Kinetics of stimulated polariton scattering in planar microcavities: Evidence for a dynamically self-organized optical parametric oscillator
A. A. Demenev, A. A. Shchekin, A. V. Larionov, S. S. Gavrilov, V. D. Kulakovskii, N. A. Gippius, S. G. Tikhodeev
aa r X i v : . [ c ond - m a t . o t h e r] D ec Kinetics of stimulated polariton scattering in planar microcavities: Evidence for adynamically self-organized optical parametric oscillator
A. A. Demenev, A. A. Shchekin, A. V. Larionov, S. S. Gavrilov,
1, 2
V. D. Kulakovskii, N. A. Gippius,
2, 3 and S. G. Tikhodeev
2, 3 Institute of Solid State Physics RAS, Chernogolovka, 142432 Russia A. M. Prokhorov General Physics Institute RAS, Moscow, 119991 Russia LASMEA, UMR 6602 CNRS, Universit´e Blaise Pascal, Aubi`ere, France (Dated: November 26, 2007)We demonstrate for the first time the strong temporal hysteresis effects in the kinetics ofthe pumped and scattered polariton populations in a planar semiconductor microcavity undera nano-second-long pulsed resonant (by frequency and angle) excitation above the lower polari-ton branch. The hysteresis effects are explained in the model of multi-mode scattering whenthe bistability of the nonlinear pumped polariton is accompanied by the explosive growth ofthe scattered polaritons population. Subsequent self-organization process in the nonlinear po-lariton system results in a new — dynamically self-organized — type of optical parametric oscillator.
PACS numbers: 71.36.+c, 42.65.Pc, 42.55.Sa
A giant stimulated polariton-polariton scattering isone of the most striking features in the optical responseof planar microcavities (MCs). The scattering was firstlyobserved in GaAs-based MCs with InGaAs quantumwells (QWs) in the active layer under a cw excitation atwave-vector k p close to the inflection point of lower po-lariton (LP) branch ω LP ( k ), when the scattering exhibitsan unusually low (smaller than 400 W/cm ) thresh-old [1, 2, 3]. Specifically, such excitation results in thestrong parametric scattering into states positioned ap-proximately on ω LP ( k ) with k s = 0 and k i = 2 k p , calledsignal and idler, respectively. The effect was theoreti-cally described in terms of four-wave mixing or paramet-ric scattering [4, 5, 6]. Subsequent studies [7, 8, 9] haveshown that the shift of the excitation from the inflectionpoint of the LP dispersion is not followed by the corre-sponding shift of the stimulated scattering along the LPbranch characteristic for the four wave mixing. Instead,the scattering goes on to the same states with k ∼ ∼ k p . The energy conservation is then fulfilled bythe shift of the signal and the idler much above the LPbranch.The stability analysis of the single macro-occupiedpump mode as well as numerical simulations of the po-lariton scattering indicate [9, 10, 11, 12] that such un-usual behavior can result from the interplay between twoinstabilities in the resonantly excited MC: bistability ofthe pumped polariton mode intensity with respect to theexternal pump, and its parametric instability with re-spect to the decay into multiple scattered polaritons in a wide range of k .The single-mode optical bistability in MC for pump-ing at k p = 0 has been observed and explained theo-retically within one-mode optical parametric oscillator(OPO) model [13]. The bistability of the scattered sig-nal at pumping near inflection point was also found and explained within three-mode OPO model [14].Different possible regimes of the above-threshold OPOhave been analyzed theoretically within the three-modeapproximation [15, 16]. However three-mode OPO modelcannot determine the signal and idler wave vectors k s,i which are selected by the parametric process abovethreshold. In the system with a specific dispersion (con-taining an inflection point), the interplay between thepump single mode bistability and its multi-mode para-metric instability can result in a regime where essentiallymulti-mode coupling between the ensemble of lower po-laritons plays the decisive role in the formation of thesignal and idler [9, 11, 17]. We will refer to this model asa dynamically self-organized (DSO) OPO.Dramatic change of LP scattering pattern fromthe figure-of-eight shape corresponding to spontaneousregime at lower pump intensity [4, 18] to that directedalong k ∼ ∼ k p for pump powers above the para-metric scattering threshold observed recently [19] sup-ports the DSO OPO model. However, the direct evi-dence of multi-mode self-organized nature of the scatter-ing demands the study of the LP system dynamics. Theprevious time-resolved studies of the LP dynamics underps-long