Propagation and amplification dynamics of 1D polariton condensates
E. Wertz, A. Amo, D. D. Solnyshkov, L. Ferrier, T. C. H. Liew, D. Sanvitto, P. Senellart, I. Sagnes, A. Lemaître, A. V. Kavokin, G. Malpuech, J. Bloch
PPropagation and amplification dynamics of 1D polariton condensates
E. Wertz, A. Amo, D. D. Solnyshkov, L. Ferrier, T. C. H. Liew, D. Sanvitto,
4, 5
P. Senellart, I. Sagnes, A. Lemaˆıtre, A. V. Kavokin,
6, 7
G. Malpuech, and J. Bloch ∗ CNRS-Laboratoire de Photonique et Nanostructures, Route de Nozay, 91460 Marcoussis, France Institut Pascal, PHOTON-N2, Clermont Universit, University Blaise Pascal,CNRS, 24 avenue des Landais, 63177 Aubi`ere cedex, France Mediterranean Institute of Fundamental Physics, 31, via Appia Nuova, 00040, Rome, Italy NNL, Istituto Nanoscienze - CNR, Via Arnesano, 73100 Lecce, Italy Istituto Italiano di Tecnologia, Via Barsanti, 73010 Lecce, Italy Physics and Astronomy School, University of Southampton, Highfield, Southampton, SO171BJ, UK Laboratoire Charles Coulomb, CNRS-Universite de Montpellier II,Pl. Eugene de Bataillon, Montpellier, 34095, cedex, France (Dated: September 13, 2018)The dynamics of propagating polariton condensates in one-dimensional microcavities is investi-gated through time resolved experiments. We find a strong increase in the condensate intensitywhen it travels through the non-resonantly excited area. This amplification is shown to come frombosonic stimulated relaxation of reservoir excitons into the polariton condensate, allowing for the re-population of the condensate through non-resonant pumping. Thus, we experimentally demonstratea polariton amplifier with a large band width, opening the way towards the transport of polaritonswith high densities over macroscopic distances.
PACS numbers: 71.36.+c, 67.85.Hj, 78.55.Cr, 78.67.Pt
Optical amplifiers are devices with an active media ca-pable of directly amplifying an optical signal traversingthem. The medium is non-resonantly pumped, and am-plification of a coherent propagating beam is obtained ina single passage by bosonic stimulated emission [1]. Suchdevices are important for the regeneration of fibred lasersignals transmitted over long distances, or for high powerlasers. Contrary to parametric amplifiers relying on fourwave mixing, optical amplifiers based on non-resonantpumping have the great advantage of a large bandwidthand absence of phase matching restrictions.Such a mechanism of optical amplification can in prin-ciple be used to regenerate a coherent population inother bosonic systems, for instance, to produce continu-ous matter-wave lasing in ultracold atomic Bose-Einsteincondensates [2]. In an atom laser, a trapped Bose-Einstein condensate is coupled to an output mode, inwhich atoms flow away while keeping the coherence prop-erties of the condensed state [3]. The duration of theatom laser is given by the size of the initial conden-sate and the outcoupling rate [4, 5], and it could be ex-tended by adding reservoir atoms feeding and amplifyingthe matter-wave condensate by bosonic stimulated relax-ation. While some progress has been done in this direc-tion [6], the continuous refilling of the ultracold atomiccloud remains experimentally challenging and, up to now,atomic lasers can only work in a pseudo-continuous modewith a finite duration.Similar strategies can be considered for condensates ofexciton-polaritons in semiconductor microcavities. Po-laritons are mixed light-matter quasi-particles behavingas bosons in the low density limit [7]. Polariton con-densates present a strong interest both for the study of the fundamental properties of quantum gases (su-perfluidity [8], quantised vortices [9] and dark [10] andbright [11] soliton formation have been reported), andfor their potential functionalities as integrated opticalelements. They present the flexibility of out of reso-nance optical excitation for their generation and, thanksto their interactions with uncondensed excitons, they canbe easily accelerated, propagating over macroscopic dis-tances [12, 13]. Several works have proposed the use ofsuch propagating polariton condensates to realize a po-lariton Berry phase interferometer [14] or various inno-vative spinoptronic devices [15–19] taking advantage oftheir