Long range correlations in a 97% excitonic one-dimensional polariton condensate
Aurélien Trichet, Emilien Durupt, François Médard, Sanjoy Datta, Anna Minguzzi, Maxime Richard
LLong range correlations in a 97% excitonic one-dimensional polariton condensate
Aurélien Trichet,
1, 2
Emilien Durupt, François Médard,
1, 3
Sanjoy Datta, Anna Minguzzi, and Maxime Richard ∗ Institut Néel, Université Joseph Fourier and CNRS, B.P. 166, 38042 Grenoble, France Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom Institut Pascal (IP), Clermont Université, Université Blaise Pascal, BP 10448, F-63000 Clermont-Ferrand, France Université Grenoble I and CNRS, Laboratoire de Physique et Modélisationdes Milieux Condensés, UMR 5493, B.P. 166, 38042 Grenoble, France (Dated: November 7, 2018)We report on the realization of an out-of-equilibrium polariton condensate under pulsed excitationin a one-dimensional geometry. We observe macroscopic occupation of a polaritonic mode with only photonic fraction, and a nature strikingly close to that of a bare exciton condensate. With thehelp of this tiny photonic fraction, the condensate is found to display first-order coherence overdistances as large as 10 microns. Based on a driven-dissipative mean field model, we find that thecorrelations length is limited by the effects of a shallow disorder under non-equilibrium conditions. PACS numbers: 71.36.+c,78.55.Et,71.35.Lk,67.85.De
By cooling down a gas of bosons below a critical tem-perature, a large fraction of the gas suddenly "condenses"into a single quantum state. This phase is character-ized by the formation of off-diagonal long range order,that reaches infinity only in the specific case of a uniformBose-Einstein condensate [1]. The transition to the quan-tum regime occurs when the average correlation length λ c within the gas becomes comparable with the mean in-terparticle distances d (cid:39) n − / , where n is the particledensity. This quantum-coherent regime is not restrictedto Bose-Einstein condensates. In more complex quantumfluids, the range of the correlations is in general large butfinite, and its decay is determined by the interactionsstrength, the dimensionality-dependent quantum fluctu-ations, the losses, and the disorder. For example, the firstquantum many-body bosonic system to be identified assuch was superfluid Helium IV. Indeed London has beenthe first in 1938 to relate its behavior with the formationof a macroscopic "condensate" [2] in spite of its liquidphase, and strong interactions.Later, the notion of quantum gases has been shown tobe a pertinent one for excitonic gas in solid state envi-ronment. Indeed, Blatt and Moskalenko predicted thatthey could undergo Bose-Einstein condensation at ele-vated temperature ( T (cid:39) K ) thanks to their very lightmass [3, 4]. Pioneering experiments have been carriedout since then, showing that it is in fact difficult to reachthe quantum regime with excitons. The most strikingexamples are exciton gases in cuprate semiconductor ma-terials [5–7] and in coupled quantum wells [8]. In thesesystems, excitons have a lifetime long enough to thermal-ize so that the correlation length is set by the thermal deBroglie wavelength λ x = (cid:112) π (cid:126) /m x k B T , where m x isthe exciton effective mass, k B is the Boltzmann constantand T the temperature. λ x typically amounts to a fewtens of nanometers at T = 1 K . The quantum regimeis in principle achievable since at condensation thresholdthe interparticle distance is still larger than the Bohr ra- dius. However, at such densities the long range nature ofCoulomb interactions causes excitons to scatter stronglywith each other, resulting in Auger driven heating of thegas, or biexciton formation [9]. Excitons can also gettrapped locally due to short range disordered potential[10]. All these mechanisms can be detrimental enoughto prevent reaching quantum degeneracy. It is only veryrecently that a first indication of long-range correlationsin a bare exciton gas has been reported in a ultra-highquality coupled quantum wells system [11]. But a clearcut