From polariton condensates to highly photonicquantum degenerate states of bosonic matterMarc 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 WilhelmConrad 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 reviewJuly 7, 2010)

Bose–Einstein condensation (BEC) is a thermodynamic phase tran-sition of an interacting Bose gas. Its key signatures are remarkablequantum effects like superfluidity and a phonon-like Bogoliubovexcitation spectrum, which have been verified for atomic BECs.In the solid state, BEC of exciton–polaritons has been reported.Polaritons are strongly coupled light-matter quasiparticles in semi-conductor microcavities and composite bosons. However, they aresubject to dephasing and decay and need external pumping toreach a steady state. Accordingly the polariton BEC is a nonequili-brium process of a degenerate polariton gas in self-equilibrium,but out of equilibriumwith the baths it is coupled to and thereforedeviates from the thermodynamic phase transition seen in atomicBECs. Here we show that key signatures of BEC can even be ob-served without fulfilling the self-equilibrium condition in a highlyphotonic quantum degenerate nonequilibrium system.

photon statistics ∣ quantum optics ∣ semiconductor photon sources

Microcavity polaritons are composite bosons, which are partlyphotons and partly excitons as quantified by the Hopfield

coefficients jC2j and jX2j giving the relative photonic and exci-tonic content (1), respectively, and are expected to condense athigh temperatures because of their light mass (2). Moreover, thephotonic and excitonic contents of polaritons can be preciselyadjusted by changing the detuning Δ ¼ Ec − Ex between the barecavity mode and the bare exciton mode. Unlike Bose–Einsteincondensates (BECs) in atomic gases, solid-state systems aresubject to strong dephasing and decay on timescales on the orderof the particle lifetimes. As a consequence, external pumping isrequired to achieve a steady state. Despite this nonequilibriumcharacter, degenerate polariton systems show several textbookfeatures of BECs (3), including spontaneous build up of coher-ence (4) and polarization (5), quantized vortices (6, 7), spatialcondensation (8), and superfluidity (9, 10). Usually this behavioris attributed to the system undergoing an equilibrium phasetransition toward a condensed state: Although the polariton gasis not necessarily in equilibrium with the lattice, it is in self-equilibrium, if the relaxation kinetics of excited carriers is fastenough compared to the leakage of the photonic componentout of the cavity, which is usually the case for positive detuningsΔ ≥ 0. In this case jX2j is larger than 50%, an effective tempera-ture can be defined and the polariton gas can be considered as athermodynamic equilibrium state, which is out of equilibriumwith the baths it is coupled to. This degenerate polariton gas isdistinguishable from a simple photon laser (11). It is oftenpointed out that this intrinsic nonequilibrium situation and thetwo-dimensional order parameter of polariton BECs give riseto interesting phenomena like half-vortices (12) and a diffusiveGoldstone mode (13, 14), which do not occur in equilibriumcondensates. Accordingly, the next interesting questions arewhether the same or even unique phenomena occur, if the systemis driven even further from equilibrium into a regime that cannotbe described by an effective temperature and whether it is indeednecessary to consider a thermodynamic equilibrium phase transi-

tion and thermalized polariton gases to have condensationeffects. It should be pointed out that the polariton system as awhole is almost never in thermal equilibrium. Usually it will con-sist of the low-energy part of the polariton distribution which maybe described by an effective temperature if this part reaches alocal self-equilibrium, the high-energy exciton-like part therma-lized at the lattice temperature, and the intermediate bottleneckregime which is almost always out of equilibrium. It has been sta-ted that such a local self-equilibrium condensed state is directlyrelated to BEC, whereas one which does not obtain a local self-equilibrium can hardly be described in the framework of BEC(15). Although this statement is true in terms of thermodynamicproperties, it is not immediately clear which of the signatures seenin local self-equilibrium polariton condensates will still prevail farfrom even local self-equilibrium. Recent theoretical analysis (16)predicts condensation effects even when the local self-equili-brium condition is not fulfilled and also not only for the equili-brium Bose–Einstein distribution, but also for a wide range ofnonequilibrium distribution functions. However, although thereare theoretical studies classifying the condensate ground statefor different detunings and mean polariton separations (17), froman experimental point of view, detailed studies of this regimewithout local self-equilibrium are still missing.

