Rapport de Stage 2012-2013 Force de traînée dans un condensat

Rapport de Stage 2012-2013Master 2 Sciences de la Matire Physique

cole Normale Suprieure de Lyonpar Grgory GREDAT

Force de trane dans un condensat depolaritons spinoriel unidimensionnel en

pompage rsonnant

m

Supervis parNicolas PAVLOFF

& Pierre-lie LARR

Laboratoire de Physique Thorique et Modles StatistiquesUniversit Paris Sud - Btiment 100

15 rue Georges Clmenceau91405 Orsay CEDEX, France

http://lptms.u-psud.fr/nicolas_pavloff/http://lptms.u-psud.fr/pierre-elie_larre/http://lptms.u-psud.fr/fr

Drag in a Resonantly Driven Two-ComponentPolariton Condensate in 1D

RsumLa dynamique dun fluide de polaritons en pompage rsonnant gnr dans un fil quantique

quasiunidimensionel est tudie au moyen dune quation de GrossPitaevskii modifie de faon rendre compte du degr de libert spinoriel ainsi que des processus dissipatifs au sein du condensat.Nous mettons en oeuvre la thorie de la rponse linaire pour, dans lapproximation des faiblesperturbations, examiner les modulations dondes engendres par un obstacle plac dans lcoulementdu fluide de polariton. Un soin particulier est port la description de la force de trane exercesur cet obstacle localis. La pertinence du critre de Landau dans ce condensat dissipatif deuxcomposantes est discute. Nous identifions deux vitesses critiques correspondant lmission dunrayonnement de type erenkov dune onde de densit (bien amortie) et dune onde densit depolarisation (comparativement faiblement amortie). La transition dun rgime de trane de typevisqueuse un rgime de rsistance donde, ainsi que les aspects spcifiques aux fluctuations depolarisation sont mis en exergue.

Mots Clefs: Condensation de Bose-Einstein Superfluidit Milieux non linaires Trans-port dissipatif dans les fluides quantiques Excitonpolariton en microcavit semiconductrice Thorie de la rponse linaire

AbstractWe study the hydrodynamics of a resonantlypumped polariton condensate in a quasi1D quan-

tum wire taking into account the spin degree of freedom and dissipation processes via a modifiedGrossPitaevskii equation. Within a weakperturbation approximation, a linear response theory isimplemented to investigate the generation of wave patterns by a polariton quantum flow incidenton a localized obstacle and the drag force exerted onto this obstacle as well. In this dissipativetwocomponent condensate, the relevance of Landaus criterion is clarified. Two erenkovlikecritical velocities are identified corresponding to the opening of different channels of radiation:one of (damped) density fluctuations and another of (weakly damped) polarization fluctuations.The transition from a viscous drag to a regime of wave resistance and the specific features of thefluctuations of polarization are pointed out.

Key Words: Bose-Einstein condensation Superfluidity Non linear media Dissipativetransport in quantum fluids Semiconductor microcavity excitonpolariton Linear response the-ory

X

ContentsIntroduction 1

I Polariton Bose-Einstein condensation overview 2I.1 Condensation in semiconductor microcavities : a recent challenge . . . . . . . . . . . 2I.2 Superfluidity and Landaus criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I.3 1D Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

II General framework for resonantly driven two-component polariton condensatesin 1D 5II.1 Phenomenological modifications of Gross-Pitaevskii equation . . . . . . . . . . . . . 5II.2 Linear response of the polariton fluid scattering against a localized obstacle . . . . . 6

II.2.1 Linearized theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6II.2.2 Drag force exerted by the fluid onto the obstacle . . . . . . . . . . . . . . . . 9

III Detailed study of the model 12III.1 Collective polariton fluid dynamics when the magnetic field is off . . . . . . . . . . . 12

III.1.1 Resolution at leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12III.1.2 Several regimes for the collective excitations . . . . . . . . . . . . . . . . . . . 12III.1.3 Kerr-type and parametric instabilities . . . . . . . . . . . . . . . . . . . . . . 13III.1.4 Density profile and drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

III.2 Polarization and dynamics when the magnetic field is turned on . . . . . . . . . . . . 19III.2.1 Elliptical polarization of the background . . . . . . . . . . . . . . . . . . . . . 19III.2.2 A richer spectrum of excitations . . . . . . . . . . . . . . . . . . . . . . . . . 19III.2.3 Density and polarization fluctuations generated by a weak impurity . . . . . 20

Conclusion 24

Acknowledgments 24

Appendix A: Heuristic argument for Landaus criterionAppendix and wave resistance analogy i

Appendix B: Critical velocity and drag divergence when tot < 0Appendix without magnetic field iii

Appendix C: Complement for the background polariton fieldAppendix and the response functions poles in presence ofAppendix a magnetic field v

References vii

List of Figures1 Exciton in a quantum well of a semiconductor layer . . . . . . . . . . . . . . . . . . 22 Polariton creation and farfield imaging method in a semiconductor microcavity . . 23 Polaritons dispersion relation sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Experimental polariton relaxation to the ground state . . . . . . . . . . . . . . . . . 35 Pseudospin S representation on the Poincar sphere. . . . . . . . . . . . . . . . . . . 76 Collective excitation spectra of the polariton condensate without magnetic field. . . 137 Internal versus incident intensity stability curve. . . . . . . . . . . . . . . . . . . . . 148 Density patterns generated by an obstacle moving in a twocomponent polariton

fluid without magnetic field in the resonant case tot = 0 . . . . . . . . . . . . . . . 179 Drag force exerted by the resonantly driven twocomponent fluid onto the obstacle

versus its relative velocity without magnetic field . . . . . . . . . . . . . . . . . . . . 1810 Density and polarization modulation patterns induced by a impurity in the reso-

nant case for = 0.1circ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111 Drag force exerted by the resonantly driven twocomponent fluid onto the obstacle

versus its relative velocity for several damping parameters and magnetic fields . . . . 2212 Graphical representation of Landaus criterion and wave emission above the critical

velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii13 Position of the four poles in the complex q plane without magnetic field. . . . . . . . iv14 Drag force versus the relative velocity of a peak obstacle for tot 0 in the limit

0 at ~ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv15 Background polariton density and polarization versus incident pumping intensity for

several magnetic field magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v16 Position of the five response functions poles in the complex q plane at resonance for

a nonzero magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi17 Position of the eight response functions poles in the complex q plane for a nonzero

magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Rapport de Stage Master 2 ENS Lyon

Introduction

Attracted to low temperatures, H. Kamerlingh Onnes liquefies Helium in 1908. Be it theintuition of finding exotic physics or just a thermodynamics challenge, his cryogenic feat came asthe anchoring point for unexpected discoveries: superconductivity and superfluidity.

The first evidence of nonclassicality below 2.2K of liquid helium hydrodynamics was evidencedby J.F Allen and A.D. Misener in 1938 [1]. In the very same journal issue, P. Kapitza introducedthe word "superfluid" in connection with the already known superconductivity [2]; however he didnot elaborate on the relation between these two phenomena.

Besides, motivated by W.H. Keesom and coworkers superfluid Helium transition studies (whocalled it a point by reference to the Greek letter shaped specific heat peak displayed at thetransition), F. London was the first to suggest [3] that superfluidity consists in a macroscopicliquid matter wave as a consequence of the macroscopic groundstate occupancy by bosons at lowtemperatures defining BoseEinstein condensation, a concept initiated by A. Einstein in 1924. Ithappened however to be very difficult to prove.

As a result of theoretical efforts, the intimate links between superfluidity and condensation areprogressively revealed. Among others, we can emphasize the contributions of N.N. Bogoliubov in1947 [4], O. Penrose and L. Onsager in 1956 [5] and also E.P. Gross and L. P. Pitaevskii in 1961[6, 7]. When Bose-Einstein condensation was discovered in cold alkali vapors in 1995 [8, 9], theexperimental evidence was immediately very clear. Since then, the opened field of ultracold dilutegases has continually enriched our knowledge on interacting manybody systems and the funda-mental processes responsible for condensation and superfluidity as well.

More than eighty years after A. Einsteins translation into German of S. Boses pioneer work onquantum statistics for photons [10], paving the way for BoseEinstein condensation but also laserphysics, both of these emerged fields have been gathered to produce a condensation of compositelightmatter bosons known as excitonpolaritons in semiconductor microcavities.

In the first section, an overview of the Bose-Einstein condensation experienced by nonequilibriumquantum gases formed by polaritons in microcavities is given. We also present Landaus super-fluidity criterion and the 1D mean field GrossPitaevskii equation accounting for condensed gasdynamics. Together with the linearized theory developed in the second section, they represent thefirst staple blocks of our study.

A great interest directed towards the specific features associated with the spin of the exciton-polaritons has been aroused, in the hydrodynamic context, in [11], where they have been revealed bythe observation of the optical spin Hall effect. Some effects due to the spin have also been pointedout in [12] where they report the observation of half vortices, or in [13] describing the nucleation ofhalf solitons . This is the reason why we focused on the description of a 1D twocomponent polaritoncondensate. In section II, a phenomenological modification of GrossPitaevskii equation aimedto take into account the dissipative nature of polariton condensates but also the experimentallycommonly used resonant pumping is first assumed. Then, within the linear response functiontheory, a theoretical framework is established for weakly perturbed polariton fluids.

As a third section, the collective excitations of the twocomponent polariton fluid with andwithout magnetic field, as well as the density / polarization fluctuations and the drag induced bya moving weak impurity constituting the spearhead of our study are investigated in details.

Grgory GREDAT Page 1/ 24 LPTMS


I Polariton Bose-Einstein condensation overview

I.1 Condensation in semiconductor microcavities : a recent challenge

Quite recently, in 2006, two research teams (in Nel Institute, Grenoble and cole Polytechniquede Lausanne) have experimentally evidenced polariton condensation in a semiconductor micro-cavity [14] (see figure 4). They observed that the photons injected, by the mean of a pumpinglaser, in an optical cavity containing some semiconductor layers (as sketched in figure 2) cou-ple to the excitons (defined in figure 1) created within these layers thus generating a polaritongas likely to relax towards a 0 momentum groundstate above a certain excitation power.

