From Polariton Condensates to Highly Photonic Quantum Degenerate States of Bosonic Matter

From polariton condensates to highly photonic quantum degenerate states of bosonic matter

Marc Aßmanna,1, Jean-Sebastian Tempela, Franziska Veita, Manfred Bayera, Arash Rahimi-Imanb, Andreas Löfflerb, Sven Höflingb, Stephan Reitzensteinb, Lukas Worschechb, and Alfred Forchelb

aExperimentelle Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany; and bTechnische Physik, Physikalisches Institut and Wilhelm Conrad Röntgen Research Center for Complex Material Systems, Universität Würzburg, D-97074 Würzburg, Germany

Edited by Peter Littlewood, University of Cambridge, Cambridge, United Kingdom, and accepted by the Editorial Board November 3, 2010 (received for review July 7, 2010)

Bose–Einstein condensation (BEC) is a thermodynamic phase tran- tion and thermalized polariton gases to have condensation sition of an interacting Bose gas. Its key signatures are remarkable effects. It should be pointed out that the polariton system as a quantum effects like superfluidity and a phonon-like Bogoliubov whole is almost never in thermal equilibrium. Usually it will con- excitation spectrum, which have been verified for atomic BECs. sist of the low-energy part of the polariton distribution which may In the solid state, BEC of exciton–polaritons has been reported. be described by an effective temperature if this part reaches a Polaritons are strongly coupled light-matter quasiparticles in semi- local self-equilibrium, the high-energy exciton-like part therma- conductor microcavities and composite bosons. However, they are lized at the lattice temperature, and the intermediate bottleneck subject to dephasing and decay and need external pumping to regime which is almost always out of equilibrium. It has been sta- reach a steady state. Accordingly the polariton BEC is a nonequili- ted that such a local self-equilibrium condensed state is directly brium process of a degenerate polariton gas in self-equilibrium, related to BEC, whereas one which does not obtain a local self- but out of equilibrium with the baths it is coupled to and therefore equilibrium can hardly be described in the framework of BEC deviates from the thermodynamic phase transition seen in atomic (15). Although this statement is true in terms of thermodynamic BECs. Here we show that key signatures of BEC can even be ob- properties, it is not immediately clear which of the signatures seen served without fulfilling the self-equilibrium condition in a highly in local self-equilibrium polariton condensates will still prevail far photonic quantum degenerate nonequilibrium system. from even local self-equilibrium. Recent theoretical analysis (16) predicts condensation effects even when the local self-equili- photon statistics ∣ quantum optics ∣ semiconductor photon sources brium condition is not fulfilled and also not only for the equilibrium Bose–Einstein distribution, but also for a wide range of icrocavity polaritons are composite bosons, which are partly nonequilibrium distribution functions. However, although there Mphotons and partly excitons as quantified by the Hopfield are theoretical studies classifying the condensate ground state coefficients jC2j and jX 2j giving the relative photonic and exci- for different detunings and mean polariton separations (17), from tonic content (1), respectively, and are expected to condense at an experimental point of view, detailed studies of this regime high temperatures because of their light mass (2). Moreover, the without local self-equilibrium are still missing. photonic and excitonic contents of polaritons can be precisely We realize a system far from equilibrium using polaritons Δ ¼ − adjusted by changing the detuning Ec Ex between the bare with high-photonic content, characterized by a negative detuning cavity mode and the bare exciton mode. Unlike Bose–Einstein Δ < 0 between the bare cavity and exciton modes, and excited by condensates (BECs) in atomic gases, solid-state systems are short laser pulses with duration of 1.5 ps. The fast decay times subject to strong dephasing and decay on timescales on the order because of the high-photonic content of the polaritons ensure of the particle lifetimes. As a consequence, external pumping is that the polariton distribution is neither in thermal equilibrium required to achieve a steady state. Despite this nonequilibrium