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arXiv:1002.2874 · Physical Review B

Stationary and time-dependent correlations in polariton condensates

P. Schwendimann, A. Quattropani, Davide Sarchi

Published Nov 24, 2010• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.82.205329

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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.
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.
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.
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.
Polariton condensates provide a versatile platform for exploring non-equilibrium phase transitions and collective phenomena in open quantum systems. Near the condensation threshold, these systems are particularly sensitive to fluctuations and instabilities, which can strongly influence the condensate formation. Using optical trapping and homodyne detection, we directly access the photon statistics and second-order correlation function g(2)(0)g^{(2)}(0)g(2)(0) of the condensate. We show that polariton condensation near the threshold is not a purely static transition, but instead undergoes a dynamical regime characterized by stochastic hopping between condensed and non-condensed states. These intermittent dynamics are accompanied by a gradual reduction of g(2)(0)g^{(2)}(0)g(2)(0) towards unity, revealing the progressive build-up of coherence even in the presence of strong temporal fluctuations. Numerical simulations, based on a stochastic Truncated Wigner description of the driven-dissipative polariton field, reproduce these dynamics and capture the essential role of noise and reservoir interactions. This work demonstrates that the observed temporal hopping is an intrinsic feature of polariton condensation, providing a dynamical perspective that goes beyond static descriptions of the condensation phase transition.
Polariton condensates provide a versatile platform for exploring non-equilibrium phase transitions and collective phenomena in open quantum systems. Near the condensation threshold, these systems are particularly sensitive to fluctuations and instabilities, yet how these dynamics govern the initial formation of the condensate remains poorly understood. To address this, we directly access the photon statistics and second-order correlation function g (2) (0) of the condensate by using optical trapping and homodyne detection. We show that polariton condensation near the threshold is not a purely static transition, but instead undergoes a dynamical regime characterized by stochastic hopping between condensed and non-condensed states. These intermittent dynamics are accompanied by a gradual reduction of g (2) (0) toward unity, revealing the progressive build-up of coherence even in the presence of strong temporal fluctuations. Numerical simulations, based on a stochastic truncated Wigner description of the driven-dissipative polariton field, reproduce these dynamics and capture the essential role of noise and reservoir interactions. This work demonstrates that the observed temporal hopping is an intrinsic feature of polariton condensation, challenging traditional static descriptions of phase transitions and offering a framework for controlling noise and coherence in driven-dissipative quantum technologies.
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.