the total length L of the cavity: ∆ ω cav = πcL .Therefore, in the particular case of a microcavity, it is of the same order of magnitudethan the cavity mode frequency, so the finesse and the Q-factor are close. .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES I intracavity I incident ≈ − R = Fπ √ R (1.4)In a standing wave microcavity, the electromagnetic field is distributed in the form ofan interference pattern, creating therefore localized maxima and minima. Consequently,in order to couple an emitter within the microcavity, it has to be placed at a maximumof the field to enhance the coupling, as we will describe more in detail later. Planar microcavity
The type of cavity that we considered in this thesis is a
Fabry-Perot cavity . It is one of the simplest optical cavity and was invented by Charles Fabryand Alfred Perot in 1899. It consists in two planar mirrors, parallel and facing each other:the light with specific wavelengths is enhanced by going back and forth between them,due to constructive interferences.Fabry-Perot microcavities are usually built using Distributed Bragg Reflectors (DBR).Those mirrors consist in a pile of alternate layers of two materials with different refractiveindices n < n . They are designed for a specific wavelength λ : each layer has a thickness d i = λ n i ( i = (1 , λ , centered at λ [31]:∆ λ = 4 λ π arcsin n − n n + n ! (1.5)Figure 1.1 presents the reflectivity of a DBR made of 20 pairs of Ga Al As/AlAswith a stop band centered at λ = 850 nm. The optical indices of the materials are n Ga . Al . As = 3 .
48 and n AlAs = 2 .
95. The reflectivity is very close to one over a hundrednanometers range.Figure 1.1:
DBR reflectivity . Reflectivity of a Bragg mirror at normal incidence madeof 20 pairs of Ga Al As/AlAs and centered in λ = 850 nm. From [32]6 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
The Fabry Perot cavity is realized by placing two DBRs in front of each other, at adistance L cav = m λ n cav , where n cav is the refractive index of the intracavity medium and m an integer. For instance, figure 1.2 shows the reflectivity of a microcavity as a functionof the wavelength, at normal incidence. The cavity consists of two DBR identical to theone of figure 1.1, parallel to each other and at a distance L cav = λ m cav . A very sharp dipis visible at λ = λ = 850 nm, which corresponds to the cavity resonance.Figure 1.2: Microcavity reflectivity . Reflectivity at normal incidence of a Fabry Perotmade with two DBR identical to the one presented in figure 1.1 and separated by adistance L cav = 2 λ n cav . From [32] ii Cavity photons The term cavity photons is due to the particular properties imposed by the cavity tothe magnetic field, that we are going to develop.
Dispersion relation
The dispersion relation arises from the fact that the effectivelength of the cavity felt by the photons depends on their angle of incidence, as shown infigure 1.3. If the photons enter the cavity with an incident angle θ , they feel a cavity oflength L θ = L cav cos θ , which shifts the resonance energy.More quantitatively, the cavity imposes a quantization condition on the z componentof the photon wavevector k γ = ( k γ || , k γz ): k γz = 2 πn cav λ (1.6)There is no constraint on the parallel component of the wavevector, so the energy ofthe photons can be written as: E γ ( k γ || ) = (cid:126) cn cav vuut(cid:18) πn cav λ (cid:19) + ( k γ || ) (1.7) .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES Photon flow inside a planar cavity . The photons travel a distance L θ = L cav cos θ between the mirrors, which corresponds to another resonance energy.Considering that the orthogonal component of the wavevector is significantly largerthan the parallel one ( k γ || (cid:28) k γz ), this expression can be approximated with a parabolicdispersion: E γ ’ hcλ (cid:18) λ k γ || πn cav (cid:19) ! (1.8)This equation leads to the definition of an effective mass of the cavity photons m ∗ γ , connected to the curvature of the parabola:1 m ∗ γ = 1 (cid:126) ∂ E∂k (1.9)which gives us the expression: m ∗ γ = n cav (cid:126) λ c (1.10)Typically, in semiconductor microcavities, it is of the order of 10 -5 times the freeelectron mass. Photon lifetime
The lifetime of the cavity photons depends on the reflectivity of themirrors. We saw that the finesse is indeed associated with the number of round-tripsa photon makes before getting out of the cavity. However, the photon lifetime of suchFabry-Perot microcavity can also be properly defined by the relation [33]: τ cav = L eff n cav π Fc (1.11)where L eff is the effective length of the cavity. Indeed, the fact that the electromag-netic field effectively enters the DBR in the form of an evanescent field has to be takeninto account, hence the definition of L eff = L cav (cid:16) n n n − n (cid:17) .8 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
Leaky modes
A further investigation about the cavity field dependence on the inci-dence angle is given in figure 1.4. Here is presented the reflectivity of the system as afunction of the in-plane wavevector k γ || and its correspondence with the incident angle θ ,for an incident light of wavelength λ = λ .Figure 1.4: Leaky modes . For small angles of incidence ( θ < ◦ ), the reflectivity is equalto one except for the sharp dip at k γ || = 0 µm -1 which coincides with the resonance. But forhigher angles, the reflectivity decreases and oscillates: is shows some dips correspondingto the leaky modes of the cavity, where a strong coupling can not be sustained anymore.From [34]The reflectivity shows different behaviours along the wavevector scan. At k γ || = 0 µm -1 ,the reflectivity sharply decreases down to zero, and illustrates the resonance of the cavityfor a light with λ . It corresponds to the main mode.By increasing the incident angle, the reflectivity stays constant at 1. The mirrorswork properly and the light is trapped inside the cavity. But when the angle becomestoo large, the reflectivity decreases and some oscillations are visible, with large dips. Itshows that the field of those modes escapes easily outside of the cavity, hence the name leaky modes . The linewidth of the dips are larger than the one of the main mode, whichshows that they have a much shorter lifetime.In the case of our cavity, the leaky modes appear for k = 2 . -1 , which correspondsto an angle of 18 ◦ [35]. But our cavity is made in GaAs, for which the total reflectionangle is 16.6 ◦ : the leaks do not happen at the interface with air. .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES i Excitons in a bulk semiconductorBand theory Quantum physics is based on the quantification of the energy in anisolated atom: its electron can only access discrete energy levels, contrary to the energycontinuum of a free electron [36]. This property is modified when several atoms are puttogether, as illustrated in figure 1.5. If a chain of N atoms are placed close together, andseparated by a distance of the order of their Bohr radius, a coupling takes place betweenthem and the degeneracy of their energy levels is removed. The electrons are not onlyconnected to one atom anymore but are delocalized over the whole chain, leading to newenergy levels closed to the ones of the isolated atom.Figure 1.5:
Development of conduction bands . Energy levels of a chain of N atom.The isolated atom (N=1) possess a few number of states, which increases with the numberof atoms (N=10) until forming continuum in a solid. From [32]Finally, the case of a solid material can be considered as a chain of a very large numberof atoms: the density of energy levels becomes so high that it can be seen as a band ofaccessible energy, hence the name band theory .The distribution of the electrons within these bands at T = 0 K imposes the electronicproperties of a material. At this temperature, the maximum level of occupation of theelectrons is given by the Fermi energy. It actually corresponds to the chemical potentialat 0 Kelvin of the material. The position of the Fermi energy compared to the energybands determines the conduction properties of a solid.The first energy band entirely below the Fermi energy is called the valence band ; atT = 0 K, it is the completely filled band with the highest energy. The band just above thevalence band is called the conduction band . If the Fermi energy is located within theconduction band, the electrons are free to evolve in the whole band, as it is not entirelyfull, and can thus propagate through the material: it is the case of a conductor . On the0
CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT other hand, if the Fermi energy is in the gap between the bands, then all the states of thevalence band are filled while all the ones of the conduction band are empty: the materialis an insulator.However, this situation can be different at room temperature. If the gap between thevalence and the conduction band is small enough, typically E g ≈ semiconductor . Direct gap semiconductor
The energy of the valence and conduction bands also de-pends on the momentum. In particular, two types of semiconductors can be defined: thedirect gap semiconductors, where the maximum of the valence band is reached at thesame wavevector than the minimum of the conduction band, and the indirect gap semi-conductors, where the maximum of the valence band and the minimum of the conductionband do not correspond to the same k .Figure 1.6: Direct gap semiconductor dispersion . The bottom parabola in redcorresponds to the valence band filled with electrons, while the upper parabola is theempty conduction band. Their extrema all take place at the same wavevector. Thered dashed line illustrates the energy of the exciton, slightly lower than the one of theconduction band: it is created by the absorption of a photon which energy coincides withthe one of the gap. On the right, a symbolic representation of an exciton in a crystal,extending over several crystal cells. From [37] .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES (cid:126) ω > E g . An electron of the valenceband is then excited to the conduction band, leaving a hole in the valence band. Theexcited electron is negatively charged while the presence of the hole imposes a positivecharge, and the whole system is called an electron-hole pair . However, if the incomingphoton has an energy slightly lower than the gap energy, this one can still be absorbed.In that case, the excited electron jumps just below the conduction band, and thus stayscoupled to the hole: the electron remains only within a couple crystal cells (see inset offigure 1.6), and the bound pair is called an exciton .An exciton is described as a single neutral quasi-particle, even though it is constitutedof two charged particles. Its effective mass is the sum of the ones of both the electronand the hole ( m X = m e + m h ), themselves extracted from their respective dispersioncurvature. Its energy follows the relation [38]: E X ( k , n ) = E g − E b ( n ) + (cid:126) k m X (1.12)where E b is the binding energy coming from the coulombian interaction and n theelectronic density. The binding energy can be written as [30]: E b ( n ) = (cid:126) m X a ∗ X n = Ry ∗ n (1.13)with Ry ∗ an effective Rydberg constant, an a ∗ X the Bohr radius of the exciton definedby analogy with the hydrogen atom as: a ∗ X = (cid:126) (cid:15)e m X (1.14)with (cid:15) the dielectric constant of the medium and e the elementary charge. Semicon-ductors like GaAs have typically a high dielectric constant, while the effective mass ofthe exciton is usually one order of magnitude lower than the mass of a free electron invacuum. We obtain a ∗ X ≈
50 Å, which is much bigger than the crystalline cell: the excitonextends over several cells, as pictured in the inset of figure 1.6.Therefore, the binding energy in such dielectric medium reaches a few meV ( E b ≈ . E th = k B T = 25meV, which means that excitons can be created only at cryogenic temperature.This type of exciton are called the Wannier-Mott excitons , in opposition to the
Frenkelexcitons that are created in material with lower dielectric constant, and which have con-sequently a higher binding energy and a smaller size, and are stable at room temperature.As we saw that the excitons can be considered as particles, a hamiltonian operator H X can be defined to describe them, with an associated creation operator ˆ a † k and annihilationone ˆ a k . We thus have: H X = X k (cid:126) ω X ( k )ˆ a † k ˆ a k (1.15)2 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
We now have all the elements to describe the excitons and their life cycle: creationthrough a photon absorption and destruction by a photon emission. However, we havedescribed here bulk excitons , i.e. excitons within a three dimensional material, where nodirection is privileged. The emitted photons are therefore randomly distributed in alldirections. In order to obtain polaritons, we want to strongly couple the excitons withthe light field: an efficient way to improve the coupling is to confine the excitons in twodimensions, so that the photons are emitted in a single direction, as we will now describe. ii Bidimensional confinement in a quantum wellBidimensional confinement Three dimensional bulk semiconductors possess a fulltranslation invariance, which imposes the momentum conservation. A recombinationof an exciton of wavevector k X thus leads to the emission of a photon with the samewavevector: k ph = k X . Therefore, the exciton can only be coupled to one mode of theelectromagnetic field. Moreover, the energy conservation must also be verified: only theexcitons which the total energy is the same than the one of the electromagnetic field canrealize a radiative relaxation.This translational invariance is broken by a bidimensional confinement. It is easilyrealized by placing a layer of semiconductor in between two layers of another semiconduc-tor with a different gap energy. For instance, in the case of our sample, a layer of InGaAsis sandwiched between two layers of GaAs, as illustrated in figure 1.7.The interesting case takes place when the central semiconductor has a smaller gapenergy than the outer one. It indeed induces a dip in the conduction band energy profileand a bump in the one of the valence band, as well as the existence of energy levels onlyaccessible within the central space layer but forbidden everywhere else, plotted in grey infigure 1.7. An exciton created between two of these levels is therefore confined along the z axis, but free to travel on the plane of the layer. Properties of 2D excitons
Such a configuration breaks the translational invariancein the z direction: the momentum conservation is only valid on the in-plane directions( x , y ). Therefore, a photon emitted after the recombination of an exciton of wavevector k X = ( k X || , k Xz ) has to satisfy only the condition k ph || = k X || while its z component remainsfree. The confinement of the exciton also influences the Bohr radius of the exciton whichgets reduced (by half in the ideal case [30]): this leads to an increase of the binding energyand an enhancement of the coupling.Again, the exciton can be treated as a quasi-particle and described with a Hamiltonianoperator. It can be written as: ˆ H DX = ˆ H k X || + ˆ H Db (1.16)with ˆ H k X || the kinetic Hamiltonian and ˆ H Db the bidimensional equivalent to the bulkexcitons Hamiltonian.The kinetic Hamiltonian has a parabolic continuum of eigenenergies: .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES Spatial and energetic structures of GaAs-InGaAs quantum well . Alayer of InGaAs is placed between two layers of GaAs. As E InGaAsg < E
GaAsg , the energyprofile of the conduction band presents a dip (respectively a bump for the valence band)where some new energy levels are accessible (represented by the gray lines). The excitonscreated between these levels are confined along the z axis.4 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT E ( k X || ) = E g + (cid:16) (cid:126) k X || (cid:17) m ∗ X (1.17)while ˆ H Db has discrete eigenvalues. Our cavity has been designed so that the groundstate is energetically well separated from the other states; by analogy with the hydrogenatom, it is called the bidimensional 1s exciton level. E D s can thus be considered thefundamental state of the exciton, and the eigenenergies of ˆ H Db can be written: E DX = E g + E D s + (cid:16) (cid:126) k X || (cid:17) m ∗ X (1.18)The bidimensionnal confinement imposes a last condition on the exciton: the momen-tum conservation during a photon-excitation interaction. This can be written as: E X + (cid:16) (cid:126) k X || (cid:17) m ∗ X = (cid:126) cn cav r(cid:16) k γ || (cid:17) + (cid:16) k γz (cid:17) (1.19)here the left hand side is the exciton energy, where we have associated E g + E D s = E X ,and the right hand side the photon one, with its wavevector decomposed as k γ = ( k γ || , k γz ).The interaction takes place when we have k X || = k γ || , thus leading to: E X + (cid:16) (cid:126) k X || (cid:17) m ∗ X (cid:62) (cid:126) cn cav | k X || | (1.20)Which for small | k X || | simplifies to: | k X || | (cid:46) n cav E X (cid:126) c (1.21)Therefore, the excitonic modes that does not verify this relation are non radiative. Inthe type of semiconductor we used, this usually corresponds to | k X || | (cid:46)
30 µm -1 .Quantum well excitons in free space are coupled to a continuum of optical modes.Therefore, the radiative emission of a photon by exciton recombination is an irreversibleprocess and the probability of finding an exciton in the excited state decreases expo-nentially with time. Typically, the associated timescale is of the order of 10 ps, whichcorresponds to the radiative broadening of the exciton linewidth.The linewidth of the exciton also broadens due to non-radiative relaxation process in thequantum wells, which consequently limit the exciton lifetime. They can happen because ofdisorder within the semiconductor structure, because of interactions with crystal latticephonons or even because of exciton-exciton interaction. Taking this phenomena intoaccount, the typical non-radiative exciton lifetimes are of the order of 100 ps. .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES We described in the previous sections two different components. First, the opticalcavity breaks the symmetry of the electromagnetic field along the z axis, which leadsto a discretization of the z component of the incoming photons. On the other hand,the quantum well confinement plays a similar role on the excitons: the symmetry whichinduces the momentum conservation along z is also broken.The starting point of this section is to combine those elements to enhance the light-matter interactions, in order to reach the strong coupling regime. i Quantum well embedded in a microcavityGeometry of the samples In order to strongly couple excitons and photons, the 2Dquantum well is placed inside a microcavity (see figure 1.8). The cavity consists of twoDBRs (piles of yellow and green layers) designed to be quasi-resonant with the excitonictransition.Figure 1.8:
Quantum well embedded in a Fabry-Perot microcavity . The semi-conductor microcavity is constituted by two piles of green and yellow layers, while thequantum well, in orange and yellow, is placed inside the cavity. The red line shows theelectromagnetic field distribution, and the white dashed line its envelope with an expo-nential decay within the DBR. Note that the quantum well is placed at an antinode ofthe electromagnetic field. Adapted from [16]To maximize the interactions, the quantum well (thin yellow layer inserted in theorange material) is placed at an antinode of the electromagnetic field, whose amplitudeis plotted in red. The white dashed line shows its envelope and the exponential decaywithin the DBRs.6
CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
At this point, an important comparison has to be made between the effective massesof both components. The cavity photons have an effective mass 4 orders of magnitudesmaller than the quantum well excitons. It results in the fact that, even though they bothhave a parabolic dispersion, they take place in a different scale of wavevectors, as shownnumerically in figure 1.9.The left image shows both dispersions scaled so that the curvature of the excitonicone is visible. The photonic dispersion is in this case very narrow around zero. The rightimage is the same, but for a much shorter range of wavevectors: the photonic curvatureis clear but the excitonic resonance seems flat. As it corresponds to the wavevectorsconsidered in this work, the excitonic curvature is neglected in the following discussionand its resonance energy is considered constant.Figure 1.9:
Comparison of photonic and excitonic resonances . Photonic and exci-tonic numerical dispersions for different momentum ranges. As we work with wavevectorsof the order of 1 µm -1 , the picture on the right is the one that we consider, and theexcitonic resonance is considered constant from now on.The interactions can be modified by putting more quantum wells inside the cavity, allof them at some maxima of the electromagnetic field. In that case, the density is sharedbetween all the quantum wells while the coupling is maintained, which allows to reachhigh densities that can lead to a condensation phenomenon as we will see later. A typicalconfiguration consists of 12 quantum wells, distributed as 3 packs of 4 wells.The quasi-resonance between the exciton and the cavity photon can also be tuned bydesign. The excitonic resonance energy is fixed by the quantum well materials, but thephotonic one depends on the length of the cavity. By implementing a slight angle (wedge)between the two DBRs, the detuning exciton-cavity photon ∆ E Xcav = E X − E γ dependson the position on the sample and can be chosen on will.Let us consider that the medium outside of the cavity is air, and neglect the linewidthbroadening. This configuration allows the selective excitation of one exciton of in-planemomentum k X || = ( k Xx , k Xy ). It is done by choosing the incident photon k γ || = ( k γx , k γy ) withan angle θ γ = ( θ γx , θ γy ) that verifies: .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES k Xx = k γ,cavx = En cav (cid:126) c sin θ γ,cavx = E (cid:126) c sin θ γx (1.22) k Xy = k γ,cavy = En cav (cid:126) c sin θ γ,cavy = E (cid:126) c sin θ γy (1.23)where k γ,cav || = ( k γ,cavx , k γ,cavy ) is the photon momentum inside the cavity, E is theexcitation energy, n cav the index of the cavity and θ γ,cav = ( θ γ,cavx , θ γ,cavy ) the photonangle inside the cavity obtained by the Snell-Descartes law.Reciprocally, the detection of a photon with an angle θ γ = ( θ γx , θ γy ) out of the cavitycomes from the recombination of an exciton k X || = ( k Xx , k Xy ) that verifies equations 1.22and 1.23. Spontaneous emission
A parameter plays a crucial role in the light matter interactionswithin our system: the spontaneous emission of photons by an exciton. This process ofenergy dissipation is described by the
Fermi golden rule , which gives the spontaneousemission rate of a photon from an initial state i to a final state f . It can be writtenas [39]: Γ i → f = 2 π (cid:126) (cid:12)(cid:12) h f | V | i i (cid:12)(cid:12) δ ( E i − E f ) ρ ( E f ) (1.24)where Γ i → f is the transition rate from state i to f per time unit, (cid:12)(cid:12) h f | V | i i (cid:12)(cid:12) is theprobability of transition from states i to f , δ ( E i − E f ) is the energy conservation conditionand ρ ( E f ) the number of final states f of energy E f . This process thus directly dependson the density of final states. The coupling matrix element is also depicted as the quantityΩ R or Vacuum Rabi frequency [40].