with the lattice nor in local self-equilibrium and a stationary character, degenerate polariton systems show several textbook situation is never reached. The thermalization times for our sam- features of BECs (3), including spontaneous build up of coher- ple agree well with values previously reported (18) where the ence (4) and polarization (5), quantized vortices (6, 7), spatial polariton thermalization time becomes shorter than their lifetime condensation (8), and superfluidity (9, 10). Usually this behavior above the degeneracy threshold for positive detunings, but is is attributed to the system undergoing an equilibrium phase always longer than their lifetime for negative detunings. Although transition toward a condensed state: Although the polariton gas there have been experiments where this regime was called out of is not necessarily in equilibrium with the lattice, it is in self- equilibrium condensation (19) or metastable condensate (15), so equilibrium, if the relaxation kinetics of excited carriers is fast far no systematic studies characterizing the condensate-like prop- enough compared to the leakage of the photonic component erties in this regime where it is not possible to define an effective out of the cavity, which is usually the case for positive detunings temperature have been performed. We will refer to highly photo- Δ ≥ 0. In this case jX2j is larger than 50%, an effective tempera- nic (HI-P) polariton states when discussing this nonequilibrium ture can be defined and the polariton gas can be considered as a regime in order to stress that it is different from common polar- thermodynamic equilibrium state, which is out of equilibrium iton condensates which are also nonequilibrium states, but can be with the baths it is coupled to. This degenerate polariton gas is described by an effective temperature. In the following we test distinguishable from a simple photon laser (11). It is often pointed out that this intrinsic nonequilibrium situation and the Author contributions: M.A. and M.B. designed research; M.A., J.-S.T., F.V., and A.R.-I. two-dimensional order parameter of polariton BECs give rise performed research; A.R.-I., A.L., S.H., S.R., L.W., and A.F. contributed new reagents/ to interesting phenomena like half-vortices (12) and a diffusive analytic tools; M.A., J.-S.T., F.V., and A.R.-I. analyzed data; and M.A. and M.B. wrote Goldstone mode (13, 14), which do not occur in equilibrium the paper. condensates. Accordingly, the next interesting questions are The authors declare no conflict of interest. whether the same or even unique phenomena occur, if the system This article is a PNAS Direct Submission. P.L. is a guest editor invited by the Editorial Board. is driven even further from equilibrium into a regime that cannot Freely available online through the PNAS open access option. be described by an effective temperature and whether it is indeed 1To whom correspondence should be addressed. E-mail: [email protected]. necessary to consider a thermodynamic equilibrium phase transi- uni-dortmund.de.

1804–1809 ∣ PNAS ∣ February 1, 2011 ∣ vol. 108 ∣ no. 5 www.pnas.org/cgi/doi/10.1073/pnas.1009847108 Downloaded by guest on September 27, 2021 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HI-P states for several common signatures of condensation like Γ Γ2 ω ð Þ¼ω ð0Þ − ½ω ð Þ − ω ð0Þ2 − build up of a macroscopic ground-state occupation, suppressed k∥ Bog i 2 Bog k∥ Bog 4 ; quantum fluctuations, and linearization of the excitation spectrum. αβ where Γ ¼ γ 1þαβ is the effective decay rate of the Bogoliubov Basic characteristics of condensation and the excitation spec- mode, γ is the bare cavity decay rate, α ¼ P − 1, where thr means trum are manifested in the polariton dispersion. Above threshold Pthr threshold, gives the relative pump strength compared to the a blueshift of the kjj ¼ 0 lower polariton (LP) is expected and the threshold value, and β quantifies the dependence of the conden- LP dispersion is supposed to change from a parabolic shape in sate amplification rate because of stimulated scattering of polar- the uncondensed case toward a different dispersion in the low- pffiffiffiffiffiffiffiffiffiffiffiffiffiℏ itons on the reservoir population. In the following we assume the momentum condensed regime jk∥ξj < 1, where ξ ¼ is β 2m gn effect of to be negligible above threshold, so that the variation LP c Γ γ the healing length