demonstration has remained elusive so far.Over the last decade, another strategy has proven morefruitful to achieve quantum phase of Bose gases in solidstate environment. In semiconductor microcavities in thestrong coupling regime, elementary excitations are notbare excitons but exciton-polaritons, which can be under-stood as excitons strongly dressed with the cavity photonfield. With polariton gases, Bose-Einstein condensation[12] and superfluid behavior [13–15] have been demon-strated unambiguously together with many interestingnew features resulting from their out-of-equilibrium sit-uation. The reason for this success is that polaritonshave a mass m p typically four orders of magnitude lighterthan that of bare excitons m x . Whether polaritons areat thermal equilibrium or not, this ultralight mass is thekey feature. Indeed, in the latter case usually referred aspolariton lasing [16], the maximum kinetic energy of thecondensate is limited by the finite polariton linewidth (cid:126) Γ instead of kT in the thermal case, i.e. (cid:126) k p / m p = (cid:126) Γ where λ p = 2 π (cid:126) /k p is the correlation length. There-fore, with a correlation length typically 100 times largerthan bare excitons, polaritons enter the quantum regimeat a density six orders of magnitude smaller. At suchsmall density, Coulomb interactions between polaritons[17] have a much less detrimental effect for the formationof long range correlations.Interestingly, the polariton effective mass m p dependson the photonic weight | G | of the dressed state according a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r to the simple relation: m p = | G | m g + | X | m x (1)where m x is the undressed exciton mass, m g is the barecavity effective mass, | X | = 1 − | G | is the excitonicweight of the dressed state. This simple equation hasa striking consequence: since typically m g /m x < − ,even excitons with a few percent of photonic dressingmight become light enough to overcome the difficultiesencountered with gases of bare excitons.In this letter, we show that in ZnO microwires featur-ing one dimensional confinement, a polariton condensatewith only | G | = 3% photonic fraction is obtained abovea threshold density by polariton lasing mechanism. Wefind that in spite of this largely dominant excitonic na-ture, the measured correlation length of this nearly ex-citonic condensate is surprisingly large, i.e. comparablewith that previously reported in condensate of ∼ photonic polaritons. Using a mean field model, we dis-cuss in detail the measured first order correlations func-tion and find that its decay is well accounted for by thenon-equilibrium features of the measurement in the pres-ence of a shallow disorder and weak interactions.ZnO microwire polaritons result from the strong cou-pling regime between the whispering gallery modes(WGMs) and the bulk ZnO excitonic transitions (usu-ally labeled A, B and C in order of increasing transitionenergy, their splitting resulting from the wurtzite uni-axial symmetry [18]). Due to momentum discretizationin the direction perpendicular to the wire axis, exciton-polaritons are intrinsically one-dimensional in this system[19]. Very large Rabi splittings ranging from meV to meV have been reported already in such microwires[19, 20] which result from the large intrinsic dipole mo-ment of ZnO exciton, and from the large integral overlapbetween the WGMs and the excitonic medium. More-over, owing to a very small excitonic Bohr radius of a B = 2 . nm , polaritons in ZnO are weakly interactingas compared to other materials.One-dimensional polariton gases have been studied al-ready in other systems and materials: long range corre-lations [21] and complete measurement of the one-bodydensity matrix [22] have been reported in linear defectsof a Telluride microcavity. Similar results have beenobtained in high-Q Arsenide-based microcavities etchedinto linear waveguides [23]. In all these works the pho-tonic fraction is | G | (cid:46) . The realization of polaritonstates with a large excitonic fraction is an experimentalchallenge