Coupling regimes
Both cavity photons and quantum well excitons have limited life-times τ which result into spectral broadening γ = 1 /τ . Various intrinsic factors areresponsible for the homogeneous broadening of both elements. For instance, the cavitymode can couple to other empty electromagnetic modes of the cavity, or to the continuumof the extra-cavity modes, which are all contained in the cavity photon broadening γ cav .On the other hand, the excitonic broadening γ X includes the coupling of the excitonswith each other [41], with the phonons [42] or with the disorder of the crystal [43].Different coupling regimes can be defined by comparing the two lifetimes with thecoupling strength through the Rabi frequency Ω R . When Ω R (cid:28) γ cav , γ X , the oscillatorstrength is not strong enough to reabsorb a photon spontaneously emitted which willbe lost in the cavity: the emission is irreversible, it is the regime of weak coupling .However, if Ω R (cid:29) γ cav , γ X , then the lifetimes are long enough so that an emitted photonstays in the cavity and can be absorbed again, which corresponds to the strong coupling regime. The phenomenon is reversible and many coherent exchange of energy can takeplace before the photon leaves the cavity, which are called the Rabi oscillations .8 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT ii Strong coupling regime
The strong coupling regime is the one that we are interested in as it results in theapparition of new eigenmodes called cavity polaritons . The first observation of suchstrong coupling in a quantum well embedded in a semiconductor microcavity has beenrealized in 1992 in the group of C. Weisbuch [6].In order to study the optical properties of the strong coupling, a simple model consists inconsidering the coupled elements as harmonic oscillators, to which is associated a creationoperator ˆ a † and an annihilation one ˆ a . An important condition of this description ishowever that both elements are bosonic, meaning that they follow the commutation law[ˆ a m , ˆ a † n ] = δ mn , with m and n two quantum states of the system.In our case, the coupling takes place between the cavity photons and the quantumwell excitons. By nature, photons are bosons, but the excitons are made of fermionicelements. Let us first then verify if they can be treated as bosons. Excitons as bosons
It is possible to associate to excitons a creation ˆ b † s and an an-nihilation ˆ b s operators [44, 45]. In that case, the commutation relation can be writtenas [42]: (cid:2) ˆ b s , ˆ b † s (cid:3) = 1 − O (cid:16) n (cid:0) a ∗ DX (cid:1) (cid:17) (1.25)where a ∗ DX is the Bohr radius of the exciton in the quantum well as defined in section1.1.2, and n the excitonic density per surface unit. It is therefore possible to consider theexciton as a boson if this density is small enough ( n (cid:0) a ∗ DX (cid:1) (cid:28) i.e. in a regime of lowlight excitation. Typically, the Bohr radius of a 2D exciton is around 5 nm, which allowsa maximal excitonic density of the order of 4 · µm -2 . In our case, we always work withlower densities, which allows us to use the bosons description for the excitons. Linear hamiltonian of the strong coupling regime
Let us simplify the notation aswe only consider the in-plane wavevector: from now on, the || notation is forgotten and k = k || .As a first step, we place the system in a configuration of low excitation. The exciton-exciton interactions are therefore negligible: the resulting hamiltonian is purely linearand can be written as:ˆ H lin = X k ˆ H k = X k (cid:18) E X ˆ b † k ˆ b k + E γ ( k )ˆ a † k ˆ a k + (cid:126) Ω R (cid:16) ˆ a † k ˆ b k + ˆ b † k ˆ a k (cid:17)(cid:19) (1.26)with ˆ a † k , ˆ a k and ˆ b † k , ˆ b k are the creation and annihilation operators for a cavity photonand a quantum well exciton with an in-plane wavevector k , respectively. E X is theexcitonic energy, E γ ( k ) the photonic one, and Ω R the Rabi frequency.The last term of equation 1.26 is particularly interesting. It contains the superpositionof two events: on one side, the creation of a photon and the annihilation of an exciton, i.e. the recombination of the exciton, and on the other side the creation of an excitonand the annihilation of a photon, so the absorption of a photon. They are connected bythe coefficient (cid:126) Ω R which represents the coupling energy . .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES exciton.In time, polaritons can therefore be seen as the succession of cavity photon, absorbedby creating an exciton, which later recombines by emitting an identical photon, that canbe absorbed once again, and so on. This image describes the polaritons as a coherentlinear combination of an electromagnetic field (the photonic part) and a polarization field(the excitonic one). iii Polariton basisEigenmodes and Hopfield coefficients The hamiltonian given by the equation 1.26can be diagonalized. It results in new eigenstates of the system that contain both thephotonic and excitonic part: the upper and lower exciton-polaritons , denoted as UPand LP respectively. The hamiltonian can be rewritten in this basis:ˆ H lin = X k (cid:126) ω LP ( k )ˆ p † k ˆ p k + X k (cid:126) ω UP ( k )ˆ u † k ˆ u k (1.27)with ˆ p † k , ˆ p k and ˆ u † k , ˆ u k the creation and annihilation operators of the upper and lowerpolaritons, respectively. They are bosonic operators and therefore follow the commutationrule: (cid:2) ˆ p † k , ˆ p k (cid:3) = δ k , k and (cid:2) ˆ u † k , ˆ u k (cid:3) = δ k , k . ω LP ( k ) and ω UP ( k ) express the dispersions associated to both of these polaritonicmodes: ω UP/LP = ω X ( k ) ± ω γ ( k )2 ± q(cid:0) ω X ( k ) − ω γ ( k ) (cid:1) + 4Ω R (1.28)The two dispersions are plotted in figure 1.10 by the solid lines: the strong couplinggreatly modifies the system (the dashed lines illustrate the system dispersions in the weakcoupling regime).The two polaritonic operators can be expressed through the photonic and excitonicones by the relation: ˆ p k ˆ u k ! = X k C k − C k − X k ! ˆ b k ˆ a k ! (1.29) X k and C k represent the excitonic and the photonic fraction of the polaritons, re-spectively, and are called the Hopfield coefficients [7]. The transformation is thereforeunitary: X k + C k = 1. They are plotted in figure 1.10 through the color scale of thedispersions, highlighting the light and matter part of the polaritons. They can also bequantified more precisely by the expressions [16]:0 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT X k = q ∆ E Xcav + (cid:126) Ω R + ∆ E Xcav q ∆ E Xcav + (cid:126) Ω R (1.30) C k = q ∆ E Xcav + (cid:126) Ω R − ∆ E Xcav q ∆ E Xcav + (cid:126) Ω R (1.31)These coefficients strongly depend on the detuning between the bare exciton energyand the cavity photon resonance ∆ E Xcav = (cid:126) ω γ ( k ) − (cid:126) ω X ( k ), as illustrated in the figures1.10.b. and c. The polaritons are fully hybrid typically if ∆ E Xcav = 0 and at zerowavevector: C k =0 = X k =0 = .An interesting interpretation of the Hopfield coefficients and the Rabi frequency con-nects them with the state of the system. As we saw, polaritons are continuously oscil-lating between their photonic and excitonic states, which are coupled through the Rabifrequency: it thus represents the average conversion rate between those two states. Asfor the Hopfield coefficients, they can be seen as the time ratio the system passes in thephotonic or excitonic state.The eigenenergies E UP ( k ) and E LP ( k ) of the polaritonic states are given by: E UP ( k ) = E X ( k ) + E γ ( k ) + q ∆ E Xcav + (cid:0) (cid:126) Ω R (cid:1) (1.32) E LP ( k ) = E X ( k ) + E γ ( k ) − q ∆ E Xcav + (cid:0) (cid:126) Ω R (cid:1) (1.33)If we now take into account the finite lifetimes of both the cavity photon and theexciton, the eigenstates are modified. A common approximation consists in adding thedecay rates γ X and γ cav to the particle energies as an imaginary part: E ∗ X ( k ) = E X ( k ) − i (cid:126) γ X (1.34) E ∗ γ ( k ) = E γ ( k ) − i (cid:126) γ cav (1.35)It is an approximation as it neglects the influence of the exciton-photon coupling onrelaxation, and does not take into account the shape of the dispersion and in particularthe dependence on k of the decay rates. However, as we focus our study on the smallwavevectors, those hypothesis are acceptable. .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES Polariton dispersions and Hopfield coefficients . a. Upper (UP) andlower (LP) polariton dispersions. The color gradient illustrates the excitonic ( X k inyellow) and photonic ( C k in blue) fractions. The dotted lines show the photon and excitondispersion in the weak coupling case. The detuning exciton-cavity photon is chosen to bezero. b. and c. Polariton dispersions with the same color scale, for an energy detuning∆ E Xcav = 2 meV and ∆ E Xcav = − CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT E ∗ UP = E X ( k ) + E γ ( k )2 − i (cid:126) γ X + γ cav s (cid:0) ∆ E Xcav − i (cid:126) ( γ cav − γ X ) (cid:1) + (cid:0) (cid:126) Ω R (cid:1) E ∗ LP = E X ( k ) + E γ ( k )2 − i (cid:126) γ X + γ cav − s (cid:0) ∆ E Xcav − i (cid:126) ( γ cav − γ X ) (cid:1) + (cid:0) (cid:126) Ω R (cid:1) E Xcav = 0, it simplifies into (cid:126) q Ω R − ( γ cav − γ x ) . In thecase of the strong coupling regime that we are concerned about, it results in an energydifference between the two branches, known as the anticrossing , and characteristic ofthe strong coupling.This behaviour is highlighted in figure 1.11, where are plotted the upper and lowerpolariton energies as a function of the detuning ∆ E Xcav . As previously, the colorscaleindicates the excitonic and photonic coefficients. The anticrossing is visible for ∆ E Xcav =0 meV, where the energy gap between the branches is equal to the Rabi energy (cid:126) Ω R .Figure 1.11: Anticrossing of the energy levels . Energies of the upper (UP) andlower (LP) polariton branches, at k = 0 µm -1 , depending on the exciton - cavity photondetuning ∆ E Xcav , with the Hopfield coefficients indicated through the colorscale. Thedashed lines are the energies of the bare photon and exciton. At ∆ E Xcav = 0 meV, thereis an energy gap equals to the Rabi energy (cid:126) Ω R . .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES γ UP ( k ) γ LP ( k ) ! = C k X k X k C k ! γ X γ cav ! (1.38)Thus, the polaritons usually have short lifetimes, typically of the order of some tensof picoseconds. Indeed, the polariton population is characterized by a continuous decaydue to the photon escape out of the cavity. The system is therefore out-of-equilibriumand needs a constant pumping to ensure a stable population density: we talk about its driven-dissipative nature.This particularity is actually a great advantage for polaritonic systems: not only arethey perfectly suited to study out-of-equilibrium quantum gases, but the photon escapeallows for the simplest optical detection. Indeed, the photons leave the cavity with allthe properties of the polaritons they are issued from: the analysis of the cavity emissiongives access to all the interesting features of the polaritons, as we will discuss in the nextsection. iv All-optical control Let us summarize the correspondence between the optical excitation and detection ofthe system and the properties of the polaritons. We saw that the selection of a specificangle and energy of the excitation field leads to the excitation of a single resonant polaritonmode within the dispersion curve. Reciprocally, the emission of a photon with a specificwavevector and energy comes from a polariton with the same properties.We therefore have an equivalence between the following quantities:• the excitation intensity and the polariton density• the incident angle of the excitation field and the in-plane wavevector of the createdpolaritons• the in-plane wavevector of the polaritons and their velocity• the density of polaritons and the emitted intensity• the polariton phase and the phase of the emitted field• the polariton wavevector and the emission angle• the polariton energy and the emission energy• the polariton lifetime and the emission linewidth4
CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT v Polaritons-polaritons interactionsNonlinearity
Since the beginning of section ii, the description of the exciton-polaritonssystem has been done in the case of low density, thus neglecting any interactions betweenthe particles. However, when the incident intensity is increased, this approximation cannot be sustained anymore and the different interactions must be taken into account. Theinteractions take place in particular between the electrons and the holes of the excitons,leaving the system in a particular regime where the excitons are still considered as bosons,but where their fermionic composition can not be neglected.The linear hamiltonian previously used needs to be completed by some second orderterms [42, 46]:• the hamiltonian of the coulombian interaction between electrons and holes of twocolliding excitons of wavevectors k and k , resulting in the creation of two newexcitons of wavevectors k + q and k − q to respect the momentum conservation.It is also know as the exchange interaction [47] and can be expressed as:ˆ H XX = 12 X k , k , q V XX ( q )ˆ b † k + q ˆ b † k − q ˆ b k ˆ b k (1.39)• the hamiltonian of the coupling saturation between exciton and photon, leading tothe saturation of the optical transition with the excitation intensity:ˆ H satXcav = − X k , k , q V sat (cid:16) ˆ a † k + q ˆ b † k − q ˆ b k ˆ b k + ˆ a k + q ˆ b k − q ˆ b † k ˆ b † k (cid:17) (1.40)A few comments can be done on those two terms. First, the coulombian interactionof a bidimensional system is defined as: V XX ( q ) = 2 πe (cid:15)Aq (1.41)with A the quantification area, i.e. the quantum well surface in our case. Under theBohr approximation, stating that πq (cid:29) a ∗ DX , which means, for our excitonic Bohr radiusof 5 nm: q (cid:28) -1 , the previous potential can be considered independent of q [42],as it is the case in our work.As for the saturation potential, it appears for a certain regime of excitation, for whichthe repulsion between the interacting excitons becomes so high that they can not stay inthe same state, which results in a fermion-like behaviour. The excitons are then describedas hard-core bosons, as the N-excitons wavefunction stays symmetric for the exchange oftwo excitons even with such a Pauli-like repulsion.It can be expressed by applying the Usui transformation [44] to quantum wells [45]: V sat = (cid:126) Ω R n sat A (1.42) .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES n sat is the exciton density from which the exciton-photon coupling saturation takesplace [46,48], typically of the order of 10 µm -2 in our cavities [40]. In the case of a gaussianexcitation spot, the saturation starts from the higher intensity, resulting in a flat densityin the center of the spot.The two previous potentials can be combined in the polariton basis under the singlepotential V pol − pol k , k , q , which depends on the Hopfield coefficients, on V XX and on V sat .Considering now only the lower branch of the polaritons, the associated hamiltonian isdefined as: ˆ H int = 12 X k , k , q V pol − pol k , k , q ˆ p † k + q ˆ p † k − q ˆ p k ˆ p k (1.43)This nonlinear hamiltonian, as well as the one expressed in 1.39, describes an interactionbetween four polaritonic modes: the annihilation of two polaritons of wavevectors k and k combined with the creation of two polaritons of wavevectors k + q and k − q . Thistype of scattering is commonly referred to as polariton four-wave mixing and leadsto the spontaneous generation of new populations of polaritons when the conservationof momentum and energy is satisfied. It is typically responsible for the polariton opticalparametric oscillator (or OPO), analog to the nonlinear optics one [13, 14, 49].Finally, the total hamiltonian of the lower polariton branch corresponds to the sum ofthe linear one and the interaction one:ˆ H = ˆ H lin + ˆ H int = X k (cid:18) (cid:126) ω LP ( k )ˆ p † k ˆ p k + 12 X k , q V pol − pol k , k , q ˆ p † k + q ˆ p † k − q ˆ p k ˆ p k (cid:19) (1.44) Energy renormalization
In order to study the effect of nonlinearity on the polaritondispersion, let us consider two modes of the lower polariton branch k and k . Theinteraction hamiltonian ˆ H int of these modes is thus the sum of the two terms ˆ p † k ˆ p † k ˆ p k ˆ p k and ˆ p † k ˆ p † k ˆ p k ˆ p k . But as ˆ p k and ˆ p k are eigenmodes of the system, they commute,therefore the two terms are the same.The evolution of the k mode is given by the Heisenberg equation: i (cid:126) ddt ˆ p k = (cid:2) ˆ p k , ˆ H (cid:3) = (cid:2) ˆ p k , ˆ H lin (cid:3) + (cid:2) ˆ p k , ˆ H int (cid:3) (1.45)and a complete calculation tells us that: (cid:2) ˆ p k , ˆ H int (cid:3) = (cid:126) g LP ˆ p † k ˆ p k ˆ p k = (cid:126) g LP ˆ N ˆ p k (1.46)where g LP = f ( V k , k , k − k ) is the polariton-polariton interaction constant of themodes k and k , and ˆ N = ˆ p † k ˆ p k the polariton population in the mode k .Now if we consider the k mode to be macroscopically populated, its density can bereplaced by its mean field value h ˆ N i . And if the previous equations are normalized bythe quantization area A , we can even introduce the k mode density n = h ˆ N i A , which6 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT works as an external potential for the polaritons in the k mode, as the equation 1.45becomes: ddt ˆ p k = − i (cid:18) E LP ( k ) (cid:126) + g LP n (cid:19) ˆ p k (1.47)where E LP comes from the linear hamiltonian. We see here that the interaction of the k mode with a pump of wavevector k introduces a shift of its energy toward the blue,therefore commonly known as blue shift :∆ E k = (cid:126) g LP n (1.48)This shift is an effect of energy renormalization of the k mode by the pump mode( k ) and thus a multimode effect. It works the same way for a continuum of modes, eventhough in that case the contributions from all the modes have to be taken into account.It happens as well on the pump mode itself, which then can be related to the self Kerreffect known in nonlinear optics.This energy renormalization has to be taken into account while considering the pumpedmode dynamics. Indeed, a typical way of creating a polariton fluid, and which will bethe case in the present work, is to excite it quasi-resonantly, which means at an energyslightly blue-detuned from its resonance (the lower polariton branch in our case). In orderto evaluate the effective detuning between the pump and the polaritons, the previouslydescribed blue shift must not be forgotten, and we have:∆ E efflasLP = E las − ( E LP k las − (cid:126) g LP n las ) (1.49)where the index las corresponds to the pumped mode of the laser.This renormalization is also responsible for the bistability phenomenon, where for ahigh enough detuning between the pump and the lower polariton branch, a certain rangeof input intensities gives access to two stable output states. However, we will not focuson it here as it will be discussed in more details in section 3.1. Spin
Until here, the description of our polariton system was entirely scalar. However, thepolariton do inherit a spin degree of freedom from their excitonic part. The spin of theexciton can indeed take two possible values of their total internal momentum, J = 1 and J = 2. Yet only the J = 1 excitons can couple to single photons, while J = 2 excitons arecalled dark-excitons. Which means that the only possible projections of the polaritonsangular momentum along the z axis orthogonal to the cavity are J z = ±
1. Polaritons andout-of-cavity photons have a one-to-one relation which allows a well defined polarizationof the emitted light. .1. LIGHT MATTER COUPLING IN SEMICONDUCTOR MICROCAVITIES vi Other geometries
The samples used in the present work are all planar microcavities as described previ-ously. However, they are not the only configuration in which the strong coupling betweena microcavity photon and a quantum well exciton can take place. This section will givea brief overview of some other geometries.Figure 1.12:
Examples of other geometry microcavitites . a. Pattern (i) and Scan-ning Electron Microscopy image (ii) of a sample combining microwires and micropillarsof different shape and size. From [50]. b. Structure of a honeycomb polariton lattice. (i)Electron microscopy image of a single micropillar, whose discrete dispersion is plotted on(ii). The pillars can also be combined as a honeycomb lattice, as sketched on (iii) andpictured on (iv). From [51].8
CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
Once the planar cavity is grown, it is indeed possible to chemically etch it in an orderpattern, to create new confinement geometries. Typically, one could create microwires,as presented in figure 1.12.a, to get an unidimensional confinement of the excitons, andtherefore study different behaviour of polariton fluids [50, 52–54].It is also possible to lower again the dimensionality and to etch micropillars, as pre-sented in figure 1.12, which can for instance be used to study the squeezing of polari-tons [55]. Those pillars can also be combined together, for example two-by two as po-lariton molecules [56], or as lattices, as picture in figure 1.12.b., typically used to observetopological phenomena [51, 57, 58].
In the previous section, we have discovered the necessary conditions for polaritons toexist and the tools to describe them. We can now explore the different techniques togenerate them and how they influence their dynamics.
The most direct way to get a strong coupling in our cavity is to inject photons at thecavity resonance. It results in a population of polaritons that we will describe in thissection. i Mean-field theory
To consider the dynamics of the polariton fluid, we now need to take into account theensemble of particles and not just one of them. To do so, we use the well known mean-field approximation [20], where we do not consider the particle operators anymore.Indeed, as the total number of particles N is high, we have N ∼ N + 1: we can replacethe particle quantum operators by their expectation values. Therefore, the creation andannihilation operators ˆ a † ( r , t ) and ˆ a ( r , t ) of a single particle at the position r and instant t can be replaced in the equations of motion by their mean field terms Ψ( r , t ) = h ˆ a ( r , t ) i and Ψ † ( r , t ) = h ˆ a † ( r , t ) i . Ψ represents then the wavefunction of the fluid and its modulussquare the density of particles: | Ψ( r , t ) | = n ( r , t ).The first description of the dynamics of a quantum gas of material particles was givenin 1961 by Gross [59] and Pitaevskii [60] while describing quantum vortices in liquidHelium. Adapted in our case by taking into account the Bogoliubov theory of the diluteBose gas [20], the equation describing the system can be written in the form: i (cid:126) ddt Ψ( r , t ) = (cid:16) − (cid:126) m ∇ r + V ext ( r ) + (cid:126) gn ( r , t ) (cid:17) Ψ( r , t ) (1.50)with V ext an external potential in which the particles are confined. This equation isreferred to as the Gross Pitaevskii equation (GPE) or nonlinear Schrödinger equation.It states in particular that the macroscopic ensemble of particles that forms the fluid .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS r , t ) acts as a classical fieldΨ( r , t ) [20, 61, 62].The previous description has been realized for a Bose gas of cold atoms in weak interac-tion, but it can be adapted to polariton fluids. If the excitonic density is low, polaritonsact as bosons, therefore the picture is coherent. Furthermore, their low effective mass (ofthe order of 10 -5 times the one of the free electron) ensures a de Broglie wavelength higherthan the polariton-polariton distance, at cryogenic temperatures. Indeed, for a typicalexcitonic density of 40 µm -1 , the mean distance between particles is about 0.1 µm, whilethe de Broglie wavelength has the expression: λ dB = s (cid:126) m ∗ LP k B T (1.51)where k B is the Boltzmann constant and T the temperature, and reaches about 1 µmfor T = 5 K.Finally, polaritons have coulombian interactions through their excitonic part, that wesaw in section ii are weak compared to the one of an hydrogenoid atom.However, a few differences need to be noticed. First of all, the definition of the phase ofthe wavefunction Ψ( r , t ) of the system. For a standard Bose gas, the phase is defined bythe chemical potential µ of the gas. In the case of resonantly pumped polaritons however,the thermodynamic equilibrium is never reached, therefore the phase is not intrinsicallydefined by the fluid but depends on the pump properties. However, it is enough for theGross Pitaevskii theory to be applied [40, 63].A second particularity of polaritons is the fact that they are composed of two types ofbosons: the cavity photon and the quantum-well excitons. The two particles feel differentexternal potentials, V c ( r ) and V X ( r ) respectively. Therefore the GPE should be writtenfor both particles, taking into account the respective potentials and the fact that theinteractions only happen between the excitons. The full Gross-Pitaevskii theory appliedto polariton fluids have been detailed by Iacopo Carusotto and Cristiano Ciuti [20, 64],and is further presented in the next section. ii Driven-dissipative Gross-Pitaevskii equation In order to apply the GPE to microcavity polariton fluids, lets us first describe the sys-tem in the exciton-photon basis by considering the associated fields Ψ X ( r , t ) and Ψ γ ( r , t ).The system losses are represented by the two linewidths γ X and γ cav . The continuouspumping F p ( r , t ) only applies to the photonic field while the interactions defined by g hap-pen between excitons. The external potentials V γ ( r ) and V X ( r ) come from the intrinsicdissipation phenomena of the microcavity, usually related to its defects.Those statements lead us to the coupled equations:0 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT i (cid:126) ddt Ψ γ ( r , t )Ψ X ( r , t ) ! = (cid:126) F p ( r , t )0 ! + " H lin ( r ) + V γ ( r ) − i (cid:126) γ cav V X ( r ) − i (cid:126) γ X + (cid:126) gn ( r , t ) ! Ψ γ ( r , t )Ψ X ( r , t ) ! (1.52)where H lin ( r ) is the linear hamiltonian written in the real space, i.e. obtained bychanging k to − i ∇ r in H lin ( k ) : H lin ( r ) = E X (cid:126) R (cid:126) R E γ ( − i ∇ r ) (1.53)The equation 1.52 can also be written in the polariton basis, and in particular if we focuson the lower branch, we obtain the driven-dissipative Gross-Pitaevskii equation forthe polariton field: i (cid:126) ∂∂t Ψ LP ( r , t ) = (cid:18) − (cid:126) m ∗ LP ∇ r + V LP ( r ) − i (cid:126) γ LP + (cid:126) g LP n ( r , t ) (cid:19) Ψ LP ( r , t )+ (cid:126) F p ( r , t ) (1.54)with m ∗ LP the effective mass of the lower polariton, V LP = | X k | V X + | C k | V γ theexternal potential felt by the lower polaritons and g LP = | X k | g the interaction constantbetween the lower polaritons in the same modes.Until the end of this section, the study is done only on the lower branch of the polariton.Therefore, in order to lighten the notation, the indices will be removed: Ψ LP ( r , t ) =Ψ( r , t ), γ LP = γ , g LP = g and m ∗ LP = m ∗ .As expected, this equation is indeed very similar to 1.50, except for the loss term − i (cid:126) γ and its necessary compensation with the pump term F p ( r , t ). It is indeed an importantdifference between our system and cold atom gases for which the theory was first written:the lifetime of our particles τ = (cid:126) πγ is of the order of some tens of picosecond, compared tothe second for atoms. Hence the fact that the losses need to be continuously compensated:our system is out of equilibrium .Let us now focus on the mean-field stationary solutions of equation 1.54 in the homo-geneous case, i.e. for an external potential equals to zero. We need to remember that thesystem is driven by the pump field F p ( r , t ) = F p ( r ) e i ( k p r − ω p t ) . Therefore, the solutionscan be written Ψ( r , t ) = Ψ ( r ) e i ( k p r − ω p t ) . It results in the mean field stationary equation: (cid:18) − (cid:126) k p m ∗ − i (cid:126) γ + (cid:126) gn ( r ) (cid:19) Ψ ( r ) + (cid:126) F p ( r ) = 0 (1.55) .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS bistability phenomenon observed in our system.As it is the starting point of most of the experimental work presented in this thesis, itwill be further detailed in chapter 3, and only quickly introduced here for the sake of thediscussion.The bistability indeed comes from the quasi-resonant pumping of the system. We needtherefore to take into account the detuning between the pump and the lower polaritonbranch ∆ E lasLP = (cid:126) ω p − (cid:126) ω LP . Multiplying the stationary equation by its complexconjugate, we obtain the equation in intensity: I ( r ) = (cid:18) ( (cid:126) γ ) + (cid:16) ∆ E lasLP − (cid:126) gn ( r ) (cid:17) (cid:19) n ( r ) (1.56)with I ( r ) the intracavity intensity. Now this equation can be derived into: ∂I∂n = 3 g n − E lasLP gn + ( (cid:126) γ ) + ∆ E lasLP (1.57)which discriminant is (2 g ) (∆ E lasLP − (cid:126) γ ) ). Therefore, the previous equation canhave two distinct roots if the detuning validates the condition ∆ E lasLP > √ (cid:126) γ .Figure 1.13: Bistability . Numerical resolution of equation 1.56 for (cid:126) γ = 0 . E lasLP = 1 meV. The green region shows the stable solutions while the red one indicatesthe unstable ones [64]. The density jumps from one branch to the other at the frontierbetween the green and red regions: we have a hysteresis cycle illustrated by the arrows.The a, b and c points corresponds to the conditions of the curves later presented in figure1.14. Adapted from [16]2 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
In that case, the system presents a bistable behaviour: the curve n ( I ) displays a rangeof intensities for which the system has two possible outputs, as plotted in figure 1.13. Thesystem is stable on the green parts of the curve, while the red region shows non-physicalsolutions [64]. The system therefore jumps from one branch to another at the limitsbetween the red and green regions. We observe a bistability cycle: within the bistablerange, the upper branch is only accessible by lowering the system intensity from a higherone, while the lower branch can only be reached by increasing the intensity from a lowerone, as indicated by the arrows. iii Landau criterion The superfluidity is a particular state of matter that was first observed in 1937 by JohnF. Allen, A. Don Misener and Piotr Kapitza [65, 66]. They independently studied liquidHelium at low temperature and observed a transition point at 2.17 K, called the lambdapoint , below which the thermal conductivity of the Helium becomes very high and theviscosity abruptly drops.A year later, Fritz London put in relation for the first time this phenomenon andthe Bose Einstein condensation, earlier theorized for atomic gases, starting from the factthat the two transition temperatures are of the same order of magnitude - 2.17 K for theHelium lambda point and 3.2 K for the condensation of a Bose gas of the same density [67].However, the main issue of this theory comes from the interaction between Helium atomsthat are too strong to consider it as a perfect gas. Indeed, later studies have shown thatonly 10% of superfluid He is actually condensed [68].The best explanation of the phenomenon was given by Lev Landau in 1941 [69], fol-lowing an idea expressed by Tisza in 1938 [70]: he developed a model based on twofluids in interaction, a classical one and a condensed one. It can explain many of theobserved behaviours in such fluids, that were not yet modeled due to the complex inter-action between He atoms. The understanding of superfluidity was completed by N. N.Bogoliubov in 1946 [71], who established a non phenomenological model of the dispersionof elementary excitations.Landau proposed several experiments to determine if a system is superfluid and to findout its superfluid fraction, as the rotating fluid experiment, later realized by Hess andFairbank [72], or the study of persistent currents, experimentally observed in Helium [73]and much later in atomic condensates [74]. However, a most used one and related to thecommonly called the
Landau criterion is based on the collective thermal excitations ina superfluid.The Landau criterion is based on the elementary excitation spectrum that we develophere. Let us thus consider a fluid of total mass m in a Galilean reference frame R , withan energy E and a momentum P . In the reference frame R moving at a speed v withrespect to R , its energy E and momentum P are given by: E = E − P · v + 12 m v (1.58) P = P − m v (1.59) .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS v and energy E , we add an elementary excitation of momentum p and energy (cid:15) ( p ). The total energyof the fluid becomes E + (cid:15) ( p ), and in the tube reference frame, we have: E = E + (cid:15) ( p ) + p · v + 12 m v (1.60) p = p + m v (1.61)Comparing these relations with the previous ones, we see that the excitation energeticcontribution is (cid:15) ( p )+ p · v . Now if this contribution is negative, i.e. if | v | > (cid:15) ( p ) | p | , it meansthat the system looses energy by getting excited: the fluid is unstable and excitations arespontaneously generated, which leads to an increase of the viscosity.On the contrary, below this critical velocity, the system is very stable in its groundstate, as the presence of an excitation would induce an energetic increase. Therefore theexcitations are strongly inhibited. There is no viscosity in this case and the system issuperfluid. The Landau criterion is thus expressed as: | v | < v c = min p (cid:15) ( p ) | p | (1.62) iv Polariton superfluid In order to apply the previous theory to the polariton fluid case, we consider a smallperturbation propagating in the fluid with a momentum p = (cid:126) k . It can be added tothe polariton stationary solutions given by equation 1.55, which leads to, using the samenotations: Ψ( r , t ) = (cid:18) Ψ ( r ) + Ae i ( kr − ωt ) + B ∗ e − i ( kr − ωt ) (cid:19) e i ( k p r − ω p t ) (1.63)with A and B ∗ the amplitudes of the perturbation. The previous wavefunction canbe inserted into the driven-dissipative Gross Pitaevskii equation (equation 1.54). Then,keeping only the linear terms, and as the second order terms cancel out, we obtain thecoupled equations: A (cid:126) ( k p + k ) m ∗ − i (cid:126) γ + 2 (cid:126) gn − (cid:126) ( ω p + ω ) ! + (cid:126) gn B = 0 (1.64) B (cid:126) ( k p − k ) m ∗ + i (cid:126) γ + 2 (cid:126) gn − (cid:126) ( ω p − ω ) ! + (cid:126) gn A = 0 (1.65)resulting in the new dispersion relation: (cid:126) ω ± = (cid:126) k p k m ∗ − i (cid:126) γ ± vuuut (cid:126) (cid:0) k p + k (cid:1) m ∗ − (cid:126) ω p + 2 (cid:126) gn ! + (cid:16) (cid:126) gn (cid:17) (1.66)4 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
Figure 1.14:
Elementary excitation dispersion . Real part of the solutions of equation1.66 of a static fluid ( k p = 0 meV) pumped quasi-resonantly (∆ E lasLP = 1 meV) fordifferent densities. The solid lines correspond to the ω + solutions while the ω − ones areplotted with the dashed lines. The black one indicates the simplified parabolic dispersionof non interacting polaritons, setting the energy origin. An energy shift appears at highdensity (a. in red, (cid:126) gn > E lasLP ), while for exactly (cid:126) gn = ∆ E lasLP (case b. inyellow) we get the Bogoliubov dispersion with a linearization for the small wavevectors.Case c. in blue is unstable [64] and corresponds to (cid:126) gn < ∆ E lasLP . Adapted from [32].The real part of this equation is plotted in figure 1.14, for different polariton densities n and for zero wavevector of the pump k p = 0 µm -1 . The solid lines illustrate the ω + solutions while the ω − is plotted in dashed lines.The black dashed line shows the polariton dispersion without interaction, simplified asa parabola. We saw that different regimes appear depending on the polariton density,directly connected also to the energy detuning between the pump and the lower polaritonbranch ∆ E lasLP = (cid:126) ω p − E LP ( k p ).Depending on the position of the system within the bistability curve presented earlier,different behaviour can be observed. The cases correspond to the ones indicated in figure1.13. The red curve corresponds to the highest density, with (cid:126) gn > E lasLP , above thebistability cycle: we observe the energy renormalization and an energy gap compared tothe initial dispersion. The yellow curve illustrates the particular case where the renormal-ization and the detuning coincide: (cid:126) gn = ∆ E lasLP , which happens at the turning pointof the bistability cycle. We are in the particular case of the Bogoliubov spectrum, witha linear dispersion for small wavevector. Finally, when (cid:126) gn < ∆ E lasLP as for the blue .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS k = 0 µm -1 , which correspondsto the behaviour of the system in the unstable branch of bistability.In the interesting case where we have exactly (cid:126) gn = ∆ E lasLP , equation 1.66 can bemuch simplified into: ω = | k | s (cid:126) gn m ∗ (1.67)An analogy with the atomic condensates can then be made [62], with the definitionof a sound velocity c s = s (cid:126) gnm ∗ (1.68)which actually corresponds to the critical velocity of the Landau criterion definedin section iii: the system keeps superfluid properties until reaching a propagation speed v f = c s . If the fluid speed gets above c s , it leaves the superfluid regime to enter theCerenkov one, where elastic scattering takes place.Figure 1.15: Superfluid and Cerenkov regimes . Elementary excitation spectra fora flowing fluid pumped quasi-resonantly. Again, the solid lines corresponds to the ω + solutions of equation 1.66 while the ω − are plotted in dashed lines. The black one is theparabolic dispersion of the bare polaritons. The densities are set so that (cid:126) gn = ∆ E lasLP .The red curve P shows the case where k p = 1 µm -1 and ∆ E lasLP = 2 meV so that theLandau criterion is satisfied. Only one state is available: the system can not scatterand is therefore superfluid. On the contrary, the P case plotted in blue illustrates theconfiguration where k p = 3 µm -1 and ∆ E lasLP = 3 meV: this time, an other state isavailable at the same energy to which the system can scatter, it is the Cerenkov regime.Adapted from [32].6 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
Both cases dispersions are illustrated in figure 1.15. It corresponds again to the solutionsof equation 1.66 but for a moving polariton fluid ( k p = 0). The ω + solutions are plottedin solid lines and the ω − in dashed lines, while the dashed black one is the dispersion ofthe non interacting lower polariton branch, to which is set the reference of energy. Thedensities have been chosen so that the relation (cid:126) gn = ∆ E lasLP is satisfied.The P case plotted in red shows the configuration where k p = 1 µm -1 and ∆ E lasLP = 2meV, which means that the Landau criterion is verified. We see that at the energy ofthe system, only one state is available: no scattering is therefore possible, the system issuperfluid.On the other hand, we have in blue the situation where k p = 3 µm -1 and ∆ E lasLP = 3meV, so that the system is supersonic. In that case, a second state is available at theenergy of the excitation: the system can thus scatter to it, we are in the Cerenkov regime.The model that we considered here simplifies the polaritonic dispersion to be parabolic,which is valid for small wavevectors. However, a more complete model considering thefull polariton dispersion can be found in [64].These two regimes have been experimentally observed for the first time in 2009 in thegroup [18], and reported in figure 1.16.The experimental transition to the superfluid regime is pictured in figure 1.16.a. Theupper line presents the real space images and the bottom line the momentum space.The polariton fluid is sent toward a defect, with an in-plane wavevector k p = − . -1 and a flow from top to bottom. The detuning between the laser and the lowerpolariton branch is ∆ E lasLP = 0 .
