In the past decade, a two-dimensional matter-light system called the microcavity exciton-polariton has emerged as a new promising candidate of Bose-Einstein condensation BEC in solids. Many pieces of important evidence of polariton BEC have been established recently in GaAs and CdTe microcavities at the liquid helium temperature, opening a door to rich many-body physics inaccessible in experiments before. Technological progress also made polariton BEC at room temperatures promising. In parallel with experimental progresses, theoretical frameworks and numerical simulations are developed, and our understanding of the system has greatly advanced. In this article, recent experiments and corresponding theoretical pictures based on the Gross-Pitaevskii equations and the Boltzmann kinetic simulations for a finite-size BEC of polaritons are reviewed.

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Exciton-polariton Bose-Einstein condensation

We develop a mean-field theory of the dynamics of a nonequilibrium Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity. The spectrum of elementary excitations around the stationary state is analytically studied by means of a generalized Gross-Pitaevskii equation. A diffusive behavior of the Goldstone mode is found in the spatially homogeneous case and new features are predicted for the Josephson effect in a two-well geometry.

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Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons.

The authors review the physics of ultrafast dynamics in semiconductors and their heterostructures, including both the observed experimental phenomena and the theoretical description of the processes. These are probed by ultrafast optical excitation, generating nonequilibrium states that can be monitored by time-resolved spectroscopy. Light pulses create coherent superpositions of states, and the dynamics of the associated phase relationships can be directly investigated by means of many-pulse experiments. The commonly used experimental techniques are briefly reviewed. A variety of different phenomena can be described within a common theoretical framework based on the density-matrix formalism. The important interactions of the carriers included in the theoretical description are the phonon interactions, the interactions with classical and quantum light fields, and the Coulomb interaction among the carriers themselves. These interactions give rise to a strong interplay between phase coherence and relaxation, which strongly affects the non equilibrium dynamics. Based on the general theory, the authors review the physical phenomena in various semiconductor structures including superlattices, quantum wells, quantum wires, and bulk media. Particular results which have played a central role in understanding the microscopic origins of the relaxation processes are discussed in detail.

Theory of ultrafast phenomena in photoexcited semiconductors

From a theoretical point of view, we discuss a variety of phenomena linked to the spin and polarization degree of freedom of exciton-polaritons in semiconductor microcavities. We start with linear optical effects including the optical spin Hall effect, formation of polarization vortices and ballistic propagation of polarized exciton-polaritons. Next, the interplay between spin-dependent dynamics and Bose condensation in the 2D system of microcavity polaritons is addressed. Theoretically, this many-body system of interacting particles is described by the spinor Gross-Pitaevskii equations. These equations provide a description of the time evolution of polarized polariton fields under different conditions of optical excitation as well as an understanding of the phenomena of superfluidity, multistability and hysteresis via renormalization of the dispersion of elementary excitations. The comprehension of polarization-sensitive dynamics can be made through the introduction of several effective fields of different nature acting on the polariton pseudospin. The theory of parametric scattering of exciton-polaritons is presented, using the second quantization formalism. It is found that the combination of nonlinearity and various mechanisms of spin reorientation leads to self-organization and the formation of polarized patterns such as polarization crosses, vortices and rings. The manipulation of polariton spins can lead to various applications in signal processing, including the construction of optical logic gates and spin memory elements; the creation of spin currents; and the control of polarized signal propagation in the microcavity plane. The concept of polariton neurons is discussed in this connection.