Energy

Quantumwell

F

e

h+

Exciton

Fig. 1: Exciton in a quantum well of a semiconductor layer. As an outcomeof the strong Coulomb coupling between an electron and its hole within theFermi sea, an exciton is a bosonic quasiparticle.

Bragg

mirro

rs

/4

x

y

Exciton

Quantumwell

5 nm

Photon

Bragg

mirro

rs

Semiconductor microcavity2

1m

z

q

qq

Fig. 2: Polariton creation and farfield imagingmethod in a semiconductor microcavity. The photonstrapped in the optical cavity couple with the excitons.The resulting bosonic quasiparticle is a polariton.Thanks to losses, since the mirrors are not perfect re-flectors, we can probe the polariton gas with the lightthat goes away from the cavity (whose wavevector isq in the sketch). This farfield method provides theadvantage of being nonintrusive.

Inside the cavity, when the semiconductorlayer absorbs and emits at the photon wave-length, the energy keeps oscillating betweenthe quantum wells excitons and the cavityphotons: as the predominant process, thisoscillation quantified by a Rabi frequencyR, mixing light and matter, originates thenew quasi-particles called polaritons. Sincephotons and excitons are bosons, polaritonsare bosons. Therefore, the macroscopicoccupancy of the groundstate as well asthe time and longrange spatial coherenceproperties they highlighted is nothing but acompelling manifestation of Bose-Einsteincondensation in a microcavity. The strongcoupling engendering polaritons producesa twobranch dispersion relation sketchedin figure 3. For small enough momenta,the lower branch, providing the groundstate, can be approximated by a parabolafrom which we define the polariton effec-tive mass. Due to the light component, po-laritons are very lightweight with typicallym . 104me .

Cavity photonsExcitons

Upper branch

Lower branch

q [a. u.]

[a.u.]

Fig. 3: Polaritons dispersion relation sketch. The upper and lower branchesof the polariton dispersion (in black) come from a mix between the cavityphotons (in orange) and the excitons (in blue) spectra.



Although they present the coherence properties of traditional Bose-Einstein condensates (suchas ultracold atomic vapors and Helium systems), polariton condensates in semiconductor microcavi-ties are intrinsically dissipative outofequilibrium systems owing to the polariton finite lifetime dueto the nonperfectly reflecting cavity mirrors. However, we can take advantage of this limitation:indeed, by optical detection of the light emitted by the gas when photons and excitons decouple,we can directly probe the intern properties of the polariton fluid. Figures 2 and 4 illustrate thisfarfield method and its experimental convenience.

Fig. 4: Experimental polariton relaxationto the ground state (from [14]). (Energy ,momentum) images displaying populationdistribution in the lower polariton branchmeasured at T = 5 K, for three laser pump-ing powers, thanks to the farfield imagingmethod. Above the condensation thresh-old power Pthr, the ground state becomesmassively occupied.

I.2 Superfluidity and Landaus criterion

One of the most striking and appealing property of various BoseEinstein condensed fluids is theirability to move with respect to an obstacle without dissipating energy: it defines superfluidity, anew state of matter. That is the reason why experimental studies about the effect of introducingan external obstacle into the flow of a quantum fluid have been on a roll since the first 4He and3He systems (see [15, 16] for instance), through ultracold atomic vapors [17, 18, 19, 20], until therecent polariton condensates [21, 22, 23, 24]. Even though a subtle link relates BoseEinstein con-densation and superfluidity, these phenomena are not identical. To avoid an abuse of language, onemay speak rather in terms of frictionless flow in the case of polariton fluids since they are intrinsi-cally dissipative systems owing to their finite lifetimes; the term of superfluidity being reserved forthe spectacular manifestation of macroscopic coherence of an interacting many-body conservativesystem.

In the 40s, L.D. Landau, willing to explain Helium superfluidity, made a major breakthroughby introducing elementary excitations as quantized collective modes which led him to the crucialprediction of a critical velocity, the sound velocity in T = 0 atomic BoseEinstein condensates,originating the breakdown of superfluidity [25, 26]. Indeed, according to his criterion (developedin Appendix A), below this critical velocity, no excitation is emitted into the quantum fluid, whatinduces no drag; yet, above it, a erenkov radiation of linear waves is emitted ahead of the obstacleand spreads through the fluid: the obstacle diffuses momentum and, as a consequence, experiences anonzero drag force. Landaus criterion is intrinsically perturbative. In 1955, R. Feynman explainedin [27] that the emergence of vortices (the circulation of their velocities being quantized as multiplesof h /m) should lead to a much lower critical velocity than Landaus one. Indeed quantized vorticesappear in quantum gases at the breakdown of superfluidity. Feynmans suggestion was relevant, itexplained the discrepancy, for 4He systems, between Landaus critical velocity prediction of 60m/sand the ten times smaller experimental measurements [28].


I.3 1D GROSS-PITAEVSKII EQUATION

We are going to question the existence of such critical velocities separating frictionless flowsand dissipative ones in 1D resonantly driven spinorial polariton condensates using a perturbativeapproach, in the spirit of Landaus criterion.

I.3 1D Gross-Pitaevskii equation

Harmonic trapping provides an experimental achievable mean to deal with a Bose-Einstein con-densed gas whose dynamic is essentially one-dimensional. In [29, 30] a theoretical framework isgiven to establish the equation governing the evolution of the condensate (position and time depen-dent) wave function (r, t) in an harmonic potential V(r) = 12 m

2 r

2 and an external potential

Uext(x). r designs the transverse direction, while x stands for the longitudinal one. Assuming aBornOppenheimer approximation (or adiabatic hypothesis) consisting in decoupling longitudinaland transverse degrees of freedom to describe the fact that transverse scale of variation of the cloudprofile is much smaller than the longitudinal one, we can write:

(r, t) = (x, t) (r; ) (1.1a)

and (x, t) =

d2r||2 = |(x, t)|2 , (1.1b)

where is the longitudinal density of the condensate and the equilibrium wave function for thetransverse motion, normalized to unity. Denoting asc the 3D swave scattering length of interatomic contact repulsion, we can derive the evolution equation through a variational principle.Extremizing the ||4 action:

S = ~2

d3r dt (t t)

dt E [] , (1.2)

where the energy functional E [] reads:

E [] =

d3r[~2

2m(||2 + 4 asc ||4

)+ (Uext + V) ||2

]; (1.3)

one comes up, in the low density limit asc 1 for the gas to be diluted enough so that the bosonsweakly interact between each others, with a meanfield GrossPitaevskii equation for the wavefunction for the longitudinal motion imposing the 1D dynamics. GrossPitaevskii equation is anonlinear Schrdinger equation used to describe motion of a BoseEinstein condensate. In ourapproximations, it reads:

~ t = ~2

2mxx +(Uext(x) + ||2

) , (1.4)

where the nonlinear coefficient is found to be:

= 2 ~ asc . (1.5)

Phenomenological modifications of this equation will be the starting point of our study ofpolariton condensates.



II General framework for resonantly driven two-component po-lariton condensates in 1D

II.1 Phenomenological modifications of Gross-Pitaevskii equation

We study a polariton condensate created in a 1D cigar shaped semiconductor micro-cavity. We willfocus on its properties in presence of a moving obstacle (in the rest frame of the polariton fluid)that is, for example, a localized impurity, a point like defect or a lithographed potential. Here isgiven the general features of our study.

Polarization As we have seen before, a polariton living in the lower branch of the dispersionrelation is an outcome of the strong coupling of a cavity photon and a quantum well heavy exciton.A heavy exciton is composed of a 1/2-spin electron and a heavy hole state which has a L = 1 orbitalangular momentum and, as a consequence, a J = 3/2 total angular momentum. Thus, there existsdifferent kinds of heavy excitons: the dark ones with JXz = 21 and the bright ones with JXz = 1.Contrary to the latter ones, as suggested by their name and as a result of the spin conservationduring this process, the dark excitons cant be optically excited. Indeed, the helicity of a photonis either +1 or 1. A given spin of the exciton leads to a single polarization state of the emittedlight since, while recombining, the exciton transfers its whole momentum to the emitted photon.We can therefore speak in terms of the pseudospin S representative of the polarization state of theemitted light and depicted by a Stokes vector on the Poincar Sphere (see pages 110-111 of [31]for a detailed discussion). For example, as shown in figure 5, when S x (respectively S x)the emitted light has a horizontal p (respectively vertical s) linear polarization; while S y (resp.Sy) depicts a /4 linear polarization. Its also worth noticing that S z (respectively Sz)corresponds to a right + (respectively left ) circular polarized photon.

Resonant pumping Since polaritons have a finite lifetime, a pumping-laser source is needed forconstant replenishment of the condensate in a scale of tens of picoseconds. Recently enough, thesuperfluid properties as well as the dynamics of a resonantly pumped polariton condensate havebeen much touted both experimentally and theoretically [32, 33]. This pumping scheme raisedinterest among the cold atoms community because, the polariton field oscillation frequency beinglocked to the one of the external tunable pump laser, our system is intrinsically driven out ofthermal equilibrium, implying that there is no equation of state relating the chemical potential tothe polariton density. Thus, the coherent excitation experienced by the structure offers, as we aregoing to see, much richer collective excitation spectra which have no counterpart in usual systemsclose to thermal equilibrium. Moreover, the polariton field phase being fixed, it seems that it ismore difficult for vortices and solitons to appear. This opens the possibility, with a defect locatedjust outside the pump spot, to observe the nucleation of vortices, vortex-antivortex pairs, arrays ofvortices or solitons [23, 34, 35].

Magnetic field Applying a magnetic field B = Bz on the system has essentially three con-sequences:2 a Landau quantification of the excitonic states, a modification of the electron/holebinding energy and a Zeeman splitting of the excitonic levels of energy. Hereupon, we will beinterested in only the Zeeman effect.

1We choose z as the quantum growth axis.2Of course, only the excitonic part is sensitive to the field.