We realize a system far from equilibrium using polaritonswith high-photonic content, characterized by a negative detuningΔ < 0 between the bare cavity and exciton modes, and excited byshort laser pulses with duration of 1.5 ps. The fast decay timesbecause of the high-photonic content of the polaritons ensurethat the polariton distribution is neither in thermal equilibriumwith the lattice nor in local self-equilibrium and a stationarysituation is never reached. The thermalization times for our sam-ple agree well with values previously reported (18) where thepolariton thermalization time becomes shorter than their lifetimeabove the degeneracy threshold for positive detunings, but isalways longer than their lifetime for negative detunings. Althoughthere have been experiments where this regime was called out ofequilibrium condensation (19) or metastable condensate (15), sofar no systematic studies characterizing the condensate-like prop-erties in this regime where it is not possible to define an effectivetemperature have been performed. We will refer to highly photo-nic (HI-P) polariton states when discussing this nonequilibriumregime in order to stress that it is different from common polar-iton condensates which are also nonequilibrium states, but can bedescribed by an effective temperature. In the following we test

Author contributions: M.A. and M.B. designed research; M.A., J.-S.T., F.V., and A.R.-I.performed research; A.R.-I., A.L., S.H., S.R., L.W., and A.F. contributed new reagents/analytic tools; M.A., J.-S.T., F.V., and A.R.-I. analyzed data; and M.A. and M.B. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. P.L. is a guest editor invited by the Editorial Board.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: marc.assmann@e2.physik.uni-dortmund.de.

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HI-P states for several common signatures of condensation likebuild up of a macroscopic ground-state occupation, suppressedquantum fluctuations, and linearization of the excitationspectrum.

Basic characteristics of condensation and the excitation spec-trum are manifested in the polariton dispersion. Above thresholda blueshift of the kjj ¼ 0 lower polariton (LP) is expected and theLP dispersion is supposed to change from a parabolic shape inthe uncondensed case toward a different dispersion in the low-momentum condensed regime jk∥ξj < 1, where ξ ¼ ℏffiffiffiffiffiffiffiffiffiffiffiffiffi

2mLPgncp is

the healing length of the condensate (20). In standard homoge-neous equilibrium Bogoliubov theory, the expected dispersion isa phonon-like linear one at small k given by

ωBogðk∥Þ ¼ ωLPðk∥Þ þ gnc þ grnr þ gncffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk∥ξÞ2½ðk∥ξÞ2 þ 2�

q:

Here g and gr are coupling constants describing the interactionbetween two condensate polaritons and between condensatepolaritons and reservoir excitons, respectively, and nc and nr aretheir densities. Using resonant excitation, the reservoir excitoncontributions are expected to be negligible. Highly photonic con-densed states are neither in thermal equilibrium, nor spatiallyhomogeneous. Their spatial extent is given by the finite size ofthe pump spot. Therefore one might expect their dispersion toshow strong deviations from the ideal homogeneous equilibriumBogoliubov dispersion. In particular, the finite pump spot sizecauses quantization and the large amount of pump and decaycauses the excitation spectrum to become diffusive at low-mo-mentum (13, 14),

ω�ðk∥Þ ¼ ωBogð0Þ − iΓ2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ωBogðk∥Þ − ωBogð0Þ�2 −

Γ2

4

r;

where Γ ¼ γ αβ1þαβ is the effective decay rate of the Bogoliubov

mode, γ is the bare cavity decay rate, α ¼ PPthr

− 1, where thr meansthreshold, gives the relative pump strength compared to thethreshold value, and β quantifies the dependence of the conden-sate amplification rate because of stimulated scattering of polar-itons on the reservoir population. In the following we assume theeffect of β to be negligible above threshold, so that the variationof Γ between zero at threshold and γ at large pump rates is gov-erned by α alone. This diffusive spectrum is expected to manifestitself as a flat region in the low-momentum region of the realpart of the excitation spectrum. The range of this flat region isexpected to increase with α, starting from an unmodified Bogo-liubov dispersion exactly at the threshold, where α is exactly zero.In Fig. 1 we compare the LP dispersion of the emission copolar-ized with the excitation for detunings of 0, −2, −4, and −7 meV tothe theoretical predictions below and above threshold. Belowthreshold (Fig. 1 A, E, I, and M) the standard quadratic LP dis-persion is observed in all cases. Increasing the excitation powernear the threshold value (Fig. 1 B, F, J, and N), a blueshift ofthe kjj ¼ 0 emission energy becomes apparent, whereas the cross-circularly polarized emission component never shows threshold-like behavior. In addition to the blueshift of the copolarizedemission, the dispersion starts to differ from the standard quad-ratic dispersion. Blue and white lines in Fig. 1 give the quadraticLP and calculated equilibrium Bogoliubov dispersions for thecondensate blueshift. We use the experimentally determinedvalues of the blueshift for the interaction energies in calculatingωBog. The prediction for the diffusive modes ω� is not shown

Fig. 1. Polariton dispersions on a logarithmic scale at several excitation densities along the threshold for detunings ofΔ 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 equilibriumBogoliubov, and diffusive Goldstone mode dispersions, respectively.