since it is difficult to obtain well-defined po-lariton mode of energy close to that of the bare excitonreservoir. To do so, one needs a very large Rabi splittingand a very low excitonic inhomogenous broadening, sothat very excitonic modes remain in the motional nar-rowing regime [24], i.e. the modes remain spectrally nar-row and well separated from the exciton reservoir. ZnO FIG. 1. (Color online) a) Measured photoluminescence inmomentum space over a large spectral range. Several onedimensional lower polariton branches are visible, each corre-sponding to a different transverse state. Dashed lines : fit ofthe dispersion branches using a semiclassical model (see text).(b) Spectral zoom over the spectral range of interest, belowthe lasing threshold, and (c) above. (d) scanning electronmicrograph of the microwire. microwires fulfill both requirements: they display thelargest Rabi splitting ever reported in inorganic semi-conductor materials and an excitonic homogeneous plusinhomogeneous broadening which we measured to be aslow as . meV at cryogenic temperature, this latter fea-ture being due to the excellent crystalline quality of thebulk excitonic medium.Experiments on cold ( T = 40 K ) single crystallineZnO microwires of hexagonal cross-section, with typically µm length and µm diameter are reported in this let-ter. A micro-photoluminescence setup is designed withspatial and angle-resolved detection capabilities. In ad-dition a modified Michelson interferometer [12, 21] is in-serted in the detection path in order to perform spatio-temporal first order correlations measurements along the1D polariton condensate. The polariton population isoptically excited along a chosen segment of the wire (typ-ically over µm ), with a frequency doubled Titanium-Sapphire picosecond laser tuned at resonance with theA exciton plus meV excess energy. Considering theexciton binding energy of meV , no free charges wereexcited in this way, nor C excitons.In a first step, the Rabi splitting of the microwire is ac-curately determined using energy and angle-resolved pho- FIG. 2. (Color online) a) Output polariton emission inten-sity (circles) and linewidth of the emission (stars) vs inputexcitation. Vertical dashed line: polariton lasing threshold.b) Blueshift of the polariton emission over the same range ofexcitation power. c) Filled blue circles: measured excitationthreshold for polariton lasing vs polariton branch energy. Hol-low circles above: theoretical gain peaks and typical linewidth(horizontal black line) of the mechanisms considered in thetext. toluminescence measurements realized under weak exci-tations, as shown in Fig.1.a. In this plot, several lowerpolariton branches are visible, each corresponding to thecoupling with a different WGM. Following a semiclas-sical approach [25] detailed in [26], the lower polaritondispersion branches are fitted with a single set of twofree parameters, i.e. the overlap integral and the Rabisplitting (cid:126) Ω . In this example, An overall Rabi splitting of (cid:126) Ω = 288 meV ± meV is found where both A and B ex-citon contributions are included; C exciton contributionis vanishing due to polarization selection rule. This valuehardly changes from wires to wires of diameter d > µm .Upon increasing excitation density, we observe the sud-den appearance of a macroscopic population of the k z = 0 polariton state in a polariton branch situated meV be-low the bare A exciton (cf. Fig.1.b and Fig.1.c). Dueto the pulsed excitation, this effect is of the polaritonlaser type. The latter is identified by studying the be-havior of the emission intensity I as a function of theexcitation power P . The results are shown in Fig.2.a.At low density, the polariton emission is linear with theexcitation intensity, as expected from phonon-assistedrelaxation of reservoir excitons. At intermediate den-sity, two-body scattering in the exciton reservoir startsto contribute to the dynamics and causes a nonlinearitywith an approximate I ∼ P . dependence. Scattering ofthe exciton bath with polaritons also causes a slight in-crease