non-linear properties. However, the finite polaritonlifetime (a few tens of picoseconds) results in the decay ofthe population along the propagation. This effect limitsthe region in which the polariton density is high enoughfor particle interactions to induce non-linear phenomena.Thus, it is important to be able to amplify polariton con-densates during their propagation.In this letter, we experimentally demonstrate polari-ton amplification by a reservoir of uncondensed excitonsin a 1D microcavity. We monitor polariton propagationin a time resolved experiment. We show that the po-lariton condensate propagates with a group velocity thatcan be controlled varying the exciton-photon spectral de-tuning. Moreover, amplification is evidenced in the spa-tial region where excitons are non-resonantly injected:strong increase of the emission is observed when polari-ton condensates traverse this region. This takes placedue to bosonic stimulated relaxation of uncondensed po-laritons, and it is well reproduced by solving a modifiedGross-Pitaevskii equation describing both the polaritoncondensate and the excitonic reservoir. Such polariton a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t FIG. 1. (Color online) Time resolved spatial distribution of the polariton emission below condensation threshold at δ = − δ = +3 meV (b), δ = 0 meV (c), and δ = −
10 meV (d). In (b)-(d), the excitationdensity is chosen to produce ejected polariton condensates with a kinetic energy of 1.5 meV. The inset in (c) shows the decayof the polariton density along the white line (the dashed line is an exponential fit). (e) Measured polariton group velocity fordifferent detunings (the dashed line is a guide to the eye). amplification opens the way to the implementation ofcascadable polariton functionalities and optical transis-tor operation under non-resonant pumping.The sample used in our experiments is described indetail in Ref. [12]. It consists in a high quality fac-tor λ /2 planar cavity ( Q ∼ µ mand a length of 200 µ m were fabricated using electronbeam lithography and reactive ion etching. The cavitywedge, present already in the planar structure, allows forthe tuning of the bare exciton-photon energy difference δ = E C − E X of each wire. Within each wire, δ re-mains constant, as they are etched in the direction per-pendicular to the wedge. Single microwires are excitednon-resonantly (typically 100 meV above the lower po-lariton energy) with a Ti:Sapphire laser delivering 2 pspulses with a 82 MHz repetition rate. The laser is focuseddown to a 3 µ m diameter spot with a microscope objec-tive. The sample emission is collected through the sameobjective and imaged on the entrance slit of a streak cam-era coupled or not to a spectrometer in order to measurethe time- and energy- or space-resolved polariton prop-agation. Momentum space is accessed by imaging theFourier plane of the microscope objective taking advan-tage of the one-to-one relation between angle of emissionand in-plane momentum of polaritons. All experimentsare performed at 10 K.Polariton condensation in such microwires has been re-ported under cw excitation [12] showing the formation ofextended polariton condensates propagating out of theexcitation region. Here, space- and time-resolved mea-surements allow direct monitoring of the propagation dy-namics, as shown in Fig. 1(a)-(d) for microwires withdifferent exciton-cavity detunings under non-resonantpulsed pumping at Y = 0, in the middle of the wire.Below the condensation threshold [Fig. 1(a)] the emis-sion dynamics is slow, reflecting the occupation of theexcitonic reservoir [20], and very little propagation is ob- served: the emission remains localized in the area be-neath the pulsed pump laser. The rise of the emissionintensity apparent in the reported time window is givenby the dynamics of exciton formation and relaxation intolow momentum states [21, 22]. Nonetheless, the totalcarrier population continuously decays in time.At carrier densities above the condensation threshold( P th ), stimulated polariton relaxation gives rise to theformation of a condensate in the region of excitationat the bottom of the lower polariton branch with zeroin-plane momentum ( k = 0). At the high excitationdensities