10 meV. The blue-circled images correspond to lowexcitation power, i.e. low polariton density. The fluid elastically scatters on the defect:parabolic wavefronts are visible in real space while a ring appears in momentum space.By increasing the excitation power (red-circled images), the fluid behaviour changes:the wavefronts start to fade away as well as the scattering ring, eventually reaching thesuperfluid regime in the last green-circled images: the viscosity has completely disap-peared, the fluid flows around the defect without any scattering, as the single excitationspot in the momentum space confirms.The Cerenkov effect is illustrated by the images of figure 1.16.b. Again, the upper lineshows the real space experimental results and the lower one the momentum space, with apolariton flow from top to bottom hitting a defect. The conditions are this time different:the pump wavevector is k p = − .
521 µm -1 for a detuning ∆ E lasLP = 0 .
11 meV, in orderto be able to reach the Cerenkov regime.As in the previous case, the excitation power increases in the images from left to right.At low density (left blue-circled image), one can again observe the elastic scattering of thefluid on the defect, with its characteristic parabolic fringes in real space and scatteringring in the far field. .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS
Superfluidity in polariton fluid . a. Experimental results of a polari-ton fluid transition to superfluidity. Top images are real space while bottom ones arethe momentum space. The flow goes from top to bottom hitting a defect, the in-planewavevector is k p = − .
337 µm -1 and the detuning ∆ E lasLP = 0 .
10 meV. The polaritondensity increases from left to right: at low density, the fluid scatters on the defect, asshown by the scattering ring in the far field and the parabolic wavefront in the densitymap. Those gradually vanish while the density increases, until they completely disappearin the superfluid regime on the right images. b. Same configuration in the supersonicregime ( k p = − .
521 µm -1 and ∆ E lasLP = 0 .
11 meV). This time, the increase of densityleads to a linearisation of the scattering fringes around the defect, also known as theCerenkov cone. From [18].8
CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT
However this time, the increase of density leads to a different behaviour of the fluid:instead of vanishing, the scattering fringes change shape: the wavefronts become lineararound the defect and we have the appearance of the so-called Cerenkov-Mach cone. Atmuch higher density (density on the right images is 5 times higher than the one on theleft), the linear shape of the fringes is maintained but widens, along with the suppressionof the elastic scattering ring.
The previous section described the dynamics of a polariton fluid when excited resonantlyor quasi-resonantly. The incoming photons were therefore sent with an energy close to theone of the lower polariton branch, and the properties of the polariton fluid were inheritedfrom the ones of the pump.This excitation scheme is the one that was mostly used in the results of this thesis,however it is not the only way to create a polariton fluid. This section gives an overviewof the non-resonant injection which can lead to a condensation of the polaritons. i Non-resonant pumping
One of the particularity of the coherent, quasi-resonant pumping of a microcavity isevidently the coherence of the generated fluid with the driving fluid, leading to the fixationof many properties of the fluid to the pump ones. To seek for a spontaneous formation ofcoherence in a polariton fluid, several configurations can be implemented, as working witha large angle optical drive [75] or by electrically injecting polaritons [12, 76]. However,the most common scheme is to pump the microcavity out of resonance, i.e. using a farblue detuned light, where the coherence of the pumping laser is lost during the relaxationprocess towards the lower polariton branch. The detuning of the excitation light mustmatch a dip of the cavity reflectivity, so that enough signal can enter the cavity and relaxinto the ground state polariton population.First, the non-resonant pumping leads to a cloud of electrons and holes that thermalizesto its own temperature through exciton-exciton interactions [30]. The system then reducesits kinetic energy by interacting with phonons, resulting in a relaxation along the polaritonbranches. It allows to obtain a photoluminescence image of the dispersion curves, whereboth polariton branches are visible, as shown in figure 1.17. However, the signal is nothomogeneously distributed and is much stronger at the bottom of the lower polaritonbranch of lower energy. The replica fringes come from multiple reflections within thesample substrate.However, for high enough polariton densities, a new relaxation phenomenon can takeplace, based on polariton-polariton collisions [20]. Two polaritons can collide and respec-tively scatter to the bottom of the LP branch and to the large wavevector region, mostlyexcitonic and with a large density of states. As polaritons have a bosonic statistics, assoon as the polariton population already at the bottom of the dispersion starts to havea phase-space density of order one, this relaxation scheme is enhanced by the bosonicstimulation process. Therefore, if the stimulation overcomes the losses, an accumulation .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS
Photoluminescence image of the polariton dispersions . Bothbranches are visible, even though there is more signal coming from the lower branchsince it is of lower energy. The replica fringes come from multireflections within thesample substrate.of a macroscopic coherent polariton population takes place in the final state which can bedefined as a condensate, that we will further study in the next section. The irreversibilityof this process is ensured by the fact that the phase-space density of the excitonic regionalways stays much lower than one. ii Bose-Einstein condensation
As we already saw in the previous sections, polaritons can be considered as bosons inthe low density regime. We can therefore wonder if they possess all the characteristics ofthis type of particles, and in particular a very typical one known as the
Bose-Einsteincondensation and initially though for cold atoms. We will first describe the theoreticalapproach of this phenomena, before looking at the polariton case.In 1924, Satyendranath Bose proposed a theoretical description of the photons statis-tics which took into account their indistinguishability and the possibility to have severalparticles in the same state [77]. The year later, Albert Einstein generalized its statisticsto bosons in general, i.e. to all particles with an integer spin, as what is now known asthe Bose-Einstein distribution and can be written as: n ( k , T, µ ) = 1 exp (cid:18) E ( k ) − µk B T (cid:19) − k is the particle wavevector, E ( k ) their dispersion function, k B the Boltzmann0 CHAPTER 1. EXCITON-POLARITONS QUANTUM FLUID OF LIGHT constant, T the temperature and µ the chemical potential, which corresponds to theamount of energy needed to add a particle to the system. In particular, Einstein predictedthat at low temperature and at thermodynamic equilibrium, non-interacting identicalbosons should all condense in the same state and become indistinguishable: it is theBose-Einstein Condensation.At low temperature, the particles can not be considered classic anymore, and they aredescribed by their wavepacket, characterized by the thermal de Broglie wavelength λ dB : λ dB = s π (cid:126) mk B T (1.70)with m the particle mass. If the temperature is sufficiently low so that the de Brogliewavelength becomes of the same order of magnitude than the interparticle distance d , thewavepackets of the particle start to overlap. A part of the gas can therefore be describedthrough a macroscopic wavefunction, which is equivalent to consider that a fraction of itis condensed and the other part not. However, at zero temperature, all of the wavepacketsmerged and the gas is described by a single macroscopic wavefunction, it is a pure BoseEinstein condensate. The first experimental observation of such a condensate has beenrealized by M. H. Anderson et al. [78] in 1995 using ultracold Rubidium atoms. Theymanaged to reach temperatures of a few hundreds nanokelvins, and the atomic gas wasdiluted enough so that the interaction could be neglected.However, the full description of the condensate dynamics must consider the particlesinteraction. To do so, the mean field approximation can also be used and the globalwavefunction of a condensate of N atoms be written ˆΦ( x , x , ..., x N ) = Q N ˆΨ i ( x i ), whereˆΨ i is the individual particle wavefunction. It is based on this approximation that Lev P.Pitaevskii and Eugene Gross established their famous Gross-Pitaevskii equation , initiallydeveloped for such atomic gases: i (cid:126) ∂ ˆΨ( x ) ∂t = (cid:18) − (cid:126) m ∇ + V ( x ) + (cid:126) g | ˆΨ( x ) | (cid:19) ˆΨ( x ) (1.71)with m the particle mass, V an external potential and g the coupling constant. g isproportional to the diffusion length between particles a s such that g = 4 π (cid:126) a s /m . Theminimization of the energy leads to the definition of a chemical potential µ = ∂E/∂N = (cid:126) gN in the homogeneous case. iii Polariton condensates As we saw in the previous sections, exciton-polaritons can be considered as bosons inthe low density regime. One can thus expect to observe a phase transition similar tothe Bose-Einstein condensation in polariton fluid as well. Imamoglu and Ram were thefirst to suggest in 1996 [79] that the bosonic properties of the polaritons could lead to acondensate state emitting a coherent laser light. They described a coherent population ofthe ground state coming from an incoherent excitonic reservoir; that could be interpretedas a phase transition, in that case similar to the Bose-Einstein condensation, or as a lasingfrom bosonic stimulated scattering. .2. SUPERFLUIDITY AND CONDENSATION IN POLARITON SYSTEMS physics soon exits the regime of weakly interacting bosons thatdescribes ultracold atoms; second, the lifetime is short enough thatwe must confront the role of non-equilibrium physics . Never-theless, the principal experimental characteristics expected for BECare clearly reported here: condensation into the ground state arisingout of a population at thermal equilibrium; the development ofquantum coherence, indicated by long-range spatial coherence, andsharpening of the temporal coherence of the emission. Experimental procedure
The sample we studied consists of a CdTe/CdMgTe microcavitygrown by molecular beam epitaxy. It contains 16 quantum wells, displaying a vacuum field Rabi splitting of 26 meV (ref. 26). Themicrocavity was excited by a continuous-wave Ti:sapphire laser,combined with an acousto-optic modulator (1- m s pulse, 1% dutycycle) to reduce sample heating. The pulse duration is sufficientlylong (by four orders of magnitude) in comparison with the charac-teristic times of the system to guarantee a steady-state regime. Thelaser beam was carefully shaped into a ‘top hat’ intensity profileproviding a uniform excitation spot of about 35 m m in diameter onthe sample surface, as shown in Fig. 4i. The excitation energy was1.768 eV, well above the polariton ground state (1.671 eV at cavityexciton resonance), at the first reflectivity minimum of the Braggmirrors, allowing proper coupling to the intra-cavity field. Thisensures that polaritons initially injected in the system are incoherent,which is a necessary condition for demonstrating BEC. In atomicBEC or superfluid helium, the temperature is the parameter drivingthe phase transition. Here the excitation power, and thus the injectedpolariton density, is an easily tunable parameter, and so we chose it asthe experimental control parameter. The large exciton bindingenergy in CdTe quantum wells (25 meV), combined with the largenumber of quantum wells in the microcavity, is crucial in maintain-ing the strong coupling regime of polaritons at high carrier density.The far-field polariton emission pattern was measured to probe thepopulation distribution along the lower polariton branch. Thespatially resolved emission and its coherence properties are accessiblein a real-space imaging set-up combined with an actively stabilized Figure 1 | Microcavity diagram and energy dispersion. a , A microcavity is aplanar Fabry–Perot resonator with two Bragg mirrors at resonance withexcitons in quantum wells (QW). The exciton is an optically active dipolethat results from the Coulomb interaction between an electron in theconduction band and a hole in the valence band. In microcavities operatingin the strong coupling regime of the light–matter interaction, 2D excitonsand 2D optical modes give rise to new eigenmodes, called microcavitypolaritons. b , Energy levels as a function of the in-plane wavevector k k in aCdTe-based microcavity. Interaction between exciton and photon modes,with parabolic dispersions (dashed curves), gives rise to lower and upperpolariton branches (solid curves) with dispersions featuring an anticrossingtypical of the strong coupling regime. The excitation laser is at high energyand excites free carrier states of the quantum well. Relaxation towards theexciton level and the bottom of the lower polariton branch occurs byacoustic and optical phonon interaction and polariton scattering. Theradiative recombination of polaritons results in the emission of photons thatcan be used to probe their properties. Photons emitted at angle v correspondto polaritons of energy E and in-plane wavevector k k ¼ ð E = " c Þ sin v : Figure 2 | Far-field emission measured at 5 K for three excitationintensities.
Left panels, 0.55 P thr ; centre panels, P thr ; and right panels,1.14 P thr ; where P thr ¼ is the threshold power ofcondensation. a , Pseudo-3D images of the far-field emission within theangular cone of ^ , with the emission intensity displayed on the vertical axis(in arbitrary units). With increasing excitation power, a sharp and intense peakis formed in the centre of the emission distribution ð v x ¼ v y ¼ Þ ; corresponding to the lowest momentum state k k ¼ b , Same data as in a but resolved in energy. For such a measurement, a slice of the far-fieldemission corresponding to v x ¼ is dispersed by a spectrometer andimaged on a charge-coupled device (CCD) camera. The horizontal axesdisplay the emission angle (top axis) and the in-plane momentum (bottomaxis); the vertical axis displays the emission energy in a false-colour scale(different for each panel; the units for the colour scale are number of countson the CCD camera, normalized to the integration time and optical densityfilters, divided by 1,000 so that 1 corresponds to the level of dark counts:1,000). Below threshold (left panel), the emission is broadly distributed inmomentum and energy. Above threshold, the emission comes almostexclusively from the k k ¼ ARTICLES
NATURE | Vol 443 |
28 September 2006 © Nature Publishing Group
Figure 1.18:
Polariton Bose-Einstein Condensation . Real space density (a.) anddispersion (b.) of a polariton BEC, obtained by increasing the intensity of the incoherentpump from left to right. A long-range coherence is reached in the real space while themomentum distribution shows an accumulation in the lower energy state. From [11].The actual equivalent for Bose-Einstein condensate in polariton has however been re-alized under non-resonant pumping in 2006 by Jacek Kasprzak et al. [11]. An importantadvantage of exciton-polaritons is their low effective mass, several order of magnitudesmaller than the exciton mass, which allows condensation at much higher temperaturesthan for the atoms (a few Kelvins while the atoms only condense at the nanokelvin scale).By exciting the system with a sufficiently high power, the incoherently injected polaritonsreach a higher density than the critical one for Bose Einstein condensation, which resultsin the creation of an actual coherent polariton condensate, as presented in figure 1.18.The long range coherence is reached in the real space while the momentum space showsan accumulation of the population in the lower energy state. hapter 2
Experimental tools
The previous chapter detailed the theory of our system, where we established our toolsto describe the polariton evolution. Before discussing the experimental results in thefollowing chapters, we will report here the tools we used in the lab to implement ourexperiment.The first section focuses on the excitation part of the experiment: after a detailedcharacterization of the sample used in this work, we will describe the two cryostats wehave in the lab and the laser source we used, as well as the Spatial Light Modulator orSLM, a device shaping the wavefront of the light beam which was very useful for thefollowing results.On the second part of the chapter, we will discover which detection tools were used toretrieve the system information. The main devices will be described, the CCD cameraand the spectrometer, and an explanation of the data analysis will be given.534
CHAPTER 2. EXPERIMENTAL TOOLS
Fabrication
The sample used for this work is a planar microcavity realized at the EcolePolytechnique Fédérale de Lausanne (EPFL) by Romuald Houdré. This semiconductorsample is very robust and has been used in the group since many years.It has been fabricated by
Molecular Beam Epitaxy (MBE) : this technology isused to grow ultrathin films with a very good control of their thickness and composition.Realized under ultra high vacuum, target materials are heated to their sublimation pointsand produce molecular beams. They can then chemically react with other species beforecondensing as a layer on the single crystal substrate. This technique has a nanometerscale precision as the atomic layers can be grown one by one [86].In our case, the sample is a stack of layers of four different compounds: Ga Al As,AlAs, GaAs and In Ga As (see figure 2.1). The substrate is a thick layer of GaAs,polished in order to allow working in transmission. Then the first Distributed BraggMirror (DBR) is grown: it is a pile of 53 alternate Ga Al As and AlAs layers, each ofthem being λ / n i thick, with λ the wavelength of the excitonic resonance ( λ = 836 nm)and n i the respective refractive index of the considered compound ( n Ga . Al . As = 3 . n AlAs = 2 . λ /n GaAs , in which are em-bedded three quantum wells of In Ga As at the antinodes of the electromagneticfield. The thickness of the cavity defines the resonance energy of the cavity photons, andtherefore the detuning between the excitonic resonance and the photonic one. In order tobe able to tune it experimentally, a small wedge is introduced between the two mirrors,in the order of 10 − radians. This way, the photonic resonance scales linearly with theposition on the sample, with an energy gradient of 710 µ eV.mm -1 leading to detuningsfrom +8 meV to -4 meV.Finally, the front mirror is grown, similar to the back one except that it only contains40 alternated dielectric layers. Reflectivity and finesse
The reflectivity of a DBR is defined with the formula [87]: R = n ( n H ) N − n ( n L ) N n ( n H ) N + n ( n L ) N (2.1)where• n is the refractive index of the medium before the mirror; for us, the inside of thecavity in GaAs: n = n GaAs = 3 . n is the refractive index of the medium after the mirror; i.e. the air for the frontmirror ( n = 1) and the GaAs substrate for the back mirror ( n = n GaAs = 3 . n H and n L the refractive indices of the two media constituting the DBR, with n L < n H . For our sample, n L = n AlAs = 2 .
95 and n H = n Ga . Al . As = 3 . .1. EXCITATION Semiconductor microcavity
Two Bragg mirrors of alternated layers ofGa Al As and AlAs of optical thickness λ / λ , in which three quantum wells ofIn Ga As are embedded at the antinodes of the electromagnetic field, which am-plitude is presented with the white line. The cavity is grown on a substrate of GaAs,previously polished to allow working in transmission. The excitation takes place on thethin mirror side, while the detection is done on the substrate side.6
CHAPTER 2. EXPERIMENTAL TOOLS • N is the number of pairs of alternated layers in the DBR: 21 for the front mirror,24 for the back oneThis leads to a reflectivity of 99,85% for the front mirror and 99,93% for the back one.From that we can extract the finesse of the cavity [30]: F = π √ R − R (2.2)with R = p R front R back the total power reflectivity. For our sample, the finesse is closeto 3000. Anticrossing
The measurement of the anticrossing of our cavity is presented in figure2.2. By pumping the sample out of resonance and at k = 0 µ m -1 , the energies of the upperand lower polariton branches can be extracted for several positions on the sample. Thisleads to the reconstruction of the photonic and excitonic resonances, in red and greendashed lines in figure 2.2. The Rabi energy corresponds to the difference between theupper and lower branches for an exciton-photon detuning equals to zero: E R = (cid:126) ω R =5 .
07 meV for this sample.Figure 2.2:
Experimental anticrossing . The blue dots represent the upper and lowerpolariton branches, obtained by non resonant pumping at k = 0 µ m -1 . Changing theposition changes the photon-exciton detuning ∆ E Xcav (horizontal axis); the exciton andphoton energies are plotted in green and red dashed line, respectively. The experimentalvalue for the Rabi splitting is extracted at zero detuning: (cid:126) Ω R = 5 .