of the condensate (20). In standard homoge- of between zero at threshold and at large pump rates is gov- α neous equilibrium Bogoliubov theory, the expected dispersion is erned by alone. This diffusive spectrum is expected to manifest itself as a flat region in the low-momentum region of the real a phonon-like linear one at small k given by part of the excitation spectrum. The range of this flat region is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expected to increase with α, starting from an unmodified Bogo- ω ð Þ¼ω ð Þþ þ þ ð ξÞ2½ð ξÞ2 þ 2 Bog k∥ LP k∥ gnc grnr gnc k∥ k∥ : liubov dispersion exactly at the threshold, where α is exactly zero. In Fig. 1 we compare the LP dispersion of the emission copolarized with the excitation for detunings of 0, −2, −4, and −7 meV to Here g and gr are coupling constants describing the interaction the theoretical predictions below and above threshold. Below between two condensate polaritons and between condensate A E I M polaritons and reservoir excitons, respectively, and n and n are threshold (Fig. 1 , , , and ) the standard quadratic LP dis- c r persion is observed in all cases. Increasing the excitation power their densities. Using resonant excitation, the reservoir exciton near the threshold value (Fig. 1 B, F, J, and N), a blueshift of contributions are expected to be negligible. Highly photonic con- the kjj ¼ 0 emission energy becomes apparent, whereas the cross- densed states are neither in thermal equilibrium, nor spatially circularly polarized emission component never shows threshold- homogeneous. Their spatial extent is given by the finite size of like behavior. In addition to the blueshift of the copolarized the pump spot. Therefore one might expect their dispersion to emission, the dispersion starts to differ from the standard quad- show strong deviations from the ideal homogeneous equilibrium ratic dispersion. Blue and white lines in Fig. 1 give the quadratic Bogoliubov dispersion. In particular, the finite pump spot size LP and calculated equilibrium Bogoliubov dispersions for the causes quantization and the large amount of pump and decay condensate blueshift. We use the experimentally determined causes the excitation spectrum to become diffusive at low-mo- values of the blueshift for the interaction energies in calculating ω ω mentum (13, 14), Bog. The prediction for the diffusive modes is not shown PHYSICS

Fig. 1. Polariton dispersions on a logarithmic scale at several excitation densities along the threshold for detunings of Δ ¼ 0 meV (Top), −2 meV (Second Row), −4 meV (Third Row), and −7 meV (Bottom). Black, blue, white, and green lines represent the calculated LP, blue-shifted LP, homogeneous equilibrium Bogoliubov, and diffusive Goldstone mode dispersions, respectively.

Aßmann et al. PNAS ∣ February 1, 2011 ∣ vol. 108 ∣ no. 5 ∣ 1805 Downloaded by guest on September 27, 2021 because at the threshold the real part of the positive branch of the h ^ 3 i ð3Þðτ ¼ 0Þ¼ :n : diffusive mode does not differ from the equilibrium Bogoliubov g 3 ; hn^i dispersion. It is apparent that at threshold the equilibrium Bogo- liubov dispersion cannot reproduce the measured dispersion where the double stops denote normal ordering of the underlying accurately for Δ ¼ 0, −2,or−4 meV. Here the measured disper- photon field operators. Photon lasers are expected to show sions lie approximately in the middle between both theoretical statistically independent emission as described by Poissonian dispersions, and the dispersions are not even symmetric with ð2Þ ð3Þ statistics [corresponding to g ð0Þ¼g ð0Þ¼1] above threshold, respect to kjj ¼ 0. The latter feature becomes more apparent with on the other hand increased fluctuations because of nonresonant increasing negative detuning and is a signature of the nonequili- scattering processes between polariton pairs (21–23) are expected brium state favoring the presence of polaritons having a wavevec- for operation well above the stimulation threshold for polaritonic tor with the same sign as the pump incidence wavevector, condensates, resulting in gð2Þð0Þ > 1. Below threshold both are although the first can be attributed to the inhomogeneity of the ð2Þð0Þ¼2 ð3Þð0Þ¼6 ¼ 0 supposed to emit thermal light with g and g . system: Although the LPs with kjj are stationary, LPs with To test these predictions we resonantly excite LPs with large ≠ 0 −1 kjj will move across the