II.2 LINEAR RESPONSE OF THE POLARITON FLUID SCATTERING AGAINST A LOCALIZED OBSTACLE

Modified (GP) equation Within this framework, the polariton condensate is well described bya meanfield equation; more precisely, by a modified GrossPitaevskii equation of the spinorialorder parameter (x, t) = (+(x, t), (x, t)):

~ t(x, t) = ~2

2m xx(x, t) + (Uext(x, t) ~ )(x, t)

+(1 |(x, t)|2 + 2 |(x, t)|2

)(x, t) + F(x, t) ,

(GP)

where = stands for the projection on the z axis.In this equation, F(x, t) is the source term accounting for the resonant pumping. It reads:

F(x, t) = fp exp [ (kp x + p t)] , (2.1)

imposing the background fluid velocity vp =~kpm

. We then assume that the wavevector kp isdirected along the x-direction of the 1D condensate: kp = kp x. The amplitude fp of the sourcecharacterizes the intensity of the pump; it can be chosen real without loss of generality. Lets precisethat m is the effective mass of a polariton at the bottom of the lower branch of the dispersionrelation sketched in figure 3. The linear term ~ t = takes into account the losses due tothe polariton finite lifetime ( 1/) in a phenomenological way. The Zeeman term reads ~ gs B B S where gs is the polariton gyromagnetic factor and B the Bohr magneton. As forthe cubic terms of (GP) equation, the 1 = (+,+) = (,) and 2 = (+,) = (,+) couplingsdescribe, within a mean field approach, the contact interactions between the different components ofthe polariton field. For instance, (+,+) accounts for the contact interaction between two polaritonswith parallel spins +. Typically, we consider a global repulsive interaction between polaritons, asobserved experimentally from the emission blueshift during the polariton condensation in [36], bytaking 1 > 0 and 2 < 0 with |2| < 1.

For the investigation of the response of the system to a weak localized potential3, we implementa perturbative analysis.

II.2 Linear response of the polariton fluid scattering against a localized obstacle

II.2.1 Linearized theory

Since the pump is coherently driving the system, it is natural, in a perturbative approach, to lookfor "stationary" solutions of the form:

(x, t) = (0 + (x, t)) e(kpxpt) , (2.2)

where |(x, t)|2 |0 |2 .Substituting (2.2) in (GP) equation, at leading order, which corresponds to a uniform (i.e.

translation invariant) background, we get:

0 = 0 + fp , (2.3)

where we define the pump detuning for a given component as:

={~p = ~p

~2 k2p2m

}+ ~

[1 |0 |2 + 2|0 |2

]. (2.4)

3By localized, we mean a potential whose spatial extend is finite: Uext(x) |x|

0.



On the other hand, at next order we get:

~ t(x, t) =[ ~

2

2m xx ~ vp x + 1 |0 |2

](x, t)

+1 0 2 (x, t) + 2[0

0

(x, t) + 0 0 (x, t)

]+ Uext(x, t)0 .

(2.5)

Leading order parametrization We define the polariton density for each component of thebackground field as: 0 = |0 |2. It follows that the total density is: 0 = +0 +

0 . It is very

useful to parametrize our spinorial order parameter 0 = (+0 (x, t), 0 (x, t)) as:

~S

p

s0

z+

x

y

+/4

/4

Fig. 5: Pseudospin S representation on the Poincarsphere. The yellow arrows represent the possible lin-ear (p, s, /4) and circular (+, ) polarizationsof the emitted light.

0 =0

[cos 2 e

2

sin 2 e2

]e . (2.6)

Indeed, using the pseudospin S repre-sentation of figure 5, noting [0, ] thepolar angle and [0, 2[ the azimuthalangle (w.r.t. the z quantization axis) weobtain:

Sx = 0 sin cosSy = 0 sin sinSz = 0 cos ,

from which we draw that:Sx = +0

0 + +0

0

Sy = +0 0 +0

0

Sz = +0 0 .

Then, we can define a totaldetuning parameter :

2tot + + = 2 ~p (1 + 2) 0 (2.7)

and also a relative detuning parameter :

2rel + = 2 ~ (1 2)Sz . (2.8)

These detuning parameters, characteristic of the resonant pumping, will be responsible for therichness of the collective excitations. They have a crucial role to play, in particular when wewill determine the stability of the solutions of equation (2.3), because, to some extent, they comedirectly from the fact that the pump drives the system out of the thermodynamic equilibriumby imposing its energy. Nevertheless, a peculiar attention will be also payed to the nondetunedresonant case where = 0.

Resonant case Taking = 0 implies that, on the one hand and from the total detuning param-eter definition, the total density is imposed by the pump: 0 = 2 ~p / (1 + 2). On the otherhand, from the relative detuning definition, the magnetic field imposes an elliptical polarization:Sz = 2 ~ / (1 2). But, in equation (2.3), we see that having an elliptical polarization is not



possible since the resonant case leads also to: +0 = 0 = fp / . It makes sense only without

magnetic field. However, if the external pump were spin dependent i.e. fp fp , there would beno problem: each component of the source would impose a component of the background polaritonfield. Hence, when we will talk about the resonant case in presence of a magnetic field, we willtake implicitly a polarized pumpinglaser beam. Lets eventually remark that, in this case, when = circ p

1 21 + 2

, the background field acquires a right (resp. left) circular polarization.

Polariton field fluctuations above the background In this perturbatively weak defects pic-ture, we consider equation (2.5) and its conjugate. We can build the following system of equations:

~ t

+(x, t)+

(x, t)(x, t)

(x, t)

= L+(x, t)+

(x, t)(x, t)

(x, t)

+ Uext(x, t)

+0+0

00

. (2.9)Using Fourier transforms, this system can be rewritten as:

L

u+(q, )v+(q, )u(q, )v(q, )

= Uext(q, )+0+0

00

, (2.10)with u

(q, )v(q, )Uext(q, )

= R2

dx dt

(x, t)(x, t)Uext(x, t)

e (qxt) (2.11)and

L =

~2 q2

2m + E 1 +0

22

0+0 2

+0 0

1 +02 ~

2 q2

2m + + E 2 +0

0 2 0

+0

2 +00 2

0

+0

~2 q2

2m E 1 0

2

2 0+0 2 +0

0 1 0

2 ~2 q2

2m + E

, (2.12)

notingE = E(q, ) = ~ + ~ vp q and = 1 |0 |2 .

The spectrum of collective excitations is obtained by diagonalizing the operator L+ ~ 1. Wecome up with a quartic equation for E(q, ) that leads to four branches for the dispersion relation.It is worth noting that, in our outofequilibrium system, the spectrum of excitation is found tobe complex, what could question the validity of Landaus criterion.

The analytic solutions are then easily derived in the momentum space by inverting system(2.10). Thanks to Cramers rule we obtain:[

u+(q, ), v+(q, ), u(q, ), v(q, )]

= 1det L

[det L1, det L2, det L3, det L4

], (2.13)

where

Li=14 is L with the ith column being Uext(q, )[+0 ,

+0, 0 ,

0]

.



II.2.2 Drag force exerted by the fluid onto the obstacle

Density fluctuations We are now interested in the density fluctuations within the linearresponse theory above the uniform background provided by 0 . Looking at the density profilearound a localized obstacle is a first step towards the study of our polariton condensate fluidity.As the local densities of each component are +(x, t) = |+(x, t)|2 and (x, t) = |(x, t)|2, thetotal density of the polaritons is (x, t) = +(x, t) + (x, t). We have therefore to compute thefollowing density fluctuations:

(x, t) = 0 (x, t) + (x, t)0 and (x, t) = +(x, t) + (x, t) .

In other terms, we must compute the integral:

(x, t) =R2

dq d(2)2 e

(qxt)[u+(q, ), v+(q, ), u(q, ), v(q, )

][+0 ,

+0, 0 ,

0]

.

Equation (2.13) enables us to write:

(x, t) =R2

dq d(2)2 Uext(q, )

(q, ) e (qxt) , (2.14)

where the response functions for each component yields:

+(q, ) = +0 det L1 + +0 det L2Uext(q, ) det L

, (q, ) = 0 det L3 + 0 det L4Uext(q, ) det L

. (2.15)

Then, the total response function is simply:

(q, ) = +(q, ) + (q, ) .

Polarization fluctuations Since we are dealing with a twocomponent polariton field, it isof paramount importance to study the polarization effects. To do so, we are interested in thepolarization fluctuations:

z(x, t) = +(x, t) (x, t) .

We can compute this quantity with equation (2.14) and define the polarization response function:

(q, ) = +(q, ) (q, ) .

Drag The most interesting configuration consists in considering either a static obstacle in acondensate that moves at constant velocity, or equivalently an obstacle moving at constantvelocity through a static condensate: the point is to move an obstacle at constant velocity in therest frame of the condensate.4 We assume from now on a -peak potential moving towards x > 0at the velocity V :

U-peakext (x, t) = (x V t) .4There may exist some cold atom systems where the Galilean invariance is broken (C.f. [37]), for example when

there is a periodic potential or a spin-orbit effect but absolutely not in our case!



It leads to the following density and polarization fluctuations:-peak(x, t) = K(x V t) , where K(X) =

R

dq2 (q, q V ) e

qX .

-peakz (x, t) = S(x V t) , where S(X) =R

dq2 (q, q V ) e

qX .

This peculiar a priori non physical obstacle is, in fact, very useful since K(X) and S(X) arethe Green functions which, by convolution, could give the fluctuations for any form of the externalpotential.

Moreover, computing these integrals, we have to evaluate the response functions at = q V .Hence, we see that it amounts to taking E(q, q V ) = ~ {V vp = Vrel} q + in the L definitiongiven by equation (2.12). So, as expected, the results will only depend on the relative velocity Vrel ofthe obstacle in the rest frame of the condensate. However, the leading order polariton field dependson kp 5, the flow velocity plays therefore implicitly another role in providing the background densityabove which we study the fluctuations induced by a weak perturbation.

In order to compute K(X) and S(X), the residue theorem appears to be very convenient: wehave therefore to study in details the behavior of the response functions poles. Then, noting themql, Res(f, ql) the residue of the function f(q) at q = ql, the Heaviside step function and = theimaginary part, one obtains:

K(X) = l

sgn (=[ql]) Res (, ql) (sgn (=[ql])X) e qlX ,

S(X) = l

sgn (=[ql]) Res (, ql) (sgn (=[ql])X) e qlX .(2.16)

The drag force exerted by the defect onto the fluid is the good quantity to probe the fluidityof our resonantly driven polariton condensate. Not only is it a theoretically interesting quantitybut, actually, it is also experimentally measurable. It is crystal clear in classical hydrodynamics[38, 39]. Yet, it is indirectly true for Bose-Einstein condensate experimental realizations throughthe measurement of the energy transfer rate: dE / dt = Fd Vrel [18, 40]. For example, in [17], acritical velocity in a Bose-Einstein condensed gas is experimentally evidenced by the mean of theestimation of the thermal fraction of noncondensed bosons whose increase can be interpreted asa result of a dissipation owed by a nonzero drag force.