07 meV. From [32] .1. EXCITATION Sample inhomogeneities
Figure 2.3 presents a picture of a large area of the sample,taken from [40]. Reconstructed from several images, this picture was realized at normalexcitation with two different wavelengths of the laser, 837.21 and 837.05 nm. Both reso-nances are clearly visible on the picture as bright lines: they indicate regions of the cavitywith the same effective length, orthogonal to the direction of the thickness gradient.Figure 2.3:
Cavity transmission . Picture of a sample area of 3.2x0.9 mm , excitedat two different wavelengths (837.05 and 837.21 nm). The resonances are clearly visibleorthogonal to the wedge, as well as the different types of defects of the cavity: the regularpattern of mosaicity, some elongated dislocations and point-like defects. From [40]Different types of irregularities are showing up in figure 2.3. First, a thin regular pat-tern of an orthogonal lattice: this indicates the mosaicity phenomenon, due to mechanicalconstraints of the crystal [88, 89]. Some irregular defects are also visible, elongated andthick, mainly on the left side of the picture: those dislocations can be attributed tomechanical shocks on the sample [30, 90, 91]. Finally, point-like defects are randomly dis-tributed over the sample. They can be bright or dark as they correspond to potentialbarriers or wells: a barrier shifts the resonance locally, preventing the nearby polaritonsto excite the states of the defect, while a well traps the polaritons, leading to a strongeremission [92, 93].The sample is glued on the holder using silver paste. The sample holder is a plate ofcopper, to insure good thermal conductivity, with a hole in its center to allow working intransmission. We have in the lab two different cryostats that will be described separately in this section.8
CHAPTER 2. EXPERIMENTAL TOOLS i Oxford Instrument Microstat
The Oxford Instrument cryostat was used for most of the results presented in this thesis.The whole cryogenic setup is sketched in figure 2.4.The cryostat has two circular windows on both sides of the cold finger end, on whichthe sample holder is screwed. The vacuum inside the cryo is done using a primary pumpand a turbopump and reaches 10 -4 mbar. As the cryostat is quite old and has a few leaksaround the cold finger, we usually keep the pump running while doing experiment, eventhough it induces small vibrations.The cooling process is realized with a continuous Helium flow. The liquid Heliumis stored in a Dewar tank, connected to the cryostat with a transfer tube. This oneis inserted inside the cold finger of the cryostat, and plugged to the university Heliumnetwork through a gas flow pump (see figure 2.4). The Helium cool down the cold fingerand thus the attached sample. Its temperature is displayed using a sensor on the sampleholder, and can be adjusted by tuning the Helium flow. We can cool the sample down to3.8 K, but we usually work a bit higher, around 4.5 K, to not use to much Helium.Figure 2.4: Cryogenic setup . The liquid Helium stored in the Dewar tank is draggedby the pump along the transfer tube. The flow goes along the cryostat, in direct contactwith the cold finger and consequently with the sample. The heated Helium flow is thenpumped to the recovery network of the university. The flow can be controlled with aflow meter, adjusting the sample temperature. This one is displayed on the temperaturecontroller, connected to a sensor on the sample holder. Adapted from [94]This system works well, but has the drawback of a high Helium consumption. Fortu-nately, the university has a Helium recovery network and its own liquefier, which ensureus an easy supply. However, we still do not let it run constantly, which limits the mea-surements to one-day experiments. .1. EXCITATION ii MyCryoFirm Optidry A second cryostat has been recently installed in the group and has been implemented inparallel of the reported work. This closed circuit cryostat is an Optidry from MyCryoFirm:this company offers cryostats with great flexibility to match the customer needs.The vacuum of the whole system is again ensured by a turbopump, preceded by aprimary pump, and can also reach 10 -4 mbar. The vacuum is well conserved in thiscryostat so we do not keep the pump running when the vacuum is reached.The MyCryoFirm products are working in closed circuit, i.e. they do not need Heliumrefueling but are connected to a compressor which ensures the Helium re-compression.The cooling of the compressor is managed by a heat exchanger, itself connected to thecold water network of the university.This system leads to a great stability of the cryostat: once the target temperature isreached, it can stay on for months with very small fluctuations of the conditions. It ismuch more convenient for a continuity of the experiment from one day to the next.Figure 2.5: New cryostat system . The left image, taken from [95], is a 3D model ofthe outside of the cryostat. The working place corresponds to its right part, composed ofthree plates at different temperatures detailed on the right picture. The lower plate, inorange, is the coldest one and can be cooled down to less than 4 K. It is the one where isattached the sample, mounted on piezo-based nanopositioners to be able to move it, andplaced at the height of the windows to be optically accessible. A second plate surroundsthe first one, in blue, at 50 K to ensure a thermal shield. Finally, the upper one in greyis a breadboard at ambient temperature, as it is directly connected to the outside wall ofthe cryostat. This is where the closest optical components can be mounted. The red linecorrespond to the optical axisOur cryostat consists of three plates at different temperatures, as illustrated in figure2.5. The coldest one is the deepest, plotted in orange: it reaches the target temperature,that can be set to less than 4K. The sample is in direct contact with it and placed at theheight of the optical axis (in red). It is mounted on two piezo-based nanopositioners fromAttocube to be scanned along the x and y axis. The temperature is controlled through a0
CHAPTER 2. EXPERIMENTAL TOOLS sensor screwed on the sample holder.A second plate, in blue in the figure, is at 50 K and works as a thermal shield. Thethird plate, in grey, corresponds also to the walls of the cryostat, and is therefore at roomtemperature. In the cryo, the horizontal plate is a breadboard where optical componentscan be mounted to be as close as possible to the sample.
To excite the polariton fluid resonantly, the light field needs to validate several require-ments. First of all, the light needs to enter the cavity: the laser should be at the resonanceenergy, around 835 nm in our case. However, we also want to be able to transfer an in-plane wavevector to the fluid, and therefore explore the dispersion curve. To do that, thelaser should be continuously tunable over the corresponding energy range, in the order of10 nm. Finally, the spectral bandwidth of the laser should be thin enough to excite onlyone excitonic mode.The more suitable laser type for those requirements are the Titanium-Sapphire lasers.The active medium of these solid state lasers is a sapphire crystal doped with titaniumions (
T i : Al O ), optically pumped. In the continuous wave configuration, they canbe tuned from about 700 to 1200 nm. In our laboratory, we use a commercial TiSa laserfrom Spectra Physics, the Matisse 2 TR. Tunable from 730 to 930 nm, it is pumped by acommercial doubled Nd:Yag green laser Verdi from Coherent, delivering 10 W at 532 nm.This way, the TiSa can emit up to 1.2 W in the T EM mode for a linewidth thinnerthan 50 kHz. A Spatial Light Modulator or SLM is a device that can shape the wavefront of a lightbeam. In our lab, we are using an LCOS model from Hamamatsu, which stands for LiquidCrystal On Silicon, and whose structure is presented on figure 2.6.Liquid crystals are a particular phase of matter, which flows and takes the shape ofits container, but whose particles still have a positional order, and therefore a preferredorientation [96]. This preferred orientation can be modulated by applying an electric field,which induces in the same time a change in the refractive index of the medium as liquidcrystals are birefringent.To use this property, LCOS SLM screen has a liquid crystal layer embedded betweenelectrodes (see figure 2.6):• a grounded transparent electrode on a glass substrate, on the side of the incominglight• a matrix of pixels electrodes, connected to an active circuit on a silicon substrate.There are 792x600 pixels, of 9.9x7.5 mm each, with a pixel pitch of 12.5 µm.A specific voltage can be applied between each pixel electrode and the grounded one,defining also the orientation of the liquid crystals in between, and with it their opticalindex. A planar incident wave experiences therefore different optical paths depending onthe pixels, and the reflected wavefront is shaped accordingly. .2. DETECTION Transverse view of the Hamamatsu LCOS-SLM screen . The siliconsubstrate supports an active matrix circuit on which pixels electrodes are connected. Aspecific voltage is applied between each pixel and a transparent grounded electrode. Thisvoltage induces a certain orientation of the liquid crystals in between the electrodes, andthus a specific optical refraction index due to their birefringence. A glass substrate finallycovers the top. An incident plane wave consequently follows a specific optical path oneach pixel, resulting in a modulated reflected wavefront.The SLM head, which contains the screen, is connected to a controller, itself connectedto a DVI input sendind a 600x800 pixels image in grey shades. The controller translatesthe greyness level (from 0 to 255, bitmap image) into voltage and reproduces the imageon the SLM screen, thus drawing it on the light wavefront.The efficiency of the SLM is not perfect: the liquid crystals do not cover the wholesurface of the screen, therefore a part of it only reflects on the back of the SLM whichacts like a standard mirror. According to its manufacturer [97], the light utilizationefficiency of our model is 97%. In order to clean the output image, a filtering is realizedin the Fourier plane of the SLM: a grating is added to the input image, so that only thelight effectively interacting with the SLM is diffracted into the first order. By blockingall the other orders, the beam is cleaned and corresponds exactly to the desired pattern.This technique allows also to use the SLM not only to shape the phase pattern, butalso the intensity profile. Indeed, the quantity of light diffracted into the first orderdepends on the contrast of the grating. If the whole spectrum of grey scale is used, all theavailable light is diffracted; if the grating only uses half of the scale, half of the light stayson the zeroth order and is therefore filtered. Some patterns can thus be drawn on thegrating, playing with the contrast, and be translated as intensity levels in the reflectedlight beam.
The detection of our system is done in real space and in momentum space. A typicalexperimental setup is presented in figure 2.7: the signal from the sample goes throughan objective with a large numerical aperture (0.42), before being separated into twoarms, which correspond to the real and momentum spaces. The momentum space passes2
CHAPTER 2. EXPERIMENTAL TOOLS through a spectrometer while the real space interacts with a reference beam, extractedfrom the initial laser beam.Figure 2.7:
Typical experimental setup . The signal out of the sample is split intotwo arms which detect the real and momentum space. The real space can interferewith a reference beam, while the momentum space goes through a spectrometer to getinformation on the signal energy.
The cameras used in this experiment are PIXIS1024BR eXcellon models from PrincetonInstruments. This charged couple device (CCD) has a back illuminated chip of 1024x1024pixels, each 13 µ m , and a quantum efficiency of 95% at 830 nm. The reading rate of theCCD is 100 kHz, while the integration time can be on the order of the millisecond. Thecamera also has a mechanical shutter.It is connected to a computer and driven through the Princeton Instruments softwareWinspec, which also manages the spectrometer. The spectrometer of the lab is also from Princeton Instruments and is an Acton SeriesSP2750 model. This device has a focal length of 750 mm and an interchangeable triplegrating turret, on which are mounted 2 gratings and one mirror. The gratings respectivelyhave 1200 and 1800 lines per millimeter, and the mirror allows to couple the entranceplane with the CCD one. The entrance plane has an adjustable slit that can be manuallytuned between 10 µm and 3 mm, i.e. to a minimal aperture smaller than the pixel size.With the 1800 lines per millimeter grating, the spectral resolution is of the order of 1 Å,which is close to 0.2 meV at 835 nm. .2. DETECTION i Image analysisSPE file format
All the measurements are pictures of the CCD, taken through theWinspec software. As it manages both the camera and the spectrometer, pictures aresaved in SPE format, which also stores information about the experimental conditions.SPE files contain a matrix corresponding to the camera pixels, with for each of themindicates the associated photon counts received during the integration time. This matrixis preceded by a header filled by many parameters, as the position and calibration of thespectrometer, the date of the measurement or the integration time. Those data are easilyaccessible and useful for the later analysis. The integration time is typically on the orderof a few millisecond, close to the shortest time accessible by the camera. We have enoughsignal to record a clean picture during this time - we usually use filters in front of thespectrometer - and a short integration time reduces our sensitivity to vibrations of thesystem induced by the pump and the cryostat.The different steps of the data analysis are presented in figure 2.8, and explained moreprecisely in the following paragraphs.
Intensity
The intensity image is a direct picture of the output plan of the cavity.Photons escape from the cavity accordingly to their decay rate, and are detected by thecamera: their spatial distribution is directly proportional to the polariton density.Thus the pictures do not need much analysis: after defining the Region-Of-Interestfor the considered measurement, the background counts are removed, as the mean valueof a region not illuminated by the polariton signal. The colorscale can also be readjusted,and chosen to be linear or logarithmic depending on what is shown. An example is givenin figure 2.8.a, where a phase modulation marked with dark lines is sent on the bottomof the picture, marked by the dark lines, with a flow from bottom to top. The intensityscale is chosen logarithmic.
Phase profile
The phase profiles are extracted from interference images of the cavitysignal with a reference beam of the same laser source, as shown in figure 2.8.b. Theinterference beam is aligned in order to have linear thin fringes to optimize the resolutionof the extracted pattern. However, as the system sightly vibrates, too thin fringes alsolead to a bad contrast: we usually work with fringes around 5 pixels width.4
CHAPTER 2. EXPERIMENTAL TOOLS
Figure 2.8:
Data Analysis . a. Real space image, signal from the sample renormalized.b. Experimental fringes obtained by interfering the signal with the reference beam. c.Fourier transform of image b: the smallest spatial frequencies are in the center, while thesignal outlined in red corresponds to the fringes that we want to keep. d. Filtering ofimage c., kept at its original position. e. Phase of the inverse Fourier transform of imaged.: the fringes are cleaned and highlighted. f. Filtering of image c., shifted towards thecenter to erase the fringes frequency. g. Phase of the inverse Fourier transform of imagef, fitted to place at 0 the phase of the area without modulation: only the phase profile iskept.The background is subtracted and the images are renormalized. In order to enhancethe interference fringes, they are filtered in the Fourier space. The Fourier transform ofthe data is computed, as presented in figure 2.8.c: the signal outlined in red correspondsto the fringes spatial frequency. The bright spot in the center of the image comes fromthe signal with spatial frequency equals to zero, i.e. the background noise that we wantto remove. Finally, the elongated spot symmetric to the selected one also comes from .2. DETECTION
Momentum space
Several pictures need to be taken in momentum space in order toaccess all the parameters of the experiment. The first one is the dispersion, which dependson the position of the sample. This one is carried by photoluminescence: by exciting thesample out of resonance, i.e. with a much higher energy than the polariton branches,the relaxation of the excess energy fills all the accessible polariton states and allows for adirect detection of the polariton dispersion.Figure 2.9:
Momentum space detection image . The brightest spot, located by thered cross, indicates the excitation momentum, while the elastic scattering of the polaritonson the cavity disorder is responsible for the shallow Rayleigh ring. In between the openslit of the spectrometer, indicated by the vertical red lines, we detect a far field imagewhich can be graduated in µm -1 , while the slit position along the wavelength axis givesus the excitation energy.The wavelength of the incoherent excitation is chosen to correspond to the first mini-6 CHAPTER 2. EXPERIMENTAL TOOLS mum of the cavity transmission, which is 810 nm in our case. The dispersion is very wellresolved with a thin entry slit of the spectrometer, even though the integration time ofthe camera should be increased to a few hundred milliseconds as the signal is quite weak.Another necessary measurement to carry out in the momentum space is the far field,which is the k x - k y image, presented in figure 2.9. This picture gives us the shape and loca-tion of the excitation in the momentum space, and its position according to the dispersion.As the excitation spot is usually collimated on the sample, it gives a main single spot inthe momentum space, whose location is extracted through a bidimensional gaussian fit ofthe image. As the sample is not perfect but presents also some structural disorder, thepolaritons can elastically scatter on the defects which results in the appearance of a ringin the momentum space, called the Rayleigh ring which intensity is usually smaller thanthe signal from the pump, and visible in figure 2.9.Those images are usually taken through the spectrometer, keeping its entry sit open,as illustrated by the two vertical red lines. The position of the slit is therefore defined bythe wavelength of the signal λ exc , giving us the excitation energy, while inside the slit theimage can be scaled in µm -1 , hence the k x - k y image. ii Parameters extractionExcitation parameters The first parameters are extracted from the dispersion of thelower polariton branch: E LP = E X − E γ ( k )2 − q(cid:0) E X − E γ ( k ) (cid:1) + 4Ω R (2.3)with E X the excitonic resonance energy, considered constant over our range of wavevec-tors; E γ ( k ) the photonic resonance energy, approximated parabolic with a curvature pro-portional to the photonic effective mass m ∗ γ , and Ω R the Rabi frequency.Four parameters need therefore to be extracted: the excitonic resonance, the cav-ity photon energy at normal incidence, the effective mass of the photon and the Rabifrequency, which is constant for a given sample.Those are obtained by fitting the lower polariton branch after a background subtrac-tion, as shown in figure 2.10: the white lines are the upper and lower polariton branches,while the yellow and green dashed lines respectively illustrate the corresponding cavityphoton and bare exciton dispersions.The polariton dispersion is necessary to calibrate the camera screen in a wavevectorscale. Indeed, the optical path is realigned each day, and with it the position of thezero wavevector. With the dispersion fit, each pixel line is assigned to its correspondingwavevector, and in particular the one of the maximum of the excitation spot from the k x - k y image, giving us the excitation wavevector k exc , in red in figure 2.10. The excitationenergy E exc is obtained as the position of the center of the spectrometer slit.In the case of a quasi resonant excitation, mainly used in the present work and il-lustrated in figure 2.10, the excitation spot is blue-shifted with respect to the polaritondispersion. Its energy is thus slightly higher than the lower branch one, by a detuning∆ E lasLP illustrated with the orange arrow in the figure. .2. DETECTION Experimental dispersion and quasi resonant excitation . The back-ground image corresponds to photoluminescence of the sample, carried out by non res-onant pumping. The relaxation of the excess energy leads to a polariton distributionalong the lower branch, much brighter than the upper one. Therefore only the lower oneis used for the numerical fit, illustrated by the white line. The corresponding excitonicand photonic dispersions are plotted in green and yellow dashed lines, respectively. Theexciton-photon detuning ∆ E Xcav is presented as the blue arrow. The excitation spot,in red, is extracted from the k x - k y image. It is slightly blue-shifted from the polaritondispersion, from an energy detuning ∆ E lasLP illustrated with the orange arrow. Derived parameters
The previously described parameters, directly extracted fromthe experimental data, are used to evaluate some derived parameters. First of all, the fluid velocity is defined as the group velocity of the polaritons, i.e. the first derivativeof the dispersion: v fluid = 1 (cid:126) ∂E∂k (2.4)In our system, the fluid velocity can be up to 1.5 µ m/ps.From the dispersion can also be extracted the effective mass of the polaritons m ∗ LP created at k exc , using the relation 1 m ∗ LP = 1 (cid:126) ∂ E∂ k (2.5)As we saw in the theoretical chapter 1, the lifetime of the polaritons is experimentally8 CHAPTER 2. EXPERIMENTAL TOOLS determined with the dispersion analysis, as it is directly connected to the linewidth of thecavity: δE = (cid:126) γ = (cid:126) τ (2.6)with γ the cavity decay rate and τ the lifetime of the polaritons. In our case, the linewidth δE is found to be 0.07 meV, which corresponds to a polariton lifetime of 9 ps.Finally, the speed of sound c sound also derived from the excitation parameters: c sound = s (cid:126) g | ψ | m ∗ LP = s ∆ E lasLP m ∗ LP (2.7)where g is the interaction constant and ψ the polariton wavefunction. The second equalityis valid under resonant excitation within the polariton linewidth and in the nonlinearregime. The speed of sound in our system is typically of the order of 0.5 µ m/ps.From the speed of the fluid and the speed of sound can be defined the Mach number of the system, M = v fluid /c sound . By comparison with unity, this quantity defines sub-or supersonic flow conditions and will be particularly used in the following work. hapter 3 Spontaneous generation oftopological defects
The first two chapters introduced all the necessary tools to understand our system: thetheoretical description first, then the experimental devices available in the lab.We present in this chapter the first part of our experimental results, the spontaneousgeneration of topological excitations. To do so, we use the property of bistability of oursystem: this one is detailed in the first part of the chapter.The experiment was implemented following a theoretical proposal published in 2017[25], suggesting to split the excitation into two beams of different intensities, the seedand the support. This configuration allows to obtain a large area of the fluid on theupper branch of bistability. Then, by giving it a flow and sending it toward a defect, wecan observe the generation of quantum turbulence, and in particular of vortex-antivortexpairs in the subsonic case. Not only does it show that the bistable regime releases thephase constraint of the quasi-resonant excitation, but also that the generated excitationsare sustained for hundreds of micrometers.Increasing the Mach number of the fluid also increases the generation rate of the vor-tices. At some point close to the subsonic-supersonic transition, vortices on one handand antivortices on the other merge together and form a pair of dark solitons. Thoseones where already observed in a different configuration [22], without the presence of thedriving field, which lead to oblique solitons vanishing over a few tens of microns. Ourimplementation greatly enhances the propagation distance, and exhibits a new behaviourof the solitons under the influence of the driving field: the dark solitons align to eachother and propagate parallel, as long as they can be sustained.690
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
The optical bistability is the ability of a system, over a certain range of input intensities,to possess two possible output values. Such a property requires nonlinearity: the outputintensity can not be linked to the input one only by a multiplicative constant. However,nonlinearity itself is insufficient to explain bistability: knowing an input intensity insidethe bistability range is not enough to determine the output one [98].The first observation of optical bistability in a passive medium have been realized in1969 by A. Szöke and coworkers [99]. They used a ring cavity filled with a saturable two-level medium: a "saturable resonator". They explain how the saturation of the absorptionleads to bleaching, and by combination with the resonator feedback induces a hysteresiscycle. This phenomenon is mainly due to the intensity-dependent absorption and thereforecalled absorptive optical bistability [100].Another configuration was observed a few years later by Gibbs et al. [101], using thistime a purely dispersive medium, with a nonlinear index but no absorption or gain,typically a Kerr medium. This dispersive bistability can be studied through the firstnonlinear index term n or the susceptibility term χ (3) [100]. It is the case we will focuson from now on as it corresponds to our system (see section 3.1.2).A theoretical model of such phenomenon has first been given by Luigi Lugiato andRodolfo Bonifacio [102] in 1978, providing an analytical treatment of optical bistability inthe absorptive and dispersive cases, that they have later develop to different configurations[103]. To understand the generation of our system’s bistability, let us consider a nonlinearmedium inside a Fabry-Perot cavity, as displayed in figure 3.1. We will consider an idealconfiguration, in which the two mirrors are identical and lossless, and follow the relations: R = | ρ | T = | τ | R + T = 1where ρ and τ are reflectance and transmittance in amplitude, and R and T the intensityones.We want to focus on the case of a dispersive bistability, so we neglect any absorption.The propagation constant k = nω/c is taken to be a real quantity with both a linear andnonlinear contributions, and spatially invariant. The different fields are thus linked bythe relations [104]: A = ρA e ikl (3.1) A = τ A + ρA (3.2)Solving the previous system by eliminating A leads to the Airy’s equation .1. OPTICAL BISTABILITY Bistable optical system
Fabry-Perot interferometer of length l filled by anideal Kerr medium without absorption. A = τ A − ρ e ikl = τ A Re iδ (3.3)where ρ = Re iφ contains its amplitude and phase while δ = δ lin + δ nonlin is the totalphase shift acquired in a round trip along the cavity. It consist of a linear part: δ lin = φ + 2 n lin l ωc (3.4)and a nonlinear one: δ nonlin = 2 n nonlin l ωc I (3.5)with I = I + I ≈ I .The equation 3.3 can be rewritten for the intensities: I = T I (1 − Re iδ )(1 + Re iδ ) (3.6)which leads after a few transformations to the relation: I I = 1 /T R/T ) sin δ/ I ,and the colored lines the ratio I /I for different values of I . On the right, figure 3.2.b.,is illustrated the shape of the evolution of I as a function of I , where the intensitiescorresponding to the colored lines on a. are reported.The blue line labeled a. illustrates the lowest value of I . It crosses the black curveonly once: only one internal intensity value I corresponds to this input intensity, as2 CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
Figure 3.2:
Intensities relation and bistability cycle . a. Graphical resolution ofequation 3.7: the black curve represents the right hand side of the equation as a functionof I , while the colored straight lines correspond to the ratio of intensities, for increasingvalues of I from line a to e. Adapted from [104]. b. Evolution of the intracavity intensityas a function of the input one. Intensities I a to I e correspond to the associated lines inpanel a. The dashed line is an unstable solution. The system presents a hysteresis cycle:the bistability occurs between I b and I d . plotted on figure 3.2.b. The yellow c. line shows the case of a higher value of I , andcrosses the black line three times. That it the case of bistability: on figure b., two statesare accessible, while the dashed line is unstable. Increasing again the incident intensity,like in the e. case, only one value of I is possible: the red curve on figure a. crosses theblack one only once.Cases b. and d. are particular: they have two intersection points with the black lineon 3.2.a. They correspond to the turning points between which the bistability occurs. Itresults in a hysteresis cycle: once the output on one branch, the system remains on it. Ittherefore does not follow the same path by increasing or reducing the input intensity, asindicated by the arrows on figure 3.2.b. The nonlinearity of a polariton fluid comes from the polariton-polariton interactions.They result in a third order term, equivalent to a Kerr-like medium in such a way thatthe polariton system behaviour is analogous to an electromagnetic wave resonant with acavity filled by a Kerr medium, as presented in the previous section.We induce bistability in the polariton system by pumping it quasi-resonantly witha laser slightly blue-detuned with respect to the lower polariton branch. The energy .1. OPTICAL BISTABILITY E lasLP = ∆ E = E las − E LP plays an important role in the state of the system, as it induces an energyrenormalization and shifts the dispersion.In the case of a red detuning, this renormalization drives the mode away from theresonance, as an increase of the pump intensity induces a decrease of absorption. Theinteresting case is however the one of a blue detuned pump: this time, a higher pumpintensity leads to a higher polariton density, and an energy renormalization closer toresonance. Thus, an intensity threshold appears, above which an increase in the polaritonpopulation self amplifies and leads to a blue shift of the dispersion up to the laser energy.This mechanism induces a bistability, sketched in figure 3.3. Let us consider our systemat the energy of the unperturbed lower polariton branch E , and a laser pump slightlyblue detuned from it, at E = E + δE ; its induced population it represented by the redgaussian curves.By gradually increasing the pump intensity, the system remains at E as long asthe intensity felt by the system stays lower than the threshold intensity I th . The jumpshappens thus at the configuration illustrated by the figure 3.3.a: the intensity at E isthe threshold one I th , while the total one is actually much higher and corresponds to theupper limit of the bistability cycle I high . By jumping to the E state, the system alsostrongly increases its density: it has now reached the nonlinear regime of the bistabilityupper branch.Once in that state, the system stays there as long as it density stays above the thresh-old I th . The figure 3.3 shows the configuration for which the system jumps back down tothe ground state E : it takes place when the total pump intensity decreases down to I th ,which therefore corresponds to the lower limit of the bistability cycle I low .However, a decrease of the pump does not induce the same path for the system.Indeed, being already in the upper state E , it stays there while the pump intensity ishigher than the threshold. The unlocking back to the E state occurs for a pump intensity I th = I low , shown in figure 3.3.b.Between I high and I low , the state of the system therefore depends on its initial position,and can be either in E or E . That is the range of intensity where the bistability occurs,resulting in a hysteresis cycle similar to the one presented in figure 3.2.b.A more quantitative way of describing this phenomena in our system is to start with thesteady-state equation. Let us place ourselves in the case of a normal incidence excitationwith k = 0 µm -1 , i.e. in the case of the degenerate four-wave mixing . The two-by-twointeraction ("four-wave mixing") indeed happens between identical and at rest polaritons,hence the term degenerate. The steady-state equation can then be written as [42]: F las ( r ) = ( i (cid:126) γ + ∆ E lasLP − gn ( r )) ψ ( r ) (3.8)where F las represents the laser pump, γ = γ LP k the relaxation term of the lowerpolaritons, g the polaritons interaction constant, and ψ and n the static wavefunctionand density: n ( r ) = ψ ( r ) ψ ∗ ( r ). By multiplying equation 3.8 by its complex conjugate,we obtain:4 CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
Figure 3.3:
Bistability in quasi-resonant pumping . E corresponds to the energy ofthe linear dispersion of the lower polariton branch, while the laser pumps the system at adetuned energy E = E +∆ E lasLP , inducing a population represented by the red gaussiancurves. On a. is illustrated the situation of an increase of the pumping intensity: initiallyat E , the system jumps to E for a pump intensity of I high , which corresponds to thethreshold value at E . Figure b. shows the inverse situation of a decreasing intensity:once the system as jump to E , it stays there until the pump intensity decreases down to I th = I low ; only then will the system go back to the unperturbed dispersion energy E .Thus, between I low and I high , depending on its initial state, the system can be either at E or E and is therefore bistable. I las ( r ) = (cid:16) ( (cid:126) γ ) + (∆ E lasLP − gn ( r )) (cid:17) n ( r ) (3.9)with I las the intracavity laser pump intensity.If the detuning between the pump and the lower polariton branch ∆ E lasLP is highenough, this equation leads to a bistable behaviour. Indeed, this equation can be derived: ∂I las ∂n = 3 g n − E lasLP gn + ( (cid:126) γ ) + ∆ E lasLP (3.10)the discriminant has the expression (2 g ) (cid:0) ∆ E lasLP − (cid:126) γ ) (cid:1) . Therefore, the deriva-tive of the intensity possesses two distinct roots if the detuning follows the conditions∆ E lasLP > √ (cid:126) γ .This equation has also been solved numerically and the results are presented on figure3.4 for parameters chosen to be similar to the typical experimental ones: (cid:126) γ = 0 .