excitation spot and experience a dif- transverse momentum kjj ¼ 5.8 μm using circularly and linearly ferent interaction energy given by the spatial pump pulse profile. polarized laser pulses. Subsequently those polaritons relax However, it is striking that the theoretical prediction again toward the ground state by acoustic phonon emission or polari- matches the experimental data well for the Δ ¼ −7 meV disper- ton–polariton scattering. If the occupation of the ground state sion recorded at threshold. Further above threshold (Fig. 1 C, G, becomes large enough, bosonic final-state stimulation is expected K, and O) the k-space region of highest intensity moves signifi- to significantly increase the scattering toward this state. We make cantly closer toward kjj ¼ 0. The green lines give the calculated use of a recently developed streak camera technique to determine diffusive Goldstone mode dispersion. At negative detunings there the second- and third-order correlation functions (24, 25). still is no complete reflection symmetry between kjj and −kjj, but Although there have been experimental investigations of the the positive wavevector half of the dispersion now shows reason- second-order coherence properties of polariton condensates able agreement with the equilibrium Bogoliubov dispersion and for GaAs (26, 27) and CdTe (28, 29) structures before, the results −1 the diffusive Goldstone mode dispersion up to kjj ≈ 0.75 μm are contradictory. We resolve this question by performing without usage of any fitting parameters. This modified dispersion systematic studies of the higher order coherence properties in ð2Þð0Þ ð3Þð0Þ indeed is a sign of collective behavior as expected in a condensed terms of g and g for different detunings and excitation and thermalized polariton gas. However, it is not immediately schemes and show that they are indeed strongly dependent clear whether the equilibrium Bogoliubov dispersion or the dif- on both. ð2Þð0Þ fusive Goldstone mode dispersion corresponds better to the ex- As can be seen in Fig. 2, g does not reach the expected perimental results because their main difference lies in the thermal value of two for any detuning as we do not single out one low-momentum regime where the excitation spectrum and the fundamental mode like in earlier publications (25), but detect condensate luminescence are superimposed. Note that the lumi- both realizations of the spin-degenerate ground state simulta- nescence from higher energy gets gradually larger with increased neously. Polaritons with different spin states will not interfere ð2Þð0Þ ð3Þð0Þ photonic content of the LP which is a clear indication of the with each other and therefore the values of g and g occupation achieving gradually a more nonequilibrium character. expected in the thermal regime of two superposed modes are Further increase of the excitation power (Fig. 1 D, H, L, and P) 1.5 and 3, respectively. Note further that if plotted on a compar- ð2Þð0Þ ð3Þð0Þ results in a more pronounced occupation of the condensate able scale g and g give results which are in quantitative ground state. Although the reflection symmetry between positive agreement. Under linear excitation all detunings between Δ ¼þ2 Δ ¼ −10 and negative kjj is still not perfect, both parts of the dispersion can and meV show a threshold, which is evidenced ð2Þð0Þ ð3Þð0Þ now be described by the same equilibrium Bogoliubov dispersion by a decrease in g and g and agrees well with the position of the threshold (shown as green dashed lines in Fig. 2) with accuracy. We interpret this behavior as a sign of effective – redistribution by polariton scattering processes. Although the evidenced in measurements of the input output curve. We define excitation spectrum is now strongly masked by the condensate the threshold as the excitation power at which the emission from the condensed state becomes stronger than the emission from the luminescence, the occupation at larger energies still varies with Δ ¼ −10 the detuning. Accordingly we are approaching a more therma- LP. In the case of strong negative detuning of meV no lized, but still nonequilibrium, regime for negative detunings. differences from a common photon lasing transition are observed Anyway, it should be noted that the shape of the excitation spec- under linearly polarized pumping. Additional dispersion measurements evidence that above threshold the photons are indeed