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because at the threshold the real part of the positive branch of thediffusive mode does not differ from the equilibrium Bogoliubovdispersion. It is apparent that at threshold the equilibrium Bogo-liubov dispersion cannot reproduce the measured dispersionaccurately for Δ ¼ 0, −2, or −4 meV. Here the measured disper-sions lie approximately in the middle between both theoreticaldispersions, and the dispersions are not even symmetric withrespect to kjj ¼ 0. The latter feature becomes more apparent withincreasing negative detuning and is a signature of the nonequili-brium state favoring the presence of polaritons having a wavevec-tor with the same sign as the pump incidence wavevector,although the first can be attributed to the inhomogeneity of thesystem: Although the LPs with kjj ¼ 0 are stationary, LPs withkjj ≠ 0 will move across the excitation spot and experience a dif-ferent interaction energy given by the spatial pump pulse profile.However, it is striking that the theoretical prediction againmatches the experimental data well for the Δ ¼ −7 meV disper-sion recorded at threshold. Further above threshold (Fig. 1 C, G,K, and O) the k-space region of highest intensity moves signifi-cantly closer toward kjj ¼ 0. The green lines give the calculateddiffusive Goldstone mode dispersion. At negative detunings therestill is no complete reflection symmetry between kjj and −kjj, butthe positive wavevector half of the dispersion now shows reason-able agreement with the equilibrium Bogoliubov dispersion andthe diffusive Goldstone mode dispersion up to kjj ≈ 0.75 μm−1

without usage of any fitting parameters. This modified dispersionindeed is a sign of collective behavior as expected in a condensedand thermalized polariton gas. However, it is not immediatelyclear whether the equilibrium Bogoliubov dispersion or the dif-fusive Goldstone mode dispersion corresponds better to the ex-perimental results because their main difference lies in thelow-momentum regime where the excitation spectrum and thecondensate luminescence are superimposed. Note that the lumi-nescence from higher energy gets gradually larger with increasedphotonic content of the LP which is a clear indication of theoccupation achieving gradually a more nonequilibrium character.Further increase of the excitation power (Fig. 1 D, H, L, and P)results in a more pronounced occupation of the condensateground state. Although the reflection symmetry between positiveand negative kjj is still not perfect, both parts of the dispersion cannow be described by the same equilibrium Bogoliubov dispersionwith accuracy. We interpret this behavior as a sign of effectiveredistribution by polariton scattering processes. Although theexcitation spectrum is now strongly masked by the condensateluminescence, the occupation at larger energies still varies withthe detuning. Accordingly we are approaching a more therma-lized, but still nonequilibrium, regime for negative detunings.Anyway, it should be noted that the shape of the excitation spec-trum shows no significant dependence on the occupation andwhether it is close to equilibrium or not. Again, it is difficultto differentiate whether the equilibrium Bogoliubov or diffusiveGoldstone mode dispersion matches the experiment better. Ob-viously there is some flat region at low momenta, especially atdetunings of −2 or −4 meV, but because the emission fromthe condensate masks the emission from the excitation spectrumeven stronger than at lower pump rates, this observation is notnecessarily conclusive evidence for the presence of the flat disper-sion in momentum space.

Because of their large photonic content, one might imaginethat the coherence properties of HI-P states are not differentfrom common photon lasers. This assumption can be tested byprobing the strength of quantum fluctuations via normal-orderedsecond- and third-order equal-time correlations of the photonnumber n defined as

gð2Þðτ ¼ 0Þ ¼ h:n̂2:ihn̂i2 ;

gð3Þðτ ¼ 0Þ ¼ h:n̂3:ihn̂i3 ;