07 meV, g = 0 .
015 meV, and energy detunings ∆ E lasLP from 0 to 0.3 meV.As expected, different behaviours can be observed. For a red-detuned pump (∆ E ≤ E > .2. SEED-SUPPORT CONFIGURATION: ENHANCEMENT OF PROPAGATION DISTANCE
Density evolution for different detunings . Numerical resolution of equa-tion 3.9 for different detunings, from -0.1 meV to 0.3 meV. The bistability only appearsfor detunings larger than √ (cid:126) γ .is different. However, the bistability occurs only for large detunings compared to thelinewidth of the cavity. The yellow curve, which corresponds to ∆ E lasLP = 0 . E lasLP = 0 . E lasLP = 0 . In 2017, Simon Pigeon and Alberto Bramati [25] suggested the use of the optical bista-bility of an exciton-polariton system to enhance the propagation of the superfluid. Inparticular, this theoretical proposal explained how the use of two different beams couldcreate a high density polariton fluid, bistable and over a macroscopic scale. This sec-tion will focus on the results of this paper, as its experimental realizations are presentedafterward.6
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
Let us consider an exciton-polariton system, offering an optical bistability between theintracavity intensities I low and I high , as presented in section 3.1.The suggested configuration of Pigeon et al. [25] is to consider two driving fields, withthe same frequency ω p and the same in-plane wavevector k p . The first one, called the seedor the reservoir, is localized in space and has a high intensity: I r > I high . It thus producesa nonlinear superfluid above the bistability cycle. The second field is the support, muchmore extended in space; for now, let us consider it theoretically as an infinitely extendedconstant field, stationary in time and homogeneous in space. Its intensity is weaker thanthe intensity of the seed one, and is placed inside the bistability cycle: I low < I s < I high .The evolution of such a system is described with the Gross-Pitaevskii equation: i∂ t Ψ C ( x , t )Ψ X ( x , t ) ! = (cid:126) F s + F r ( x )0 ! e − i ( k p · x − ω p t ) + (cid:126) ω C ( k ) + V ( x ) − iγ C Ω R Ω R ω X + g | Ψ X ( x , t ) | − iγ X ! × Ψ C ( x , t )Ψ X ( x , t ) ! (3.11)with Ψ C ( x , t ) the cavity field and (cid:126) ω C ( k ) the cavity mode dispersion; Ψ X ( x , t ) theexcitonic field and (cid:126) ω X the exciton energy considering an infinite mass. F s is the ampli-tude of the support driving field ( I s = | F s | ) and F r ( x ) the one of the seed. V ( x ) is thephotonic potential, Ω R the Rabi frequency, g the exciton-exciton interaction term and γ C and γ X the decay rates of the cavity and the exciton, respectively.The goal of this configuration is to enhance the propagation and density of the polaritonfluid by combining the properties of both beams. Numerical simulations were done tounderstand their combination, presented in figure 3.5. They have been realized for acavity without any defects ( V = 0) such that: (cid:126) ω C ( k = ) = 1602 meV, (cid:126) ω X = 1600meV, (cid:126) γ X = (cid:126) γ C = 0 .
05 meV, (cid:126) Ω R = 2 . (cid:126) g = 0 .
01 meV/µm . The drivingfield parameters are δE = (cid:126) ω p − (cid:126) ω LP ( k = k p ) = 1 meV and | k p | = (0 . T µm -1 .The field intensity is shown as a function of space, plotted in logarithmic scale. Theflow of particles goes from left to right. The driving intensity sent in the system is pic-tured with the red dashed line, while the black solid line shows the steady-state photonicintracavity density.On the upper panel, only the seed is sent on the left, illustrated with the red highlightedregion ( F r ( x ) = 0; F s = 0). Given the presence of an in-plane wavevector, the photonicdensity expands a bit, but with an exponential decay due to the finite polariton lifetime.Even with the present best quality samples, the propagation distance in this configurationis limited to around 50 ps. Moreover, as the density is decreasing all along propagation,all related parameters are also constantly changing. .2. SEED-SUPPORT CONFIGURATION Field density in seed only or seed support configuration . The blacksolid line illustrates the intracavity intensity (logarithmic scale), while the red dashedone is the driving intensity sent to the system. The colored regions delimits the differentdriving fields: the seed (or reservoir) in red and the support in green. The polariton flowgoes from left to right. On the upper panel, only the seed is sent: it creates a high densitypolariton fluid, decaying exponentially out of the pumping region. The propagation lengthis limited by the polariton lifetime. In the lower panel, the support field is added: even ifits density is two order of magnitude lower than the seed one, the total polariton densityis maintained at its higher level all along the presence of the support. From [25]The lower panel shows what happens when the support, in green, is added to theprevious configuration. The support intensity is two order of magnitude lower than theseed one, but infinitely extended in space for this simulations. The presence of the supportfield has a strong impact on the total polariton density: its high level created from thereservoir is maintained without any decay all over the region where is the support, despiteits much weaker intensity. Some modulations can be observed at the frontier between thesupport and the reservoir, due to the sharp designed profile of F r ( x ). They were notobservable in the seed-only configuration due to the quick decay of the photonic density,which does not occur with a support field.Those simulations do not exactly match an experimental situation. In particular, theextent of the support beam can obviously not be infinite: the fluid propagation lengthis limited by the pump available power. However, thanks to the localization of the seedand the low intensity of the support, it can easily reach macroscopic scale of hundreds ofmicrons, and this independently of the polariton lifetime.8 CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
Bistability is essential to explain this behaviour. The seed is placed outside the bista-bility cycle, where only one stable state is accessible (see figure 3.6). It therefore ensuresthe system on the upper branch of the cycle, the nonlinear branch. On the other hand,the support is chosen to be inside the bistability cycle, where two states are reachable.The support itself can not reach the upper branch and, if alone, would only drive thesystem in the linear configuration. Yet, as the seed places the system in the nonlinearbranch and touches the support illuminated region, by extension, the neighbour regionjumps also on the upper branch which expands to all the support area.Figure 3.6:
Positions of seed and support intensities compared to the bistabilitycycle . The seed is chosen to be above the hysteresis cycle. Only one stable state isavailable at this place, on the upper nonlinear branch of bistability. The support hasmuch lower intensity, inside the bistable region: the support alone would not be enoughto place the system on the nonlinear branch , and would remain linear at low density.However, if both fields are sent, the seed presence ensures the system to stay on the upperbranch, and this everywhere sustained by the support.
The idea of the seed-support configuration is not only to obtain an extended fluidof polaritons, but also to study its properties, and in particular its ability to generatetopological excitations, such as dark solitons and quantized vortex-antivortex pairs. Thedifferent hydrodynamic regimes of a polariton fluid and their effects on the generation oftopological excitations have been previously studied in the case of a single intense andlocalized pump, placed just upstream of a structural defect [22, 105]. The presence of thedefect creates some turbulence along the flow, which evolves differently depending on theratio between the speed of the fluid and the sound velocity, i.e. the Mach number of thesystem: M = v f /c s . However, the speed of the fluid that needs to be taken into accountis the one around the defect, therefore always a bit higher than the one extracted far from .3. SUBSONIC FLOW: VORTEX STREAM GENERATION To numerically reproduce the effect of a cavity structural defect, a large potentialphotonic barrier is introduced ( V = 0). The seed is placed upstream to it, localized andwith a fixed intensity above the bistability cycle, illustrated in point P in figure 3.7.Figure 3.7: Distribution of the support intensities presented in figure 3.8 . Thered dots are obtained with the support field only, while the blue ones were computedwith both the seed and the support. The seed intensity is placed above the bistabilitycycle, represented by the black dot labeled P. The label O corresponds to a case withoutsupport, the cases A to E are in the bistable regime and the F one is just above it.Adapted from [25]The goal of exploring the bistable regime and its influence on the vortex generation isachieved by tuning the intensity of the support field. Figure 3.7 presents the bistability0
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS system of the considered system, and the positions of the support intensities shown infigure 3.8. The other cavity and driving parameters are identical to the one presented inthe section 3.2.1.Figure 3.8:
Density and phase snapshots of a polariton superfluid around aphotonic defect for different support intensities . Case O corresponds to a seedwithout any support field: the polariton population decays exponentially. Cases A to Fpresent different configurations for an increasing support intensity (detailed in figure 3.7):the non-linear fluid is extended over hundreds of microns, and vortex-antivortex pairsare spontaneously generated on the wake of the defect for bistable support intensities.Adapted from [25]The situation labeled O corresponds to a localized seed pump in the absence of thesupport one. As previously explained, the polariton density downstream of the pumpexponentially decays due to the short lifetime of the polaritons. When reaching thedefect, it is already too low to observe any non-linear interactions, and no topologicalexcitations can be observed.Intensities A to E are placed within the bistability cycle. The first conclusion to getfrom those pictures is that the non linear fluid is effectively sustained for macroscopiclength, even in the region where only the support is sent, which would not be enough if .3. SUBSONIC FLOW: VORTEX STREAM GENERATION I low , the vortex density is quite high. But with theincrease of the support intensity, from case B to E, the number of vortex pairs observablein the stream is gradually decreasing, until they almost do not propagate but annihilatequickly as in case E, even if still inside the bistability cycle. The vortex density is plottedon the upper panel of figure 3.9, and actually scales inversely with the support intensity.On case E in particular, one can see that the vortex pairs are not sustained long in thestream but recombine close to the potential barrier.Figure 3.9: Support intensity influence on the vortex stream . The upper panelshows the vortex density in the stream decreasing inversely to the support intensity. Onthe lower panel is plotted the evolution of the vortex pair velocity v p , normalized tothe fluid velocity v f , with the support field. The red curves are a linear fit of the data,depending on the support field amplitude for the upper panel and on the support intensityin the lower one. Adapted from [25]The lower panel of figure 3.9 illustrates how the velocity of the vortex pairs v p evolveswith the driving intensity. The velocity has been renormalized to the fluid one v f . Even at2 CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS smallest support intensity, the vortex speed is a bit lower than the fluid one, but remainsclose. However, it strongly decreases with the support increase, down to half of its initialvalue. It actually scales as the inverse of the amplitude of the support field: this showsthe coherence of the phenomenon. The vortices are slowed down by the support fieldwhich acts as a friction force on them. This could be interesting as the tuning of thesupport intensity could have a direct control on the properties of the vortices.The images presented previously in figure 3.8 are time snapshots of a simulated polari-tons fluid. They enable to precisely locate the vortices, and to observe the fork shapeof the phase pattern typical of vortices. However, those pictures can not be reproducedexperimentally, as it would require a single-shot picosecond time resolution that we donot have in the lab. We only have access to millisecond resolution: numerical simulationshave been made in order to predict how those phenomena would look like in a time inte-grated picture. Figure 3.10 shows the comparison between one of the previous snapshotimage on the left, with the same parameters, and on the right what it would look like ifthe picture was taken with 1 ms integration time.Figure 3.10:
Comparison between a time snapshot and a time integrated imageof a vortex stream . On the left, the time snapshot shows the position of vortex andantivortex flowing in the wake of the defect. On the right, the time integrated image hasblurred the positions of the vortices and results in a thick shadow along the flow.A time average flow of vortex pairs appears as a thick line of lower density, which dip .3. SUBSONIC FLOW: VORTEX STREAM GENERATION
Implementation
We have implemented this proposal to experimentally verify it andour work got published in [26]: most of the figures of this section are adapted from theones of the paper, as the setup displayed in figure 3.11. The initial laser source is ourcontinuous Titanium Sapphire Matisse laser. It is split up a first time into the main beamand the reference beam, later used to realize interferograms and get information on thephase. The main beam is later split a second time to generate the seed and the supportbeams. The seed is then focused on the sample into a spot of 30 microns diameter, for anintensity I r = 10 . . On the other hand, the support is elongated on the verticalaxis through two cylindrical lenses, then collimated and sent to the sample as an ellipticalspot of 400 microns length, with an intensity of 5.8 W/mm . The inset of figure 3.11gives a representation of the relative position of the beams; the seed is not centered to thesupport so that the topological excitations can be studied on the flat part of the gaussiansupport beam.Figure 3.11: Experimental setup . The continuous Titanium-Sapphire laser beam (cwTi:Sa) is split into three using half waveplates (HWP) and polarizing beam-splitter (PBS).A reference beam, in light red, is first set aside for interferogram on the detection part; theseed beam in red is focused on the sample to a spot of 30 microns diameter; the supportin orange is extended and elongated in the vertical direction through cylindrical lenses(CL) before being collimated and sent into the microcavity. The seed and the supportare not centered to one another (see inset) but share the same wavevector. The detectionis done in real space, from which we can get information on the density and phase maps,and also in momentum space through the spectrometer.The sample used for this experiment works in transmission: an objective was placedjust after the cryostat with a large numerical aperture of 0.42. The detection is done in4
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS two spaces: the real space gives a copy of the polariton density in the plane of the cavity.The reference beam previously extracted from the initial laser beam can also interact withthe signal of the cavity to get interferograms, used to reconstruct the phase map. All theother parameters of the excitation are coming from the momentum space data, acquiredthrough a spectrometer. Details about the data extraction and analysis have been givenin section 2.2.3.
Vortex stream generation
The first step of the experiment is to align the seed andthe support beam independently. Figure 3.12 shows the momentum and density maps ofeach field: the seed is on the left and the support on the right. On top are presented the k x - k y images, centered around k = 0. Both excitation spots are at the same position,the beams enter with the same in-plane wavevector in the cavity ( | k y | = 0 . -1 and | k x | = 0 µm -1 ). The support beam is however collimated on the sample while the seed isfocused: in the momentum space, the support is therefore more localized than the seed.Figure 3.12: Seed and support beams separately . The left column presents themomentum and density maps of the seed alone, while the right one corresponds to themaps of the support beam. The momentum maps show the excitation spot of both beamsat the same position. The support being collimated on the sample, its momentum spaceshows a very localized spot on panel b. The Rayleigh scattering ring appears due tosample inhomogeneities. The density maps display the range of the areas excited by bothbeams: on panel c., the seed is localized on the lower part of the image whereas thesupport on panel d. is everywhere. Note that the intensity scale of the picture c is tentimes higher than the one of d. The defect used to generate vortices is circled in orange:in both cases, no topological excitations are observable. .3. SUBSONIC FLOW: VORTEX STREAM GENERATION k = 0,and with a radius slightly larger than the excitation vector. It is the Rayleigh scatteringring due to the sample structural disorder and the elastic scattering on defects. Thedetuning between the excitation and the lower polariton branch, here δE = 0 .
26 meV isvisible in the figure.The density maps show us the size and location of each beam on the sample. The flowgoes from bottom to top, at a velocity v f = 0 . CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
Figure 3.13:
Vortex stream generation . a. Time integrated density map of a flowof vortex pairs generated in the wake of a defect. As the vortices move along the flow,the time integrated image results in a blurry density dip in the wake of the defect. b.Interferogram of the previous image, showing phase irregularities along the vortices prop-agation. c. Visibility map extracted from the interferogram, displaying a lower visibilityalong the vortex stream.that in this region, vortices merge together and become gray solitons. Indeed, the vortexcore size is controlled by the hydrodynamic properties of the system, and in particular itshealing length ξ , defined as [20]: ξ = (cid:126) √ m ∗ (cid:126) gn with m ∗ the effective mass of the polariton and the product gn the polariton density.On the pictures of figure 3.13, one can see that we are at the edge of the polariton fluid,and that the dark region on top of the density map corresponds to a linear fluid. Butbefore reaching it, the non linear fluid on the upper bistability branch sees its densityslowly decreasing. This density decrease has a direct impact on the healing length whichinversely increases, leading to vortex cores bigger and bigger. At some point, vortices onone hand and antivortices on the other hand reach their respective neighbor, and mergetogether into a gray soliton. A precise phase jump is indeed visible at the correspondingposition of the interferogram. It can also be explained it terms of Mach number, as thedecrease of density means also a decrease of the sound speed and therefore an increase ofthe Mach number: the flow becomes supersonic there. .3. SUBSONIC FLOW: VORTEX STREAM GENERATION k y , the support (picture 3.14b.) is displaced ofroughly 40° compared to the seed, and has components in both k x and k y directions.Density maps of each beams alone are presented in images c. (seed) and d. (support).As previously, the seed is localized, and induces a light vertical flow of polariton in itswake. The support is much more extended, and due to its k x component, it creates adiagonal polariton flow. The considered defect is positioned through the orange dashedcircle. Again, the scales of the density maps are different by a factor ten.Figure 3.14: Vortex stream propagation direction . The first column presents theresults of the seed alone, sent with a vertical wavevector. a. corresponds to the momentumspace image and c. is the density map. The second column is the support, tilted of about40° compared to the seed; again, b. is the k x - k y image and d. is the real space. Note thatthe scales of c. and d. have a factor 10 between them. The position of the defect is circledby the dashed orange line. e. shows the density map resulting in the superposition ofseed and support. The vortex stream line is again visible, this time tilted in the directionof the flow.The density map obtained when both beams are sent together is presented in picture3.14e. As before, the surface of the nonlinear fluid is greatly enhanced, and the vortexstream is generated after the defect. This time however, the stream direction has changedand follows the flow imposed by the support orientation. The vortex direction can thusbe easily controlled by tuning the excitation wavevector of the support beam.8 CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
A last theoretical prediction was experimentally realized: the confirmation that thevortices are generated only if the support intensity is on the left side of the bistabilitycycle. Figure 3.15 displays two images between which only the support intensity varies:on a., the support power density is 5.8 W/mm , while it increased to 6 W/mm on pictureb. The first picture exhibits the typical dark shadow associated with the vortex streamgeneration. But in the second one, the support intensity has become too high to allowthe topological excitation to generate and propagate: the phase is fixed by the drivingfield and therefore flat. No shadow appears, the vortex stream is inhibited.Figure 3.15: Vortex stream suppression . a. Vortex stream in the wake of a defectunder 5.8 W/mm support power density. b. Suppression of the vortex stream for asupport power density of 6 W/mm The subsonic case inducing the generation of vortices and their sustainability for macro-scopic distances, the supersonic configuration is now investigated. According to previousstudies in polaritons superfluid [22], higher Mach numbers, around or above 1, allow forthe generation of dark solitons. However, the system configuration did not allow forlong propagation: the quasi-resonant pump fixing the phase of the fluid, it was sent onlyupstream of the defect and the solitons were observed within the exponential decay ofthe solitons propagation. Therefore, they could only be seen for short distances (around30 micrometers, limited by the polariton lifetime) and with changing conditions alongtheir propagation as the density exponentially decreased. The goal of this part is thus touse the previous configuration of seed-support excitation, and to generate dark solitonssustained over long distances to be able to study their hydrodynamic behaviour. After a .4. SUPERSONIC FLOW: PARALLEL DARK SOLITON PAIR GENERATION
Solitons are nonlinear solitary waves which have the property of conservation of theirshape and velocity during their propagation. The first reported observation of such phe-nomenon dates from 1834 and was realized by the Scottish engineer John Russel whodescribed it as such [107]:
I was observing the motion of a boat which was rapidly drawn along a narrow channelby a pair of horses, when the boat suddenly stopped—not so the mass of water in thechannel which it had put in motion; it accumulated round the prow of the vessel in a stateof violent agitation, then suddenly leaving it behind, rolled forward with great velocity,assuming the form of a large solitary elevation, a rounded, smooth and well-defined heapof water, which continued its course along the channel apparently without change of formor diminution of speed. I followed it on horseback, and overtook it still rolling on at a rateof some eight or nine miles an hour [14 km/h], preserving its original figure some thirtyfeet [9 m] long and a foot to a foot and a half [30-45 cm] in height. Its height graduallydiminished, and after a chase of one or two miles [2–3 km] I lost it in the windings ofthe channel. Such, in the month of August 1834, was my first chance interview with thatsingular and beautiful phenomenon which I have called the Wave of Translation.
Even though Russel passed many years focusing on the study of such solitary waves,the scientific community did not realized at the time the significance of his work. Thefirst theoretical understanding of their dynamics was made by Korteweg and de Vries[108] in 1895 who derived a simple equation of propagation of wave on water surfacetraveling only in one direction, the KdV equation. However, it is only in the 1960’sthat researchers understood that this equation’s solutions had the form of solitary waves,which exhibit properties of particles, as their shape is conserved during propagation oreven after collision with each other. From that came the name solitons as "particles of asolitary wave".For long, the term "solitons" was only used to describe such nonlinear stationary shape-maintaining waves, but as their study expanded, it now refers to a wider range of phe-nomena which can evolve during propagation, sometimes accelerate, decompose or formcoupled states by interacting [109]. Two families of solitons can be easily distinguished,depending on their impact on the system they evolve in. If the system is not impacted bythe crossing of the wave, and keeps the same energetic state before and after it, we talkabout non topological solitons, described by the KdV equation. If however the systemhas a different state on both side of the wave, it is a topological solitons which can bedescribed by different equations as for instance the non-linear Schrödinger equation orthe Sine Gordon equation, as it is the case in our system with a phase jump across thesoliton [109].0
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
In the last few decades, solitons have been studied in many different field of physics.Obviously, the study started on water surface and shallow water for fluid dynamics [110–113]. Later on, solitons were created as temporal pulses in optical fibers [114, 115] oras spatial structures in waveguides [116]. They were also studied in thin magnetic films[117, 118] as well as in semiconductor microcavities [119]. Liquid Helium [120] or complexplasma [121] were also able to sustain those features. Theoretical works were realizedstarting from the 70s, in particular in the context of Bose-Einstein condensates [59, 60,122–124], which were followed a few decades later by experimental observations after thefirst realization of atomic condensates [78, 125, 126]. Research focused much on matter-wave solitons [127–131] as one of the first purely nonlinear states experimentally realizedin BECs [132].In any cases, all those phenomena result in the balance between the wave dispersion,which tends to spread it out, and the nonlinear properties of the medium, which onthe contrary compresses the wave. Nonlinearity is therefore essential for the solitons tobe sustained. In optics in particular, two cases need to be separated: the self-focusingand the self-defocusing nonlinearity. Self-focusing results from a positive correction tothe refractive index, proportional to the intensity: the beam’s divergence is suppressedand the beam collapses. This phenomenon allows for the generation of bright solitons,characterized by an intensity peak above a continuous background [133]. On the otherhand, self-defocusing supports the propagation of dark solitons as an intensity dip on acontinuous background.On polariton fluids, the first observation of solitons was realized in our group tenyears ago [22]. Sending the polariton flow toward a structural defect with suitable size,solitons were generated in its wake. The system was resonantly pumped: as previouslyexplained, a strong localized pump fixes the phase of the fluid and prevents the formationof topological excitations. Adopting a theoretical proposal [105], a half circle shapedpump was implemented just upstream of the defect and the solitons were observed alongthe free propagation of the polaritons.In the case of a spontaneous formation in an unpumped region, the two generatedsolitons are oblique: they are formed just after the defect, then propagate straight withan aperture angle α between them. This angle can be linked to the Mach number ofthe system through the theory of dark solitons [20, 62]: it predicts that solitons arewell sustained for supersonic speeds [134]. At subsonic speeds, they are subject to snakeinstabilities that can lead to their decay into vortices [130], even though their propagationis kinematically not prohibited. Indeed, the experiment of Amo et al. presents stablesolitons at subsonic speeds: it can be explained as a result of the finite lifetime of thepolaritons [135].The main restriction of this configuration is the fact that the solitons propagate in anunpumped region, and consequently in a region where the polariton density exponen-tially decays. Not only does it limits the propagation length of the solitons, it also limitstheir study as the hydrodynamic conditions are constantly changing along the propaga-tion. Even though, many researches have been pursued since then to understand betterthe kinematic of solitons in polaritons fluid, like the study of their stability [23, 136], .4. SUPERSONIC FLOW: PARALLEL DARK SOLITON PAIR GENERATION Experimental implementation
The implementation of this experiment is similar tothe one presented in section 3.3.2. The experimental setup is the same, with two colinearbeams sent to the cavity: the seed, localized and intense, and the support, extended andwith an intensity within the bistability cycle.Figure 3.16:
Seed and support beams distribution . The support beam is elongatedalong the vertical direction and its intensity is within the bistability cycle. The flow is frombottom to top, and a structural defect is placed in the center of the support. Upstreamto it is sent the seed, localized and with an intensity above the bistability cycle. Thedistribution of the total intensity along the vertical axis is displayed on the right as wellas the upper and lower bistability limits.2
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
The position of the seed and the support beams on the sample are shown in figure3.16. The seed is shifted compared to the support center, to be placed just upstream ofthe considered structural defect - the flow is from bottom to top. The curve on the rightshows the distribution of the intensity in the cavity along the vertical axis. The regionclose to the seed is above the upper bistability limit, while the main part of the supportis inside the cycle. The combination of the two beams ensures the fluid to be bistable andon the upper branch of the cycle.
