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Physical Review B

Coherence of condensed microcavity polaritons calculated within Boltzmann-Master equations

T. D. Doan, Huan Cao, D. B. Tran Thoai, H. Haug

Published Nov 6, 2008• 55 citations
Publisher

Abstract

The first-order spatial and the second-order temporal coherences of condensed microcavity polaritons are calculated in terms of Boltzmann equations for the excited states and the ground state supplemented by a Master equation for the probability to find a given number of particles in the condensate. The resulting first-order spatial coherence agrees both in its pump power dependence and in its variation with distance with the results of a recent double-slit experiment. Inserting the calculated rates between the excited states and the condensate with various saturation and depletion models, we solve for stationary situations the Master equation for the ground-state population. The resulting second-order correlations for the various models are compared with recent measurements.

DOI
10.1103/physrevb.78.205306

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The first-order spatial and the second-order temporal coherences of condensed microcavity polaritons are calculated in terms of Boltzmann equations for the excited states and the ground state supplemented by a Master equation for the probability to find a given number of particles in the condensate. The resulting first-order spatial coherence agrees both in its pump power dependence and in its variation with distance with the results of a recent double-slit experiment. Inserting the calculated rates between the excited states and the condensate with various saturation and depletion models, we solve for stationary situations the Master equation for the ground-state population. The resulting second-order correlations for the various models are compared with recent measurements.
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.
The semiclassical approach consisting of Boltzmann equations for the excited states and the ground state supplemented by a Master equation for the probability distribution of the condensate population is solved for picosecond pulsed microcavity polaritons. With the simple birth- and death-type Master equation for the condensate population that disregards the condensate phase, one gets results, which, compared to quasistationary solutions, are in better agreement with the experiments. First, the time-dependent solutions show the influence of the change to the coherent state already at pump powers close above the condensate threshold. Second, and even more important for the interpretation of corresponding pulsed experiments, the time-dependent solution give results for the second-order correlation function with seemingly larger correlations above threshold, although no polariton-polariton interaction in the ground state has been included.
The demonstration of coherent polariton emission from a CdTe microcavity structure [1] has provided a new system for the study of condensate physics. Unlike atomic condensates, the microcavity system is not in equilibrium; the polariton life-time in the cavity is only ∼ 2ps, so the population has to be maintained by pumping. This raises interesting questions about how the properties of such a non-equilibrium condensate differ from those of an equilibrium system. Recent experimental measurements of the first (g(1)) and second (g(2)) order correlation functions of the emitted light have shown that the decay times of both are ∼ 100−200ps, much longer than the cavity life-time. In this work, we present a theroetical treatment which provides a quantitative explanation of these results, and shows that the ultimate cause of the coherence decay is interactions between polaritons in the condensate. The microcavity polariton condensate is a mesoscopic quantum system, consisting of N ∼ 102 po-laritons within the finite size excitation spot. As a result, number fluctuations of ∼ N 1 2 are expected. For a finite equilibrium condensate of interacting particles, it is straight forward to show that the decay
The dynamics of the expansion of the first order spatial coherence g(1) for a polariton system in a high-Q GaAs microcavity was investigated on the basis of Young's double slit experiment under 3 ps pulse excitation at the conditions of polariton Bose-Einstein condensation. It was found that in the process of condensate formation the coherence expands with a constant velocity of about 10(8) cm/s. The measured coherence is smaller than that in a thermal equilibrium system during the growth of condensate density and well exceeds it at the end of condensate decay. The onset of spatial coherence is governed by polariton relaxation while condensate amplitude and phase fluctuations are not suppressed.
The statistics of the condensed polaritons is described in terms of the Wigner function. In the framework of the truncated Wigner method, the Wigner function obeys a Fokker-Planck equation, which is solved analytically. The second-order correlations in the stationary state are in excellent agreement with those obtained from the numerical solution of the master equation and show a qualitative and, well above threshold, also quantitative agreement with recent experiments. Furthermore, the contributions of the different noise effects that influence the polariton ground-state statistics are explicitly defined. Exploiting the equivalence between Fokker-Planck and Langevin descriptions of stochastic processes, the time-dependent correlations of the polaritons close to the stationary state are derived. An explicit expression for the polariton linewidth is obtained, whose numerical values reproduce qualitatively the experimental ones. Finally, the limit of validity of the truncated Wigner method in the present model is discussed.
We analyze the decay of the first- and second-order correlations, g(1)(t){g}^{(1)}(t)g(1)(t) and g(2)(t){g}^{(2)}(t)g(2)(t), using a Langevin equation of the order parameter for polaritons in a microcavity, pumped nonresonantly with a noise-free laser diode. We consider a coupling of the condensate to the excited states by two types of scattering mechanism both derived from the polariton-polariton interaction: (a) one particle is scattered from an excited state to the ground state, while another excited-state particle is scattered to a higher-energy state, and (b) two ground-state particles are scattered into low-lying excited states with opposite momenta. This nonresonant scattering rate increases with the condensate population. We calculate from the Langevin equation for the order parameter the temporal decay of g(1)(t){g}^{(1)}(t)g(1)(t) and the linewidth $\ensuremath{\kappa}$ analytically. Our results make close contacts with the linewidth enhancement factor well known from semiconductor lasers. A semiclassical evaluation for the polariton kinetics yields a first-order correlation function, which is in good agreement with experimental results and with the results of Schwendimann and Quattropani, although our formulation is more complete because it contains also the dispersive effects of both types of scattering processes and uses no fitting parameters. Our analysis also provides an understanding of the rather different pump dependencies of the linewidth reported in the literature. The calculated second-order correlation function g(2)(0){g}^{(2)}(0)g(2)(0) is shown to stay for larger pump values as observed on a plateau above the coherent limit again due to the nonresonant scattering processes.