of the linewidth in this regime. Above threshold,in spite of the time integrated nature of the measurement,a much stronger nonlinearity shows up accompanied by an abrupt narrowing of the linewidth, whose size is lim-ited by the apparatus resolution of µeV .This is the usual behavior expected in the case of po-lariton lasing, with two unusual characteristics. First,unlike microcavities made of other materials like Ar-senides, Tellurides or Nitrides, the polariton emissionhardly blueshifts at all at threshold until the excitationintensity becomes ten times larger. This is due to thevery small excitonic Bohr radius in ZnO which resultsin lowered inter-polariton Coulomb interaction. It alsoguarantees a fortiori that lasing occurs in the strong-coupling regime. Second, we find that the photonic frac-tion of the condensate is as low as | G | = 3% . We extractthis value from the polariton laser energy (cid:126) ω p and Rabisplitting, using the expression of the electromagnetic en-ergy density in dispersive media [27]: | G | = 1 + Ω ω p − ω x ) , (2)where the A and B excitonic transition energies (cid:126) ω xA and (cid:126) ω xB have been merged into a single one since Ω (cid:29) ω xB − ω xA . Another estimate | G | = 2 . is con-sistent with the previous one and does not require theknowledge of Ω . It is obtained from the measurementof the effective masses and using Eq.(1) which simplifiesinto | G | (cid:39) m g /m p . The polaritonic effective mass isdeduced from the measured curvature of the dispersionrelation in Fig.1.b, and the bare photonic mass amountsto m g = n (cid:126) ω p /c with n = 2 . , the background di-electric constant in ZnO and c the speed of light.Among the numerous available lower polariton modes,why does the system choose a mode for lasing with such alarge excitonic fraction? In fact, as usually assumed withpolariton gases, the polariton laser is fed by a stimulatedinelastic scattering mechanism where free excitons in thereservoir are transferred towards the polariton states. Inorder to find out at which polariton energy the lowestthreshold occurs, we measured the excitation thresholdfor lasing versus the polariton energy, and compared theenergy of this lowest threshold with various possible scat-tering mechanisms. The results are shown in Fig.2. Itfollows that due to the large Rabi splitting all possiblemechanisms would provide laser gain in a spectral re-gion where the polaritons have a large excitonic fraction.Moreover, according to our measurement (blue rectanglein Fig.2.c), a gain mechanism based on biexciton (’X A X A ’in Fig.2.c) recombination is likely to be involved, as re-ported previously in Nitride planar microcavity [28]. Amechanism based on scattering between a free A exci-ton and a bound exciton (’BX’ in Fig.2.c) could also bea good candidate [29]. On the other hand, some othermechanisms, although popular in the domain of photonlasing in bulk ZnO slab, can be unambiguously ruled outby this comparison. These include exciton-LO phononscattering (’LO’ in Fig.2.c), and free A exciton-exciton FIG. 3. (Color online) Measured g (1) ( x, − x ) functions on twopoints of two different wires, below (red circles) and above(blue squares) threshold. Condensate energies: meV (a)and . meV (b). Solid lines: theoretical fit of the data:below (red line) and above (blue line) threshold (details in thesupplemental material). c) and d) microphotoluminescenceimages obtained below threshold. Dashed blue line: potentialexperienced by the polariton condensate of (a) and (b). scattering resulting in the so-called P-band ( P (1) and P (2) in Fig.2.c) [30].In order to determine the nature of this quasi-excitonicpolariton condensate confined in a one-dimensionalwaveguide, we performed spatial correlations measure-ments at zero time delay. This is obtained by inter-ferometric measurements performed with the condensateemission [12]. Since the excitation is pulsed, we measuredthe time-integrated normalized correlation function g (1) ( x, − x ) = (cid:82) T ψ ( x, t ) ψ † ( − x, t ) dt (cid:113)(cid:82) T | ψ ( x, t ) | dt (cid:82) T | ψ ( − x, t ) | dt (3)where ψ is the condensate wavefunction and the point x = 0 is chosen in the middle of the polariton conden-sate, homogeneously excited over µm by an ellipsoidallaser spot. Since in an out-of-equilibrium situation two orthree modes can sometime contribute to the condensate[31], the obtained interferogram is spectrally resolved inorder to select only the dominant transverse mode (Thefull g (1) ( x, − x, ω ) spectra are shown in the supplemen-tal information). The results are shown in Fig.3.a andFig.3.b in two different wires of similar diameters for thesake of generality. Below threshold, the correlations arevery short, far below our experimental resolution of µm ,and entirely determined by the numerical aperture of ourdetection objective [32]. Above threshold, in the lasingregime, we observe that a significant amount of correla-tions build up over ∼ µm length. This measurementshows that in spite of a largely excitonic nature, the en-hanced interactions and the much larger mass (by a factorof ∼ ) with respect to 50% photonic polaritons do notprevent the formation of correlations over long distances.The measured g (1) ( x, − x ) also exhibits modulations anddecay versus x slightly different for both condensates. By analyzing in detail these features, we can determine thenature of the condensate, i.e. whether the correlationslength is limited by the one dimensionality, the disorder,or the interactions. Note that in our experiment, unlikein most previous works, the correlations length is notlimited by the excitation spot size.To understand the role of the disorder and interac-tions, the condensate wavefunction ψ ( x, t ) is simulated[33] by a mean-field generalized Gross-Pitaevskii equa-tion including loss and gain, coupled via the gain termto a rate equation governing the dynamics of the exci-tonic reservoir density n R ( x, t ) [34], which in the case ofa delta-pulsed pump can be solved analytically [35]. Thesimulations include a Gaussian random disorder of corre-lation length l c and amplitude V , and take into accountthe finite aperture of the objective. We show in Fig.3 theresults for the first-order correlation function (3) withparameters close to the experimental conditions and twodifferent realizations of disorder for each condensate. Wefind that the mean-field model well captures the mainfeatures observed in the experiment [36]. The decay ofthe correlations in the model, due to phase fluctuations,originates from the out-of-equilibrium nature of the con-densate, under the combined effect of the random poten-tial which breaks spatial inversion symmetry and the ef-fect of the temporal average in Eq.(3). The modulationsare due to a shallow disorder of amplitude ∼ meV , i.e.comparable with the polariton linewidth. In the model,the disorder sets also the width of the central peak, whichin our measurements is of the order of the experimentalresolution. In this regime the interactions only have amarginal effect on the correlations. Simulations predicta peak broadening at stronger interactions or weaker dis-order.In conclusion, we have shown that polariton lasing inZnO microwires yields the formation of a one-dimensionalweakly-interacting condensate of quasi-excitonic nature.This peculiar feature is achieved thanks to weak excitonicinteractions together with a large Rabi splitting, and anegligible excitonic disorder of the bulk semiconductormaterial. In spite of the tiny photonic fraction, the cor-relations length of this condensate is found to be in theten micron range, i.e. comparable to that of a 50% pho-tonic polariton condensate. According to our mean-fieldanalysis, the first-order coherence is dominated by dis-order while interactions have a negligible effect. In thisregime, the decay of correlations seems to be weakly af-fected by driven-dissipative induced noise [37] and quan-tum fluctuations [38]. Therefore, even heavier and moreexciton-like - i.e. above - ZnO microwire polaritonsshould still be able to easily condense.A.T. acknowledges financial support by the "Fondationnanoscience" RTRA contract No.FCSN 2008-10JE. E.D.,F.M., A.M. and M.R. acknowledge financial support bythe ERC grant "Handy-Q" No.258608. Enlightening dis-cussions with Le Si Dang and M. Wouters are warmlyacknowledged. We thank Prof. Zhanghai Chen for pro-viding us with high quality samples. ∗ [email protected][1] O. Penrose and L. Onsager, Phys. Rev. , 576 (1956).