Dark soliton pairs generation
To generate dark solitons in the wake of a defect,supersonic conditions need to be created. The in-plane wave vector is chosen to be high( k = 1 . -1 ) in order to ensure a high velocity of the fluid: in this case, v f = 1 .
52 µm/ps.The sound of speed is extracted through the energy renormalization and is measured tobe c s = 0 . Spontaneous generation of a pair of parallel dark solitons . Intensity(left) and interferogram (right) of a dark soliton pair spontaneously generated in the wakeof a structural defect. They propagate away from each other for a short distance, beforealigning and staying parallel for over a hundred microns.The figure 3.17 presents the observation of a spontaneous generation of dark solitonsin the wake of a defect: intensity on the left and interferogram on the right. The firstconclusion that can be done from those results is that the propagation length is indeedgreatly enhanced. The scale bar illustrates 20 microns: the solitons themselves are sus-tained for more than a hundred microns, one order of magnitude more than previouslyreported. .4. SUPERSONIC FLOW: PARALLEL DARK SOLITON PAIR GENERATION π all along the propagation, confirming their transverse velocity is close to zero. Oblique grey solitons generation
In order to have a compared analysis of the dif-ferent cases, we performed an experiment in a configuration close to the one of Amo etal. [22], with a strong pumping upstream of the defect. The results are presented infigure 3.18: again, the left image corresponds to the intensity map while the right one isthe phase interferogram. In this case, only the seed is sent: it is localized about twentymicrons before the defect, in order to surely avoid the phase fixing of a strong coherentpumping. It is therefore outside of the field of view of figure 3.18 in order to enhancecontrast in the picture.Figure 3.18:
Spontaneous generation of a pair of gray solitons . Intensity (left)and interferogram (right) of the hydrodynamic generation of oblique gray solitons in thewake of a structural defect. The strong and localized pump is placed about 20 micronsupstream of the defect, outside of the field of view, hence the good contrast of the picture.The generated solitons have a different shape than in the previous configuration, asthey stay oblique for their whole propagation. They are generated dark with a phasejump of π but do not stay that way. They vanish along the propagation, and with it theirphase jump gradually reduces. Their propagation length is also much shorter than in thepresence of the support field: they have completely vanished after 40 microns propagation.Finally, their width is large: about 10 µm full width at half maximum (FWHM) for theoblique solitons while the parallel one was fitted at 2.8 µm FWHM.4 CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS
A direct comparison between the two cases previously presented has been done, in par-ticular about the separation distance. Figure 3.19 presents the evolution of the separationdistance between the solitons along their propagation, for both cases.Figure 3.19:
Spontaneous solitons separation distance . The dotted lines show theexperimental values of the separation distance between the solitons along their propaga-tion: the orange line corresponds to the oblique solitons presented in figure 3.18, whilethe black one is extracted from the parallel solitons in figure 3.17. The solid black line isa fit of the parallel solitons data.The dotted lines show the experimental results: in black the parallel solitons withboth seed and supports fields, and in orange the oblique case with only the seed. Theblack solid line is a fit of the experimental points, in order to get a value of the equilibriumseparation distance. The fitting function is chosen to be 2 d (1 − exp ( − x/d l )) in order toget a transient and a stationary region. The transient region corresponds to the first partof the propagation where the solitons are oblique, with a asymptotic length of d l = 14µm. The solitons then reach their stationary region where they propagate parallel, at anequilibrium separation distance of 2 d = 8 µm.The oblique solitons always follow the same trend: they continuously go further toone another, before completely vanishing after 40 microns propagation. To fully understand the phenomenon taking place in the system, numerical simulationsin collaboration with the group of Guillaume Malpuech in Clermont-Ferrand have beenrealized to reproduce the solitons behaviour. They used the coupled equations of theexcitons and cavity photons fields, ψ X and ψ γ : i (cid:126) ∂ψ γ ( r , t ) ∂t = (cid:20) − (cid:126) ∇ m ∗ γ + V ( r ) − i Γ cav (cid:21) ψ γ ( r , t ) + V ψ X ( r , t ) + ( S ( r ) + R ( r )) e − iω t (3.12) i (cid:126) ∂ψ X ( r , t ) ∂t = [ V ( r ) + g X | ψ X ( r , t ) | − i Γ X − ∆ X ] ψ X ( r , t ) + V ψ γ ( r , t ) (3.13) .4. SUPERSONIC FLOW: PARALLEL DARK SOLITON PAIR GENERATION m ∗ γ the effective cavity photon mass, ω the laser energy, Γ cav and Γ X the photonand exciton lifetimes respectively, V the half Rabi splitting and ∆ X the cavity-excitondetuning. All these parameters have been taken the same as the experimental ones. V ( r ) is a 10 meV potential barrier that reproduces the presence of the structuraldefect, using a Gaussian shape of 10 µm width. The spatial profiles of the seed andsupport are modeled through R ( r ) and S ( r ) respectively, with adjustable magnitudes.Figure 3.20: Simulation of the spontaneous generation of dark parallel solitons .Using the driven-dissipative Gross-Pitaevskii equation, intensity (left) and interferogram(right) simulation of a dark soliton pairs spontaneously generated in the wake of a defect.The results of those simulations are presented in figure 3.20. They offer a perfectagreement with the experimental pictures of figure 3.17. The left image is the intensitymap, switched on to the upper branch of the bistability cycle by combination of boththe localized intense seed and the bistable extended support. The interference pattern onthe right exhibits a clear phase jump all along the propagation, confirming the solitonicnature of the intensity dips. As expected, the solitons are sustained for a macroscopicdistance and stay parallel to one another during their propagation.The use of the numerical simulations allows for a better resolution in time, and inparticular to understand the establishment of the steady state. The seed and the supportneed to be sent simultaneously, but it does not matter which one is turned on first.Once the two beams enter the system, the flow created by the seed is strong enough topropagate to all the support illuminated area and switch it to the upper branch of thebistability. During this expansion phenomenon, the high density region is delimited by adomain wall, separating it from the low density region. When this wall hits the defect,6
CHAPTER 3. SPONTANEOUS GENERATION OF TOPOLOGICAL DEFECTS it got split into a V shape around the defect, still pushed by the expansion of the highdensity area which tends to reduce the angle between its two branches. Different casescan occur depending on the velocity of the walls. The first possibility is that they spreadaway and vanish soon, leaving only a blurry gray region in the wake of the defect. Theycan also move quickly toward one another, and switch everything to the upper branch ofbistability, which means a flat high intensity everywhere. Finally, if they go toward thecenter but with a lower velocity, this one can be compensated by the repulsive interactionbetween the nonlinear shock waves associated with the domain walls. The stabilizationof the soliton pair can therefore be seen as a balance between the repulsion of the darksolitons and the pressure of the domain walls sending them toward one another.Another important result to notice is the fact that the solitons come as pair. Indeed,the phase jump across one soliton is π , and the total phase jump across the fluid mustremain null. The total number of solitons have consequently to be even.Numerically, the propagation distance of the dark solitons is not limited, as the supportbeam is modeled to be everywhere. It does not correspond exactly to the experimentalconfiguration, as the wedge of the cavity and the support finite extension should be takeninto account. hapter 4 Impression of bound soliton pairsin a polariton superfluid
We described in the previous chapter how to greatly enhance the propagation length ofa polariton superfluid, and simultaneously getting rid of the phase constraint of the pump.This allowed us to observe the spontaneous generation of topological excitations and theirpropagation for over a hundred microns. However, their generation was not controlled,and depended on parameters out of our reach; in particular, the necessary presence of astructural defect to induce the turbulence leading to the topological excitations [26, 27].The starting point of this chapter is to get rid of this limitation and to be able toartificially create solitons in the fluid. It is realized by imprinting a phase modulationon the system, leading to the formation of dark solitons that can evolve freely on thenonlinear fluid. Such impression of solitons offers many tools for the detailed study ofthis phenomenon, as their shape and position can be tuned on demand. It results onceagain in the unexpected binding mechanism between the imprinted solitons, leading tothe propagation of a dark soliton molecule [28].After an introduction of the experimental implementation and of different solitons con-figurations, the hydrodynamic behaviour of these structures is studied, as well as theirconnection with the driven-dissipative nature of the system.978
CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS
To be able to properly imprint solitons on the polariton fluid, some specific arrange-ments have been used in the excitation part. In particular, the pump beam has beendesigned to simultaneously allow a phase impression and a free evolution of the systemto sustain the solitons propagation.After a detailed description of the setup, the experimental results will be presented,as well as the different shapes and configurations tried on the fluid. i Intensity and bistability
The main difficulty of this experiment is to combine an impression of a phase mod-ulation, i.e. a region of the system whose phase is imposed by the pump; and a freepropagation area, i.e. a region where the phase of the pump beam is flat but where thesystem can still sustain topological excitations.Figure 4.1:
Beam intensity and bistability . Left, spatial distribution of the excitationbeam. The white dashed line illustrates the position of I high , upper intensity threshold ofthe bistability cycle, as shown in the intensity profile on the center image. On the right,a reminder of the theoretical bistability cycle with the definition of I high and I low
It is achieved by using the properties of the system’s optical bistability (see section3.1). Indeed, the input intensities above the bistability cycle impose their phase on thefluid, while this constraint is released for the input intensities within the bistability cycle. .1. EXPERIMENTAL IMPLEMENTATION I high (see right picture). All the area inside this circle is above the bistabilitycycle, as shown in the profile in the center. Therefore, it fixes the phase of the fluid andcorresponds to the effective impression region. Outside of this circle however, most ofthe beam intensities are within the bistability cycle: the phase is not imposed anymoreand the system is able to readjust it and to sustain the free propagation of topologicalexcitations. The beam is elongated in the y direction in order to flatten its profile andextend the bistable region where the solitons free propagation will be studied. ii Phase profile design As the solitons induce a phase jump on the system, their implementation can be doneby modeling the phase of the excitation. To do so, we use a Spatial Light Modulator, aliquid-crystal based device that can shape the wavefront of an incident light (see section2.1.4).The phase modulation induced by dark solitons is a phase jump of π : the phase profilecorresponding to a pair of dark solitons is thus an elongated region π -shifted comparedto the background beam. The solitons must not be imprinted everywhere but only in theimpression region: a rectangular shape in the center of the beam should be created.The SLM is very convenient for the design of the soliton phase profile. Furthermore,as it is entirely controlled via a software, the position of the solitons can be finely tunedto match ideally with the excitation beam. The figure 4.2 shows a typical image sentto the SLM to create the solitons pattern. The SLM reproduces it on its screen whichtransfers it to the beam wavefront: it results, as desired, in a rectangular phase jump of π which consequently induces an intensity dip. The dashed line illustrates the incidentbeam position on the screen of the SLM.Figure 4.2: SLM profile . Typical image sent to the SLM to create a π -shifted rectangularshape on the excitation beam. Its position and shape are easily tunable. The dashed linecorresponds to the position of the incident beam on the SLM screen.00 CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS
The efficiency of the SLM is not perfect (90% ): in order to get rid of the residual lightnot shaped by the device, we implement a Fourier filtering. In addition to the desiredphase pattern, we add to the image a grating, so that the light effectively shaped by theSLM is diffracted in the first order, while the rest stays in the zeroth one. In the Fourierplane of the SLM, all the orders of the grating focus in different spots: only the first ordersignal is kept, which cleans the beam of all impurities due to the reflection on the SLM. iii Spatial filtering
The filtering in the Fourier space is actually not only for cleaning the beam from SLMimpurities: we also use it to finely tune the shape of the phase jumps. Indeed, we want toimprint the solitons vertically, along the y axis: we do want the horizontal phase jumpsto be as sharp as possible, in order not only to induce dark solitons well imprinted onthe system, but also such that they can freely continue their propagation along y throughthe bistable region of the fluid. However, the rectangular shape created by the SLMalso induces short phase jumps along the y direction, that we want to avoid in the fluid.Indeed, they join the two horizontal phase jumps, limiting the region in phase opposition:on the fluid, if they are too well defined, they prevent the y -solitons to follow their pathand connect them together.To avoid this situation, the Fourier filtering is used to blur this vertical phase jump.Practically, the filter is a slit of tunable width along y . It thus cuts the frequency com-ponents in the y direction, while leaving the x ones unchanged.The figure 4.3 presents the effect of the slit on the filtered signal. The left column(figure 4.3.a) shows the signal in the Fourier space of the SLM (zoomed on the first orderof the grating, i.e. the one we keep), and the opening of the slit is marked by the redhorizontal lines. The second and third columns show the top part of the correspondingreal space signal after the slit, in intensity (fig. 4.3.b) and phase (fig. 4.3.c).The Fourier space of the SLM has some frequency components in the x and in the y directions, corresponding to the horizontal and vertical phase jumps. On the upper line,the slit is quite open and only a few components of the vertical phase jump are removed:the filtered real space is therefore very similar to the signal designed by the SLM, andboth the intensity dip and the phase jump along the y direction are well defined.By reducing the slit opening (see medium line), more components of the vertical phasejump are cut, which blurs its definition in intensity as well as in phase.Finally, on the bottom line, the slit is almost closed: in that case, most of the verticalcomponents are filtered out, which leads to an extended and shallow intensity dip and asmooth phase jump. Meanwhile, all of the horizontal components are kept: therefore thehorizontal phase jump keep their good definition and are unchanged by the filtering.The result of the filtering procedure on the fluid impression is illustrated in figure 4.4.The images are a density map of the cavity plan in logarithmic scale, in a high power .1. EXPERIMENTAL IMPLEMENTATION Filtering in the SLM Fourier space . Fourier space of the SLM (a.) and thedifferent filtering indicated by the slit opening in red. The top part of the correspondingfiltered signal is shown in intensity (b.) and phase (c.). The vertical phase jump definitionis defined by the slit opening, while the horizontal ones are left unchanged.They are centered on the top part of the imprinted solitons. The fluid is created witha flow, from bottom to top on the pictures.On the left, the slit is open, filtering out the grating components but keeping most ofthe ones of the soliton pattern. Due to the presence of the flow, the horizontal soliton isstretched, but join the vertical solitons together on a short distance along y .On the right, the slit is much more closed, filtering out most of the components of thevertical phase jump of the pump. As this one is therefore sent very smooth to the sample,it induces in the fluid an extension of the vertical solitons along y : they use all the large y CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS
Figure 4.4:
Effect of the slit on the fluid . Images if the top end of the imprintedsolitons on the fluid, at high pump intensity preventing any free propagation. On the left,the solitons obtained with a wide slit. Because of the presence of the flow from bottomto top, the solitons are slightly stretched, but join together quickly. On the right, theslit is much more closed: the solitons are very elongated and their respective transversevelocity is low when the join together.range of the blurred phase jump to join, finally meeting each other with a low transversevelocity. That configuration should benefit the further propagation of the solitons. iv Global experimental setup
The experimental setup is sketched in figure 4.5. The laser source is a Titanium Sap-phire Matisse, and its spot is elongated in the y direction by two cylindrical lenses (CL).The beam is then split in two by a polarizing beam splitter (PBS) preceded by a half-wave plate (HWP). The main beam is shaped as desired: the SLM draws the phase front,later filtered by the slit. It is sent collimated to the cavity, imaging the SLM plan sothat the phase jumps are well defined on the sample. The inset illustrates the excitationbeam configuration on the sample: the solitonic pattern is placed in the center, wherethe intensity is above the bistability limit, delimited by the white dashed line. The beamenters the cavity with an upward in-plane wave vector that gives a flow to the polaritons.The black rectangle is the detection field of view: it is shifted on top of the illuminatedregion to observe the solitons free propagation through the bistable area.As usual, the detection is done in both real and momentum space. The real space givesthe intensity map of the cavity plan, as well as information on the phase pattern throughthe interferences with the reference beam previously separated from the laser beam. Theexperimental conditions of the system are extracted from the momentum space images(see section 2.2.3). .1. EXPERIMENTAL IMPLEMENTATION Experimental setup . The excitation beam is designed by the SLM andfiltered through the slit. It is collimated on the sample, imaging the SLM plan. Theinset illustrates the beam configuration: the solitonic pattern is in the center, where theintensity is above the bistability limit, located at the white dashed line. The detectionfield of view is delimited by the black rectangle, shifted from the center in order to focuson the bistable region and the solitons free propagation. The detection is done in realand momentum space, so that the experimental conditions can be associated with thecorresponding intensity and phase maps. i Parallel propagation
The essential role of the bistability in the propagation of the solitons is explained infigure 4.6. The figure a. reminds the S shape of the bistability curve and the threeassociated intensity regions: below the cycle, the low density region in grey, denoted asLD; above the cycle, the high density region highlighted in red and denoted as HD; andthe bistable cycle left blank.On b. is plotted the phase pattern designed by the SLM, with the same field of view asthe detection: we can see that the phase modulation is only present on the bottom partof the images. Figures c. and d. were realized in the exact same conditions except for thetotal intensity of the excitation. In c., the total laser power is high, which puts almostall the illuminated area above the bistability cycle: the red HD region expands above themajor part of the picture. The high density region fixes the properties of the fluid: it istherefore a replica of the driving pump field. Indeed, the solitons are artificially createdonly in the bottom part of the picture, while on top, phase and intensity are flat.Figures d. are obtained from the c. ones by gradually decreasing the input intensity.04
CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS
Figure 4.6:
Impression of dark parallel solitons . a. Theoretical bistability profileand the three associated intensity ranges. The color code is used all along this chapter.b. SLM phase pattern with the same field of view as the detection: the phase modulationare only sent on the bottom part of the images. c. High power density and phase maps.Almost all of the illuminated area is in the HD regime: the phase is fixed by the drivingfield and replicates the pattern designed by the SLM. d. Same configuration as c. atlower power. The bistable region has extended toward the beam center, and reached thetop part of the solitons. The system has readjusted its phase to let the solitons propagatethrough the bistable area, until the low density region where the nonlinear interactionsare too low to sustain dark solitons.The bistable region expands toward the center of the beam, and eventually reaches thetop part of the imprinted solitons. The fluid then readjusts its phase and lets the solitonspropagate through the bistable region, even though the region between the solitons isout of phase with the driving field. Indeed, the dark solitons in d. are clearly visiblewithin the bistable region, inducing a phase jump of π all along their propagation. Thepropagation is sustained as long as the system is in the bistable regime. As the illuminatedregion is finite, the solitons will reach its border: in the low density region, the nonlinearinteraction are too low to sustain dark solitons. ii Influence of the intensity To find the good configuration for the solitons to propagate through the bistable region,several parameters need to be finely tuned. In particular, the total intensity of the pumphas an important impact on the soliton propagation, as they need a bistable fluid toreadjust the phase of the fluid.To qualitatively study the influence of the pump intensity, a scan is realized and theresults are presented in the figure 4.7. The SLM phase pattern is presented on a., againwith the same field of view as the detection images, and the experimental images areshown in b. The top line shows the density maps and the bottom one the interferograms.The input power is gradually decreased from picture (i) to (vi). The flow is from bottomto top, and the red colored regions indicates the area above the bistability cycle. .1. EXPERIMENTAL IMPLEMENTATION
Scan of the input power . a. Phase pattern designed by the SLM withthe same field of view as the detection. b. Density (top) and phase (bottom) maps of ascan of the total input intensity. On (i), the power is maximum and the phase is fixedeverywhere. The intensity gradually decreases and with it the size of the HD region in red:the solitons propagate and open, until the bistable part reaches the imprinted solitons in(vi) where they align and propagate parallel.On picture (i), the total power is high: the fluid is above bistability on the wholepicture. Its phase is therefore fixed, which shows that the solitons are sent only on thebottom part of the image. From picture (ii) to (v), as the power decreases, the bistableregion expands. The solitons propagate further and further but the system still can notperfectly sustain them: they are grey as their phase jump is lower than π and they openand vanish along the flow. The phase maps confirm as well that the phase modulationinduced by the solitons vanishes with them.Finally, on picture (vi), the bistable area joins the top part of the imprinted solitons.They are then able to align to each other, and to remain dark and parallel all along theirpropagation. This time, their phase is π and stays constant. They are sustained throughthe whole bistable region, and vanish only at its edge, where the system jumps to the lowdensity regime.This set of measurement scans the bistability cycle, and confirms the necessity to bein this regime to achieve the free propagation of dark solitons in a resonantly pumpedpolariton fluid.06 CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS iii Influence of the filtering
A second crucial parameter for the good propagation of the solitons is the spatialfiltering in the Fourier plan of the SLM. As explained in section 4.1.iii, a slit is placedalong the x axis in order to clean the beam after the SLM, but also to smooth the phasejump along y to stimulate the solitons propagation.The qualitative study of the slit width influence is shown in figure 4.8. From a. to e.,the slit is progressively closed, while all other parameters remain the same. In particular,the pump intensity is unchanged: the imprinting region highlighted in red is the same inall the pictures.Figure 4.8: Scan of the spatial filtering . Density (top) and phase (bottom) maps ofa scan of the spatial filtering: on a., the slit is widely open, while its width is graduallyreduced from b. to e. It reduces the respective lateral velocity of the solitons, which alignonly on e. where they meet with a small enough angle.The images a. present the density and phase maps corresponding to a fully opened slit.The phase pattern imprinted in the fluid has therefore a sharp profile. The soliton alongthe x axis is however not visible: the polariton flow, from bottom to top, prevents a welldefined impression. The solitons propagation is anyway inhibited: they vanish soon afterthe HD region and the phase jump is quickly erased. .1. EXPERIMENTAL IMPLEMENTATION π phase jump all along. The setup previously implemented gives a large flexibility on the shape of the imprintedphase pattern. Even though the first goal of the experiment was to reproduce the genera-tion of a dark solitons pair, the SLM pattern is easily tuned and some other configurationshave been tried, in order to study the solitons behaviour with different initial conditions. i Four solitons
Figure 4.9:
Four solitons imprinting . a. SLM phase pattern corresponding to thedetection region. b. Density map of the fluid. The two solitons pairs are imprinted onthe red region and propagate through the bistable white one. c. Interferogram of thefluid. The phase jump propagates with the solitons.The first tested pattern, shown in figure 4.9, is a double pair of solitons. On the left,figure a. is a scheme of the SLM phase pattern. It does not represent the whole pattern,08
CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS but only the one corresponding to the top part of the beam, in order to coincide with thecorresponding detection pictures. Those ones are plotted on figure b. and c. and showthe density map and the interferogram, respectively.The double pair of solitons is realized by sending two rectangular shapes in phaseopposition with the background thanks to the SLM. The imprinted phase pattern isdesigned so that each of the four solitons is equidistant from its neighbor. It stays likethat in the high density region of the fluid, where the solitons are imprinted (red partof figure 4.9.b.), but during their free evolution, they get closer to their respective pair,so that the area in phase with the driving expands while the one in phase opposition isreduced. However, the solitons do propagate and maintain a phase jump on the fluid.This configuration gives an overview of the scalability of our method, and opens theway of the study of soliton lattices in polariton superfluids. ii Opening solitons
As the tendency of the solitons seems to be to get closer to one another, we triedto compensate this effect by sending them with an opposite direction and a positiverespective lateral velocity.Figure 4.10:
Opening solitons imprinting . a. Imprinted phase pattern inducingsolitons moving away from each other. b. and c. Fluid density map and interferogram ofthe impression.The results are presented in figure 4.10: the left figure shows the SLM phase patterncorresponding to the fluid density and phase maps plotted in figures b. and c. Thesolitons are well defined in the imprinting region, at the bottom of the figure. However, .1. EXPERIMENTAL IMPLEMENTATION π , in agreement with their color. They vanish when they meet, as grey solitons arenot stable through collision [109, 123]. iii Closing solitons Figure 4.11 presents the opposite case where the solitons are sent toward each other.The solitons are sent oblique with a relatively high opposite velocity. As before, the threepictures illustrate the phase pattern designed by the SLM, the density map of the fluidand the corresponding interference pattern. The flow is from bottom to top and theimpression is realized in the high density (HD) region highlighted in red.Figure 4.11:
Closing solitons imprinting . Imprinted phase pattern (a.), density map(b.) and interferogram (c.) of solitons sent toward each other.The collision of the solitons induces a breaking of the pair into several solitons whichthen spread though the fluid. They indeed meet with a too high velocity to smoothlyalign or bounce, and split into different pieces, each of them inducing a phase modulation.However, they are not completely black, and not perfectly sustained by the system: theyvanish along their propagation.10
CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS iv Single soliton
A different configuration has been tested in figure 4.12, where a single soliton is im-printed on the fluid. The phase pattern designed by the SLM is now different than theprevious ones. The impression region is split into two parts in phase opposition, betweenwhich the soliton will appear. In order to have a completely symmetrical pattern, thebistable part of the beam is chosen to be at a π /2 phase difference from both domains ofthe high density region, so that none of them is favored by the system.Figure 4.12: One soliton imprinting a. The soliton is imprinted on the bottom partof the images, inducing a single delimitation between two area in phase opposition. Thebistable part of the beam is sent with a π /2 phase to ensure the symmetry. b. On thefluid, the single soliton is not sustained in its initial form and splits into two (b.), as aphase switch of half the fluid is not supported by the system. c. The two resulting greysolitons have a phase jump between them and stay quite close to each other.The fluid density is presented in figure b. The impression is done in the HD region: thesingle soliton is well imprinted, with a clear phase jump. But when it enters the bistableregion, the soliton is not supported anymore and splits into two. A single soliton indeedimplies a phase difference between the two half-parts of the system, so compared to thedriving field, either half of the fluid is in total phase opposition, or all the fluid has a π /2phase difference. This seems to be energetically too expensive for the system, even in thebistable regime.The splitting of the soliton is anyway interesting as it shows that its propagation is nottotally inhibited, and even preferred to its halt. The propagation can be separated into .2. PARALLEL ALIGNMENT: EQUILIBRIUM SEPARATION DISTANCE The study of those different configurations indicates that the dark parallel solitons arethe most stable in the polariton fluid in the bistable regime. It is thus the configurationthat is going to be further studied in the next section. In order to better understandthe characteristic behaviour of such phenomenon, several runs of measurements havebeen performed varying the initial inter-soliton distances as well as the hydrodynamicconditions. The results of this analysis will be reported in this section.