trum shows no significant dependence on the occupation and emitted from the bare cavity mode in this case. Under circularly whether it is close to equilibrium or not. Again, it is difficult polarized pumping the threshold is not even reached for a detun- to differentiate whether the equilibrium Bogoliubov or diffusive ing of −10 meV. This result is in accordance with previous results Goldstone mode dispersion matches the experiment better. Ob- showing that polariton relaxation is less efficient under circularly viously there is some flat region at low momenta, especially at polarized pumping, which in turn leads to a higher threshold detunings of −2 or −4 meV, but because the emission from excitation power P (30). For all other detunings significant the condensate masks the emission from the excitation spectrum thr deviations from a simple photon laser behavior emerge. Even even stronger than at lower pump rates, this observation is not at high-excitation powers the ground-state emission has lower necessarily conclusive evidence for the presence of the flat disper- energy compared to the bare cavity mode with an energy differ- sion in momentum space. ence of at least 4 meV. Although this behavior is not necessarily a Because of their large photonic content, one might imagine proof that the strong coupling regime is still intact (31), it is at that the coherence properties of HI-P states are not different least very likely. The second- and third-order correlation func- from common photon lasers. This assumption can be tested by tions give further evidence that the system is not a simple photon probing the strength of quantum fluctuations via normal-ordered laser in the range of detunings between þ2 and −7 meV, corre- second- and third-order equal-time correlations of the photon 2 sponding to photonic contents in the range from jC j ≈ 44–70%. number n defined as Although at first a decay toward one is seen for linearly polarized −7 h ^ 2 i excitation, especially for a detuning of meV, an increase is ð2Þðτ ¼ 0Þ¼ :n : g 2 ; evidenced for further increased excitation densities. Depending hn^i on the detuning gð2Þð0Þ can reach values even higher than the

1806 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1009847108 Aßmann et al. Downloaded by guest on September 27, 2021 Fig. 2. Measured gð2Þð0Þ (black dots) and gð3Þð0Þ (red triangles) determined by simultaneous two- and three-photon detections of the whole fundamental mode emission for a wide range of excitation powers and detunings of −10 meV ðjC2j ≈ 77%Þ, −7 meV ðjC2j ≈ 70%Þ, −4 meV ðjC2j ≈ 62%Þ, −2 meV ðjC2j ≈ 56%Þ, 0 meV ðjC2j ≈ 50%Þ, and þ2 meV ðjC2j ≈ 44%Þ under linearly polarized (Upper) and circularly polarized (Lower) excitation. Red (blue) lines denote the coherent (thermal) limit. Green dashed lines give the position of the degeneracy threshold determined by measurements of the dispersion. Below threshold the necessary integration time for the correlation measurements increases drastically. Only data points for excitation powers where the results are not limited by the long-term stability of the setup are considered.

thermal value of 1.5. For further increased excitation power range. Instead gð2Þð0Þ decreases monotonically toward a value be- a smooth decrease back toward one is observed. For circularly tween 1.3 and 1.4. There is a trend toward further decrease at polarized excitation the general behavior of the correlation high-excitation power, however, the dependence on pump power functions is similar to the linearly polarized case as a decrease is very weak. Going to more negative detunings, the dip already and a reoccurrence of the degenerate mode quantum fluctuations seen without polarization-sensitive detection still occurs. Appar- can be identified for detunings between þ2 meV and −7 meV. ently, the excitation power corresponding to the occurrence of However, in this case the correlation functions can also show the dip takes on smaller values compared to Pthr for larger nega- increased fluctuations slightly above threshold as can be nicely tive detuning. Moreover, the dip in gð2Þð0Þ is most pronounced for −4 seen for a detuning of meV. This increase is caused by the the most negative detuning (−7 meV). In fact, for a detuning of build up of polarization. The thermal regime value of 1.5 is just −7 meV almost complete coherence is reached at an excitation valid for unpolarized