https://iopscience.iop.org/article/10.1088/0268-1242/25/1/013001/pdf

Polariton polarization-sensitive phenomena in planar semiconductor microcavities

The application of femtosecond spectroscopy to the study of ultrafast dynamics in semiconductor materials and nanostructures is reviewed with particular emphasis on the physics that can be learned from it. Excitation with ultrashort optical pulses in general results in the creation of coherent superpositions and correlated many-particle states. The review comprises a discussion of the dynamics of this correlated many-body system during and after pulsed excitation as well as its analysis by means of refined measurements and advanced theories. After an introduction of basic concepts—such as coherence, correlation and quantum kinetics—a brief overview of the most important experimental techniques and theoretical approaches is given. The remainder of this paper is devoted to specific results selected in order to highlight how femtosecond spectroscopy gives access to the physics of coherences, correlations and quantum kinetics involving charge, spin and lattice degrees of freedom.First examples deal with the dynamics of basic laser-induced coherences that can be observed, e.g. in quantum beat spectroscopy, in coherent control measurements or in experiments using few-cycle pulses. The phenomena discussed here are basic in the sense that they can be understood to a large extent on the mean-field level of the theory. Nevertheless, already on this level it is found that semiconductors behave substantially differently from atomic systems. Subsequent sections report on the occurrence of coherences and correlations beyond the mean-field level that are mediated either by carrier–phonon or carrier–carrier interactions. The corresponding analysis gives deep insight into fundamental issues such as the energy–time uncertainty, pure dephasing in quantum dot structures, the role of two-pair or even higher correlations and the build-up of screening. Finally results are presented concerning the ultrafast dynamics of resonantly coupled excitations, where a combination of different interaction mechanisms is involved in forming new types of correlations. Examples are coupled plasmon–phonon and Bloch–phonon oscillations.The results reviewed in this paper clearly reveal the central role of many-particle correlations and coherences for the ultrafast dynamics of dense semiconductor systems. Both the presence of strong correlation effects and the formation of coherences in a genuine many-particle system have important implications for the controllability of optical signals from this class of materials, which is of utmost importance for applications in present-day and future optoelectronic devices.

https://iopscience.iop.org/article/10.1088/0034-4885/67/4/R01/pdf

Femtosecond spectroscopy in semiconductors: a key to coherences, correlations and quantum kinetics

We present numerical studies of the exciton polariton condensation kinetics in microcavities for the combined action of polariton-polariton and polariton-acoustic phonon scattering both for quasi-stationary and picosecond pulse excitation, respectively. For excitations of nearly resonant polaritons mainly the polariton-polariton scattering mechanism results in a condensation on the minimum of the lower polariton branch at relatively low areal polariton densities of the order of ${10}^{9}--{10}^{10}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$. The simultaneously acting polariton-phonon scattering increases the number of condensed particles considerably by providing a heat dissipation from the polariton gas to the lattice. For nonresonant excitations at large $k$ values the considered scattering mechanisms cannot support a polariton condensation. Above (below) laser threshold there exists a (no) polariton condensate and the polariton distribution of the noncondensed particles can be fitted by a Bose-Einstein (Maxwell-Boltzmann) distribution. Our results for $3\phantom{\rule{0.3em}{0ex}}\mathrm{ps}$ pulse excitation are in very good agreement with recent experimental observations of Yamamoto et al.

Condensation kinetics of microcavity polaritons with scattering by phonons and polaritons

A recently developed quantum kinetic description for the electrons in a two-band polar semiconductor coupled to LO phonons and excited by a femtosecond laser pulse is extended to include excitonic effects. Our numerical treatment yields a description of the relaxation of the electron-hole pairs which is valid on a time scale where the semiclassical Boltzmann description is no longer applicable. Oscillations with the period of the LO lattice oscillation are superimposed on the resulting polarization decay under excitation close to the band edge. The polarization oscillations are quantum beats between the direct interband transitions and their LO-phonon sidebands

Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors

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.

Coherence of condensed microcavity polaritons calculated within Boltzmann-Master equations

The ground state binding energy, the energy shift and the electric dipole moment of a hydrogenic impurity in a quantum-well wire subjected to an external electric field have been calculated. The variational ansatz for the wave function of the electron, that takes into account the effect of the finite barrier height, has been used. Numerical results for the case of GaAs quantum-well wire show a large influence of the electric field, the finite barrier height and the impurity position on the binding energy, the energy shift and the electric dipole moment.

Effect of the electric field on a hydrogenic impurity in a quantum-well wire

The kinetics of the boson condensation for micro-cavity (MC) polaritons interacting with thermal acoustic phonons is studied within rate equations, taking into account the finite MC cross-section. For a smoothly switched on cw excitation and finite polariton lifetimes we find a build-up of a large population in the lowest state of the polariton spectrum for sufficiently large deformation potential coupling at sufficiently high densities, which are still below the saturation density. The critical influence of the value of the heavy-hole deformation potential is illustrated, as well as the favorable influence of positive values of the detuning. This spontaneous polariton condensation would result in a perpendicular emission out of the micro-cavity.

D. B. Tran Thoai

Papers

Condensation kinetics of microcavity polaritons with scattering by phonons and polaritons

Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors

Coherence of condensed microcavity polaritons calculated within Boltzmann-Master equations

Effect of the electric field on a hydrogenic impurity in a quantum-well wire

Condensation kinetics of cavity polaritons interacting with a thermal phonon bath