where the double stops denote normal ordering of the underlyingphoton field operators. Photon lasers are expected to showstatistically independent emission as described by Poissonianstatistics [corresponding to gð2Þð0Þ ¼ gð3Þð0Þ ¼ 1] above threshold,on the other hand increased fluctuations because of nonresonantscattering processes between polariton pairs (21–23) are expectedfor operation well above the stimulation threshold for polaritoniccondensates, resulting in gð2Þð0Þ > 1. Below threshold both aresupposed to emit thermal light with gð2Þð0Þ ¼ 2 and gð3Þð0Þ ¼ 6.To test these predictions we resonantly excite LPs with largetransverse momentum kjj ¼ 5.8 μm−1 using circularly and linearlypolarized laser pulses. Subsequently those polaritons relaxtoward the ground state by acoustic phonon emission or polari-ton–polariton scattering. If the occupation of the ground statebecomes large enough, bosonic final-state stimulation is expectedto significantly increase the scattering toward this state. We makeuse of a recently developed streak camera technique to determinethe second- and third-order correlation functions (24, 25).Although there have been experimental investigations of thesecond-order coherence properties of polariton condensatesfor GaAs (26, 27) and CdTe (28, 29) structures before, the resultsare contradictory. We resolve this question by performingsystematic studies of the higher order coherence properties interms of gð2Þð0Þ and gð3Þð0Þ for different detunings and excitationschemes and show that they are indeed strongly dependenton both.

As can be seen in Fig. 2, gð2Þð0Þ does not reach the expectedthermal value of two for any detuning as we do not single out onefundamental mode like in earlier publications (25), but detectboth realizations of the spin-degenerate ground state simulta-neously. Polaritons with different spin states will not interferewith each other and therefore the values of gð2Þð0Þ and gð3Þð0Þexpected in the thermal regime of two superposed modes are1.5 and 3, respectively. Note further that if plotted on a compar-able scale gð2Þð0Þ and gð3Þð0Þ give results which are in quantitativeagreement. Under linear excitation all detunings betweenΔ ¼ þ2 and Δ ¼ −10 meV show a threshold, which is evidencedby a decrease in gð2Þð0Þ and gð3Þð0Þ and agrees well with theposition of the threshold (shown as green dashed lines in Fig. 2)evidenced in measurements of the input–output curve. We definethe threshold as the excitation power at which the emission fromthe condensed state becomes stronger than the emission from theLP. In the case of strong negative detuning of Δ ¼ −10 meV nodifferences from a common photon lasing transition are observedunder linearly polarized pumping. Additional dispersion mea-surements evidence that above threshold the photons are indeedemitted from the bare cavity mode in this case. Under circularlypolarized pumping the threshold is not even reached for a detun-ing of −10 meV. This result is in accordance with previous resultsshowing that polariton relaxation is less efficient under circularlypolarized pumping, which in turn leads to a higher thresholdexcitation power Pthr (30). For all other detunings significantdeviations from a simple photon laser behavior emerge. Evenat high-excitation powers the ground-state emission has lowerenergy compared to the bare cavity mode with an energy differ-ence of at least 4 meV. Although this behavior is not necessarily aproof that the strong coupling regime is still intact (31), it is atleast very likely. The second- and third-order correlation func-tions give further evidence that the system is not a simple photonlaser in the range of detunings between þ2 and −7 meV, corre-sponding to photonic contents in the range from jC2j ≈ 44–70%.Although at first a decay toward one is seen for linearly polarizedexcitation, especially for a detuning of −7 meV, an increase isevidenced for further increased excitation densities. Dependingon the detuning gð2Þð0Þ can reach values even higher than the

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thermal value of 1.5. For further increased excitation powera smooth decrease back toward one is observed. For circularlypolarized excitation the general behavior of the correlationfunctions is similar to the linearly polarized case as a decreaseand a reoccurrence of the degenerate mode quantum fluctuationscan be identified for detunings between þ2 meV and −7 meV.However, in this case the correlation functions can also showincreased fluctuations slightly above threshold as can be nicelyseen for a detuning of −4 meV. This increase is caused by thebuild up of polarization. The thermal regime value of 1.5 is justvalid for unpolarized two-mode emission. As the excitation powerreaches the threshold, the emission will also start to polarize andthe two modes will not contribute equally to the correlation func-tions anymore (Materials and Methods). This onset of polarizedemission will lead to an increase in gð2Þð0Þ whereas the buildup of coherence will lead to a decrease. Although these resultsare in fair qualitative agreement with mean-field and reservoircalculations of the second-order correlation function of a polar-iton BEC (21, 22), they are not sufficient evidence for deviationsfrom a photon laser as the nonmonotonous behavior of the cor-relation function can also be a result of the interplay of coherenceand polarization.