Furthermore, the drag force is given by [41] as:

Fd =R

dx (x, t) xUext(x, t) = R

dxUext(x, t) x(x, t) . (2.17)

As a consequence, with a peak obstacle one gets (see, e.g., [42] for an analogous treatmentwith a scalar polariton field in a non-resonant pumping scheme):

F -peakd = 2

2l

sgn (=[ql]) ql Res (, ql) . (2.18)

Finite size obstacle We can consider a finite size defect modeled by a Gaussian potential movingtowards x > 0 at the velocity V :

UG-peakext (x, t) =

exp[ (x V t)2 /2

].

5Indeed, in equation (2.3), the detuning parameters involve the kinetic energy of the flow.



From equation (2.17), one obtains:

F G-peakd = 2

2l

ql Res (, ql) e2 q2l /2

[sgn (=[ql]) + erf

( ql

2

)], (2.19)

where erf stands for the Gauss error function.



III Detailed study of the modelIn this section, we are going to delve into the solutions of the equations previously given in orderto determine the collective excitations of our resonantly driven two-component polariton fluid aswell as the response to a weak defect.

III.1 Collective polariton fluid dynamics when the magnetic field is off

III.1.1 Resolution at leading order

What is clear from equation (2.3), is that, when the magnetic field is off, nothing breaks really thesymmetry between the components because the pump does not distinguish between a polarizationand another. As a consequence, we can immediately put = /2 and = 0, so that +0 =

0 = 0/2

and, since S lives in the (x, y) plane, the system is linearly polarized: Sz = 0. Yet, there existalso mathematical solutions leading to an elliptic polarization Sz 6= 0. In this first part, we are notgoing to focus on these former solutions for pedagogical reasons.

From now on, we work with 0 = 0 for which we have to solve a noncoupled complex cubicequation. On the one hand, the combination (2.3)0 (2.3)

0 gives:

sin = fp

0/2 , (3.1)

while the combination (2.3) (2.3) leads to the cubic equation:

30 +B20 + C0 +D = 0 , (3.2)

where

B = 4 ~p(1 + 2), C = 4

2 + ~22p(1 + 2)2

and D = 8 f2p

(1 + 2)2.

As it is drawn in figure 7, the number of real solutions of equation (3.2) depends on the pumpintensity Ip f2p . Thanks to equation (3.1), 0 is then entirely determined.

III.1.2 Several regimes for the collective excitations

Solving det L = 0 where L is given by (2.12), we obtain the following dispersion relations, for which,as depicted in figure 6, we observe five regimes depending on the (total) pump detuning:6

~1,2(q) = ~ vp q (q) ((q) + 0 (1 + 2))

~3,4(q) = ~ vp q (q) ((q) + 0 (1 2)) ,

(3.3)

with (q) = ~2q2

2m tot .

6The squareroot notation is only formal, its argument can be negative.


III.1 COLLECTIVE POLARITON FLUID DYNAMICS WHEN THE MAGNETIC FIELD IS OFF

-4 -2 0 2 4

-14

-7

0

7

14

-4 -2 0 2 4

0

-0.5

-1

-1.5

-2

-4 -2 0 2 4

-14

-7

0

7

14

-4 -2 0 2 4

0

-0.5

-1

-1.5

-2

-4 -2 0 2 4

-14

-7

0

7

14

-4 -2 0 2 4

-0.5

-1

-1.5

-2

-4 -2 0 2 4

-10

-5

0

5

10

-4 -2 0 2 4

0

-0.5

-1

-1.5

-2

-2.5

-4 -2 0 2 4

-10

-5

0

5

10

-4 -2 0 2 4

0

-0.5

-1

-1.5

-2

-2.5

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

0

-2 -1 0 1 2

-1

-0.5

0

0.5

= [1,2(q)]

q

-2 -1 0 1 2

-0.6

-0.4

-0.2

0

0.2

0.4

= [1,2(q)]

q

-2 -1 0 1 2

-0.4

-0.2

0

0.2

=[

1,2

(q)]

q-2 -1 0 1 2

-0.2

-0.1

-0.15

-0.05

0

=[

1,2

(q)]

q

q0 q0

-2 -1 0 1 2

-0.2

-0.1

-0.15

-0.05

0

=[

1,2

(q)]

q

-2 -1 0 1 2

-0.2

-0.15

-0.1

-0.05

0

=[

1,2

(q)]

q-2 -1 0 1 2

-0.2

-0.1

0

=[

1,2

(q)]

q

Fig. 7: Internal versus incident intensity stability curve. For sake of clarity, we consider a onecomponent condensate (2 = 0). The plain (dashed) red line represents the dynamically (un)stablebranch of the polariton background density. From top to bottom, the insets show how the Kerrtype to parametric instabilities set in as the lobe formed by = [1,2(q)] crosses the 0 line at q = 0,then splits into two lobes that cross once more the 0 line at q0(P0 ). We took = 0.110 and,typically, ~p > /

3. The eigenfrequencies 1,2 are expressed in units of 1 0/~ and q in units

of 1/. When the branch is unstable, the total detuning parameter tot is positive.

Kerr-type instabilities In a Kerr medium7, the propagation equation of the electromagneticfield is very similar to GrossPitaevskii equation, provided we replace the time variable by thepropagation direction of the light. Indeed, the nonlinear term comes from the refraction indexwhich depends on the photon number, comparably to our case where it depends on the number

7A Kerr medium is a nonlinear birefringent optical medium.



of polaritons. This gives rise to a well known bistable behavior of the field intensity with theexcitation intensity. When the instability involves only the q = 0 mode, it will be referred to, byanalogy, as a Kerr-type instability. This gives a part of the unstable branch of 0 as a function ofIp = f2p shown in figure 7 for min

(? (1)0 ,

? (2)0

)< 0 < max

(K (1)0 ,

K (2)0

)where

K,? (1)0 (1 + 2) =

4 ~p3

23

~22p 3 2

andK,? (2)0 (1 + 2) =

4 ~p3 g

23

~22p 3 2 2 g (1 g)

[~22p + 2

],

with g = 21

.

Note that K,? (1)0 coincide exactly with the local extrema of Ip as a function of 0, what wouldlead to a perfectly Kerrlike bistability if no parametric instability were displayed.

Parametric instabilities Also known as modulational instabilities, the parametric instabilitieswe point out involve the two q0 and q0 modes.

They are the source of the unstable branch with min(P (1)0 ,

P (2)0

)< 0 < min

(? (1)0 ,

? (2)0

)where

P (1)0 (1 + 2) = 2

andP (2)0 (1 + 2) = 2

1 + g1 2 g (1 g)

.

In fact, since 1 < g < 0, we have clearly P (2)0 < P (1)0 .

In [45], an experimental investigation is led to demonstrate the spin bistability of polaritons ina semiconductor microcavity under resonant optical pumping. Contrary to their study, we focusedmainly on nonpolarized excitation beams (excepted for the resonant case in presence of a magneticfield, as we have seen in subsection II.2.1). Carrying on the study, we will have to make sure thatwe work with dynamically stable solutions.

III.1.4 Density profile and drag

After calculation, when +0 = 0 , one finds from equation (2.15):

(q) = 0(q)

(~ q V ~1(q)) (~ q V ~2(q)). (3.4)

+ = implies immediately that moving an obstacle in our two-component polariton conden-sate in the presence of a linearly polarized background does not induce polarization fluctuations:S(X) = 0. The response function found this way is the exact same as in [32] (where they workin 2D). We therefore deal with a onecomponent like polariton field where the intracomponentscoupling is 1 + 2.

The poles of the response function are thus given by:

4 q4 4 2 q2m(V 2rel c2(d)

)+tot

1 0 8 ~ q Vrel

(1 0)2+ 4

[tot

(tot 2mc2(d)

)+ 2

](1 0)2

= 0 , (3.5)



where the sound velocity of the density wave c(d) is found to be:

mc2(d) = (1 + 2)02 . (3.6)

The length which appears in equation (3.5) is related to the background density through itsdefinition = ~/m1 0. We are going to explain how c(d) is found to be the sound velocity.First, assuming an infinite lifetime for the polaritons (i.e. doing 0), we recover a kind of weaklyinteracting Bose gas described by a "true" Gross-Pitaevskii equation. Then, we must consider thefact that, in a Bose gas, the chemical potential, giving a characteristic energy for the condensate,should impose the polariton density. Here it would read = mc2(d). Although is not definedin the resonant pumping scheme, we can mimic this relation by focusing on the resonant case8where tot = 0: indeed, it means that the external pump equivalently imposes the polaritondensity ~p = mc2(d). As shown in panel (c) of figure 6, the resonant case gives, as expected, aBogoliubov-like dispersion that is quadratic for |q| 1 and linear for |q| 1. In the phononregime i.e. in the limit of the large wavelengths ( |q| 1), we get: 1,2(q)

(vp c(d)

)q.

Hence, in the rest frame of the fluid, the slope c(d) of the collective excitations as a function ofq is nothing but the characteristic sound velocity.

A few words about equation (3.5): it is an equation of the 4th degree in q without any cubicterm (so, the sum of the four poles is zero!). Moreover, the odd term has a Vrel factor in itscoefficient, involving the parity q q .

Density patterns around the obstacle Once the quartic equation (3.5) is solved, the poles qlof the response function are determined. Then, thanks to relation (2.16), we can plot the densitymodulations generated by a pointlike defect moving towards x > 0 in the polariton fluid and by afinite size obstacle as well (C.f. figure 8). In the resonant case, when 0, there is clearly a criticalvelocity, given simply by c(d) above which, according to Landaus criterion discussed in subsectionI.2, a erenkov wave precedes the obstacle. When we put a small enough lifetime for the polaritons,such a scenario is still qualitatively consistent, however doing so, the density pattern experiences adamping indicative of the more dissipative nature of the system. This implies that we cannot speakrigorously about a superfluid regime below a critical velocity anymore. The erenkov radiationejected from the obstacle is damped as increases. The envelope of these damped oscillations isgiven by e=(ql)X and, in the equations, the imaginary parts are well associated with .

