Abstract
Superfluidity, the ability of a quantum fluid to flow without friction, is one of the most spectacular phenomena occurring in degenerate gases of interacting bosons. Since its first discovery in liquid helium4 (refs 1, 2), superfluidity has been observed in quite different systems, and recent experiments with ultracold trapped atoms have explored the subtle links between superfluidity and Bose–Einstein condensation^{3,4,5}. In solidstate systems, it has been anticipated that exciton–polaritons in semiconductor microcavities should behave as an unusual quantum fluid^{6,7,8}, with unique properties stemming from its intrinsically nonequilibrium nature. This has stimulated the quest for an experimental demonstration of superfluidity effects in polariton systems^{9,10,11,12,13}. Here, we report clear evidence for superfluid motion of polaritons. Superfluidity is investigated in terms of the Landau criterion and manifests itself as the suppression of scattering from defects when the flow velocity is slower than the speed of sound in the fluid. Moreover, a Čerenkovlike wake pattern is observed when the flow velocity exceeds the speed of sound. The experimental findings are in quantitative agreement with predictions based on a generalized Gross–Pitaevskii theory^{12,13}, and establish microcavity polaritons as a system for exploring the rich physics of nonequilibrium quantum fluids.
Main
Bound electron–hole particles, known as excitons, are fascinating objects in semiconductor nanostructures. In a quantum well with a thickness of the order of a few nanometres, the external motion of the exciton is quantized in the direction perpendicular to the well, whereas it is free within the plane of the well. When the quantum well is placed in a highfinesse microcavity, the strongcoupling regime between excitons and light is easily reached^{14}, giving rise to exciton–photon mixed quasiparticles called polaritons, which are an interesting kind of twodimensional composite boson. Thanks to their sharp dispersion, polaritons have a small effective mass (of the order of 10^{−5} times the freeelectron mass) that allows the building of manybody coherent effects, such as Bose–Einstein condensation^{15,16}, at a lattice temperature of a few kelvins. Furthermore, their partially excitonic character results in strong interactions between polaritons, which are expected to lead to the appearance of superfluid phenomena. Indirect evidence of superfluid motion in polariton systems has recently been reported through the observation of pinned quantized vortices^{9}, Bogoliubovlike dispersions^{10} and pioneering experiments on polariton parametric oscillators^{11}. Despite these remarkable works, a direct demonstration of exciton–polariton superfluidity is however still missing. In this Letter, we report the observation of superfluid motion of a quantum fluid of polaritons created by a laser in a semiconductor microcavity.
In our experiments, to probe superfluidity we study the perturbation that is produced in an optically created moving polariton fluid when a static defect is present in the flow path, as proposed in refs. 12, 13. This procedure is a direct application to the polariton system of the standard Landau criterion of superfluidity^{5}, originally developed for liquid helium and recently applied to demonstrate superfluidity of atomic Bose–Einstein condensates (refs 17, 18). The flow without friction characteristic of a superfluid is demonstrated in the case of polaritons as a flow without scattering.
To explore the quantum fluid regime a complete control of three key parameters is needed: the inplane momentum of polaritons (that is, the polariton flow velocity), the oscillation frequency of the polariton field, and its density. In this respect polaritons constitute an ideal system from the experimental point of view. The strength of the polariton–polariton interaction within the fluid can be controlled through the particle density, which in turn is changed in a precise way by adjusting the incident laser power. Their partially photonic character also allows the creation of polariton fluids with a welldefined oscillation frequency, ω_{p}, which is that of the excitation laser, and with a welldefined linear momentum, k_{p}, by choosing the angle of incidence θ_{p} (k_{p}=(ω_{p}/c)sinθ_{p}, where c is the light speed). The possibility of controlling the polariton fluid oscillation frequency is in stark contrast with equilibrium systems, such as atomic condensates, where the oscillation frequency of the condensate is fixed by the equation of state relating the chemical potential to the particle density^{3}: owing to their relatively short lifetime (of the order of few picoseconds), the steady state of an excited microcavity results from the interplay between the pumping rate and the radiative as well as nonradiative losses. This feature results in much wider possibilities in the structure of the system’s spectrum of elementary excitations than in the equilibrium case^{12,13,19,20,21,22}.