excitation pulses [18] have revealed a well pro-nounced figure-of-eight distribution of the final LP scat-tering states in low excitation regime, but only a smallnarrowing of the LP momentum distribution with timeat higher excitation.In this Letter we report the optical studies of kineticsof the LP system under ns-long pump pulses and dis-cover for the first time the strong hysteresis effects in thekinetics of the optical response of the pumped as wellas of the scattered MC polaritons, thus directly demon-strating the behavior, predicted within the DSO OPOmodel [9, 11, 17].The MC structure has been grown by a metal or-ganic vapor phase epitaxy. The Bragg reflectors are com-posed of 17 (20) repeats of λ/ . Ga . As/AlAs lay-ers in the top (bottom) mirrors. The 3 / λ GaAs cavitycontains six 10-nm thick In . Ga . As/GaAs quantumwells (QWs). The Rabi splitting is Ω ∼ E C and, accordingly, in the detuning ∆ between the exciton ~ ω X ( k = 0) and photon ~ ω C ( k = 0) mode energies. Ex-periments have been carried out on several spots of thesame sample with ∆ in the range from -1.5 to -2 meV.The sample was placed into an optical cryostat withcontrolled temperature. A pulsed Ti-sapphire laser withpulse duration of ∼ ∼ . ◦ relative to the cavity normal. The pump beam hasbeen focused onto a spot with diameter of 100 µ m. Thekinetics of angular distribution of PL signal I ( k , t ) fromthe MC has been detected in a wide solid angle aroundthe cavity normal by the streak camera with spectral,angular, and time resolution of 0.28 meV, 1 ◦ , and 70 ps,respectively. The transmission signal I tr ( k p , t ) has beendetected with the same streak camera.The pump pulse profile I P ( t ) is shown in Fig. 1a(dashed curve). The magnitude of I P ( t ) determines theintensity of the external electric field outside the MC | E ext ( t, k p ) | ∝ I P ( t ). The intensity builds up during thefirst 100 ps and then decreases monotonically (by about3 times to t = 1 ns). The pulses with a circular ( σ + )polarization and a spectral FWHM of ∼ − k p = ( k px , k py ) = (1 . , µ m − .Figure 1b shows the recorded kinetics of MC emissionnormal to its plane, I ( k = 0 , t ). The spectra are recordedin the σ + polarization. The emission in σ − is about twoorders of magnitude smaller.The MC emission I ( k = 0 , t ) is proportional to | E QW ( k = 0 , t ) | , the intensity of the k = 0 har-monic of the electric field on QW inside MC. Figure 1bclearly shows that I (0 , t ) and thus the time dependence of | E QW (0 , t ) | differs significantly from the exciting pulseshape. At low P = 11 . , the signal reachesits maximum only slightly later (by ∼
50 ps) than thepumping pulse and then decreases quickly, by one orderof magnitude at t ∼ .
35 ns when the pump intensity isstill about 60% of the maximum. At
P > . ,the signal behavior changes drastically. After its markeddecrease (together with the pump) in the range of t =0 . − .
35 ns the signal starts to grow and reaches thesecond maximum at t ∼ .
85 ns already at the excitationpulse fall off. The intensity of this maximum grows bymore than two orders of magnitude in the range of P between 14.9 and 17.2 kW/cm , i.e., threshold-like.The recorded time dependences of the MC transmis-sion at the pump angle I tr ( k p , t ) are shown in Fig. 1a T r a n s m i ss i on i n t e n s it y ( a r b . un it s ) pump17.316.214.911.50 0.5 1 1.510 −3 −2 −1 P L i n t e n s it y ( a r b . un it s ) Time (ns) (b)(a)
FIG. 1: (Color online) (a) Time dependences of the pumppulse I P ( k p , t ) and of the transmission I tr ( k = k p , t ) at dif-ferent excitation densities. (b) Kinetics of the LP emissionintensity I ( k = 0 , t ) at different excitation densities. Num-bers show the peak pump intensity P in kW/cm . (solid curves). I tr ( k p , t ) is proportional to the inten-sity of the k = k p harmonic of the QW electric field | E QW ( k p ) , t | inside MC. Again we see clearly that I tr ( k p , t ) and thus | E QW ( k p ) | do not directly followthe exciting field | E ext ( k p , t ) | ∝ I P ( t ). At P . . I tr and, thus, | E QW ( k p ) | is a monotonous su-perlinear function of | E ext ( k p ) | on both the up anddown going parts of the excitation pulse, the maxima of I tr ( t ) and I P ( t ) and, hence, those of | E QW ( k p ) , t | and | E ext ( k p , t ) | nearly coincide with each other. However,the monotonous dependence I tr ( I P ) becomes distortedwith increasing P . Figure 1a shows that I tr starts todemonstrate a narrow second peak in the range of nearlyconstant exciting field | E ext ( k p ) | at t ∼ . | E QW ( k p ) | starts at t = 0 . ± .