reported in Fig. 1(b)-(d) (50 P th ) the conden-sation takes place few picoseconds after the arrival ofthe excitation pulse ( t = 0). The presence of an un-condensed excitonic reservoir in the excitation region in-duces a blueshift of the polariton energy (4 meV) causedby repulsive exciton-polariton interactions [12, 23]. Sincethe excitonic reservoir remains localized in the region ofexcitation (because of the large exciton effective mass),no blueshift is generated outside this area. In this way,the reservoir induces an interaction energy that can betransformed into kinetic energy when the condensate ex-its the excitation region, resulting in the acceleration andpropagation of the polariton condensate on both sides ofthis region, as observed in Fig. 1(b)-(d). In the highexcitation density case reported in Fig. 1, part of thecondensate energy ( ∼ ∼ . µ m · ps − , and it isdirectly given by the kinetic energy. The group velocityis inversely proportional to the square root of the polari- FIG. 2. (Color online) (a) Time resolved spatial distributionof the polariton emission at 5 P th in a wire with δ = − µ m from the rightedge of the wire.(b) Simulation considering polariton conden-sates propagating and interacting with the excitonic reservoir.(c) Measured amplification gain at each of the successive re-populations of the condensate; the green dotted line is anexponential fit with a decay of 300 ps. The modulation of theintensity observed both in (a) and (b) at Y > ton mass, which is fixed by the curvature of the polari-ton dispersion [12]. Thus, by changing this curvature wecan control the group velocity of the expelled polaritonpackets. This is shown in Fig. 1(b)-(d) for three differ-ent detunings of +3, 0 and −
10 meV, respectively. Inall three cases we selected excitation densities giving riseto the same kinetic energies for the ejected condensates(1.5 meV). When the detuning is decreased towards neg-ative values, polaritons get lighter and the propagationspeed increases, as summarized in Fig. 1(e).The inset of Fig. 1(c) shows the characteristic decayof the density as the polariton wavepackets propagateaway from the excitation region (propagation length of ∼ µ m). This decay is given by the photon lifetime inour etched microcavities, which is on the order of 15 ps,and evidences the need for an amplification or repumpingof the condensate if propagation over longer distances isdesired. We propose an amplification mechanism similarto that of optical amplifiers. It is based on the stimu-lated relaxation of the non-resonantly excited reservoirof excitons/polaritons to the bottom of the lower polari-ton branch, induced by the presence of a large polaritonpopulation in that state.In order to experimentally prove polariton amplifica-tion, we place the excitation spot 35 µ m away from oneof the edges of the wire, as schematically represented inFig. 2(a). Above threshold [5 P th , Fig. 2(a)], the con-densate is formed at t = 25 ps after the arrival of theexcitation pulse, and and two condensates are ejectedfrom the excitation region, one departing to the left (la-beled i ) and another one to the right (labeled i’ ). Whencondensate i’ reaches the edge of the wire, it is reflected FIG. 3. (Color online) Energy of the successive condensatesas they pass through Y=-25 µ m on Fig. 2(b): (a) measuredexperimentally and (b) simulated. The black dashed line isthe measured energy at k = 0 under the excitation spot. and propagates back. Remarkably, when it gets back tothe pumped area (at t = 50 ps) a strong increase of theemission signal is observed, accompanied by the prop-agation of two new condensate packets ( ii and ii’ ), tothe right and to the left of the region where the reser-voir is located. The regeneration of the condensate takesplace five times in our observation window. To evaluatethe gain of the condensate amplification we first mea-sure the intensity of the signal close to the edge of thewire (Y=32 µ m), and then we estimate from this valuewhat the signal intensity would be without amplificationat Y=-15 µ m (considering only the polariton decay). Fi-nally we take the ratio of this estimated intensity to themeasured one at this very same position. The results arereported on Fig. 