Amplification and quantum statistics of microcavity polaritons under nonresonant excitation

Abstract

A model describing amplification and quantum statistics of the exciton polaritons with k = 0 in a non-resonantly excited semiconductor quantum well embedded in a microcavity is presented. Exploiting the bottleneck effect for exciton polaritons, it is assumed that the polaritons with k ≠ 0 act as a reservoir. The time evolution of the k = 0 polaritons is described by a master equation, in which two-and one-polariton transitions between the mode with k = 0 and the reservoir are accounted for. The k = 0 mode exhibits a threshold depending on the material parameters and on the injected exciton density. Below threshold the quantum statistics of the emission is characteristic of an incoherent process, while above threshold it approaches that of a laser.71.36.+c, 73.21.Fg I INTRODUCTIONIn the last few years, non-linear scattering between exciton polaritons has been observed in semiconductor quantum wells embedded in microcavities under resonant as well as non-resonant excitation 1, 2 . Resonant non-linear effects have been investigated in experiment 3-5 and in theory [6][7][8] . It has been shown that for resonant excitation the experiments may be understood in terms of parametric processes (parametric amplification and oscillation) both in a pump and probe configuration and in a configuration in which only the resonant pump is present. It has been also shown that a spontaneous transition from an incoherent emission regime to a coherent one occurs under resonant excitation 9,10 The emission under non-resonant excitation has also been investigated in experiments. As early as 1998 it has been shown that an amplification regime is present in a CdTe quantum well when the non-resonant pump energy is sufficiently large 11 . The amplified emission is observed in correspondence to the polariton mode with k=0. It has also been shown 12, 13 that the amplification is obtained in the strong coupling regime. In more recent years other characteristics of this amplification effect have been investigated experimentally. In particular, the onset of a transition from incoherent to coherent emission has been demonstrated 14 and the quantum statistics of the emitted radiation has been measured 15,16 . This transition has analogies with the transition characterizing the onset of laser action. Furthermore, some exciting results on the spatial coherence of the emission have been recently published 17 . However, in spite of the large number of existing experimental results, the theoretical investigation of the non-resonant amplified emission is still in progress. The theoretical description of the amplification process has to reproduce the threshold effect observed in the experiments and has to allow calculating the photon statistics of the emitted radiation i.e. the quantum polariton statistics. It should include both the exciton-phonon interaction that is responsible for the energy losses of the excitons generated by non-resonant pumping as well as the exciton-photon and the excitonexciton interaction leading to the formation of a macroscop...

Unconditional generation of bright coherent non-Gaussian light from exciton-polariton condensates

“…In the work of Schwendimann et al, a master equation for exciton-polariton condensates was derived, where the "system" is considered to be the k = 0 condensate, and the "reservoir" are all the modes with k = 0 [27,28]. The equations are essentially a variation of standard lasing equations, where there are a combination of one and two polariton loss and gain terms, which under suitable conditions gives rise to a condensate population.…”

“…Here are the net gain and diffusion coefficients. The precise expressions for G i and D i may be calculated using the expressions in [27][28][29]. While different materials give rise to different parameters, the integral expressions given in the above references possess certain properties which are common to all systems.…”

“…[27,28], where a master equation for the polariton condensate is derived. Including the injection locking terms, the density matrix of the condensate evolves in time according to…”

Excitations in a Nonequilibrium Bose-Einstein Condensate of Exciton Polaritons

“…The amplification rate of the condensate due to stimulated scattering of polaritons from the reservoir is a monotonically growing function R (n R ) of the reservoir density n R . Interactions between the condensate and the reservoir polaritons are modelled by the interaction constantg, generally different from the condensate one g. With respect to single-mode theories [17,18], the generalized GPE has the important advantage of fully taking into account the multi-mode nature of the spatially extended polariton field. Moreover, unlike kinetic approaches based on the Boltzmann equation [15,19,20], the present model is able to describe the coherent dynamics of the condensate.…”

Thermodynamics and linear response of a Bose‐Einstein condensate of microcavity polaritons

“…However, due to the peculiarity of the polariton system, in particular the finite polariton lifetime, the intrinsic 2-D nature and the presence of interface disorder, the existing theoretical frameworks rather interpret the phenomenon in strict analogy either with laser physics 4,5 or with the BCS transition of Fermi particles 6,7,8 . In particular, the problem of the polariton kinetics and of the non-equilibrium effects, due to the finite polariton lifetime and the relaxation bottleneck, have been investigated in many works 5,8,9,10,11 . From these studies, the main effect of non-equilibrium seems to be a significant depletion of the condensatewith corresponding loss of long-range coherence - 12 and the appearance of a diffusive excitation spectrum at low momenta 8,11 .…”