To make sure the onset of polarized emission is not the maininfluence on the observed behavior of the correlation function westudied gð2Þð0Þ under circularly polarized excitation also for thecocircularly polarized emission only. As shown in Fig. 3, herethe expected values of gð2Þð0Þ ¼ 2 are approximately reached inthe limit of low-excitation power for all detunings exceptþ2 meV. At this detuning the emitted intensity below thresholdis too small to perform sensible measurements using our setup.Above threshold the shape of gð2Þð0Þ shows qualitative agreementwith theoretical results (21). On resonance it is apparent that fullcoherence is not reached within the available excitation power

range. Instead gð2Þð0Þ decreases monotonically toward a value be-tween 1.3 and 1.4. There is a trend toward further decrease athigh-excitation power, however, the dependence on pump poweris very weak. Going to more negative detunings, the dip alreadyseen without polarization-sensitive detection still occurs. Appar-ently, the excitation power corresponding to the occurrence ofthe dip takes on smaller values compared to Pthr for larger nega-tive detuning. Moreover, the dip in gð2Þð0Þ is most pronounced forthe most negative detuning (−7 meV). In fact, for a detuning of−7 meV almost complete coherence is reached at an excitationpower of approximately 1.1 Pthr and a steep rise to gð2Þð0Þ > 1.6 is

Fig. 2. Measured gð2Þð0Þ (black dots) and gð3Þð0Þ (red triangles) determined by simultaneous two- and three-photon detections of the whole fundamentalmode 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 meVðjC2j ≈ 56%Þ, 0 meV ðjC2j ≈ 50%Þ, andþ2 meV ðjC2j ≈ 44%Þ under linearly polarized (Upper) and circularly polarized (Lower) excitation. Red (blue) lines denotethe coherent (thermal) limit. Green dashed lines give the position of the degeneracy threshold determined by measurements of the dispersion. Below thresh-old the necessary integration time for the correlation measurements increases drastically. Only data points for excitation powers where the results are notlimited by the long-term stability of the setup are considered.

Fig. 3. Measured gð2Þð0Þ of the cocircularly polarized fundamental mode fora wide range of detunings and excitation powers under circularly polarizedexcitation. Red (blue) lines denote the coherent (thermal) limit. Green dashedlines give the position of the degeneracy threshold determined by measure-ments of the dispersion.

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evidenced at 1.5 Pthr. Calculations of the second-order correla-tion function are generally done using one of two different ap-proaches (21): Mean-field calculations predict a decrease ofgð2Þð0Þ toward approximately 1.2 at the threshold, followed bya short rise for increasing excitation power until a constant valueof about 1.3 is reached. This prediction is in agreement with ourresults for no or small negative detuning. A two-reservoir modelpredicts a sharp decrease of gð2Þð0Þ at the threshold, a strong re-currence of the photon bunching up to values of gð2Þð0Þ ¼ 1.6 forexcitation powers moderately larger than the threshold value anda slow drop for even higher excitation powers. This modelbetter reproduces our results for large negative detuning. Weconclude that because of the increasing relaxation bottleneck anddecreased scattering rate expected for large negative detuning thetwo-reservoir model appears to be a valid description in the HI-Pregime and the dip seen for several detunings is a sign of ineffi-cient scattering between polaritons from the degenerate groundstate and those in excited states. Because the photon statisticsof emission depends strongly on the detuning we suppose thatone might also find variations depending on the Rabi splittingand the excitonic and photonic decay constants. Therefore,microcavities operated in the HI-P regime may open up thepossibility to introduce a high-intensity light source with tunablephoton statistics.

In conclusion we have driven a microcavity polariton systeminto a completely out of equilibrium degenerate HI-P state by in-creasing its photonic content up to jC2j ≈ 70% and examinedwhether the system bears more similarities to an inverted system;i.e., a laser or a thermalized polariton BEC. Surprisingly, althoughthe matter component is small and the degeneracy transitionwe have demonstrated cannot be considered a thermodynamicequilibrium phase transition by anymeans, the system showsmanyfeatures one would expect from a BEC. In particular we haveobserved a modified dispersion and compared it to standard equi-librium Bogoliubov theory predicting a linearized spectrum andnonequilibrium Bogoliubov theory predicting a diffusive Gold-stone mode and demonstrated the detuning dependence of theemitted photon statistics. We believe our results can open upthe road for tunable photon statistics light sources and deepenthe understanding of nonequilibrium phase transitions.