In our experiment, a polariton fluid is excited in a microcavity sample (see the Methods section), cooled at 5 K, with a circularly polarized beam from a frequencystabilized, singlemode continuouswave titanium:sapphire laser. The laser field continuously replenishes the escaping polaritons in the fluid. The beam is focused onto the sample in a spot of ∼100 μm in diameter with angles of incidence between 2.6^{∘} and 4.0^{∘} (Fig. 1a). The wavelength of the pump laser is around 836 nm, close to resonance with the lower polariton branch. The images of the surface of the sample (nearfield emission) and of the far field in transmission configuration are simultaneously recorded on two different highresolution CCD (chargecoupled device) cameras. With the use of a spectrometer and at lowpower, offresonance excitation, the characteristic parabolic lowerpolariton dispersion can be observed, as shown in Fig. 1b.
To study the propagation properties of the injected polariton fluid, the centre of the excitation spot is placed on top of a natural pointlike defect present in the sample. Defects of different sizes and shapes appear naturally in the growth process of microcavity samples (see Supplementary Information). At low excitation power and quasiresonant excitation of the lower polariton branch, polariton–polariton interactions are negligible: in the nearfield (realspace) images, the coherent polariton gas created by the laser is scattered by the defect and generates a series of paraboliclike wavefronts around the defect, propagating away from it, mostly in the upstream direction (Figs 2cI and 3bI). They result from the interference of an incident polariton plane wave with a cylindrical wave produced by the scattering on the defect. In momentum space, polariton scattering gives rise to the wellknown Rayleigh ring^{23} that is observed in the farfield images (Figs 2cIV,3bIV).
As the laser intensity is augmented, polariton–polariton interactions increase, resulting in the singlepolariton dispersion curves being shifted towards higher energies (blueshift due to the repulsive interactions) and also becoming strongly distorted as a consequence of collective manybody effects^{12,13}. In a simplified picture, for a specific density ψ_{c}^{2}, from parabolic (Fig. 1c) the dispersion is predicted to become linear in some kvector range with a discontinuity of its slope in the vicinity of the pump wavevector k_{p} (see Fig. 1d and refs. 12, 13). Under these conditions, a sound velocity can be attributed to the polariton fluid, being given by where g is the polariton–polariton coupling strength and m is the effective mass of the lower polariton branch. If the flow velocity v_{p} of the polariton fluid (given by v_{p}=ℏk_{p}/m) is chosen such that the sound speed c_{s}>v_{p}, then the Landau criterion for superfluidity is satisfied, as shown in ref. 13. In such a case, as no states are any longer available for scattering at the frequency of the driving polariton field (see Fig. 1d), the polariton scattering from the defect is inhibited and the fluid is able to flow unperturbed.
This situation is observed in Fig. 2, where the real (cIII) and momentum (cVI) space images of the polariton fluid in the presence of a ∼4μmdiameter defect are shown for a pump angle of incidence of 2.6^{∘}, corresponding to a low inplane momentum of k_{∥}=−0.337 μm^{−1} (v_{p}=6.4×10^{5} m s^{−1}, point A in Fig. 1b). The superfluid regime is first attained only in the centre of the Gaussian excitation spot for the excitation density corresponding to Fig. 2cII. As the intensity of the excitation laser is increased, the superfluid condition extends to the rest of the spot (Fig. 2cIII), whereas the density in its central part hardly changes. Simulations based on the solution of polariton nonequilibrium Gross–Pitaevskii equations (see ref. 13 and the Methods section) are shown in Fig. 2d. The calculations have been carried out by fitting the size and depth of the defect and by adjusting the values of g and ψ^{2} around the experimentally estimated values (ψ^{2} is obtained from the experimental emitted intensity and g is estimated from the aperture of the Čerenkov fringes as discussed later on). Whereas at low excitation density (Fig. 2cI,IV,dI,IV) the fluid presents parabolic density wavefronts in real space and a scattering ring in momentum space as mentioned above, at higher excitation density the scattering ring collapses (Fig. 2cV,VI,dV,VI), showing that any scattering of the polariton fluid by the defect is inhibited and that unperturbed flow is eventually attained. In real space (Fig. 2cIII,dIII), a complete suppression of the density modulation is observed (see also Supplementary Video S1). In all these figures, we can see an excellent agreement between the observed effects and the theory.