Drag force We focused on the stable branch for the polariton field background by consideringeither tot = 0 or tot < 0. Taking tot > 0 is restrictive since the parametric instabilitiesstarting when the background density reaches P (2)0 prevent us from taking the limit 0. Infigure 9, the drag force is plotted as a function of the relative velocity of a localized obstacle,which is modeled both by a peak and a Gaussian peak, in the flow of the polariton fluid forseveral values of the damping parameter . In the resonant case, it is worth noting that, whenVrel c(d), the drag force exerted onto the obstacle is very similar to the Stokes viscous frictionforce: F -peakd /2 Vrel (as pointed out in [32] and [46]). By analogy, represents the viscosity ofthe fluid. For slow flows, the drag force is related to the momentum diffusion in the fluid since the

8Note that, in this resonant case, 0 depends only on the ratio fp/. When 0, we have to take fp forthe polariton background density not to diverge. In a way, the gain and the losses have to be balanced for a realisticconfiguration.



(a)

-5 0 5 10 15

-1.6

-1.2

-0.8

-0.4

0

0.4

(b)

-5 0 5 10 15

-1.6

-1.2

-0.8

-0.4

0

0.4

(c)

-5 0 5 10 15

-2

-1

0

1

2

(d)

-5 0 5 10 15

-1.5

-0.75

0

0.75

1.5

K(X

)[1/1]

K(X

)[1/1]

= 0.21 0

= 0.0011 0

X []

X []

X []

X []

G-peak(X

)[/1]

G-peak(X

)[/1]

Fig. 8: Density patterns generated by an obstacle moving in a twocomponent polariton fluidwithout magnetic field in the resonant case tot = 0. While (a) (resp. (b)) stands for a subsonicmoving peak (resp. Gaussian) obstacle at Vrel = 0.2 c(d), (c) (resp. (d)) stands for a supersonicone at Vrel = 2 c(d). The gaussianpeak shaped obstacles (b) and (d) whose widths are = 0.75 induce essentially a smoothing of the profile. In (c) and (d), we observe the ejection of a erenkovwave, damped when increases. We took g = 0.2.

poles are first pure imaginary numbers. When the flow is faster and reaches a critical value (c(d)in the resonant case and vcrit for tot < 0 as developed in Appendix B), a drag is emitted aheadof the obstacle: it is the onset of the previously mentioned erenkov radiation. By analogy withfluid mechanics, these threshold velocities originate a wave resistance regime, i.e. a regime wherea resistance is generated by the energy loss owing to the emission of density waves created by themoving obstacle. Moreover, we can compare our results to the drag force exerted onto objects whichare moving on a viscous fluid surface. In [38], a continuous transition towards the wave resistanceregime is evidenced as observed in our case and shown in figure 9 for finite polariton lifetimes, beit for tot < 0 or in the resonant case. Furthermore, when the viscosity decreases, the drag force,as a function of the relative velocity of the obstacle, is found to become quasi-discontinuous in [39].A rigorous discontinuity of the drag force is also pointed out in [47] which investigates superfluityin a perturbative framework. Raphalde Gennes theory of wave resistance devoted to the studyof capillarygravity waves (see [48]) gives another example of a rigorous drag force discontinuityin a nonviscous fluid. In [49], it is shown how the discontinuity vanishes when the fluid becomesviscous.

While, for a nonviscous = 0 polariton fluid, the drag force displays a discontinuity at c(d)in the resonant case, taking a negative detuning parameter induces a divergent behavior at vcrit



(detailed in Appendix B). This is somewhat consistent with the 2D results of [32]: whereas, in theresonant case, the drag force as a function of the velocity is a ramp, it is discontinuous for negativedetunings. Curiously enough, when Vrel exceeds the threshold values, the drag force diminishesas the damping parameter increases independently of the value of tot. This counter-intuitivephenomenon has been also highlighted in a study of liquid nitrogen drops floating at the surface of aliquid bath [49]. In fact, the viscosity of the fluid shortens the range of drag and, as a consequence,diminishes the wave resistance that is the main source of drag above the critical velocities. At largevelocities, the drag force exerted onto a impurity saturates at value 2m0 2 / ~2, independentlyof the damping strength, at resonance and for negative detunings as well (one can check it fromequation (B4) in the limit of nonviscosity). This saturation comes artificially from the peakmodel: indeed, taking a Gaussian potential (see figure 9), we notice that, at large velocities, thedrag force vanishes. Actually, if the flow has a large kinetic energy compared to the amplitudeof the external potential ( / here), one expects the flow to be negligibly inflected around theobstacle; what implies no drag.

(a)

0 1 2 3 4

0

0.5

1

1.5

2

(b)

0 1 2 3 4

0

0.5

1

1.5

2

(c)

0 1 2 3 4

0

2

4

6

(d)

0 1 2 3 4

0

1

2

3

F-peak

d

[ m02/~2]

F-peak

d

[ m02/~2]

c(d) c(d)

= 0.021 0

= 0.11 0

vcrit vcrit

Vrel

[1 0 /m

]

Vrel

[1 0 /m

]

Vrel

[1 0 /m

]

Vrel

[1 0 /m

]

tot = 0

tot < 0

tot = 0

tot < 0

FG-peak

d

[ m02/~2]

FG-peak

d

[ m02/~2]

Fig. 9: Drag force exerted by the resonantly driven twocomponent fluid onto the obstacle versusits relative velocity without magnetic field. In panel (a) (resp. (b)), the drag force exerted onto apeak (resp. Gaussian) obstacle is represented for different values of the damping parameter inthe resonant case. In panels (c) and (d), for a negative tot = 1 0 detuning parameter, thedrag force displays a divergent behavior at vcrit which is nonetheless smoothed as increases. Oncemore, g = 0.2.


III.2 POLARIZATION AND DYNAMICS WHEN THE MAGNETIC FIELD IS TURNED ON

III.2 Polarization and dynamics when the magnetic field is turned on

III.2.1 Elliptical polarization of the background

Switching the magnetic field on, the background field 0 acquires two independent components:the system becomes elliptically polarized.

The combination (2.3)0 (2.3) 0 gives now: 0 =

fp= [0 ]. This enables us to find:

tan 2 =sin ( /2)sin (+ /2)

, (3.7)but also

0 =(fp

)2 [sin2 (+ /2) + sin2 ( /2)

](3.8a)

and Sz =(fp

)2 [sin2 (+ /2) sin2 ( /2)

]. (3.8b)

Then, the combination (2.3) (2.3) gives:

cot (+ /2) = 1

[1

0 + 2 0 ~ (p + )

]. (3.9)

Using (3.7) and (3.15a), equation (3.9) leads to the system: 21 +X2 =

11 + Y 2

(

fp

)2[~ (p ) + Y ]

21 + Y 2 =1

1 +X2 (

fp

)2[~ (p +) + X] ,

whereX = cot (+ /2) and Y = cot ( /2) .

After numerical resolution of a polynomial equation of the 9th degree in X, we have simply to take:

= 12

(arctan 1

X+ arctan 1

Y

)and = arctan 1

X arctan 1

Y.

By replacing these values in (3.7) and (3.15a), the solutions for the background field are entirelydetermined. An overview of the behavior with the pump intensity and the magnetic field is givenin Appendix C.

III.2.2 A richer spectrum of excitations

Once more, the quartic equation for E(q, ) leads to four branches for the dispersion relation:

~14(q) = ~ vp q X (q)

Y(q) , (3.10)

with

X (q) = (q) ((q) + 1 0) +rel (rel 1 Sz)Y(q) = (q)2

[(21 22

)S2z + 22 20 + 4rel (rel 1 Sz)

]+2 (q)1 0rel (2rel 1 Sz) +2rel

[(21 22

)20 + 22S2z

].

When ~ = 0 = Sz, we recover equation (3.3). But we also see that, even if the magnetic fieldis 0, the excitation spectra are modified when the background is nonlinearly polarized. For eachresult presented hereafter, we checked the stability of 0(Ip) as it is done in subsection III.1.3.



III.2.3 Density and polarization fluctuations generated by a weak impurity

This time, the response functions have again a global 1/q2 behavior but with a 6th degree polynomial

numerator over a 8th degree polynomial denominator. Writing (q, ) = N(q, )D(q, ) , we find:

N (q, ) = (Sz + 0) (rel (q)){

(2 1)Sz (rel + (q))

E2(q, ) + (rel + (q)) [rel + (q) + (1 2) 0]}

(3.11)

and, with no surprise,9

D(q, ) = E4(q, ) 2X (q) E2(q, ) + X 2(q) Y(q) . (3.12)

As a consequence, we obtain:

D(q, ) (q, ) /2 = {Sz [(2 1) Sz +rel] + 0 [(q) + (1 2) 0]}[2rel (q)2

]+ (0 (q) Szrel) E2(q, )

(3.13)and

D(q, ) (q, ) /2 = (Sz (q) + 0rel)[2rel (q)2

]+ (Sz (q) 0rel) E2(q, ) . (3.14)

Without magnetic field and when the background is linearly polarized, we recover, from (3.14), theresult of the previous section: there is no polarization modulation induced by the moving obstacle.Otherwise, we can expect the emission of polarization waves.

Polarization and density sound velocities in the resonant case When = 0 and = 0,the response function denominator is much simpler. In the phonon regime, we exhibit, from theBogoliubovlike dispersion, two sound velocities: c(d) (for "density") and c(p) (for "polarization").They read:

c(d)~p /m

= 11 + g

11 (1 g2) (1 2

2circ

)(3.15a)

andc(p)

~p /m= 11 + g

1 +1 (1 g2) (1 2

2circ

). (3.15b)

These sound velocities, which, by the way, are such as c(d) < c(p), can be defined only if circ.At = circ, the dispersion relation becomes exactly quadratic for the two branches associatedwith c(d). Exceeding the value circ implies the emergence of a parametric instability = [(q0)] > 0.When ~ = 0, we recover equation (3.6)10.