two-mode emission. As the excitation power power of approximately 1.1 P and a steep rise to gð2Þð0Þ > 1.6 is reaches the threshold, the emission will also start to polarize and thr the two modes will not contribute equally to the correlation functions anymore (Materials and Methods). This onset of polarized emission will lead to an increase in gð2Þð0Þ whereas the build up of coherence will lead to a decrease. Although these results are in fair qualitative agreement with mean-field and reservoir calculations of the second-order correlation function of a polariton BEC (21, 22), they are not sufficient evidence for deviations from a photon laser as the nonmonotonous behavior of the correlation function can also be a result of the interplay of coherence and polarization. To make sure the onset of polarized emission is not the main PHYSICS influence on the observed behavior of the correlation function we studied gð2Þð0Þ under circularly polarized excitation also for the cocircularly polarized emission only. As shown in Fig. 3, here the expected values of gð2Þð0Þ¼2 are approximately reached in the limit of low-excitation power for all detunings except þ2 meV. At this detuning the emitted intensity below threshold Fig. 3. Measured gð2Þð0Þ of the cocircularly polarized fundamental mode for is too small to perform sensible measurements using our setup. ð2Þð0Þ a wide range of detunings and excitation powers under circularly polarized Above threshold the shape of g shows qualitative agreement excitation. Red (blue) lines denote the coherent (thermal) limit. Green dashed with theoretical results (21). On resonance it is apparent that full lines give the position of the degeneracy threshold determined by measure- coherence is not reached within the available excitation power ments of the dispersion.

Aßmann et al. PNAS ∣ February 1, 2011 ∣ vol. 108 ∣ no. 5 ∣ 1807 Downloaded by guest on September 27, 2021 evidenced at 1.5 Pthr. Calculations of the second-order correla- objective with a numerical aperture of 0.26. The signal was then focused tion function are generally done using one of two different ap- on a streak camera for correlation measurements, or the Fourier plane proaches (21): Mean-field calculations predict a decrease of was imaged on a monochromator for dispersion measurements. gð2Þð0Þ toward approximately 1.2 at the threshold, followed by a short rise for increasing excitation power until a constant value Details of Correlation Measurements. The basics of our photon correlation of about 1.3 is reached. This prediction is in agreement with our method were already explained in the references given in the text. Accessing extremely short pulses, like the emission from HI-P states in the degenerate results for no or small negative detuning. A two-reservoir model regime, introduces additional difficulties. In this regime the timing jitter predicts a sharp decrease of gð2Þð0Þ at the threshold, a strong re- ð2Þð0Þ¼1 6 becomes comparable to the emission pulse duration. Therefore the momen- currence of the photon bunching up to values of g . for tary intensity detected at a certain position of the CCD will be determined by excitation powers moderately larger than the threshold value and the photon statistics as well as by the momentary influence of timing jitter, a slow drop for even higher excitation powers. This model which will cause deviations from the real value of gð2Þ. The deviation is better reproduces our results for large negative detuning. We systematic, so it is possible to correct for this jitter effect. From a theoretical conclude that because of the increasing relaxation bottleneck and point of view, the effect of jitter can be modeled as a Gaussian distribution decreased scattering rate expected for large negative detuning the having a certain FWHM describing the peak position of the emitted pulse on two-reservoir model appears to be a valid description in the HI-P the CCD. This timing jitter will cause a broadening of the duration of the regime and the dip seen for several detunings is a sign of ineffi- averaged intensity compared to the real emission pulses. Accordingly gð2Þð0Þ will be overestimated whereas gð2ÞðτÞ will be underestimated for large cient scattering between polaritons from the degenerate ground ð2Þ state and those in excited states. Because the photon statistics enough τ. The exact measured g depends on the ratio of the timing