2(c), showing a gain of more than 4, anda time decay of the gain on the order of 300 ps (fitteddashed lines), compatible with that of the reservoir.In order to understand in detail the amplification dy-namics of the polariton signal, we study the reservoirinduced blueshift and the emission energy of the sub-sequently expelled condensates. This is experimentallyexplored in Fig. 3(a), which shows the energy of the con-densates traversing the region at Y=-25 µ m on Fig. 2(a).At t = 25 ps condensation takes place at k = 0 inthe pumped region with an energy that is blueshiftedby an amount ∆ E i = 1 . P th ) polariton-parametric relaxation is negligible (contrary to the caseof Fig. 1). ∆ E i corresponds to the condensate energy inthe excitation region, given by the total reservoir popu-lation at t , and sets the kinetic energy of condensates i and i’ [12]. The reservoir induced interaction energyis thus fully converted into kinetic energy. At t , thereservoir population has partly decayed resulting in adecrease of the k = 0 energy in the region of excita-tion (∆ E ii = 1 . i’ induces a re-stimulation ofthe condensate from the reservoir, amplifying the con-densate density. This re-stimulation takes place simul-taneously with the energy relaxation of condensate viapolariton-reservoir pair scattering, down to the local po-tential energy set by the reservoir density at the timedelay t (orange arrow in Fig. 3).The mechanism we have described presents strong sim-ilarities to that of an optical amplifier, in which bosonicstimulation takes place exactly to the same energy stateas that of the incoming field, which acts a seed. In ourcase, stimulation takes place simultaneously with the in-elastic relaxation of the incoming condensate. The cas-cade observed in Fig. 3 reflects the time evolution of the k = 0 energy in the excitation region [measured in dashedlines in Fig. 3(a)], sampled by the passage of the suc-cessive condensates. The relaxation of the condensatetraversing the reservoir region is, thus, essential to ac-count for this energy cascade. As ∆ E sets the kineticenergy of the expelled condensates, the energy relaxationresults in the slow down of the subsequent emitted con-densates, as it can be observed from the increasing slopeof the dashed lines in Fig. 2(a). We can model the cou-pled dynamics of the condensate and the reservoir, withthe use of a modified Gross-Pitaevskii equation includ-ing terms for the re-stimulation and inelastic relaxationof the condensate [27, 28] depending on the local reser-voir density [25]. This model allows to fully reproduceour results, as evidenced in Figs. 2(b) and 3(b), includingthe slow down of the successively amplified condensates.The flexibility of the out-of resonant excitation pro-vides supplementary control of the propagating conden-sates. If the initial injected density is high enough,polariton-polariton interactions close to the excitationarea induce polariton scattering towards lower- energystates [24], as discussed for the case of Fig. 1. The situ-ation with excitation close to the wire edge is shown inFig. 4(a), where the condensate that is initially ejectedtowards the right has a smaller energy than the reservoir-induced barrier [29] and gets trapped between the excita-tion region and the edge of the wire [12, 30], reboundingcontinuously between the two without appreciable energyloss [25]. The spatial oscillation of this polariton wave-packet is well reproduced by our simulations when runwith a high initial density [Fig. 4(b)].We have evidenced the amplification of polariton con-densates under non-resonant excitation. The propaga-tion with controlled velocity along with the re-populationprocess shown here, are two key ingredients to achieve FIG. 4. (Color online) Time resolved spatial distribution ofthe polariton emission at very high pump powers (50 timesthe condensation threshold) for a wire with δ = − transport of polariton condensates over macroscopic dis-tances with a high density. Our scheme provides a largeenergy bandwidth for the re-populating beam, rangingfrom the gap energy of the employed quantum wells tothe absorption energy of the materials constituting theBragg mirrors of the cavity (from ∼ RTRA Tri-angle de la Physique ”Boseflow1D”, by the contractANR-11-BS10-001 ”Quandyde”, by the FP7