The existence of polarization and density sound velocities open the possibility of an experimentalsignature of the specific effects of spin in the presence of an obstacle: the observation that nopolarization wake is emitted at the velocity for which the density wake first appears.

9Indeed, it is det L as there is no simplification with the numerator.10Lets recall that 1 < g < 0 implies

g2 = g.



Wave patterns In the resonant case, the response functions are not only simpler but also displayfive and no longer eight poles due to a simplification between numerator and denominator. When = 0, the number of poles is further decreased to four. For an obstacle moving into the polaritonfluid flow with a relative velocity Vrel < c(d), the four poles are pure imaginary numbers implyingthat excitations are evanescent waves localized around the obstacle. However, when c(d) < Vrel c(p), the two last poles acquirereal parts: a erenkov radiation of polarization waves is emitted. For nonzero damping parameters,three velocity regimes are noticeable as well and we note v(d) , (p)crit the analogous threshold velocitiesin the resonance case but also for other detuning parameters values where eight poles have to betaken into account. In figure 10, the ejection of such waves is evidenced in the resonant case fortwo values of the viscosity when v(d)crit < Vrel < v

(p)crit and Vrel > v

(p)crit as well. We also notice (see panel

(d)) that the polarization wave (in gray) is much weakly damped than the density one (in orange).However, it is not clear that v(p)crit originates a z oscillation ahead of the obstacle. Panel (a) showsthat the onset of a density wave can also coincide with an oscillation of S(X). Yet, the oscillationmodulation of S(X) displayed above v(p)crit in panel (c) suggests that the eigenmodes leading to thepolarization wave emission we have mentioned correspond rather to a linear combination between and z.

= 0.021 0 = 0.21 0

(a)

-10 0 10 20 30

-2

0

2

(b)

-10 0 10 20 30

-1

-0.5

0

0.5

1

(c)

-10 0 10 20 30

-2

0

2 (d)

-10 0 10 20 30

-1

-0.5

0

0.5

1

K(X

),S(X

)[1/1]

K(X

),S(X

)[1/1]

X []

X []

X []

X []

K(X

),S(X

)[1/1]

K(X

),S(X

)[1/1]

Fig. 10: Density and polarization modulation patterns induced by a impurity in the resonantcase for = 0.1circ. While panels (a) and (b) account for v(d)crit < Vrel < v

(p)crit, panels (c) and

(d) stand for Vrel > v(p)crit. The figures are drawn in the frame where the fluid is at rest and theobstacle moves from left to right at velocity Vrel m / ~ = 0.75 (a), 0.7 (b), 1 (c), 1.5 (d). Wetook g = 0.2.



0 1 2 3 4

0

0.5

1

1.5

2

F-peak

d

[ m02/~2]

Vrel

[1 0 /m

]c(d) c(p)

= 0.0011 0 = 0.021 0 = 0.11 0 = 0.21 0

0 1 2 3 4

0

0.5

1

1.5

2

FG-peak

d

[ m02/~2]

Vrel

[1 0 /m

]c(d) c(p)

0 1 2 3 4

0

2

4

6

F-peak

d

[ m02/~2]

Vrel

[1 0 /m

]v(d)crit v(p)crit

= 0.0011 0 = 0.021 0 = 0.11 0 = 0.21 0

0 1 2 3 4

0

1

2

3

FG-peak

d

[ m02/~2]

Vrel

[1 0 /m

]v(d)crit v(p)crit

0 1 2 3 4

0

0.5

1

1.5

2

F-peak

d

[ m02/~2]

Vrel

[1 0 /m

]

= 0.25 circ

= 0.5 circ

= 0.75 circ

= circ

0 1 2 3 4

0

0.5

1

1.5

2

FG-peak

d

[ m02/~2]

Vrel

[1 0 /m

]

0 1 2 3 4

0

2

4

6

F-peak

d

[ m02/~2]

Vrel

[1 0 /m

]

= 2 circ

= 3 circ

= 4 circ

0 1 2 3 4

0

1

2

3

FG-peak

d

[ m02/~2]

Vrel

[1 0 /m

]

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 11: Drag force exerted by the resonantly driven twocomponent fluid onto the obstacle versusits relative velocity for several damping parameters and magnetic fields. Whereas left panels aredrawn for a peak potential, right panels stand for a Gaussian one. Panels (a), (b), (e) and (f)are plotted in the resonant case; (c), (d), (g) and (h) for tot < 0. This negative total detuning isobtained for 1 Ip = 0.21 [a.u.], ~p = 0.67 (1 Ip)1/3 and g = 0.2. While panels (a) and (b) (resp.(c) and (d)) depict the behavior of the drag as increases when = 0.5circ (resp. = 2circ),panels (e), (f), (g) and (h) depict the evolution with the magnetic field when = 0.0011 0. Forimpurities and small damping, when tot < 0, we no longer deal with the twostep drag found atresonance (with c(d) and c(p) as threshold velocities), but rather with a twicediverging drag (withv

(d)crit and v

(p)crit as threshold velocities). For both, we recover the usual value at large velocities.



Drag When the magnetic field is switched on, owing to the supplementary polarization waveemission above v(p)crit, one expects the drag to be twice enhanced. In figure 11, such a behavior iswell evidenced. In the resonant case, impurities generate a twostep drag. These two steps, theonset of which are c(d) and c(p) in the nonviscous limit, are nonetheless smoothed by the dampingparameter (panel (a)). Moreover, the first one vanishes as we increase the magnetic field (panel(e)). It is worth noting that, when = circ and, as a consequence, when the background iscircularly polarized (0 = 0 or

+0 = 0), the peak obstacle faces a drag force which displays only

one step (see the gray curve of panel (e)): it is as if there were only one component, by analogywith the previous section, but this time the threshold velocity reads: mc2(p) = 1 0.

The two behaviors we have just pointed out in the resonant case, namely the smoothingand the first step dying out as the magnetic field is increased, account also for the tot < 0 casewe studied (and for which, by the way, we checked the dynamical stability). There is however adifference: the density wave ejection threshold velocity v(d)crit increases with (panels (g) and (h))whereas in the resonant case, it decreases. The tot < 0 case offers, in presence of a magneticfield, two drag divergences at the threshold velocities of polarization and density waves ejectionswhen 0. The replacement of the discontinuities displayed in the resonant case by divergences isreminiscent of the previous study without magnetic field and a similar scenario as the one detailed inAppendix B is assumed (see Appendix C for the analogous behavior of the eight response functionspoles).

Finally, notice that in any cases, for fast moving peak scatterers, the artificial saturation atusual value 2m0 2 / ~2 for 1D BoseEinstein condensates is still verified. As expected, the dragalso cancels for finitesize obstacles.



ConclusionDuring this internship, I studied dissipative and superfluid properties of the flow of a resonantly-

pumped 1D polariton condensate past a localized obstacle. We described the dynamics of thequantum fluid by a phenomenological modification of GrossPitaevskii equation. The model takesinto account spin degrees of freedom of polaritons, interactions between them, resonant pumpingprocesses and effects due to the presence of the localized obstacle and a possible external magneticfield as well.

We treated the problem in the framework of linear response theory, that is when the obstacleinduces weak modulations in the parameters of the flow. We identified three regimes of flowsseparated by two specific values of the gas velocity. The first threshold originates erenkov radiationof linear density waves and the second one, the ejection of polarization modulations. Even thoughgas excitations are localized around the obstacle for low velocities, we evidenced that the drag forceexperienced by the obstacle is nonzero, what demonstrates that in such quantum fluids, there is nolowvelocity genuine superfluid behavior in contrast to what is typically observed in conservativeBoseEinstein condensates.

These results are in good agreement with previous studies led for coherently driven scalarpolariton condensates (see for example [32]) and nonresonantly pumped polariton fluids (in [42, 50]for instance). Finally, this work naturally calls for further investigations and developments. Inparticular, it is important to check if the drag force divergences analytically pointed out in thecase where the detuning parameter (which characterizes resonant pumping effects) is nonzero isexperimentally observable. It is also natural to apply the perturbative approach described insubsection II.2.1 to higher dimensions.

?

Acknowledgments

It is of paramount importance for me to thank Pierre-lie LARR genuinely, for the time hedevoted to me, his precious explanations, his philanthropy but also for all the things I had thepleasure to share with him: passion for science, music, cigarettes, coffees and jokes. I also wantto warmly thank Nicolas LEVERNIER, who largely contributed to the pleasant atmosphere ofthe office and my brother Damien GREDAT for his generous final read-through. Moreover, I amgrateful to Nicolas PAVLOFF, who granted me his trust for this internship, for the interesting topicshe broaches. Finally, I would like to thank all Ph.D. students and postdocs for their welcome andkindness.


Appendix A: Heuristic argument for Landaus criterion and wave resistance analogy

[ Appendix A \Heuristic argument for Landaus criterion

& wave resistance analogy

Landaus criterion Following Feynmans discussion in [27], we can draw a simple picture toillustrate Landaus criterion. Lets consider a pointlike mass M moving at the velocity V into aquantum fluid at rest. A dissipative behavior would correspond to the diffusion of momentum intothe fluid. Thus, we note p the momentum transferred to the fluid, (p) the energy associated withthese excitations and V the new velocity of the obstacle after such an emission of excitations.

Obstacle

excitations

V

p(p)

V

Energy conservation during the process yields: 12 M V2 = 12 M V

2 + (p), while momentumconservation gives: V = V + p /M . Though, one must admit that the quasiparticles whichcorrespond to the excitations should have a much lower mass m than the obstacle one. As aconsequence, forM m the energy transferred to the fluid reads: (p) ' V p = V p cos < V p.This way, we point out that the emission of excitations through the quantum fluid is possible onlyif:

V > vcrit = min[(p)p

]. (A1)

Equation (A1) is nothing but a formulation of Landaus criterion.

Wave resistance analogy In [48], a study is led to describe the dispersive properties of gravity-capillary waves generated by a moving obstacle onto the free surface of a calm classic liquid. Infact, in 1871, Lord Kelvin already highlighted in [51] that above vcrit = (4 g / )1/4 (where is theliquid density, the liquidair surface tension and g the acceleration due to gravity), the obstacleexperiences a wave resistance. The dispersion relation of these excitations reads:

G.-C.(q) =q

(g +

q2), (A2)

implying that Kelvins critical velocity is well the one equation (A1) would predict. The minimumvelocity in water above which we expect a stationary wake of gravitycapillary waves to be engen-dered ahead of the obstacle is vcrit = 23 cm/s, in very good agreement with the experimental resultof 25.33 cm/s.