The most intriguing interrogation about the solitons alignment is to identify the differ-ent parameters that play a role in the establishment of such an equilibrium distance. Thefirst idea is to try to modify the initial conditions of the system to see if it has any effecton its behaviour. This has been realized through the use of the SLM, changing the shapeof the phase pattern to modify the solitons imprinted on the fluid. The first parameterwe decided to study is the separation distance between the solitons.Figure 4.13:
SLM patterns corresponding to different separation distances
Thedistance between the imprinted solitons is tuned by changing the width of the rectangular π -shifted rectangle.It can be easily controlled by the shape designed on the driving beam by the SLM. Byenlarging the π -shifted rectangular shape of the pattern (see figure 4.13), the solitons are12 CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS imprinted further apart.The results of a typical run of measurements are exposed in figure 4.14. The topimages present the intensity of the fluid while the bottom ones show the correspondinginterferograms. For this set of experiments, the fluid velocity is 1.00 µm/ps, the soundvelocity c s = 0 .
67 µm.ps, the effective polariton mass m ∗ = 1 . · − kg and the energydetuning ∆ E lasLP = 0 .
34 meV. Surprisingly, even though the initial separation distanceis reduced from 24 µm in figure a. to 15 µm in figure e., we observe that the behaviourof the solitons is identical in all the cases. When they enter the bistable part of the fluidand are not artificially imprinted anymore, the separation distance between the solitonsreduces until reaching the same equilibrium value, from which the solitons align and stayparallel.Figure 4.14:
Experimental scan of the initial separation distance . The distancebetween the imprinted solitons is tuned from 24 µm in a. to 15 µm in e. by changingthe width of the rectangular π -shifted rectangle. When they enter the bistable fluid, thesolitons propagation is maintained, but not their separation distance: they get closer toone another until reaching an equilibrium separation distance from which they align.In the case of figure 4.14, the equilibrium separation distance is about 5 µm for all theplotted figures. After reaching this configuration, they keep this separation distance aswell as their associated phase jump and continue their propagation for more than 50 µm, .2. PARALLEL ALIGNMENT: EQUILIBRIUM SEPARATION DISTANCE ψ X ) and the cavity photon ( ψ γ ) fieldsrespectively: i (cid:126) ∂ψ γ ∂t = " − (cid:126) ∇ m ∗ γ − i Γ cav ψ γ + V ψ X + F ( r ) e − iω t (4.1) i (cid:126) ∂ψ X ∂t = (cid:2) g X | ψ X | − i Γ X − ∆ E Xcav (cid:3) ψ X + V ψ γ (4.2)where m ∗ γ is the effective cavity photon mass, ω the laser energy, Γ cav the photonlifetime, g X the exciton interaction constant, V the half Rabi splitting and ∆ E Xcav thecavity-exciton energy detuning. In order to best match the experimental configuration,all those parameters have been taken the same as in the experiment (the exciton lifetimeΓ X was set equal to 150 ps). F ( r ) represents the pump and contains both its amplitudeand phase. The steady-state solutions are presented in figure 4.15. As usual, the flow isfrom bottom to top, the top images show the density map of the fluid while the bottomimages are the associated interferograms, giving access to the fluid phase.The driving field used in the simulations has been chosen to reproduce the experimentalone: it has been taken as a Gaussian beam whose center is shifted towards the bottomof the images plotted on figure 4.15, so that the region where the intensity is above thebistability cycle is highlighted in red. In this high density (HD) region, the solitons areimprinted: a π phase jump is imposed; the experimental filtering operated by the slit ismodeled here through a smoothing of the phase jump along the y direction.The numerical simulations are in good agreement with the experiment: the solitonsfollow exactly the same behaviour. No matter how far away from each other they areimprinted, as soon as they are released within the bistable fluid, they get closer to oneanother before aligning at a specific equilibrium separation distance. This alignmenttakes place for solitons at around 5 µm from each other, which reproduce accurately theexperimental configuration.To get a quantitative idea of the separation distance evolution, it has been studied allalong the solitons propagation. To do so, the transverse solitons profile has been fittedby its expression [62]: | ψ ( x ) | = tanh x − d sep / A ! tanh x + d sep / A ! (4.3)where A corresponds to the full width at half-maximum (FWHM) of one soliton and d sep to their separation distance, as illustrated in the inset of figure 4.16.14 CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS
Figure 4.15:
Numerical simulations of the scan of the initial separation distance .As for the experimental set (figure 4.14), the imprinted separation distance is numericallyscanned from 24 µm (a.) to 15 µm (e.). The same behaviour is observed: when thesolitons enter the bistable fluid, their separation distance reduces until the solitons alignat a specific equilibrium distance from each other.The fit results of the experimental set are reported in figure 4.16. The hydrodynamicconditions are kept constant during the whole set, the only parameter which is variedbetween each picture is the initial separation distance. The color lines correspond to thedifferent images, and show the evolution of the fitted separation distance d sep along thepropagation axis y . As usual, the red HD region illustrates the impression area wherethe input intensity is above the bistability, while in the white bistable region, the solitonspropagates freely through a flat driving field.The graph confirms the tendency: despite the change of the initial separation distance,the freely propagating solitons get closer to another, and align at a specific separationdistance. This distance is the same for all the cases of the set, at 4.8 µm illustrated bythe black dashed line. It confirms the existence of a equilibrium between two oppositeforces, which push the solitons in opposite directions and only equilibrate when they areat a specific distance from each other.On hand hand, dark solitons have repulsive interactions [132,141], which pushed them .2. PARALLEL ALIGNMENT: EQUILIBRIUM SEPARATION DISTANCE Experimental scan of the initial separation distance . Each color linecorresponds to one image of the experimental set presented in figure 4.14. Each dot illus-trates the fitted separation distance at the corresponded position along the propagationaxis y . No matter the initial separation distance, they all converge to the same value of4.8 µm in this case, illustrated by the black dashed line.away from each other. However, they are also propagating within a pumped fluid, evenif it is a bistable one. So even if phase readjustments are possible, all the area of thefluid with a different phase than the pump one remain effectively unpumped. Sustaininga large out-of-phase region has therefore a high energetic cost for the system, which tendsto reduce this area and pushes the solitons toward one another.By imprinting the solitons far away from each other, their propagation is first dominatedby the force induced by the driving, and they propagate towards each another. Howeverby getting closer to one another, their repulsion increases, until reaching an equivalentvalue to the driving-induced force. Prevented to go closer together or further apart, thesolitons propagate parallel.This parallel propagation of a soliton pair corresponds to the one observed in the spon-taneous case, reported in section 3.4, but remains in full opposition with the previouslyexpected [132] and reported [22] soliton behaviour. However, this artificial implementa-tion has high control and scalability: we will use those advantages to lead for a deeperstudy of the phenomenon. The first interrogation from those results is the origin of this equilibrium separationdistance, and on which parameters it depends. The size of a vortex core is known to beof the order of the healing length ξ of the system [142]. By analogy, the FWHM of the16 CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS solitons ( A in equation 4.3) should also be connected to ξ . The equilibrium separationdistance seems also to be the minimal one between the solitons, and therefore shouldbe connected to their width so that both solitons can be effectively distinguished. Thehealing length is thus the parameter that will be considered in the following.The healing length of a polariton fluid is defined by the hydrodynamic conditions of thesystem, and in particular by the effective polariton mass m ∗ , the polariton interactionconstant g and by the polariton density n [142]. An approximation can however bemade, as we are in the bistable regime, and in particular in the left part of the bistabilitycycle. Indeed, at the turning point of the cycle, on the upper bistability branch at thelow intensity threshold, the laser-polariton detuning verifies the relation ∆ E lasLP = gn .We can thus write the healing length as: ξ = s (cid:126) m ∗ gn = (cid:126) √ m ∗ ∆ E lasLP (4.4)The hydrodynamic parameters can be tuned experimentally:• the effective mass of the polaritons m ∗ is defined through the curvature of thedispersion, and can be tuned by changing the detuning between the exciton and thecavity photon ∆ E Xcav . Experimentally, this is done by changing the working pointon the sample: the wedge between the two DBR modifies the length of the cavityand thus the cavity photon resonance. The induced modification of the dispersionconsequently changes the polariton mass and influences the hydrodynamic behaviourof the system.• the detuning between the laser and the lower polariton branch ∆ E lasLP isdirectly tunable by shifting the laser frequency. It also affects the bistability curve ofthe system: a different detuning leads to a different intensity profile on the fluid andhas a direct influence on the impression and propagation of the solitons. The hy-drodynamic conditions must be adjusted in order to find a favourable configurationagain, by tuning the fluid and sound speeds.• the fluid velocity v fluid is the derivative of the dispersion: v fl = 1 (cid:126) ∂E∂k (4.5)which changes with the in-plane wavevector of the driving field. Therefore, theresonance of the lower polariton branch is also affected, as well as the detuninglaser-lower polariton ∆ E lasLP and with it the bistability cycle.• finally, the speed of sound c sound depends on the square root of both the detuninglaser - lower polariton ∆ E lasLP and on the effective mass m ∗ : c sound = s ∆ E lasLP m ∗ .2. PARALLEL ALIGNMENT: EQUILIBRIUM SEPARATION DISTANCE Mach number : M = v fluid c sound (4.6)which contains not only m ∗ and ∆ E lasLP through the speed of sound, present in thehealing length definition too, but also the fluid velocity v fluid . It is thus the parameterchosen to compare the different results.The influence of the hydrodynamic parameters on the equilibrium separation distancebetween the solitons is investigated by analyzing the propagation of imprinted solitons onpolariton fluids with different Mach numbers. In the experiment, the cavity is spatiallyexplored, and on several positions which presents a low enough disorder, solitons areimprinted. Their free propagation is then achieved by alternatively playing with thewavevector, the laser energy and the pump power to reach an intensity pattern where thebistable region coincide with the top part of the imprinted solitons.The experimental results have all been gathered in figure 4.17, where each spot corre-sponds to one set of measurements: it shows the mean equilibrium separation distance ofall the solitons pictures taken in the same hydrodynamic conditions, but changing onlythe initial separation distance. They have been plotted as a function of the Mach number:the conditions have been widely scanned as it goes from M = 1 to more than 6. All theparameters have also been reported in table 4.1, where are listed the detuning laser -lower polariton branch ∆ E lasLP , the speed of sound c sound , the effective polariton mass m ∗ and the Mach number M .∆ E lasLP [meV] c sound [µm/ps] m ∗ [kg] v fluid [µm/ps] M d sep [µm]0.24 0.60 1.05 · − · − · − · − · − · − · − · − · − Parameters of the experimental sets plotted in figure 4.17 .The results shown in figure 4.17 are very surprising. Indeed, the expected dependence ofthe separation distance on the hydrodynamic of the system is not observed at all: on thewhole range of conditions, the equilibrium separation of the solitons is almost constant,slightly fluctuating around the mean value of 4.75 µm shown by the black dashed line.18
CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS
Figure 4.17:
Experimental equilibrium separation distances as a function of theMach number . Each spot corresponds to the mean equilibrium distance for one set ofmeasurements with the same hydrodynamic conditions. Even though the Mach numberrange is quite large, the equilibrium distance is almost constant around a mean value of4.75 µm, illustrated by the black dashed line.This result is quite unexpected, as the behaviour of dark solitons is usually governedby the hydrodynamic conditions of the system where they are generated. On the otherhand, as explained earlier, in this experimental study all the parameters are connected.By trying to modify the global conditions, all of the parameters are changed between eachpoints in order to find each time a favorable situation for the solitons propagation. Theycould eventually compensate each other, leading to a similar configuration.The scan of one single parameter is experimentally very difficult to realize, as they areall connected. In order to independently study the influence of each parameter, accuratenumerical simulations have been done, presented in the next section.
An important advantage of the numerical simulations is the possibility to tune thesystem parameters independently. It has been very useful for the verification of our as-sumption that the equilibrium separation distance is set by the hydrodynamic parametersthrough the healing length. To do so, several sets of simulations have been realized, wherethe system conditions are kept constant, and only one parameter is tuned. This way, thehealing length is also gradually tuned and its influence on the separation distance shouldbe clearly visible. .2. PARALLEL ALIGNMENT: EQUILIBRIUM SEPARATION DISTANCE E lasLP , while the other ones are left unchanged, similar to the typicalexperimental ones. In the definition of the Mach number, ∆ E lasLP is present only in thesound velocity; all others parameters being kept constant, the Mach number is thereforetuned as the inverse of √ ∆ E lasLP . As the healing length ξ also follows a relation on1 / √ ∆ E lasLP , M and ξ are linearly connected.The results of such simulations are presented in figure 4.18: as for the experimental ones,no trend appears in the equilibrium separation distance, it stays the same independentlyof the hydrodynamic conditions. The black dashed line shows the mean value of thedifferent points, of 3.76 µm in this case.Figure 4.18: Numerical equilibrium separation distances as a function of theMach number . Each spot corresponds to the equilibrium distance between two parallelsolitons for a different detuning ∆ E lasLP ; all the other parameters are kept constant.Some remarks need to be done concerning the figure 4.18. First of all, the Mach numberrange is shorter than the experimental one. It is due to the fact that only the detuning ischanged on this scan: also numerically, a large enough detuning is necessary to implementa bistability (∆ E lasLP > √ (cid:126) γ , with γ the lower polariton decay rate), itself required forthe propagation of the solitons through the fluid. On the other hand, a too high detuningleads to unstable solitons [143, 144]: instead of staying parallel and dark, they oscillateand blur, making impossible the definition of a separation distance.The second remark, connected with this behaviour, is the presence of error bars innumerical results. They come from the previously mentioned oscillations: the dots offigure 4.18 are the mean value of the separation distance between the solitons over 50 µmpropagation. If the conditions are not exactly favourable to the solitons propagation, they20 CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS slightly oscillate, which is represented through the error bars. Such behaviour is presentedon figure 4.19, where we can see a breathing along the propagation of the soliton pair.Figure 4.19:
Numerical simulation of the impression of unstable solitons. . Theconditions do not allow a parallel propagation of the soliton pair, which induces a breath-ing behaviour and an uncertainty on the separation distance definition.Even though unexpected at first, those numerical results confirm the experimentalones: the equilibrium distance between the parallel dark solitons is independent of thehydrodynamic conditions. But the question remains: what does it depend on ?Another specificity of our system that could explain this unusual behaviour is its driven-dissipative nature. In our configuration, the pump is sent everywhere; that is exactlywhat led the solitons to align in the spontaneous configuration (see section 3.4). Thisdriven-dissipative character is imposed by the decay rate of the polariton γ polaritons , itselfdefined by the cavity mirrors reflectivity. Therefore, the equilibrium distance could begoverned by the dissipation of the system. From the experimental point of view, a scanof the polariton decay rate would mean a collection of different samples, with exactly thesame properties except for their DBR thickness. As we do not have them in the lab, theverification of the effect of the decay rate has been studied numerically.To do so, specific simulations have been made, where this time all the hydrodynamicconditions were kept constant. Only the decay rate of the polaritons has been tuned, sothat any observed trend could be associated with the dissipation. The parameters have .2. PARALLEL ALIGNMENT: EQUILIBRIUM SEPARATION DISTANCE E Xcav = E X − E cav = 1 meV, 2 (cid:126) Ω R = 2 . E lasLP = 0 . v fluid = 1 µm/ps.Figure 4.20: Numerical simulations of the equilibrium separation distance asa function of the decay rate of the polaritons . The red dots correspond to theequilibrium separation distance of the parallel solitons for different polariton decay rate,while fluid and sound speeds are kept constant at typical experimental values. The bluedot illustrates our experimental results: the mean value of all the extracted separationdistances for the decay rate of our cavity.The results are presented in figure 4.20. The red dots correspond to the mean separationdistance of each simulation realized for a specific polaritons decay rate. The error barscome again from the slight breathing of the solitons in the region where the separationdistance is averaged. This time, a clear trend is observable: the separation distancestrongly depends on the polaritons decay. A higher decay rate, i.e. a shorter polaritonlifetime, induces a smaller separation distance.This is indeed concordant with the fact that the inner region of the solitons is effectivelyunpumped by the driving field because it is out of phase: the refilling is only done throughtunneling across the solitons. Therefore, for a certain distance between the solitons, ashorter polariton lifetime leads to a lower density in between the solitons: the equilibriumtakes place for solitons closer together.The linewidth of our cavity is (cid:126) γ polaritons = 0 .
07 meV, which corresponds to a lifetimeof 10 ps. We illustrated our experimental results through the blue dot, which correspondsto the mean separation distance of all our results, the 4.75 µm extracted from figure 4.17.It stands within the error bar of the numerical results and allows to conclude for a goodagreement between the theory and the experiment.The work of this chapter has resulted in the implementation of a new technique ofphase pattern impression. The phase constraint imposed by the quasi-resonant pumping22
CHAPTER 4. IMPRESSION OF BOUND SOLITON PAIRS has been released using the bistability properties, and the design of the driving fieldthrough the SLM allows to artificially generate dark solitons on demand. The presenceof the driving field along the propagation of the solitons enhances its distance, but is alsoresponsible for an interesting binding mechanism between the solitons. They propagateparallel, as a form of dark-soliton molecule in a local nonlinear medium. The deeper studyof this behaviour lead us to notice that the characteristic separation distance between thesoliton is governed by the driven-dissipative nature of our system. This technique is verypromising for the control and manipulation of such collective excitations, and could forinstance be used in the generation of multiple solitonic pattern, or the study of the solitonsmutual interaction. hapter 5
Generation of solitonic patterns instatic polariton fluid
In the two previous chapters, we studied the generation and propagation of paralleldark solitons in a supersonic polariton fluid. However, we did not focus on the stability ofthose solitons, which was assured by the high speed of the polariton flow. Indeed, solitonsare unidimensional objects, which, when placed in bi- or tridimensional environments,develop transverse modulations known as "snake instabilities" [133]. Those ones havebeen studied theoretically and experimentally in different systems, such as self-defocusingnonlinear media [145–147] or ultracold bosonic and fermionic gases [129, 130, 148].Snake instabilities lead solitons to decay into quantized vortex-antivortex pairs, result-ing into quantum vortex streets which can be seen as quantum equivalent of the vonKarman vortex streets. They were not observed in our driven dissipative quantum fluids,as under resonant excitation, solitons were always generated within a flow as we saw, andunder non-resonant excitation, the very fast relaxation of the solitonic structures shouldprevent the observation of those instabilities.This chapter is based on a theoretical proposal made as part of a collaboration with thegroup of Guillaume Malpuech in Clermont-Ferrand [144]. It follows the previous resultson bistability and suggests a new configuration to generate solitonic pattern, but thistime within a static polariton fluid. It uses a transverse confinement within an intensitychannel to create a pair of dark solitons, which decays into vortex streets due to thedisorder of the system. 12324
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Before presenting our experimental results in the next section, we give here an overviewof the theory [144] and of the first numerical realizations.
We use again in this section a polariton fluid pumped quasi-resonantly, but the pumpis not spatially uniform and can be decomposed into two parts. A low intensity pump issent everywhere, and is called the support, labeled as S . On top of it, some regions arealso pumped by a strong power, named the pump and labeled P ( r ).This configuration is very similar to the one we presented in chapter 3, except thatthe support intensity is this time scanned also below the bistability cycle. The position ofthe support intensity plays an important role on the behaviour of the illuminated fluid,as we will discover.The dynamics of the fluid is still described by the driven-dissipative Gross-Pitaevskiiequation (equation 1.54), that we can rewrite in our context, considering only the lowerpolariton branch: i ∂ Ψ ∂t = (cid:16) − (cid:126) m ∗ ∇ − iγ + g | Ψ | (cid:17) Ψ + ( S + P ) e − iω t (5.1)where m ∗ is the effective mass of the polaritons, γ their decay rate, g their interactionconstant and ∆ E lasLP = (cid:126) ω the detuning between the pump energy and the lowerpolariton resonance.It is important to notice at this point that both S and P are sent at normal incidence:there is no in-plane wavevector, and consequently no velocity of the fluid which is at rest.If we first consider only the support, the polariton fluid is homogeneously pumped, anda bistability takes place. This one is illustrated on figure 5.1 in green and red: the greenline shows the evolution of the polariton density for an increasing support field, and thered one for a decrease of the support, which shows different behaviours.Now we can add the pump on a specific region of the fluid: figure 5.2 illustrates in redthe laser profile. The support is sent everywhere, while the pump P ( r ) is only presentfor x <
30 µm. The polariton density is plotted in black. As in chapter 3, we see thatit does not follows the same profile as the pump. Indeed, the pump sets the polaritondensity in the upper branch of the bistability, which stays there even in region where onlythe support is sent (30 < x <
140 µm in figure 5.2). At some point however, the highdensity is not sustained anymore, and the polariton density abruptly drops. As a discretesymmetry has been spontaneously broken, this sudden drops can be defined as a domainwall [149], and is followed by small density oscillations (from x ≈
140 µm). The decayof the density takes place within one healing length ξ = (cid:126) / √ m ∗ gn ( m ∗ the polaritoneffective mass, g their interaction constant and n = | Ψ | their density). .1. THEORETICAL PROPOSAL Polariton bistability and domain wall velocity . Polariton density asa function of the support amplitude, in green for increasing support and red for thedecreasing one. The black line illustrates the domain wall velocity ( y axis on the right).The typical support amplitude S c corresponds to zero velocity of the domain wall. From[144]The domain wall can propagate along the x direction, and is stable against instabilitiesalong the y one. The propagation of such domain walls have been previously studied inpolaritons [150, 151] and, more largely, in optics [152–154].The velocity of this domain wall is of particular interest as it gives the direction ofthe moving wall and the condition of stability. It has been computed numerically as afunction of the support intensity, and plotted in black line in figure 5.1. We can see theparticular support intensity S c for which the domain wall velocity is zero. Its value canbe evaluated by the Maxwell construction [144, 155] for γ →
0. It is done by consideringthe stationary driven-dissipative Gross-Pitaevskii equation in which we neglect the lossterm, resulting in: (cid:18) − (cid:126) ω − (cid:126) m ∗ ∇ + g | Ψ | (cid:19) Ψ + S = 0 (5.2)Since all coefficients are real, the wave function Ψ and the pumping S can only takereal values as well. The previous equation can thus be rewritten as a Newton’s equationof motion for a material point ( x = Ψ) with a mass m = (cid:126) / m ∗ : m d xdt = F ( x ) (5.3)This material point is thus governed by a position-dependent force F ( x ) = gx − (cid:126) ω x + S , to which can be defined a potential U , which the two maxima correspondto the two stable domain of the system, i.e. the high and low density branch of the26 CHAPTER 5. SOLITONS IN STATIC FLUID
Figure 5.2:
Pump-support configuration and domain wall . Pumping profile in red:the weak support is everywhere while the strong pump is sent only for x <
30 µm. Theblack line shows the polariton density: it remains high even in region where only thesupport is sent, before abruptly dropping around x = 135 µm, creating a domain wall.From [144]bistability. The system remains stationary only if these two maxima are equal, otherwisethe domain wall propagates. The analytical solution of this equation leads finally to thecritical value of the support intensity S c : S c ≈ (cid:126) ω ) / √ g (5.4)The velocity of the domain wall evolves as a function of S − S c . Indeed, for a supportintensity higher than S c , the wall propagates to the right with a speed proportional to S − S c : v ≈ S − S c S c ξτ (5.5)Then, the high density region expands and eventually fills up the whole space. Thehealing length ξ has been chosen to be 1.8 µm for these calculations. On the contrary, forsupport intensity lower than S c , the domain wall move backward and reach the limit ofthe pump region. Finally, if the support intensity is close to S c , the speed of the domainwall is close to zero and the domain wall is localized in space. In these conditions, theGross-Pitaevskii solutions bifurcate to dark soliton multiplets, as we will see in the nextsection.The moving property of this domain walls explains the results we presented in chapter3. At that time, we wanted to extend the region of the fluid with high polariton density,which is why we placed the system above the critical support intensity S c , placing the .1. THEORETICAL PROPOSAL S c . The next step is now to add another high density region in front of the other. In thebidimensional fluid, it results in a central channel where only the support is present, whilethe lateral regions are illuminated by both the support and the pump. The width of thechannel is fixed constant at L = 23 µm, which corresponds to 13 ξ of the high densityregion.The opposition of the two domain walls can lead to the development of solitons. Indeed,with an adequate ratio of the pump/support intensities, their interaction can create aphase jump and thus two regions in phase opposition, resulting in two parallel solitonsguided by the low density channel. The channel is indeed in the lower branch of thebistability cycle, where stationary phase defects exists: the phase is not fixed as theparticles diffuse from the high density regions [144].The numerical results of this configuration are presented in figure 5.3, for a supportintensity S = 0 . S c . Images a. present the ideal case density map: the presence of twodomain walls in front of each other gives rise to two dark solitons in the channel. Thesystem is invariant along y so is effectively 1D. One can notice that, as the high densityregions impose their identical phase to the fluid, only a even number of solitons can besustained inside the channel.These solitons are however created in a static fluid: they are subject to instabilitiesalong the y direction. They can be studied through the imaginary part of the energyof weak excitations, obtained through the Bogoliubov-de Gennes equations [17, 156, 157].This one is plotted in figure 5.3.b, for two different pump intensities: the upper linecorresponds to the case of P = 1 . S c , while for the lower one P = 2 S c . The green andred colors show the symmetric and antisymmetric modes.Modulational instabilities, known as snake instabilities and well known in conservativecondensates [158], are linked to the presence of a positive imaginary part of the energy.Those ones are plotted in figure 5.3.b: we can see that they can be decomposed into twocomponents (in red and green) which correspond to modes with different symmetries.The highest one defines the symmetry of the instabilities: on the upper line of figure 5.3,the final instabilities have an antisymmetric pattern, while on the lower line, the highestmode is the red one, which leads to symmetric instabilities. More details about the modescompetition are given in [144].The instabilities in the simulations are generated by adding small noise or fluctuationsas this breaks the translational symmetry along the y axis. The results of figure 5.3have been obtained with a weak Gaussian disorder with a correlation length of 2 µm andamplitude γ = 0 .