02 ns, con-tinues about 0.1 ns, and gives way to its sharp decreaseat t = 0 . ± .
02 ns. The duration of the increase anddecrease in | E QW ( k p ) | is close to an available time reso-lution of our detecting system of 70 ps. The second peakof | E QW ( k p ) | grows quickly with P and shifts slightlytowards the pulse onset.Time dependences I P ( t ), I ( k = 0 , t ), and I tr ( k p = 0 , t )in Fig. 1 can be redrawn as implicit functions I tr ( I P )and I S ( I tr ) presenting the dependences of the inner field | E QW ( k = k p ) | on the exciting field | E ext ( k = k p ) | andof the k = 0 harmonic of the QW electric field on | E QW ( k = k p ) | , respectively, for each P . Figures 2a andb show the resulting dependences at P = 16 . >P thr . Both the experimentally measured dependences ofthe k = k p electric field inside MC on the external field E QW | k = k p ( E ext | k = k p ) and of the k = 0 electric QW field T r a n s m i ss i on i n t e n s it y ( a r b . un it s ) Pump intensity (arb.units)
Transmission intensity (arb.units) S i gn a l i n t e n s it y ( a r b . un it s ) t m (d)calc.(b)exp.(a)exp.(c)calc. t m t m + 0.7 nst m + 0.6 ns FIG. 2: (Color online) Measured (top panel) and calcu-lated (bottom panel) dependences of transmission intensityon pump intensity and PL intensity on transmission intensityat P = 16 . . Circles mark the characteristic pointswith maxima of transmission (at t = t m ) and k = 0 emission(approximately 650 ns later). inside MC on that at k = k p E QW | k =0 ( E QW | k p ) acquirejumps and hysteresis behaviour.These experimental results find their qualitative expla-nation in the framework of the system of a semi-classicalGross-Pitaevskii type equation for QW excitonic polar-ization P ( k , t ) and a Maxwell equation for E QW ( k , t )in response to the driving external field far from theMC [9, 11, 17]. According to this theoretical model, thedynamics of the stimulated parametric scattering in theplanar MCs has a following scenario. Its start is initi-ated by a single-mode instability of the pumped modeat k = k p , which results in the jump of | E QW ( k p ) | andtransfers this mode into the region of its strong instabilitywith respect to the parametric LP-LP scattering at onceinto a large range of k . That provides an explosive growthof LP population in a wide k -space region, mainly around k = 0, on one hand, and causes the abrupt decrease inthe driven mode population, on the other hand. Theformation of an OPO with a three dominating macrooc-cupied modes with signal and idler at k = 0 and 2 k p occurs due to a dynamical self-organization in the multimode scattering and takes a long – hundreds of ps – time.The calculated dynamics of the MC optical responsewithin the model of Refs. [9, 11, 17] at the excita-tion slightly above the hard excitation threshold are dis-played in Fig. 2c and d. The system of Gross-Pitaevskiiand Maxwell equations has been numerically solved forexperimental-like time dependence of E ext ( t ). As seenfrom Fig. 2c,d, the model demonstrates the hysteresis independences of I tr on I P (panel c) and of I ( k = 0) on I tr (panel d) similar to the experimental ones. The con-sidered model takes into account only coherent scatter-ing processes (nondiagonal components of the LP densitymatrix) and neglects important scattering processes likeLP–phonon or LP–free carriers. Nevertheless, it demon-strates the hysteresis in a qualitative agreement with the T i m e ( n s ) x ( µ m −1 ) T i m e ( n s ) −1 −0.5 000.51 k y ( µ m −1 ) P=4.5 kW/cm P=10.5 kW/cm (a) (c)(f)(d) (e)(b) FIG. 