In uniform Bose-Einstein condensates, the excitations sprectrum is given by the Bogoliubovdispersion relation:

~B(q) =

~2 c2s q2 +(~2 q2

2m

)2, (A3)

Grgory GREDAT page i LPTMS

Appendix A: Heuristic argument for Landaus criterion and wave resistance analogy

where cs is the sound velocity (essentially related to the contact interaction term between atomsand the condensate density) and m the mass of the atoms.

For a qmode to be excited, be it for the Bose-Einstein condensate or the classic liquid we justmentioned, the energy transferred to the fluid by a moving obstacle must correspond to a frequencyV q higher than the threshold vcrit q. This condition is graphically translatable (left graphs of figure12): the slope of the line V q has to reach vcrit to intercept the excitation spectrum. For Bogoliubovspectrum, it corresponds to cs. The linear waves thus created ahead of the obstacle are strikinglysimilar as we can notice in figure 12.

q

B(q)

0

{vcrit = cs} q

q

G.-C.(q)

0

vcrit q

g

(a)

(b)

Fig. 12: Graphical representation of Landaus criterion and wave emission above the criticalvelocity. Whereas (a) shows a gravitycapillary waves emission in water, (b) shows the erenkovlinear waves emission in a Bose-Einstein condensate. The top right image is taken from [52] andthe bottom right one from [38].

X

Grgory GREDAT page ii LPTMS

Appendix B: Critical velocity and drag divergence when tot < 0 without magnetic field

[ Appendix B \Critical velocity and drag divergence when tot < 0 without

magnetic field

Critical velocity when tot 0 without magnetic field First, we can notice that, when 0, equation (3.5) is solvable as a biquadratic equation. We get:

~2 q22m = m

(V 2rel c2(d)

)+tot

(Vrel) , (B1)

where(V ) = m2

(V 2 c2(d)

)2+ 2totmV 2 . (B2)

The condition for the four poles to be reals gives in fact the critical velocity vcrit above whichwe expect erenkov radiations to appear: (Vrel) > {(vcrit) = 0} , evidencing the critical velocity:

mv2crit =(mc2(d) tot

)+tot

(tot 2mc2(d)

). (B3)

Detailed study of the response functions poles We put = 0. The response function givenby (3.4) can be rewritten as:

(, q) = 2 0(q)

~2 (2 2(q)) ,

where~2 2(q) = (q) [(q) + 0 (1 + 2)] .

For the integrals giving the density fluctuations and, consequently, the drag force to be welldefined above the critical velocity, the poles of the response function must have a nonzero imaginarypart. This job is naturally done by . However, working with an infinite lifetime for the polaritonsimplies taking some precautions. In this case, we have to remove the poles from the real axis inthe complex plane. But should we move them up or down ? To address this issue, we can beginby questioning the validity of the perturbative approach we implement.

The main idea is to adiabatically bring the external potential on with a e t factor and take 0+. One shows that the adiabatic boundary condition used to define the response function isequivalent to doing the replacement + . This way, the causality is respected as explainedon page 101 of reference [53] and we deal with a delayed response function.

Working with a peak potential, we have to take = q Vrel to perform the integral. We nowsearch the poles as q = q0 + with q0 = {q+,q+, q,q} and || |q0|. We find:

(q0) = q0 Vrel

(q0) q (q)|q = q0 q0 V2

rel= mVrel

~2 q202m tot +m

(c2(d) V

2rel

) .As a result, we get:

(q+) =mVrel(Vrel)

Vrel >vcrit

0+ and (q) = mVrel(Vrel)

Vrel >vcrit

0 .

Grgory GREDAT page iii LPTMS

Appendix B: Critical velocity and drag divergence when tot < 0 without magnetic field

Hence, we see that, above the critical velocity, the poles q+ (resp. q) have to be movedtogether in the upper (resp. lower) half of the complex q plane as sketched in panel (a) of figure 13.(Note that (q0) does not depend on the sign of q0, what is consistent with the fact that equation(3.5) has the parity q0 q0).

To sum up, thanks to relation (B1), we draw the global behavior with Vrel of the poles in thecomplex plane. Figure 13 offers a comparison between the behavior of the poles when = 0 andwhen the polariton lifetime is finite (in a significant damping case). When mV 2rel < mc2(d) totthe four poles are purely imaginary quantities. After their collision on the imaginary axis theyacquire a real part. Finally, above vcrit, when = 0, the poles q+ (resp. q) stick to the realaxis with an infinitesimal positive (resp. negative) imaginary shift. While q+ and q+ move awayfrom the origin, q and q converge to 0.

q+ q+

q q

= (q)

< (q)

Vrel < vcrit

Vrel > vcrit

0 0+{} 0

= (q)

< (q)

6= 0Vrel < vcritVrel > vcrit

0

(a) (b)

Fig. 13: Position of the four poles in the complex q plane without magnetic field. While panel(a) shows how we have to remove the poles from the real axis when Vrel > vcrit and = 0 (whatis equivalent to taking a very long but finite polariton lifetime), panel (b) shows that putting asignificant damping does not change qualitatively the behavior of the poles. The arrows indicatethe direction of motion of the poles when Vrel increases.

0 1 2 3 4

0

2

4

6

F-

pea

kd

[ m0

2/~2]

Vrel

[1 0 /m

]

tot = 11 0tot = 0.51 0tot = 0.11 0tot = 0

Fig. 14: Drag force versus the relative veloc-ity of a peak obstacle for 0 in the limit 0 at ~ = 0.

Drag force divergence Ultimately, the dragforce yields:

F -peakd (Vrel) =

2m0 2

~2m(V 2rel c2(d)

)(Vrel)

(Vrel vcrit) . (B4)

This drag force as a function of the relative velocityVrel of the obstacle moving in the polariton fluid isrepresented in figure 14. When tot < 0, the dragdisplays a divergent behavior at vcrit. Below vcrit,the drag force cancels: we have a superfluid regime.Well above the critical value of the velocity, the dragforce saturates at 2m0 2 / ~2 and we recover thesame behavior as the resonant case.

Grgory GREDAT page iv LPTMS

Appendix C: Complement for the background polariton field and the response functionspoles in presence of a magnetic field

[ Appendix C \Complement for the background polariton field and the response

functions poles in presence of a magnetic field

Background density and polarization Thanks to the resolution described in subsection III.2.1,we can understand how the background polariton density as a function of the external pump inten-sity behaves when we increase the magnetic field. In figure 15, it is shown that, for low magneticfields, low pump intensities lead to up to nine leading order solutions for +0 and

0 whereas high

pump intensities imply a unique solution (panels (a), (b), (c) and (d)). At higher values of themagnetic field, there is no longer nine possible solutions but three at most (panels (e) and (f)), rem-iniscent of the cubic relation found when = 0. As displayed in the orange curves, for high enoughpumping intensities, the polariton background becomes elliptically polarized and Sz increases as increases.

1

0

1Sz

1 Ip [a.u.] 1 Ip [a.u.] 1 Ip [a.u.]

= 0.01 circ = 0.1 circ = circ

0 0.01 0.02 0.03 0.04 0.05

0

0.5

1

1.5

2

0 0.01 0.02 0.03 0.04 0.05

0

0.5

1

1.5

2

0 0.1 0.2 0.3

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04 0.05

-0.5

-0.25

0

0.25

0.5

0 0.01 0.02 0.03 0.04 0.05

-0.5

-0.25

0

0.25

0.5

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

(a) (c) (e)

(b) (d) (f)

Fig. 15: Background polariton density and polarization versus incident pumping intensity forseveral magnetic field magnitudes. While the red curves show the polariton background density,the orange ones show its pseudospin projection on the z axis for three values of the magnetic field.The blue circles correspond to the linearly polarized case tot = 0. We took ~p = 0.4 [a.u.], = 0.02 [a.u.] and g = 0.2.

The linearly polarized case tot = 0 From (2.7) definition, we start immediately imposingthe polariton background density, the same way a chemical potential would do:

tot = 0 0 =2 ~p

(1 + 2). (C1)

Grgory GREDAT page v LPTMS

Appendix C: Complement for the background polariton field and the response functionspoles in presence of a magnetic field

Using equations (3.15a) and (3.15b), one finds:

cot (+ /2) = cot ( /2) . (C2)

Hence, we show that when we consider the case tot = 0, the polarization of the background hasto be linear: Sz = 0. In figure 15, we verify that relation (C1) leads also numerically to a linearlypolarized background (blue circles). However, we can wonder whether +0 and

0 are equal or not.

In fact, carrying out the calculation, we find:

= 0, = 2 arctan(

~

)and = 2 . (C3)

so, +0 and 0 have opposite phases. Moreover, a condition on the pump intensity is imposed:

f2p = ~p

(~22 + 2

1 + 2

)so, 0 =

2 f2p~22 + 2

. (C4)

We can therefore understand how tot = 0 is obtained for higher pump intensities when themagnetic field increases as numerically evidenced in figure 15.

Detailed study of the response functions poles In the resonant case, we show that thereis a simplification in the response function, leading to five poles instead of eight. In figures 16 and17, the evolution of the poles is sketched in each case for impurities moving from left to right ina static polariton fluid. Whereas, in the resonant case, two poles collide on the imaginary axis engendering a wave emission at the critical velocities v(p)crit and v

(d)crit (orange with blue ones and

green with red ones), the evolution displayed in figure 17 for tot < 0 is much more complicatedand has to be compared with the well described scenario of panel (b) of figure 13.

= (q)

< (q)

= 0

0

Fig. 16: Position of the five response func-tions poles in the complex q plane at reso-nance for a nonzero magnetic field.

= (q)

< (q)0

Fig. 17: Position of the eight response func-tions poles in the complex q plane for anonzero magnetic field.

Grgory GREDAT page vi LPTMS

References[1] J.F. Allen. Nature, 141(75), 1938.

[2] P. Kapitza. Nature, 141(74), 1938.