ITNs ”Cler-mont4” (235114) and ”Spin-Optronics” (237252). T.L.was supported by the FP7 Marie Curie project EPO-QUES (298811). ∗ [email protected][1] E. Desurvire, J. R. Simpson, and P. C. Becker, Opt.Lett. , 888 (1987).[2] M. Holland, K. Burnett, C. Gardiner, J. I. Cirac, andP. Zoller, Phys. Rev. A , R1757 (1996).[3] A. ¨Ottl, S. Ritter, M. K¨ohl, and T. Esslinger, Phys. Rev.Lett. , 090404 (2005).[4] E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmer-son, S. L. Rolston, and W. D. Phillips, Science , 1706(1999).[5] I. Bloch, T. W. H¨ansch, and T. Esslinger, Phys. Rev.Lett. , 3008 (1999). [6] N. P. Robins, C. Figl, M. Jeppesen, G. R. Dennis, andJ. D. Close, Nature Phys. , 731 (2008).[7] J. Kasprzak, M. Richard, S. Kundermann, A. Baas,P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H.Szymanska, R. Andre, J. L. Staehli, V. Savona, P. B. Lit-tlewood, B. Deveaud, and L. S. Dang, Nature , 409(2006).[8] A. Amo, J. Lefr`ere, S. Pigeon, C. Adrados, C. Ciuti,I. Carusotto, R. Houdr´e, E. Giacobino, and A. Bramati,Nature Phys. , 805 (2009).[9] K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas,I. Carusotto, R. Andre, L. S. Dang, and B. Deveaud-Pledran, Nature Phys. , 706 (2008).[10] A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet,I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre,E. Giacobino, C. Ciuti, and A. Bramati, Science ,1167 (2011).[11] M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gor-bach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez,K. Biermann, R. Hey, and P. V. Santos, Nature Phot. , 50 (2012).[12] E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. San-vitto, A. Lemaitre, I. Sagnes, R. Grousson, A. V. Ka-vokin, P. Senellart, G. Malpuech, and J. Bloch, NaturePhys. , 860 (2010).[13] G. Christmann, G. Tosi, N. G. Berloff, P. Tsotsis, P. S.Eldridge, Z. Hatzopoulos, P. G. Savvidis, and J. J.Baumberg, Phys. Rev. B , 235303 (2012).[14] I. A. Shelykh, G. Pavlovic, D. D. Solnyshkov, andG. Malpuech, Phys. Rev. Lett. , 046407 (2009).[15] T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, Phys.Rev. Lett. , 016402 (2008).[16] R. Johne, I. A. Shelykh, D. D. Solnyshkov, andG. Malpuech, Phys. Rev. B , 125327 (2010).[17] A. Amo, T. C. H. Liew, C. Adrados, R. Houdre, E. Gi-acobino, A. V. Kavokin, and A. Bramati, Nature Phot. , 361 (2010).[18] I. A. Shelykh, R. Johne, D. D. Solnyshkov, andG. Malpuech, Phys. Rev. B , 153303 (2010).[19] H. Flayac, D. D. Solnyshkov, and G. Malpuech, Phys.Rev. B , 125314 (2011).[20] J. Bloch and J. Y. Marzin, Phys. Rev. B , 2103 (1997).[21] T. C. Damen, J. Shah, D. Y. Oberli, D. S. Chemla, J. E.Cunningham, and J. M. Kuo, Phys. Rev. B , 7434(1990).[22] J. Szczytko, L. Kappei, J. Berney, F. Morier-Genoud,M. Portella-Oberli, and B. Deveaud, Phys. Rev. Lett. , 137401 (2004).[23] L. Ferrier, E. Wertz, R. Johne, D. D. Solnyshkov,P. Senellart, I. Sagnes, A. Lemaitre, G. Malpuech, andJ. Bloch, Phys. Rev. Lett. , 126401 (2011).[24] D. Tanese, D. D. Solnyshkov, A. Amo, L. Ferrier,E. Bernet-Rollande, E. Wertz, I. Sagnes, A. Lemaˆıtre,P. Senellart, G. Malpuech, and J. Bloch, Phys. Rev.Lett. , 036405 (2012).[25] See Supplementary Information for a complete descrip-tion of the model and additional results under high den-sity excitation.[26] T. Freixanet, B. Sermage, A. Tiberj, and R. Planel,Phys. Rev. B , 7233 (2000).[27] S. Choi, S. A. Morgan, and K. Burnett, Phys. Rev. A , 4057 (1998).[28] M. Wouters, T. C. H. Liew, and V. Savona, Phys.Rev. B , 245315 (2010); M. Wouters and V. Savona, arXiv:1007.5453v1 (2010).[29] T. Gao, P. S. Eldridge, T. C. H. Liew, S. I. Tsintzos,G. Stavrinidis, G. Deligeorgis, Z. Hatzopoulos, and P. G.Savvidis, Phys. Rev. B , 235102 (2012).[30] G. Tosi, G. Christmann, N. G. Berloff, P. Tsotsis, T. Gao,Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg,Nature Phys. , 190 (2012).[31] D. Ballarini, M. D. Giorgi, E. Cancellieri, R. Houdr,E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, andD. Sanvitto, arXiv:1201.4071v1 (2012).[32] A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. delValle, M. D. Mart´ın, A. Lemaitre, J. Bloch, D. N.Krizhanovskii, M. S. Skolnick, C. Tejedor, and L. Vi˜na,Nature , 291 (2009).[33] C. Adrados, T. C. H. Liew, A. Amo, M. D. Mart´ın,D. Sanvitto, C. Ant´on, E. Giacobino, A. Kavokin, A. Bra-mati, and L. Vi˜na, Phys. Rev. Lett. , 146402 (2011).[34] M. Galbiati, L. Ferrier, D. D. Solnyshkov, D. Tanese,E. Wertz, A. Amo, M. Abbarchi, P. Senellart, I. Sagnes,A. Lemaˆıtre, E. Galopin, G. Malpuech, and J. Bloch,Phys. Rev. Lett. , 126403 (2012). SUPPLEMENTARY INFORMATIONCondensation and relaxation model
