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arXiv:0910.2978 · Physical Review B

Higher order coherence of exciton-polariton condensates

Tomoyuki Horikiri, P. Schwendimann, A. Quattropani, Sven Höfling, A. Forchel, Y. Yamamoto

Published Jan 25, 2010• 67 citations
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Abstract

The second- and third-order coherence functions ${g}^{(n)}(0)$ ($n=2$ and 3) of an exciton-polariton condensate are measured and compared to the theory. Contrary to an ideal photon laser, deviation from unity in the second- and third-order coherence functions is observed, thus showing a bunching effect, but not the characteristics of a standard thermal state with ${g}^{(n)}(0)=n!$. The increase in bunching with the order of the coherence function, ${g}^{(3)}(0)>{g}^{(2)}(0)>1$, indicates that the polariton condensate is different from a coherent state, a number state, or a thermal state. The measurement of third-order coherence has the advantage, compared to the second-order one, that the difference between a thermal state and a coherent state is more pronounced. The experimental results are in agreement with the theoretical model where polariton-polariton and polariton-phonon interactions are responsible for the loss of temporal coherence.

DOI
10.1103/physrevb.81.033307

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The second- and third-order coherence functions g(n)(0){g}^{(n)}(0)g(n)(0) (n=2n=2n=2 and 3) of an exciton-polariton condensate are measured and compared to the theory. Contrary to an ideal photon laser, deviation from unity in the second- and third-order coherence functions is observed, thus showing a bunching effect, but not the characteristics of a standard thermal state with g(n)(0)=n!{g}^{(n)}(0)=n!g(n)(0)=n!. The increase in bunching with the order of the coherence function, g(3)(0)>g(2)(0)>1{g}^{(3)}(0)>{g}^{(2)}(0)>1g(3)(0)>g(2)(0)>1, indicates that the polariton condensate is different from a coherent state, a number state, or a thermal state. The measurement of third-order coherence has the advantage, compared to the second-order one, that the difference between a thermal state and a coherent state is more pronounced. The experimental results are in agreement with the theoretical model where polariton-polariton and polariton-phonon interactions are responsible for the loss of temporal coherence.
A second-order coherence function of an exciton-polariton condensate is calculated in the time domain by the quantum Monte-Carlo wavefunction method.We fully consider polariton-polariton interactions within the five lowest quantized exciton-polariton branch in a one-dimensional model.Our calculated second-order coherence function qualitatively agrees with the experimental results obtained by H. Deng et.al, 1) and it approaches to one at well above threshold like an ideal photon laser.However, the polariton number distribution is broad compared with Poissonian distribution, which indicates that the polariton condensate is different from a coherent state.
Second-order time correlations of polaritons have been measured across the condensation threshold in a CdTe microcavity. The onset of Bose-Einstein condensation is marked by the disappearance of photon bunching, demonstrating the transition from a thermal-like state to a coherent state. Coherence is, however, degraded with increasing polariton density, most probably as a result of self-interaction within the condensate and scatterings with noncondensed excitons and polaritons. Such behavior clearly differentiates polariton Bose condensation from photon lasing.
The first-order coherence properties of strongly coupled systems supporting polariton lasing and Bose-Einstein condensation (BEC) have been thoroughly studied and show consistent results. In contrast, second-order coherence properties have been found to vary for different systems, and the second-order coherence function can deviate from the value of one, typical for atomic BEC, revealing super-Poissonian fluctuations even above the BEC/polariton lasing threshold. This calls for a deeper understanding of the emission statistics and dynamics in light-based condensates, especially in the non-equilibrium regime. Here we demonstrate a coherent state of polariton emission from a plasmonic lattice measured as the second-order coherence function value separately for Γ\mathrmΓΓ- and non-Γ\mathrmΓΓ-point radiation (ground state and high-energy tail). We observe a high degree of coherence with the value g(2)(τ=0)−1<10−4g^{(2)}(τ=0)-1<10^{-4}g(2)(τ=0)−1<10−4. We also demonstrate the ultrafast ($>$1 THz) nature of the dynamics. The process can be interpreted as a BEC with ultrafast, sub-50-fs thermalization, but we also suggest an alternative explanation as superradiance in a system where the emitters have vibrational degrees of freedom.
Second-order time correlation measurements with a temporal resolution better than 3 ps were performed on a CdTe microcavity where spontaneous Bose-Einstein condensation is observed. After the laser pulse, the nonresonantly excited thermal polariton population relaxes into a coherent polariton condensate. Photon statistics of the light emitted by the microcavity evidences a clear phase transition from the thermal state to a coherent state, which occurs within 3.2 ps after the onset of stimulated scattering. Following this very fast transition, we show that the emission possesses a very high coherence that persists for more than 100 ps after the build-up of the condensate.
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We investigate with a Boltzmann-Master equation approach the coherence properties of mi-crocavity (mc) polaritons and compare the results with corresponding experiments in GaAs mc’s. The spatial first-order coherence function is determined by the polariton population of the ground state and the excited states which are known from the solution of the Boltzmann equation. Fig. 1 shows how the spatial correlation increases with increasing pump powers. The results are in very good agreement with the double-slit experiment by Deng et al. [1] for GaAs mc’s with a cavity life time of 2ps. The results do not leave much room for depletion effects in these mc’s which have been predicted for CdTe mc’s with stronger interactions[2]. The second-order correlation function g(2) of the polariton condensate is calculated by a Master equation for the probabilty to find n particles in the ground state. The rates between the excited states and the ground state known from the solution of the Boltzmann equation are corrected by a gain saturation [3]. The resulting g(2)(P/Pth) show a smooth transition from the thermal limit (g(2) = 2) below threshold to the coherent limit (g(2) = 1) not to far above treshold. In order to explain the observed large correlations above the coherent limit which persist for a wide range of pump powers [4, 5, 6], we study the non-resonant two-polariton scattering processes introduced by [7] and also a kinetic depletion model. A fully satisfactory understanding is however up to now not reached.