Materials and MethodsExperimental Details. In our experiments the sample was kept at 8 K in ahelium flow cryostat. The microcavity structure is basically identical to thatused in earlier measurements (20), but was fabricated in another growth run.It consists comprehensively of 12 GaAs/AlAs quantum wells embedded in aplanar microcavity with 16 (21) AlGaAs/AlAs mirror pairs in the top (bottom)distributed Bragg reflector. Reflectivity measurements gave results of1.6158 eV for the bare exciton energy and 13.8 meV for the Rabi splitting.Using a Ti:Sapphire laser with a pulse duration of 1.5 ps and a repetitionrate of 75.39 MHz the pump was focused to a spot approximately 30 μmin diameter on the sample at an angle of 45° from normal incidence. Thepump was resonant with the LP branch at an in-plane wavenumber of kjj ¼5.8 μm−1 for resonant polariton injection or tuned to the first minimum ofthe cavity reflectivity curve at a wavelength of 744 nm for nonresonant andincoherent pumping. We collected the emitted signal using a microscope

objective with a numerical aperture of 0.26. The signal was then focusedon a streak camera for correlation measurements, or the Fourier planewas imaged on a monochromator for dispersion measurements.

Details of Correlation Measurements. The basics of our photon correlationmethod were already explained in the references given in the text. Accessingextremely short pulses, like the emission from HI-P states in the degenerateregime, introduces additional difficulties. In this regime the timing jitterbecomes comparable to the emission pulse duration. Therefore the momen-tary intensity detected at a certain position of the CCD will be determined bythe photon statistics as well as by the momentary influence of timing jitter,which will cause deviations from the real value of gð2Þ. The deviation issystematic, so it is possible to correct for this jitter effect. From a theoreticalpoint of view, the effect of jitter can be modeled as a Gaussian distributionhaving a certain FWHM describing the peak position of the emitted pulse onthe CCD. This timing jitter will cause a broadening of the duration of theaveraged intensity compared to the real emission pulses. Accordinglygð2Þð0Þwill be overestimated whereas gð2ÞðτÞwill be underestimated for largeenough τ. The exact measured gð2Þ depends on the ratio of the timing jitterFWHM J to the pulse FWHM P. This situation can be simulated theoreticallyand shows that the measured gð2Þ

m ð0Þ and gð3Þm ð0Þ depend on the real gð2Þð0Þ

and gð3Þð0Þ corresponding to the following relation,

gð2Þm ð0Þ ¼ gð2Þð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�JP

�2

s; gð3Þm ð0Þ ¼ gð3Þð0Þ½1þ

�JP

�2

�:

Accordingly it is possible to extract the real values from the measured ones ifthe emission pulse widths and the jitter widths are known. The jitter widthcan be determined by measuring gð2Þ

m for a light source with well-defined Pand gð2Þ. We used a laser pulse with P ¼ 1.5 ps as determined by using anautocorrelator and gð2Þ ¼ 1. The resulting gð2Þ

m ðτÞ is discussed in detail in ref. 32and is well-reproduced by a simulation using J ¼ 1.8 ps.

P can be determined directly from the integrated streak camera data,which give the jitter-broadened emission pulse widths. Because all essentialparameters are known, it is possible to calculate gð2Þð0Þ from gð2Þ

m ð0Þ.In the case of a superposition of two noninterfering modes A and B, the

resulting measured gð2ÞABð0Þ is given by

gð2ÞABð0Þ ¼ gð2ÞA ð0ÞR2A þ gð2ÞB ð0ÞR2

B þ 1RARB;

where RA and RB are the relative intensity ratios of mode A and B to the totalintensity. Therefore it is possible to calculate gð2Þ

A ð0Þ from gð2ÞABð0Þ if the relative

intensity ratios and gð2ÞB ð0Þ are known:

gð2ÞA ð0Þ ¼ gð2ÞABð0Þ − gð2ÞB ð0ÞR2B − 1RARB

R2A

:

Comparing single-mode gð2Þ measurements to the values obtained bytwo-mode measurements shows that for circularly polarized excitation thetwo modes are indeed independent and the cross-circularly polarized modestays thermal.

ACKNOWLEDGMENTS. The Dortmund group acknowledges support throughthe Deutsche Forschungsgemeinschaft (DFG) research grant DFG 1549/15-1. The group at Würzburg University acknowledges support by the Stateof Bavaria.

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