[2] F. London, Nature , 643 (1938).[3] J. M. Blatt, W. Brandt, and K. W. Boer, Physical Re-view , 1691 (1962).[4] S. A. Moskalenko, Soviet Physics - Solid State , 199(1962).[5] D. Hulin, A. Mysyrowicz, and C. BenoitalaGuillaume,Phys. Rev. Lett. , 1970 (1980).[6] D. Snoke, J. P. Wolfe, and A. Mysyrowicz, Phys. Rev.Lett. , 827 (1987).[7] J. L. Lin and J. P. Wolfe, Phys. Rev. Lett. , 1222(1993).[8] L. V. Butov, A. L. Ivanov, A. Imamoglu, P. B. Little-wood, A. A. Shashkin, V. T. Dolgopolov, K. L. Camp-man, and A. C. Gossard, Phys. Rev. Lett. , 5608(2001).[9] J. I. Jang and J. P. Wolfe, Phys. Rev. B , 045211(2006).[10] L. V. Butov, A. Imamoglu, A. V. Mintsev, K. L. Camp-man, and A. C. Gossard, Phys. Rev. B , 1625 (1999).[11] A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler,L. V. Butov, A. V. Kavokin, K. L. Campman, and A. C.Gossard, Nature , 584 (2012).[12] 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).[13] A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti,I. Carusotto, R. Houdre, E. Giacobino, and A. Bramati,Nat. Phys. , 805 (2009).[14] K. G. Lagoudakis, T. Ostatnicky, A. V. Kavokin, Y. G.Rubo, R. Andre, and B. Deveaud-Pledran, Science ,974 (2009).[15] A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet,I. Carusotto, F. Pisanello, G. Leménager, R. Houdré,E. Giacobino, C. Ciuti, and A. Bramati, Science ,1167 (2011).[16] J. Kasprzak, D. D. Solnyshkov, R. Andre, L. S. Dang,and G. Malpuech, Phys. Rev. Lett. , 146404 (2008).[17] M. Vladimirova, S. Cronenberger, D. Scalbert, K. V. Ka-vokin, A. Miard, A. Lemaitre, J. Bloch, D. Solnyshkov,G. Malpuech, and A. V. Kavokin, Phys. Rev. B , 075301 (2010).[18] C. Klingshirn, J. Fallert, H. Zhou, J. Sartor, C. Thiele,F. Maier-Flaig, D. Schneider, and H. Kalt, Phys. Stat.Sol. (b) , 1424 (2010).[19] A. Trichet, L. Sun, G. Pavlovic, N. A. Gippius,G. Malpuech, W. Xie, Z. Chen, M. Richard, and L. S.Dang, Phys. Rev. B , 041302 (2011).[20] L. Sun, Z. Chen, Q. Ren, K. Yu, L. Bai, W. Zhou,H. Xiong, Z. Q. Zhu, and X. Shen, Phys. Rev. Lett. , 156403 (2008).[21] F. Manni, K. G. Lagoudakis, B. Pietka, L. Fontanesi,M. Wouters, V. Savona, R. Andre, and B. Deveaud-Pledran, Phys. Rev. Lett. , 176401 (2011).[22] F. Manni, K. G. Lagoudakis, R. Andre, M. Wouters, andB. Deveaud, Phys. Rev. Lett. , 150409 (2012).[23] 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, Nat.Phys. , 860 (2010).[24] V. Savona, C. Piermarocchi, A. Quattropani, F. Tassone,and P. Schwendimann, Phys. Rev. Lett. , 4470 (1997).[25] V. Savona, C. Piermarocchi, A. Quattropani,P. Schwendimann, and F. Tassone, Phase Transi-tions , 169 (1999).[26] A. Trichet, F. Médard, J. Zuñiga-Pérez, B. Alloing, andM. Richard, New Journal of Physics , 073004 (2012).[27] T. G. Philbin, Phys. Rev. A , 013823 (2011).[28] P. Corfdir, J. Levrat, G. Rossbach, R. Butte, E. Feltin,J.-F. Carlin, G. Christmann, P. Lefebvre, J.-D. Ganiere,N. Grandjean, and B. Deveaud-Pledran, Phys. Rev. B , 245308 (2012).[29] B. Hönerlage and U. Rässler, Journal of Luminescence , 593 (1976).[30] C. Klingshirn and H. Haug, Physics Reports , 315(1981).[31] D. N. Krizhanovskii, K. G. Lagoudakis, M. Wouters,B. Pietka, R. A. Bradley, K. Guda, D. M. Whittaker,M. S. Skolnick, B. Deveaud-Pledran, M. Richard, R. An-dre, and L. S. Dang, Phys. Rev. B , 045317 (2009).[32] See details in the Supplemental Material.[33] P. Muruganandam and S. Adhikari, Computer PhysicsCommunications , 1888 (2009).[34] M. Wouters and I. Carusotto, Phys. Rev. Lett. ,140402 (2007).[35] See details in the Supplemental Material.[36] A description beyond mean field is currently being devel-oped, starting from the disorderless case [37].[37] A. Chiocchetta and I. Carusotto, (2013),arXiv:1302.6158.[38] F. D. M. Haldane, Phys. Rev. Lett.47