[3] F. London. The -Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy. Nature,141, 1938.doi: 10.1038/141643a0.

[4] N. N. Bogoliubov. On the theory of superfluidity. Journal of Physics (Union of SovietSocialist Republics), 11, 1947.

[5] O. Penrose and L. Onsager. Bose-Einstein Condensation and Liquid Helium. PhysicalReview, 104, 1956.doi: 10.1103/PhysRev.104.576.

[6] L.P. Pitaevskii. Vortex lines in an imperfect Bose gas. Soviet Physics, Journal of Ex-perimental and Theoretical Physics, 13, 1961.

[7] E. P. Gross. Structure of a quantized vortex in boson systems. Nuovo Cimento, 20, 1961.

[8] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observationof Bose-Einstein Condensation in a Dilute Atomic Vapor. Science, 269, 1995.doi: 10.1126/science.269.5221.198.

[9] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, andW. Ketterle. Bose-Einstein Condensation in a Gas of Sodium Atoms. Physical ReviewLetters, 75, 1995.doi: 10.1103/PhysRevLett.75.3969.

[10] S. Bose. Plancks gesetz und lichtquantenhypothese. Zeitschrift fr Physik, 26, 1924.

[11] C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V.Kavokin, G. Malpuech, and A. Bramati. Observation of the optical spin Hall effect. NaturePhysics, 3, 2007.doi: 10.1038/nphys676.

[12] K. G. Lagoudakis, T. Ostatnick, A.V. Kavokin, Y. G. Rubo, R. Andr, and B. Deveaud-Pldran. Observation of half-quantum vortices in an exciton-polariton condensate. Science,326, 2009.doi: 10.1126/science.1177980.

[13] R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese, T. Boulier, D. Andreoli, E. Giacobino,J. Bloch, A. Bramati, G. Malpuech, and A. Amo. Half-solitons in a polariton quantum fluidbehave like magnetic monopoles. Nature Physics, 8, 2012.doi: 10.1038/nphys2406.

[14] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M.Marchetti, M. H. Szymaska, R. Andr, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud,and Le Si Dang. BoseEinstein condensation of exciton polaritons. Nature, 443, 2006.doi: 10.1038/nature05131.

http://www.nature.com/nature/journal/v141/n3571/pdf/141643a0.pdfhttp://link.aps.org/doi/10.1103/PhysRev.104.576http://www.sciencemag.org/content/269/5221/198http://link.aps.org/doi/10.1103/PhysRevLett.75.3969http://www.nature.com/nphys/journal/v3/n9/abs/nphys676.htmlhttp://www.sciencemag.org/content/326/5955/974.fullhttp://www.nature.com/nphys/journal/v8/n10/full/nphys2406.htmlhttp://www.nature.com/nature/journal/v443/n7110/full/nature05131.html

[15] D. R. Allum, P. V. E. McClintock, and A. Phillips. The Breakdown of Superfluidity in Liquid4He: An Experimental Test of Landaus Theory. Philosophical Transactions of theRoyal Society of London A, 284, 1977.doi: 10.1098/rsta.1977.0008.

[16] C. A. M. Castelijns, K. F. Coates, A. M. Gunault, S. G. Mussett, and G. R. Pickett. Landaucritical velocity for a macroscopic object moving in superfluid 3B: Evidence for gap suppressionat a moving surface. Physical Review Letters, 56, 1986.doi: 10.1103/PhysRevLett.56.69.

[17] C. Raman, M. Khl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ket-terle. Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas. Physical RewiewLetters, 83, 1999.doi: 10.1103/PhysRevLett.83.2502.

[18] R. Onofrio, C. Raman, J. M. Vogels, J. R. Abo-Shaeer, A. P. Chikkatur, and W. Ketterle.Observation of Superfluid Flow in a Bose-Einstein Condensed Gas. Physical Rewiew Let-ters, 85, 2000.doi: 10.1103/PhysRevLett.85.2228.

[19] D. Dries, S. E. Pollack, J. M. Hitchcock, and R. G. Hulet. Dissipative transport of a Bose-Einstein condensate. Physical Review A, 82, 2010.doi: 10.1103/PhysRevA.82.033603.

[20] P. Engels and C. Atherton. Stationary and Nonstationary Fluid Flow of a Bose-EinsteinCondensate Through a Penetrable Barrier. Physical Review Letters, 99, 2007.doi: 10.1103/PhysRevLett.99.160405.

[21] A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D. Martin, A. Lematre,J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, C. Tejedor, and L. Via. Collective fluiddynamics of a polariton condensate in a semiconductor microcavity. Nature, 457, 2009.doi: 10.1038/nature07640.

[22] D. Sanvitto, S. Pigeon, A. Amo, D. Ballarini, M. De Giorgi, I. Carusotto, R. Hivet, F. Pisanello,V. G. Sala, P. S. S. Guimaraes, R. Houdr, E. Giacobino, C. Ciuti, A. Bramati, and G. Gigli.All-optical control of the quantum flow of a polariton condensate. Nature Photonics, 5,2011.doi: 10.1038/nphoton.2011.211.

[23] A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Lemnager,R. Houdr, E. Giacobino, C. Cuiti, and A. Bramati. Polariton Superfluids Reveal QuantumHydrodynamic Solitons. Science, 332, 2011.doi: 10.1126/science.1202307.

[24] A. Amo, J. Lefrre, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdr, E. Giacobino,and A. Bramati. Superfluidity of polaritons in semiconductor microcavities. Nature Physics,5, 2009.doi: 10.1038/nphys1364.

http://rsta.royalsocietypublishing.org/content/284/1320/179http://link.aps.org/doi/10.1103/PhysRevLett.56.69http://link.aps.org/doi/10.1103/PhysRevLett.83.2502http://link.aps.org/doi/10.1103/PhysRevLett.85.2228http://link.aps.org/doi/10.1103/PhysRevA.82.033603http://link.aps.org/doi/10.1103/PhysRevLett.99.160405http://www.nature.com/nature/journal/v457/n7227/full/nature07640.htmlhttp://www.nature.com/nphoton/journal/v5/n10/full/nphoton.2011.211.htmlhttps://www.sciencemag.org/content/332/6034/1167.shorthttp://www.nature.com/nphys/journal/v5/n11/full/nphys1364.html

[25] L. D. Landau. The theory of superfluidity of Helium II. Journal of Physics USSR, 5,1941.

[26] L. D. Landau. On the theory of superfluidity of Helium II. Journal of Physics USSR, 11,1947.

[27] R. P. Feynman. Progress in Low temperature Physics, volume 1. 1955.

[28] J. Wilks. The Properties of Liquid and Solid Helium. 1967. Clarendon Press, Oxford.

[29] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in clouds of Bose-Einsteincondensed atoms. Physical Rewiew A, 58, 1998.doi: 10.1103/PhysRevA.58.2417.

[30] P. Leboeuf and N. Pavloff. Bose-Einstein beams: Coherent propagation through a guide. Phys-ical Rewiew A, 64, 2001.doi: 10.1103/PhysRevA.64.033602.

[31] M. Romanelli. Mlange quatre ondes de polaritons dans des microcavits semi-conductrices.PhD thesis, Universit Pierre et Marie Curie, Laboratoire Kastler Brossel, 2005.

[32] A. C. Berceanu, E. Cancellieri, and F. M. Marchetti. Drag in a resonantly driven polaritonfluid. J. Phys.: Condens. Matter, 24, 2012.doi: 10.1088/0953-8984/24/23/235802.

[33] C. Ciuti and I. Carusotto. Quantum fluid effects and parametric instabilities in microcavities.Phys. Status Solidi B, 242, 2005.doi: 10.1002/pssb.200560961.

[34] S. Pigeon, I. Carusotto, and C. Cuiti. Hydrodynamic nucleation of vortices and solitons in aresonantly excited polariton superfluid. Physical Review B, 83, 2011.doi: 10.1103/PhysRevB.83.144513.

[35] G. Nardin, G. Grosso, Y. Lger, B. Pietka, F. Morier-Genoud, and B. Deveaud-Pldran.Hydrodynamic nucleation of quantized vortex pairs in a polariton quantum fluid. NaturePhysics, 7, 2011.doi: 10.1038/nphys1959.

[36] D. Bajoni, P. Senellart, E. Wertz, I. Sagnes, A. Miard, A. Lematre, and J. Bloch. PolaritonLaser Using Single Micropillar GaAs-GaAlAs Semiconductor Cavities. Physical ReviewLetters, 100, 2008.doi: 10.1103/PhysRevLett.100.047401.

[37] S. Stringari. Novel superfluid features in ultra cold atomic gases, June 2013. 4th lecture atthe Collge de France.url: superfluidity in ultracold atomic gases (supercurrents and superstripes).

[38] T. Burghelea and V. Steinberg. Onset of Wave Drag Due to Generation of Capillary-GravityWaves by a Moving Object as a Critical Phenomenon. Physical Rewiew Letters, 86,2001.doi: 10.1103/PhysRevLett.86.2557.

http://link.aps.org/doi/10.1103/PhysRevA.58.2417http://link.aps.org/doi/10.1103/PhysRevA.64.033602http://iopscience.iop.org/0953-8984/24/23/235802/http://onlinelibrary.wiley.com/doi/10.1002/pssb.200560961/abstracthttp://link.aps.org/doi/10.1103/PhysRevB.83.144513http://www.nature.com/nphys/journal/v7/n8/full/nphys1959.htmlhttp://link.aps.org/doi/10.1103/PhysRevLett.100.047401http://www.college-de-france.fr/site/antoine-georges/guestlecturer-2013-05-14-17h00.htm#|q=../antoine-georges/guestlecturer-2012-2013.htm|p=../antoine-georges/guestlecturer-2013-06-04-17h00.htm|http://link.aps.org/doi/10.1103/PhysRevLett.86.2557

[39] J. Browaeys, J.-C. Bacri, R. Perzynski, and M. I. Shliomis. Capillary-gravity wave resistancein ordinary and magnetic fluids. Europhysics Letters, 53, 2001.doi: 10.1209/epl/i2001-00138-7.

[40] T. Paul, M. Albert, P. Schlagheck, P. Leboeuf, and N. P

Documents

Rapport de Stage 2012-2013 Force de traînée dans un condensat