In this supplementary, we describe the theoreticalmodel used to simulate polariton propagation, amplifi-cation, and relaxation in 1D wire microcavities. As amain tool, we have used the Gross-Pitaevskii equationfor polaritons in the parabolic approximation valid atlow wavevectors, which is the regime observed in the ex-periments. This Gross-Pitaevskii equation has been ex-tended to take into account the stimulated replenishingof the propagating condensate and its energy relaxationdue to the interaction of polaritons with the reservoir ofuncondensed excitons. The latter are strongest when thecondensate crosses the region of space where the reser-voir is present, localized under the excitation spot. The relaxation mechanism assumes a thermalized reservoir inwhich excitonic lower energy states are more populatedthan higher energy ones. In that case, dissipation of thecondensate energy takes place because the probability forcondensate polaritons to scatter down is larger than toscatter up, since it is easier to find a low-energy excitonin the reservoir which would go up (conserving the to-tal energy), than to find a high energy one able to godown. Once the polariton-reservoir exciton scatteringhas occurred, the extra energy gained by the reservoiris quickly dissipated by phonons, which interact stronglywith excitons. For the condensed polaritons, interactionswith phonons have been shown to play a negligible rolein their relaxation (Ref. 12 of the main text).The 1D Gross-Pitaevskii equation for polaritons withphenomenologically included lifetime reads: i (cid:126) ∂ψ ( x, t ) ∂t = − (cid:126) m pol ∆ ψ ( x, t ) + α (cid:16) | ψ ( x, t ) | + n R ( x, t ) (cid:17) ψ ( x, t ) − i (cid:126) τ pol ψ ( x, t ) (1)where m pol = 4 × − m is the polariton mass ( m isthe free electron mass ), α = 3 E b a B /W is the polariton-polariton interaction constant ( E b = 6 meV is the excitonbinding energy, a B = 10 nm is the exciton Bohr radiusand W = 3 . µ m is the width of the wire), τ pol = 30 psis the polariton lifetime.The stimulation of reservoir excitons towards the con-densate can be phenomenologically described by a termproportional to the condensate wavefunction and to thereservoir density n R , whose profile is determined by thatof the pump: + iγn R ( x, t ) ψ ( x, t ) (2)The energy towards which the condensate relaxes isdetermined by the local blueshift created by the reser-voir. While crossing the reservoir zone, the polaritonpacket relaxes towards the top of the potential created bythe reservoir, which is exponentially decaying with time.The relaxation of the condensate can be described in the lowest-order approximation by adding a small imaginarypart to the Hamiltonian [27, 28]: i (cid:126) ∂ψ ( x, t ) ∂t = (1 − i Λ ( x, t )) ( H − µ ( x, t )) ψ ( x, t ) (3)where Λ ( x, t ) (cid:28) x, t ) describes thereforenot the decay of the particles, but the decay of the ex-cess energy and its convergence towards µ ( x, t )). In ourcase, Λ ( x, t ) is a function of space and time, and of thereservoir density, since the relaxation takes place onlyin the region of the reservoir. The same applies also tothe chemical potential µ ( x, t ) = αn R ( x, t ) (interactionsin the condensate can be neglected in this regime withrespect to the potential created by the reservoir [23]).The final Gross-Pitaevskii equation for the condensatedynamics reads: i (cid:126) ∂ψ∂t = − (cid:126) m pol ∆ ψ + iγn R ( x, t ) ψ − i (cid:126) τ pol ψ + α (cid:16) | ψ | + n R ( x, t ) (cid:17) ψ + i Λ n R ( x, t )max ( n R ( x, t )) (cid:126) m pol ∆ ψ (4)where the relaxation parameter Λ ( x, t ) =Λ n R ( x, t ) / max ( n R ( x, t )) contains the reservoirdensity explicitly (the max ( n R ( x, t )) normalizes the re- laxation to the maximal density of the reservoir at t = 0.In our simulations, we have taken the fitting parameterΛ = 0 .