jitter FWHM J to the pulse FWHM P. This situation can be simulated theoretically of emission depends strongly on the detuning we suppose that ð2Þ ð3Þ ð2Þ and shows that the measured gm ð0Þ and gm ð0Þ depend on the real g ð0Þ one might also find variations depending on the Rabi splitting ð3Þ and g ð0Þ corresponding to the following relation, and the excitonic and photonic decay constants. Therefore, microcavities operated in the HI-P regime may open up the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 possibility to introduce a high-intensity light source with tunable ð2Þ ð2Þ J ð3Þ ð3Þ J g ð0Þ¼g ð0Þ 1 þ ;gð0Þ¼g ð0Þ½1 þ : photon statistics. m P m P In conclusion we have driven a microcavity polariton system into a completely out of equilibrium degenerate HI-P state by in- 2 Accordingly it is possible to extract the real values from the measured ones if creasing its photonic content up to jC j ≈ 70% and examined the emission pulse widths and the jitter widths are known. The jitter width ð2Þ whether the system bears more similarities to an inverted system; can be determined by measuring gm for a light source with well-defined P i.e., a laser or a thermalized polariton BEC. Surprisingly, although and gð2Þ. We used a laser pulse with P ¼ 1.5 ps as determined by using an ð2Þ ð2Þ the matter component is small and the degeneracy transition autocorrelator and g ¼ 1. The resulting gm ðτÞ is discussed in detail in ref. 32 we have demonstrated cannot be considered a thermodynamic and is well-reproduced by a simulation using J ¼ 1.8 ps. equilibrium phase transition by any means, the system shows many P can be determined directly from the integrated streak camera data, features one would expect from a BEC. In particular we have which give the jitter-broadened emission pulse widths. Because all essential ð2Þð0Þ ð2Þð0Þ observed a modified dispersion and compared it to standard equi- parameters are known, it is possible to calculate g from gm . In the case of a superposition of two noninterfering modes A and B, the librium Bogoliubov theory predicting a linearized spectrum and ð2Þ resulting measured g ð0Þ is given by nonequilibrium Bogoliubov theory predicting a diffusive Gold- AB stone mode and demonstrated the detuning dependence of the ð2Þ ð0Þ¼ ð2Þð0Þ 2 þ ð2Þð0Þ 2 þ 1 emitted photon statistics. We believe our results can open up gAB gA RA gB RB RARB; the road for tunable photon statistics light sources and deepen the understanding of nonequilibrium phase transitions. where RA and RB are the relative intensity ratios of mode A and B to the total ð2Þð0Þ ð2Þð0Þ intensity. Therefore it is possible to calculate gA from gAB if the relative ð2Þð0Þ Materials and Methods intensity ratios and gB are known: Experimental Details. In our experiments the sample was kept at 8 K in a ð2Þ ð2Þ helium flow cryostat. The microcavity structure is basically identical to that ð0Þ − ð0Þ 2 − 1 ð2Þð0Þ¼gAB gB RB RARB used in earlier measurements (20), but was fabricated in another growth run. gA 2 : It consists comprehensively of 12 GaAs/AlAs quantum wells embedded in a RA planar microcavity with 16 (21) AlGaAs/AlAs mirror pairs in the top (bottom) ð2Þ distributed Bragg reflector. Reflectivity measurements gave results of Comparing single-mode g measurements to the values obtained by 1.6158 eV for the bare exciton energy and 13.8 meV for the Rabi splitting. two-mode measurements shows that for circularly polarized excitation the Using a Ti:Sapphire laser with a pulse duration of 1.5 ps and a repetition two modes are indeed independent and the cross-circularly polarized mode rate of 75.39 MHz the pump was focused to a spot approximately 30 μm stays thermal. in diameter on the sample at an angle of 45° from normal incidence. The ¼ pump was resonant with the LP branch at an in-plane wavenumber of kjj ACKNOWLEDGMENTS. The Dortmund group acknowledges support through −1 5.8 μm for resonant polariton injection or tuned to the first minimum of the Deutsche Forschungsgemeinschaft (DFG) research grant DFG 1549/ the cavity reflectivity curve at a wavelength of 744 nm for nonresonant and 15-1. The group at Würzburg University acknowledges support by the State incoherent pumping. We collected the emitted signal using a microscope of Bavaria.

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Aßmann et al. PNAS ∣ February 1, 2011 ∣ vol. 108 ∣ no. 5 ∣ 1809 Downloaded by guest on September 27, 2021