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physica status solidi (b)

The Statistical Properties of the Coherently Excited Polariton Wave in a Small Volume Crystal

Vyacheslav R. Misko, S. A. Moskalenko, Anatol Rotaru, Yu. M. Shvera

Published May 1, 1990• 6 citations
Publisher

Abstract

Abstract A selfconsistent steady state solution is given of the Fokker‐Planck equation for polaritons with wave vector ko and the kinetic equations for the scattered ones are found. The statistics of the coherently excited polaritons ko differs in a most general case from the pure coherent state. It depends on the crystallite volume V, the frequency detuning Δω between the polariton bare frequency ω(ko) and the laser one ωL, and the driving field energy flow SL. In the case of small values of the product V · SL polariton bunching takes place when Δω < 0, and antibunching in the opposite case. Polariton squeezed states can arise in both the detuning cases.

DOI
10.1002/pssb.2221590156

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Abstract A selfconsistent steady state solution is given of the Fokker‐Planck equation for polaritons with wave vector ko and the kinetic equations for the scattered ones are found. The statistics of the coherently excited polaritons ko differs in a most general case from the pure coherent state. It depends on the crystallite volume V, the frequency detuning Δω between the polariton bare frequency ω(ko) and the laser one ωL, and the driving field energy flow SL. In the case of small values of the product V · SL polariton bunching takes place when Δω < 0, and antibunching in the opposite case. Polariton squeezed states can arise in both the detuning cases.
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.
Abstract A quantum behavior of the light emitted by exciton polaritons excited in a pillar semiconductor microcavity with embedded quantum well is investigated. Considering the bare excitons and photon modes as coupled quantum oscillators allows for an accurate accounting of the nonlinear and dissipative effects. In particular, using the method of quantum states presentation in a quantum phase space via quasiprobability functions (namely, a P-function and a Wigner function), we study the effect of the laser and the exciton-photon detuning on the second order correlation function of the emitted photons. We determine the conditions for the phenomena of bunching, giant bunching, and antibunching of the emitted light. In particular, we predict the effect of a giant bunching for the case of a large exciton to photon population ratio. Within the domain of parameters supporting a bistability regime we demonstrate the effect of bunching of photons.
The coherent and nonlinear properties of a polaritonic crystal (PolC), formed by trapped two-level atoms in an optical cavity array and interacting with an optical field, are analyzed. The nonlinear Schrödinger equation is considered for the dynamics of coupled atom-light states and low-branch polaritons associated with PolCs in the continuous medium limit. The existence of a stable ground-state PolC wave function is predicted using the variational approach. For negative scattering lengths, the wave function collapses in the presence of small quintic nonlinearity.
The quantum theory of the statistical properties and the photocount statistics of intense radiation in Raman scattering are presented making use of the coherent-state techniques. Both the Heisenberg-Langevin and Fokker-Planck equations approaches are developed. The existence of the conditional Glauber-Sudarshan quasi-distribution is also discussed. The time dependences of the photocount distribution and its factorial moments are demonstrated in the case of stimulated scattering and anticorrelation and antibunching of scattered photons are found.
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.
Coherent properties of a two-dimensional spatially periodic structure, polaritonic crystal (PolC) formed by trapped two-level atoms in an optical cavity array interacting with a light field, are analyzed. By considering the wave function overlapping for both photonic and atomic states, a cubic-quintic complex nonlinear Schr\"odinger equation is derived for the dynamics of coupled atom-light states, wave function of low-branch polaritons, associated with PolC in the continuous limit. A variational approach predicts that a stable ground-state wave function of PolC exists but is accompanied by an oscillating width. For a negative scattering length, the wave function collapses in the presence of a small quintic nonlinearity appearing due to a three-body polariton interaction. By studying the nonequilibrium (dissipative) dynamics of polaritons with adiabatic approximation, we have shown that the collapse of PolC wave function can be prevented even in the presence of small decaying of a number of polariton particles.