The collapse of the scattering ring in the superfluid regime is clearly summarized in Fig. 2b, where the ratio between the polaritons scattered to a constant area in momentum space (dashed yellow rectangle in Fig. 2cIV) and the total polariton density is plotted as a function of the excitation density: when the superfluid regime is obtained, the scattered light drops by a factor of four. Note that this factor is limited here by the finite size of the excitation spot and is expected to attain much higher values if larger pump spots are used.
It is important to stress that the evidence of superfluidity presented here is substantially different from what was observed in ref. 11, which shows the dispersionless propagation of a polariton bullet even when crossing a defect. Although these results strongly suggest a superfluid character, the propagation of the sizeable polariton bullet on top of a homogeneous fluid under parametric scattering conditions cannot be described in terms of the Bogoliubov theory of a weakly perturbed fluid. In contrast, the experimental study reported in the present Letter is consistent with the Landau criterion and allows a clear demonstration of superfluidity in a quasihomogeneous polariton fluid. It has also been suggested to exploit the Landau criterion to investigate photonic superfluidity in nonlinear optical systems^{24,25,26}, but no experimental studies have yet been reported.
The qualitative shape of the perturbation created in the fluid by the defect is studied in Fig. 3 in a different regime (see also Supplementary Video S2). For this purpose, a polariton fluid with a higher momentum is created using a laser beam with a larger incidence angle, of 4.0^{∘} (k_{p}=−0.521 μm^{−1}, point B in Fig. 1b). This allows us to enter a regime where the Bogoliubov dispersion of collective excitations has a soundlike nature, with a sound speed that is now lower than the flow speed (c_{s}<v_{p}). This is shown in Fig. 1f, where the calculated spectrum of excitations for the density at which a sound velocity is well defined presents a linearized dispersion together with available states at the same and lower energies than the pump. The generalized Landau condition for superfluidity is not fulfilled and the defect is able to generate collective Bogoliubov excitations in the fluid. They manifest as a Čerenkovlike density modulation pattern with characteristic straight wavefronts in the realspace images (Fig. 3bII,III) and as a strongly reshaped Rayleigh scattering ring in the farfield emission pattern^{12,13} (Fig. 3bV,VI). A similar density pattern was recently observed in atomic Bose–Einstein condensates propagating against a localized optical potential at supersonic velocities^{27}. At lower pump power (Fig. 3bI,IV), the Bogoliubov excitations go back to singleparticle ones (Fig. 1e) and the parabolicshaped modulation corresponding to the standard Rayleigh scattering ring is recovered, as in Fig. 2cI,IV. The calculated images also reproduce these observations as shown in Fig. 3c: again, the agreement with the experimental observations is excellent.
Let us note that the observation of the linear Čerenkovlike wavefronts in Fig. 3cII is a direct indication of the existence of a welldefined sound speed in the system. The measure of the angle of aperture φ between the linear waves generated by the defect allows the precise determination of the sound speed in the system, given by sin(φ)=c_{s}/v_{p}. In the conditions of Fig. 3cII we find c_{s}=8.1×10^{5} m s^{−1}. With the use of equation (1) the polariton–polariton coupling constant ℏg can be estimated, being of the order of 0.01 meV μm^{2}, which is consistent with previously predicted values on the exciton–exciton interaction^{28}.
We have reported a direct experimental demonstration of superfluidity in a fluid of exciton–polaritons flowing in a semiconductor microcavity. By varying the pump intensity, we have observed that the system goes from a nonsuperfluid regime in which a static defect creates a substantial perturbation in the moving fluid to a superfluid one, in which the polariton flow is no longer affected by the defect. In a supersonic regime, superfluid propagation is replaced by the appearance of a Čerenkovlike perturbation produced by the defect, in agreement with theoretical predictions and detailed calculations. Our observations pave the way towards the investigation of a rich variety of quantum fluid effects associated with the nonequilibrium nature of the microcavity polariton system.
Methods
Sample.