3: (Color online) Time dependences of a k -distributionof LP emission at P = 4 . . Left (a,d) and right (c,f) panels displaythe k x dependence at k y = 0 and k y -dependence at k x = 0,respectively. The thick and thin solid lines in the centralpanels (b,e) show, respectively, the pulse and MC emissionprofiles. experiment.The hysteresis of the dependence of I tr on I p resultsfrom the blue shift of LP eigenenergy. The increase ofan overall LP population shifts the driven mode fre-quency towards the pump frequency, and results in in-creased transmissivity. Even when the pump intensityhas been fallen down, the system retains considerablesignal population that keeps it close to the resonance.At the same time, the hysteresis of E QW ( k s ∼
0) vs. I tr ∝ | E QW ( k p ) | reveals a more complicated nature ofthe studied system. Indeed, the model does not pre-sume any characteristic time rather than the lifetime ofcavity polaritons ( ∼ τ ∼ ps) might come from onlythe collective phenomena caused by numerous inter-modescattering processes. Since the evolution to the “three-mode” ( k = { , k p , k p } ) state involves a lot of modes,that state appears as an essentially collective formation.Moreover, the eventual state differs from stable“three-mode” OPO solution even in the case of stationary pump,which may be proved by performing the stability analy-sis similar to that discussed in [15]. The actual stabil-ity of the three-mode pattern is maintained by the pres-ence of numerous weak “above-condensate” modes, sothe whole system occurs to be highly correlated, i.e., itdemonstrates a new – dynamically self-organized – typeof OPO.To support this scenario of DSO OPO experimentally,the LP scattering dynamics in a wide range of k hasbeen measured by recording time dependences of an an-gle distribution of MC emission. Panels (a,d) in Fig. 3display k x -distribution at k y = 0 whereas panels (c,f)display k y distribution at k x =0 for the point on the MCsample with a smaller P thr ∼ .
75 kW/cm for two exci-tation densities P = 4 . , i.e., slightlyand well above P thr . I P ( t ) and I ( k = 0 , t ) are givenin panels (b,e). The k − distribution of the emission issymmetric in k y direction but shows a well pronouncedasymmetry in the direction of exciting pulse ( k x ). Themaximum emission in the very beginning of the pulse isat k x ∼ − . µ m − and then during t ∼ . k = 0. This behavior is well expected. The phonon-assisted scattering dominating at low LP densities cannotprovide the LP relaxation to k = 0 because of comparablemagnitudes of LP life time and phonon assisted scatter-ing time [20, 21]. The effective LP relaxation at higherdensities appears due to the onset of LP-LP scatterings.Figures 3a and c show that dynamics at P = 4 . ( ∼
20 % above P thr ). The scattering devel-oping after the bistable transition at t ∼ . k x between − . . µ m − , k y between ∓ . µ m − . With increasingtime the signal intensity increases about two times andreaches its maximum at t ∼ k -distribution specific for the stimulatedpolariton scattering under cw excitation.The marked narrowing of the signal in the k -space dur-ing the excitation pulse duration of ∼ . P . That is illustrated in Figs. 3b and d display-ing the LP scattering dynamics at P = 10 . ∼ P thr . The bistable transition of the driven mode at this P occurs earlier, at t ∼ . k − distribution of the LP emission bothin k x and k y directions. Figure 4 displaying the de-pendences of a FWHM of k -distribution shows that thedynamical self-organization of the parametric scatteringtakes