2. It is clearly seen that the kinetic energy of thecondensate decays because of the relaxation governedby the last term in Eq. 4, while the potential energyresulting from the condensate interactions (previous tothe last term) does not decay.Finally, the reservoir dynamics is described by theequation ∂n R ∂t = D ∆ n R − n R τ R − γn R | ψ | + P ( t ) exp (cid:32) − ( x − x ) σ (cid:33) where D = 5 × − µ m ps − is the diffusion coeffi-cient (fitting parameter obtained from experimental im-age at low pumping), τ R = 300 ps is the reservoir life-time, P = P δ ( t ) is the pumping, which directly createsthe reservoir population at t = 0 with P = 1200 µ m − ,and σ = 4 µ m is the width of the pump. We completethe initial conditions by imposing the presence of a con-densate at t = 0 possessing the same spatial profile asthe reservoir: ψ ( x,
0) = 1 × exp( − ( x − x ) /σ ). Thisinitial condition in our simulations accounts for the ini-tial relaxation of polaritons forming the condensate state.Once it is formed, the dominant mechanism populatingthe condensate is the stimulated scattering described byEq. 2, in which γ is a fitting parameter with a value of3 × µ m s − .The configuration of the present experiment does notallow to apply the self-consistent mixed Boltzmann-Gross-Pitaevskii model for polariton relaxation withrenormalization of the condensed states developed re-cently for microcavity micropillars [34], because in thepresent case the condensation occurs on free propagatingstates, which are not confined. This is why we used thesimplified model corresponding to T = 0 K, with first or-der phenomenological relaxation terms. The importanceof both re-stimulation and relaxation is clear from Figs.2 and 3 of the main text. Polariton-polariton scatteringwithin the condensate is fully taken into account by theGross-Pitaevskii Eqs. 1 or 4. Time evolution of the condensate energy at highexcitation density
Figure S1 shows the measured time and energy re-solved emission corresponding to the high excitation den-sity case (50 P th ) shown in Fig. 4(a). The upper traceshows the energy measured at the position of the exci-tation spot, coming from the emission from the k = 0polaritons that condense in that region. The initialblueshift arises from the strong exciton-polariton inter-actions, with a decay in time corresponding to the life- time of the excitonic reservoir. The lower trace showsthe emission from a region to the right of the excitationspot, between this and the edge of the microwire. Inthis case of very high excitation density and large reser-voir induced blueshift in the excitation region ( ∼ FIG. 5. (Color online) Experimentally measured time and en-ergy resolved emission in the conditions of Fig. 4 of the maintext. The upper trace shows the emission from the region atY=0, where the excitation spot is located. The lower traceshows the energy measured at Y=15 µ m, where the conden-sate is trapped and rebounds elastically between the edge ofthe microwire and the potential barrier created by the reser-voir.m, where the conden-sate is trapped and rebounds elastically between the edge ofthe microwire and the potential barrier created by the reser-voir.