01 meV.28
CHAPTER 5. SOLITONS IN STATIC FLUID
Figure 5.3:
Instabilities of guided solitons . Solitons generation inside a channel oflow intensity. The channel has a width L = 25 µm and the support intensity within thechannel is s = 0 . S c . The pump on the lateral regions is P = 1 . S c for the upper lineand P = 2 S c for the lower one. a. Intensity map of the ideal case of stable parallel darksolitons generation. b. Imaginary part of the energy of weak excitations of the stationarysolutions of a. The green and red colors represent the symetric and antisymetric modes. c.Intensity map of the stationary solutions with weak disorder, leading to the generation ofmodulational instabilities. d. Phase map corresponding to the bordered region of picturec. From [144]Figure 5.3.c. and d. show the density and phase map of the instable solitonic pattern.Figure d. corresponds to a zoom on the central region of figure c. The position of theinstabilities along y are determined by the disorder, however it does not affect the patternshape if the disorder is low enough ( γ (cid:28) (cid:126) ω ). In both cases, the dark parallel solitonsbreak into two stationary vortex streets. Further analysis [144] show the stability of thesefinal patterns: the snake instability has been frozen by the confining potential.We have seen that depending on the pump intensity, the instability symmetry canchange. In order to summarize the different configuration conditions, a phase diagramhave been realized and is presented in figure 5.4. The colorscale corresponds to the valueof the maximal instability wavevector, while the x and y axis represent respectively thesupport intensity and the total intensity, normalized by S c . The length of the channel L and the detuning laser-LP (cid:126) ω have been kept constant for the whole diagram.Several regions can be delimited, with different color shades: each of them correspondsto a different configuration within the channel, whose profile is plotted in the insets. Thewhole diagram is divided in three regions delimited by two blue lines. They highlight thenumber of solitons: on the bottom left angle, four solitons are generated; two solitons canbe observed in the center of the diagram, in between the two lines, while no solitons aresustained on the top right angle, where the support intensity is high. .1. THEORETICAL PROPOSAL Phase diagramm as a function of S and P intensities . The colorscale indicates the maximal instability wavevector, and therefore the configuration ofthe instabilities (orange-red is symmetric and green is antisymmetric). The blue linesseparates the regions with different numbers of solitons. On the right, the purple regionpossess a high intensity inside the channel and therefore no solitons, while the dark greyregion delimits a non-stationary steady state. The red and green dots correspond to theconditions shown in figure 5.3, while the blue one indicates the conditions of the mazesolving of figure 5.5. From [144]The four solitons are sustained for lower pump intensity. The large majority of it corre-sponds to the symmetric configuration in orange, but a tiny light green region also showsthe possibility of four solitons in the antisymmetric configuration. The small elongatedlight gray region shows the position where a pattern of four solitons is generated. Ithappens for low pump and high support, which therefore induces only a weak transversepolariton flow towards the channel which favors the stability of solitons pattern.The two solitons region is subdivided in three parts. The orange part correspondsto the symmetric instabilities defined in figure 5.3, while the green one indicates theantisymmetric case. The dark grey region highlights a non-stationary steady state (limitcycle). This happens in the particular conditions of the simulations, i.e. without energyrelaxation and with low disorder. In that case, a pair of breathing solitons is generatedwhich oscillates in time, as shown in the Supplemental Material Video of [144].On the right, the purple region illustrates a high density in the channel, without anysolitons nor vortices. This is comparable to the polariton neuron picture depicted in [150],where logic gates are implemented in polariton fluids using similar intensity channels.30 CHAPTER 5. SOLITONS IN STATIC FLUID
Another configuration can also be observed by working with no support. In that case,it is possible to tune the phase between the two high intensity regions. In particular, byputting them in opposition of phase, the number of solitons created in the channel is odd,in order to conserve the phase [144]. This can be seen as a 2D generalization of the phaseconservation described in [53].
This channel configuration can now be extended in more complex geometries, and inparticular to a complicated maze. Maze solving studies have started with the work ofLeonhard Euler [159] which is now considered as the starting point of topology. Morerecently, the problem is commonly solved using the potential method [160], where apotential is assigned to the destination, and the choices are determined by its gradient.This method is used in many different fields, as in robotics [161, 162], in biology throughthe motion of biological organisms [163, 164], in microfluidics [165] or in plasma physics[166]. Some chemical methods based on the velocity map of the reaction have also beendeveloped [167], as well as optical ones using the diffusion of a wavepacket [168]. In ourcase, we suggest an algorithm based on the dead-end filling of instabilities, using an alloptical control and a picosecond time resolution.The previous configuration is now developed into an arrangement of low intensity chan-nels forming a maze, as pictured in figure 5.5. The S and P intensities correspond tothe blue dot of figure 5.4. All the channels do not have the same configuration: some ofthem are open-ends, connected to the outside of the maze, while the others are dead-ends,surrounded by high density walls.Figure 5.5: Maze resolution . Intensity maze for S and P conditions indicated by theblue dot in figure 5.4. a. Snapshot of the system 20 ps after the pump and support havebeen switched on: the heads of the solitons start to be repelled by the dead-ends. b.Stationary final distribution after 1 ns: the solitons fill only the path providing the mazesolution. From [144] .1. THEORETICAL PROPOSAL S and P is set, just after the switching of the high densityregions on the upper bistability branch, the corridors of the maze get filled by solitons.However, the solitons present in the dead-end channels quickly start to withdraw: figure5.5.a shows the system distribution 20 ps after the support and the pump have beenswitched on. The heads of the solitons move on the opposite direction of the channel end,as indicated by the black arrows.Eventually, they reach the open-end channels: instead of following their movement untilall channels are empty, those ones stay fixed. Figure 5.5.b shows the configuration of thesystem 1 ns after being switched on: the only filled corridors are the one solving the maze, i.e. the open end ones.The dead end channels have indeed a different behaviour than the open channels astheir head can be seen as a domain wall by itself [144]. However, the motion conditionsare not the same as for the lateral domain wall that we studied previously. Figure 5.6shows the thresholds of the support intensities that start the motion of the domain wallsin both cases, dead ends in blue and open ends in red, as a function of the channel width L .Figure 5.6: Threshold support intensities . Critical support intensities for the motionof the two types of solitons, the dead ends in blue and the open ones in red, as a functionof the width of the channels. The dead ends start to move for a lower intensity than theopen ones. From [144]Below L = 14 µm, the channels are to thin to be filled, hence the sharp decrease of thecritical support intensities. Above 14 µm however, the open ends always need a highersupport intensity to be emptied of their solitons than the dead ends. Therefore, as longas we choose the support intensity to be in the range between those critical intensities,the dead end solitons will move and thus be removed of the channels, while the open endones are stable: the maze is solved.32 CHAPTER 5. SOLITONS IN STATIC FLUID
This theoretical proposal has been followed by its experimental realization. After a briefdescription of the setup, we report the results we obtained for the different configurationswe tried.
The setup we used for this experiments is shown in figure 5.7 and is quite similar to theone of the soliton impression, presented in chapter 4. Two main differences have yet tobe noticed. First of all, to facilitate the development of the instabilities, the experimentneeds a static fluid: the excitation is therefore sent at normal incidence, and the in-planewavevector is zero. Then, the solitons are generated through intensity channels: theexcitation beam is shaped in intensity and not in phase anymore.Figure 5.7:
Experimental setup . As for the soliton impression, the beam is shapedthrough the SLM, this time in intensity: a darker channel is designed in its center. Thebeam is sent at normal incidence to the sample, and the signal is detected in real andmomentum space.As we saw in chapter 2 and used in chapter 4, the Spatial Light Modulator (SLM) canshape the wavefront of the incoming light beam. However, a trick can be employed touse it as a intensity modulator, and thus design an intensity pattern on the beam. Todo so, we work again with a grating from which we keep only the first order, but witha specific, controlled contrast. Indeed, if the full range of the gray scale is used in theSLM pattern, all the light is diffracted to the first order. However, by using only partof the scale, only a portion of light is diffracted, resulting in a darker region. This way, .2. EXPERIMENTAL REALIZATION
The starting point of the experiment is to explore the numerical phase diagram pre-sented in figure 5.4. To do so, we should scan the intensity along both axis, i.e. verticallyby tuning the total intensity, and horizontally by tuning only the support one.Figure 5.8 illustrates the evolution of the system with a tuning of the support intensity S only. The top line shows the density maps while the corresponding phase maps areplotted on the bottom line. The support density increases from left to right: the firstpicture corresponds to a completely dark channel. Nothing happens then as the vortexstreets need a support to develop.On the second picture however, for S = 0 . P , the expected vortex streets are clearlyvisible: the symmetric pattern is clear, and the pinned vortex-antivortex are spotted onthe phase map by the red and blue circles, respectively.By increasing gradually the intensity within the channel, the pattern elongates ( S =0 . P then S = 0 . P ) while looses in definition. At S = 0 . P , it has already vanished inthe center, only remains the pinned ends of the vortex street.This behaviour is quite surprising as it does not correspond to the numerical sim-ulations, where the solitons vanished from the end of the channels towards the center.However, it can be explained by the gaussian shape of our excitation beam, while theprevious simulations were realized for a flat intensity. The conditions are therefore notthe same along the channel, the density in particular is lower at the ends, which explainswhy the instabilities remain there longer.On the last picture, where S = 0 . P , the instabilities have completely vanished. Thedifference between the support and the pump intensities are not high enough for thedomain walls to be generated, which therefore prevents the generation of the vortexstreets.Those first results coincide well with the numerical simulations, but only the symmetricpattern has been observed. In the phase diagram however, the antisymmetric region hasan elongated shape toward the horizontal axis: in order to see them, we then tried totune the total intensity S + P to scan the diagram vertically.34 CHAPTER 5. SOLITONS IN STATIC FLUID
Figure 5.8:
Scan of the support intensity S . The density within the channel increasesfrom left to right: it starts completely dark ( S = 0 . P ) and raises up to S = 0 . P .The red dashed rectangle indicates the channel position. The symmetric solitonic patternappears at S = 0 . P , elongates a bit and vanishes with the intensity increase. The phasemaps show the appearance of vortex streets, where vortex and antivortex are marked bythe blue and red circles for S = 0 . P .Figure 5.9 presents the images of the system for several values of the total intensity S + P . Again, the top line shows the density maps, and the phase maps are presented inthe bottom line. The intensity increases from left to right until a maximal total power P max . The red dashed rectangle indicates the position of the low intensity channel: it iscentered on the figure, with a horizontal width of 15 µm and a vertical length of 150 µm.The first picture, where P = 0 . P max , shows a linear fluid. Indeed even the regions ofthe pump do not reach the threshold intensity value for the fluid to jump into the uppernonlinear branch of the bistability.When the intensity becomes high enough so that the high density regions jump intothe nonlinear regime, the vortex streets appear ( P = 0 . P max ). The pattern is againclearly symmetric. The instabilities are maintained over a small range of intensities, thengradually vanish from the center, starting from P = 0 . P max . At the maximum laser .2. EXPERIMENTAL REALIZATION P max , the instabilities are gone and only remains the grey shadow of the channel.Figure 5.9: Scan of the total intensity S + P . Density (upper line, logarithmic scale)and phase (lower line) maps for different values of the total intensity S + P . The laserpower increases from left to right until the maximum value P max (0.36 P max , 0.69 P max ,0.71 P max , 0.79 P max and P max , respectively). At low intensity, the whole fluid is linear.When the pump regions jump into the nonlinear regime, the solitonic pattern appears,best defined for P = 0 . P max , then vanishes for higher intensities.This time again, only the symmetric configuration has been observed: no clear tran-sition from one symmetry to another is visible. The experiment was repeated severaltimes with different conditions ( S/P ratio, laser-LP detuning, channel length...) but theantisymmetric instabilities were never found.In order to understand our experimental difficulties, new simulations have been made,to plot a new phase diagram corresponding more accurately to our sample properties.36
CHAPTER 5. SOLITONS IN STATIC FLUID
The main difference between the previous theory and the experiment comes from thedisorder of the sample. The numerical results presented in section 5.1 were done in thequasi-ideal case: a slight disorder is necessary to generate the instabilities, but it waschosen to be small, of 0.01 meV amplitude. This way, the solitonic pattern is not affectedby the disorder position and thus get a regular shape.Experimentally, our sample has a more important disorder amplitude: it was evaluatedaround 0.1 meV. In this case, it plays a role in the shape of the instabilities as somevortex/antivortex can be pinned by the disorder speckle, leading to irregularities in thevortex street pattern. Therefore, a new phase diagram has been realized, plotted in figure5.10The impact of a stronger disorder to the instability shape also makes it more difficultto numerically determine the dominant mode, and therefore the symmetry of the insta-bilities. The new color scale of figure 5.10 phase diagram is thus based on the solitonmass center X c ( y ), numerically easy to define.Indeed, in the symmetric configuration, the soliton mass center stays quite aligned alongthe y axis, while in the antisymmetric case, it follows a sinusoidal trajectory. Its standarddeviation (cid:10) ∆ (cid:11) = D(cid:0) X c ( y ) − h X c ( y ) i y (cid:1) E y can therefore give us useful information: thehigher it is, the more antisymmetric the pattern is.Figure 5.10: Phase diagram in the presence of disorder . The color scale illustratesthe standard deviation of the center between the solitons h ∆ i . The different instabilityconfigurations are plotted in the insets. The antisymmetric pattern corresponds to thegreen area, much smaller than previously. The light grey region indicates where thesolutions are not stationary in time, while the dark grey area shows where no solitons arevisible and the channel is entirely filled with polaritons. From [144] .2. EXPERIMENTAL REALIZATION The first dimension of the channel that we have scanned is its length L . Again, theoperation is very easy as it consists in changing the image send on the SLM screen. Theresults are presented in figure 5.11.This set of experiment has been realized for a channel with a horizontal width of 15µm. It expands vertically along the images, from 40 µm length up to 240 µm in the lastimage on the right.When the channel is short, the instabilities are very clear, as well as their symmetricpattern. Indeed, the dimension of the channel imposes boundary conditions on theirspatial period and position [144], and therefore on the number of vortex-antivortex pairsthat can appear in a channel of finite length. Besides, one can see that in all the images, theinstabilities are better defined at the ends of the channel, where the boundary conditionsare strong. In the center, however, they have a relative translational symmetry along thechannel axis and therefore get a larger flexibility.A particularly interesting image in this set of measurements is the last one on the right.Here, the top end of the channel is connected to the low density region, while the lowerone is still surrounded by high density fluid. The instability pattern is therefore different.On the bottom, as before, the presence of the horizontal wall breaks the symmetry along y and pins the instabilities: the vortex streets are very clear in the phase map, indicatedthere by the red and blue circles. On the top end however, the connection to the lowdensity region maintains the translational symmetry, allowing the instabilities to slightlymove along the y axis. In this time integrated image, it results in blurred parallel solitons.38 CHAPTER 5. SOLITONS IN STATIC FLUID
Figure 5.11:
Scan of the length L of the channel . Density (upper line) and phase(bottom line) maps of the system for different lengths L of the channel. The lengthincreases from 40 µm (left) to 240 µm (right). The symmetric instabilities are well definedfor short channels, while they blurry in the center of the long ones. In the last picture onthe right, the top end of the channel has reached the low density regions.The horizontal dimension of the channel has also been tuned, as presented in figure5.12. This time, the channel is sent over the whole beam, so is connected to the lowdensity region on both ends. Three different widths have been studied: 25 µm, 40 µmand 55 µm. The dimensions of the channel are represented by the red dashed rectangles. .2. EXPERIMENTAL REALIZATION Scan of the width W of the channel . Density and phase maps of thesystem for different width of the channel (25 µm, 40 µm and 55 µm, respectively). Thesolitons fill the entire channel and therefore increase their number. On a., the two highdensity regions have the same phase: the soliton number is even; while on b, the highdensity regions are in opposition of phase, which leads to an odd number of solitons.40 CHAPTER 5. SOLITONS IN STATIC FLUID
For each channel width, we see that the instability patterns fill all the surface of thechannel: the soliton number increases with the width. In a. for instance, we observe2, then 4 and eventually up to an array of 6 solitons. We only have an even number ofsolitons, as the wavefront of the pump has a flat phase and the total phase jump must beconserved.Now we can also combine the wavefront shaping property of the SLM with the intensitymodulation: on panel 5.12.b, the two high density regions have a phase shift of π betweenthem. Therefore, the number of solitons inside the channel has to be odd, so that thetotal phase jump in conserved: we can see one, three and five solitons depending on thetotal width. Those results are analog to the one presented in [53] in the unidimensionalcase, with in our case a translational symmetry along y .This one is responsible for the inhibition of the modulational instability. However, forwide channels, we can see the breaking of the solitons: the confinement induced by thehigh density region is not strong enough to sustain stable solitons, hence the developmentof instabilities.Now that we have studied the single channel configurations, we can study the effectof a modification of the shape of the corridor, as a starting point of a maze realization.Experimentally however, the optical impression is limited by the cavity bandwith: inthe momentum space, the bandwidth of the cavity filters out the signal with too highcomponents in k-space. Therefore, if the shape of the incoming beam is too large inmomentum space, the high spatial frequency components will not enter the cavity andthe signal will be blurred out on the fluid, hence the difficulty to imprint complicatedpatterns.This is why, as a first observation, we considered a cross, where the vertical corridorpossess dead-ends surrounded by high density fluid, but where the horizontal channel isopen and connected to the low density region. The results are presented in figure 5.13, fora channel width of 25 µm and for different intensity ratios, from S = 0 . P to S = 0 . P .The red dashed line indicates the shape and position of the corridors.The instabilities fill the corridors for low support intensities. However, looking at thephase maps, we can see that their best definition is around S = 0 . P , where the phasejumps are really clear. The shape of the instabilities also confirms what was alreadyobserved: the dead end channels pin the instabilities and the symmetric pattern is visible,while they appear parallel on the open horizontal channel.Increasing again the support intensity, we reach the value S = 0 . P , where the dead-end channel has exceeded the critical support intensity S C , contrary to the open-end one.Therefore, the solitons have been pushed away from the vertical corridor, but remains inthe horizontal one, as we can clearly see in the phase map. A few instabilities can howeverstill be guessed in the upper part of the vertical channel due to disorder of the samplewhich locally pin them. These results demonstrate that the proof of the resolution of thissimplified maze is experimentally achieved. .2. EXPERIMENTAL REALIZATION Cross shaped channel . Scan of the support intensity of a cross shapedchannel, from S = 0 . P to S = 0 . P . The vertical corridor have dead-ends, surroundedby high density walls, while the horizontal corridor is connected to the low density region.For S = 0 . P , the solitons still fill the horizontal channel while they have vanished in thevertical one: the maze is solved.Those results are very promising on the possibility of an actual maze solving withpolaritonic devices. For now, the experimental limitation stays the implementation ofthe maze shape, which in our all optical configuration and with our kind of sample cannot be develop to complicated shapes. It would indeed require better quality sample andhigher laser power to implement a larger scale maze. Other configurations could also beconsidered, as the etching of the maze directly on the sample, or the use of shaped mask. onclusion The starting point of this work came from the study of the polariton optical bistabilityand its related properties [25]. The ability to release the phase constraint imposed bythe quasi-resonant pump particularly caught our attention, as well as the proposal of theseed-support configuration, leading to an extended high-density bistable fluid.The experimental implementation showed to be in excellent agreement with the an-nounced theory [26, 27]. Indeed, topological excitations, such as vortex-antivortex pairsand dark solitons, were spontaneously generated in the wake of a structural defect, withinthe high density quasi-resonantly pumped fluid. Therefore, the system did not only getrid of the pump phase constraint, but were also able to sustain those topological excita-tions over hundreds of microns, greatly enhancing their propagation length compared tothe previous observations [22].This experiment also revealed a very unexpected behaviour of the solitons. Indeed,the presence of the driving field imposed by the bistable pump prevents the dark solitonsto propagate away from each other, as they usually do in an undriven system [22, 138].In our configuration, they tend to align to each other and propagate parallel as a bindingmechanism.Following those results, we tried to get rid of the last constraint we had on the solitongeneration, namely their spontaneous formation in the wake of structural defect. To doso, we decided to artificially imprint them on the fluid and observe their further freepropagation. This was implemented by wisely designing the excitation beam using aSpatial Light Modulator, and we managed to generate on demand dark solitons on apolariton fluid [28]. Once again, due to the presence of the driving field, we observed thisbinding between the solitons, propagating parallel as a dark soliton molecule.The flexibility of our method allowed us to explore the system parameters in orderto better understand this surprising behaviour. After a careful study, we realized it isdirectly connected to the driven-dissipative nature of our system, and that the separationdistance between the solitons is governed by the decay rate of the microcavity. The easydesign and implementation of this technique opens the way to deeper studies of quantumturbulence phenomena.It was done in the last part of this thesis work, where the previous technique wasdeveloped to implement intensity modulations on the polariton fluid. This time, the ideawas to generate solitonic structures in guided low-density channel in a static polariton14344
CHAPTER 5. SOLITONS IN STATIC FLUID fluid [144]. The absence of flow destabilizes the solitons, which break into vortex streetsdue to transverse snake instabilities. Those instabilities present different symmetries thatwe were able to study, as well as their different behaviour depending on the dead- oropen-end nature of the channel. This property was then applied to offer an all-opticalfast analog maze solving algorithm, even though some technical limitations still preventsus to implement too complicated shapes.This thesis work has been focused on the study of topological excitations, namely vor-tices and dark solitons. This topic is an important subject of research in the field ofexciton-polaritons and more generally quantum fluids, and we have concentrated our in-vestigations on their control. By implementing novel techniques to generate and sustainthem at will, we were able to greatly facilitate their control and the one of their environ-ment parameters. It offers many possibilities of developments, going from the study ofinteraction in multiple solitons pattern or their robustness against modulational instabil-ities, the improvement of ultrafast all-optical maze solving algorithms and more generallythe controlled and quantitative study of quantum turbulence in driven-dissipative fluidsof light. ommunications
Conferences • ICSCE10 , Melbourne, Australia, January 2020, oral presentation•
TeraMetaNano-4 , Lecce, Italy, May 2019, oral presentation•
Optique Toulouse 2018 , Toulouse, France, July 2018, poster presentation•
QFLM , Les Houches, France, June 2018, poster presentation
Publications • Giovanni Lerario, Sergei V. Koniakhin,
Anne Maître , Dmitry Solnyshkov, Alessan-dro Zilio, Quentin Glorieux, Guillaume Malpuech, Elisabeth Giacobino, Simon Pi-geon, and Alberto Bramati. Parallel dark soliton pair in a bistable two-dimensionalexciton-polariton superfluid.
Physical Review Research , , 042041, 2020.• Ferdinand Claude, Sergei V Koniakhin, Anne Maître , Simon Pigeon, GiovanniLerario, Daniil D Stupin, Quentin Glorieux, Elisabeth Giacobino, Dmitry Sol-nyshkov, Guillaume Malpuech, and Alberto Bramati. Taming the snake instabilitiesin a polariton superfluid.
Optica , 7:1660-1665, 2020.•
Anne Maître , Giovanni Lerario, Adrià Medeiros, Ferdinand Claude, Quentin Glo-rieux, Elisabeth Giacobino, Simon Pigeon, and Alberto Bramati. Dark-solitonmolecules in an exciton-polariton superfluid.
Physical Review X , , 041028, 2020.• Thomas Boulier, Maxime J. Jacquet, Anne Maître , Giovanni Lerario, FerdinandClaude, Simon Pigeon, Quentin Glorieux, Alberto Amo, Jacqueline Bloch, AlbertoBramati, and Elisabeth Giacobino. Microcavity Polaritons for Quantum Simulation.
Advanced Quantum Technologies , page 2000052, 2020.• M. J. Jacquet, T. Boulier, F. Claude,
A. Maître , E. Cancellieri, C. Adrados,A. Amo, S. Pigeon, Q. Glorieux, A. Bramati, and E. Giacobino. Polariton fluidsfor analogue gravity physics.
Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences , (2177), 2020.14546 CHAPTER 5. SOLITONS IN STATIC FLUID • Giovanni Lerario,
Anne Maître , Rajiv Boddeda, Quentin Glorieux, Elisabeth Gi-acobino, Simon Pigeon, and Alberto Bramati. Vortex-stream generation and en-hanced propagation in a polariton superfluid.
Physical Review Research , , 023049,2020.• S. V. Koniakhin, O. Bleu, D. D. Stupin, S. Pigeon, A. Maitre , F. Claude, G.Lerario, Q. Glorieux, A. Bramati, D. Solnyshkov, and G. Malpuech. StationaryQuantum Vortex Street in a Driven-Dissipative Quantum Fluid of Light.
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