a long time: the narrowing of the angle distributiontakes place in the whole time range of the strong scat-tering signal up to t ∼ k -distribution FWHMdecreases in this time interval from ∼ µ m − ,which is still markedly larger than the FWHM in thecase of cw excitation ( . . µ m − ). These experimentalresults clearly prove the DSO OPO model.To conclude, the strong hysteresis effects in the kinet-ics of the pumped and scattered polariton populationshave been observed for the first time in a planar semi-conductor MC under a nano-second-long pulsed resonantexcitation slightly above the LP branch. The hystere-sis effects are explained in the model of a hard regimeof the onset of parametric scattering, when the bistabil-ity of the nonlinear pumped LP mode is accompaniedby the explosion-like growth of the scattered LP popula-tion and subsequent dynamical self-organization processin the open polariton system resulting in dynamicallyself-organized OPO. F W H M ∆ k ( µ m − ) ∆ k x ∆ k y FIG. 4: (Color online) Recorded kinetics of a FWHM of k -distribution of LP emission ∆ k x ( k y = 0) and ∆ k y ( k x = 0)at P = 16 . . We thank M. S. Skolnick for rendered samples. Thiswork was supported by the Russian Foundation for BasicResearch, the Russian Academy of Sciences and the ANRChair of Excellence Program. [1] R. M. Stevenson, V. N. Astratov, M. S. Skolnick, D. M.Whittaker, M. Emam-Ismail, A. I. Tartakovskii, P. G.Savvidis, J. J. Baumberg, and J. S. Roberts, Phys. Rev.Lett. , 3680 (2000).[2] A. I. Tartakovskii, D. N. Krizhanovskii, and V. D. Ku-lakovskii, Phys. Rev. B , R13298 (2000).[3] J. J. Baumberg, P. G. Savvidis, R. M. Stevenson, A. I.Tartakovskii, M. S. Skolnick, D. M. Whittaker, and J. S.Roberts, Phys. Rev. B , R16247 (2000).[4] C. Ciuti, P. Schwendimann, and A. Quattropani, Phys.Rev. B , 041303 (2001).[5] D. M. Whittaker, Phys. Rev. B , 193305 (2001).[6] P. G. Savvidis, C. Ciuti, J. J. Baumberg, D. M. Whit-taker, M. S. Skolnick, and J. S. Roberts, Phys. Rev. B , 075311 (2001).[7] V. D. Kulakovskii, A. I. Tartakovskii, D. N.Krizhanovskii, N. A. Gippius, M. S. Skolnick, and J. S.Roberts, Nanotechnology , 475 (2001).[8] R. Butte, M. S. Skolnick, D. M. Whittaker, D. Bajoni,and J. S. Roberts, Phys. Rev. B , 115325 (2003).[9] N. A. Gippius, S. G. Tikhodeev, V. D. Kulakovskii, D. N.Krizhanovskii, and A. I. Tartakovskii, Europhys. Lett. , 997 (2004).[10] V. D. Kulakovskii, D. N. Krizhanovskii, A. I. Tar-takovskii, N. A. Gippius, and S. G. Tikhodeev, Physics– Uspekhi , 967 (2003), [Uspekhi Fiz. Nauk , 995(2003)].[11] N. A. Gippius and S. G. Tikhodeev, J. Phys.: Condens.Matter , S3653 (2004).[12] N. A. Gippius, S. G. Tikhodeev, L. V. Keldysh, and V. D.Kulakovskii, Physics – Uspekhi , 306 (2005) [UspekhiFiz. Nauk , 327 (2005)].[13] A. Baas, J. P. Karr, H. Eleuch, and E. Giacobino, Phys.Rev. A , 023809 (2004).[14] A. Baas, J.-P. Karr, M. Romanelli, A. Bramati, andE. Giacobino, Phys. Rev. B , 161307(R) (2004).[15] D. M. Whittaker, Phys. Rev. B , 115301 (2005).[16] M. Wouters and I. Carusotto, Phys. Rev. B , 075332(2007). [17] S. S. Gavrilov, N. A. Gippius, V. D. Kulakovskii, andS. G. Tikhodeev, Zh. Eksp. Teor. Fiz. , 819 (2007),[JETP , 715 (2007)].[18] W. Langbein, Phys. Rev. B , 205301 (2004).[19] D. N. Krizhanovskii, S. S. Gavrilov, A. P. D. Love,D. Sanvitto, N. A. Gippius, S. G. Tikhodeev, V. D. Ku- lakovskii, D. M. Whittaker, M. S. Skolnick, and J. S.Roberts, submitted (2007).[20] J. Bloch and J. Y. Marzin, Phys. Rev. B , 2103 (1997).[21] F. Tassone, C. Piermarocchi, V. Savona, A. Quattropani,and P. Schwendimann, Phys. Rev. B56