The sample used in this experiment is an AlGaAs microcavity grown by molecular beam epitaxy. The length of the cavity is 2λ and it contains three In_{0.04}Ga_{0.96}As quantum wells of 80 Å placed at the maxima of the electromagnetic field of the cavity mode. The two Bragg mirrors forming the cavity are made of alternated λ/4 layers of AlAs and GaAs, with reflectivities between 99.85 and 99.95%. In the strongcoupling regime, we measured a Rabi splitting of 5.1 meV. The high finesse of this microcavity leads to polariton linewidths of ∼100 μeV at low power excitation. In the cavity under study the substrate has been polished to make transmission measurements. All discussed experiments have been done at a cavity–exciton detuning of −1.1 meV.
Experimental setup.
To pump polaritons in our microcavity, we use a frequencystabilized, continuouswave, singlemode Ti:sapphire laser (linewidth: 1 MHz). The narrow linewidth and stability of the excitation source is a key parameter in our experiment as it allows the excitation of a welldefined polariton mode.
The beam is spatially filtered by coupling to an optical fibre, it is circularly polarized using zeroorder half and quarterwave plates and it is focalized on the sample by a lens. To avoid thermal effects caused by the laser illumination, the beam is chopped with a 3.2% duty cycle, at a frequency of about 200 Hz. The light emitted by the microcavity is collected in transmission geometry with a widenumericalaperture ocular. The polarizationresolved detection line allows us to simultaneously image the near field (image of the sample) and the far field (the Fourier plane of the collecting ocular, momentum space of the emission). In the latter case, the use of an imaging spectrometer in the detection path allows for the energy–momentum resolution of the emission, giving access to the dispersion curves, such as that shown in Fig. 1b.
Gross–Pitaevskii equation.
Figures 2d and 3c show the result of simulations based on the solution of the polariton nonequilibrium Gross–Pitaevskii equations: where ψ_{X(C)} is the exciton (cavityphoton) wavefunction, x the inplane twodimensional space vector, t time, F_{p}, ℏk_{p} and ℏω_{p} the amplitude, momentum and energy of the pump field, respectively, ℏω_{X(C)} the energy of the excitons (cavity photons) at k=0, γ_{X(C)} the decay rate of the excitons (cavity photons), 2Ω_{R} the vacuum Rabi splitting between the polariton modes, V_{X(C)} the excitonic (photonic) potential that accounts for the defects present in the sample and g the exciton–exciton interaction potential. x_{0} indicates the bidimensional coordinates of the centre of the Gaussian spot on the sample, and δ_{x} is the radial width of the circular Gaussian spot. Let us note that in our simulations we have just considered photonic defects with a depth of 1 meV, that is, V_{X}=0, but similar results are expected in the case of considering excitonic defects. In the simulations, the size of the defects is 4 μm and 5 μm, respectively, for Figs 2 and 3.
Only for the calculation of the spectra of excitations presented in Fig. 1, the Gross–Pitaevskii equation was used for a spatially homogenous excitation. In this case, the equations have been linearized around the steadystate solution, allowing us to calculate the Bogoliubovlike excitation spectrum as discussed in refs 12 and 13.
Note: When originally published online, the supplementary information for this article was incomplete. The new complete version is now available to download.
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Acknowledgements
This work was partially supported by the Ile de France programme IFRAF. A.A. and S.P. were funded by the Agence Nationale pour la Recherche (ANR07NANOGEMINI); A.B. is a member of the Institut Universitaire de France. I.C. acknowledges financial support from the Italian MIUR and the EuroQUAMFerMix programme.
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Affiliations
Laboratoire Kastler Brossel, Université Pierre et Marie Curie, Ecole Normale Supérieure et CNRS, UPMC Case 74, 4 place Jussieu, 75252 Paris Cedex 05, France
 Alberto Amo
 , Jérôme Lefrère
 , Claire Adrados
 , Elisabeth Giacobino
 & Alberto Bramati
Laboratoire Matériaux et Phénomènes Quantiques, UMR 7162, Université Paris DiderotParis 7 et CNRS, 75013 Paris, France
 Simon Pigeon
 & Cristiano Ciuti
BECCNRINFM and Dipartimento di Fisica, Universita di Trento, I38050 Povo, Italy
 Iacopo Carusotto
Institut de Photonique et d’Electronique Quantique, Ecole Polytechnique Fédérale de Lausanne, Station 3, CH1015 Lausanne, Switzerland
 Romuald Houdré
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All authors contributed equally to this work.
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Correspondence to Alberto Amo or Alberto Bramati.
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