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Physical Review A

Maxwell-Bloch formulation for semiconductors: Effects of coherent Coulomb exchange

Charles M. Bowden, Govind Prasad Agrawal

Published May 1, 1995• 38 citations
Publisher

Abstract

A generalized Bloch-Maxwell formulation for laser-field-coupled semiconductors is derived from a two-band model which includes direct Coulomb interactions. The momentum-dependent, microscopic, electron-hole equations of motion in the time-dependent Hartree-Fock approximation and neglecting interband exchange interactions form the starting point for the formulation. A self-consistent set of coupled equations in four dynamical variables for the medium, together with the electric-field amplitude coupled through the Maxwell wave equation in a semiclassical approximation, are obtained to lowest order in the coherent Coulomb exchange interaction and the density-of-states distribution. Intrinsic optical bistability is predicted in a steady state due to a carrier density-dependent redshift of the band edge due explicitly to coherent Coulomb exchange. Integration of the dynamical equations for conditions which correspond to the ultrafast time regime exhibit intrinsic adiabatic inversion, adiabatic following, anomalous Rabi cycling, and unique, as well as fast, optical switching, all of which depend upon the coherent Coulomb exchange interaction.

DOI
10.1103/physreva.51.4132

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A generalized Bloch-Maxwell formulation for laser-field-coupled semiconductors is derived from a two-band model which includes direct Coulomb interactions. The momentum-dependent, microscopic, electron-hole equations of motion in the time-dependent Hartree-Fock approximation and neglecting interband exchange interactions form the starting point for the formulation. A self-consistent set of coupled equations in four dynamical variables for the medium, together with the electric-field amplitude coupled through the Maxwell wave equation in a semiclassical approximation, are obtained to lowest order in the coherent Coulomb exchange interaction and the density-of-states distribution. Intrinsic optical bistability is predicted in a steady state due to a carrier density-dependent redshift of the band edge due explicitly to coherent Coulomb exchange. Integration of the dynamical equations for conditions which correspond to the ultrafast time regime exhibit intrinsic adiabatic inversion, adiabatic following, anomalous Rabi cycling, and unique, as well as fast, optical switching, all of which depend upon the coherent Coulomb exchange interaction.
The availability of femtosecond pulses in the optical-frequency regime allows the study of coherent effects in the interaction between semiconductors and light. Because the properties of the exciting pulses become important ingredients in spectroscopy, the theoretical analysis must treat the complete time dependence of the system. For this purpose we use the semiconductor Bloch equations (SBE),1-3 which are the coupled equations of motion for the interband polarization and the electron and hole populations for an interacting electron-hole system in the random-phase approximation. The equations describe the electron-hole Coulomb attraction leading to excitonic features; they contain phasespace filling and band-gap renormalization effects, as well as the dipole interaction with the applied light field.
Summary form only given. Coupled semiclassical Maxwell-Bloch equations are originally derived, based on the generalized pseudospin formalism for description of the dynamics of the resonant coherent interactions of ultrashort light pulses with degenerate three-level quantum system in 2D. The full-wave vector Maxwell equations are considered to be coupled to the three-level atom model for the active medium, therefore two distinct sets of pseudospin equations are derived corresponding to the TM and TE optical waves. In order to account for the polarization induced both along the propagation axis and in transverse direction in two spatial dimensions, we introduce a dipole-coupling interaction Hamiltonian allowing Rabi flopping of the population difference along and perpendicular to the propagation axis with frequencies depending on the corresponding field components. Relationship between the polarisation induced and the state vector components, describing the population difference dynamics, is derived in order to couple the quantum system equations to the Maxwell's curl equations. The pseudospin equations are phenomenologically extended to include relaxation effects.
A microscopic many-body theory for the optical and electronic properties of semiconductors is reviewed with an emphasis on the role of correlation effects. At the semiclassical level, the semiconductor Bloch equations include many-body effects via bandgap and field renormalization as well as correlation contributions representing two electron-hole pair amplitudes, excitonic populations, and coupled interband and intraband coherences. These Coulomb interaction induced carrier correlations lead to characteristic signatures in nonlinear semiconductor spectroscopy. At the fully quantum mechanical level the dominant light-matter correlations are described by coupled semiconductor Bloch and luminescence equations. Excitonic emission properties of quantum well and microcavity systems are discussed, including effects such as coherent signatures in the secondary emission and coherent control of the emitted light.
We theoretically demonstrate the population switching of quantum dots (QD's), modeled as two-level atoms in idealized one-dimensional (1D) and two-dimensional (2D) photonic crystals (PC's) by self-consistent solution of the Maxwell-Bloch equations. In our semiclassical theory, energy states of the electron are quantized, and electron dynamics is described by the atomic Bloch equation, while electromagnetic waves satisfy the classical Maxwell equations. Near a waveguide cutoff in a photonic band gap, the local electromagnetic density of states (LDOS) and spontaneous emission rates exhibit abrupt changes with frequency, enabling large QD population inversion driven by both continuous and pulsed optical fields. We recapture and generalize this ultrafast population switching using the Maxwell-Bloch equations. Radiative emission from the QD is obtained directly from the surrounding PC geometry using finite-difference time-domain simulation of the electromagnetic field. The atomic Bloch equations provide a source term for the electromagnetic field. The total electromagnetic field, consisting of the external input and radiated field, drives the polarization components of the atomic Bloch vector. We also include a microscopic model for phonon dephasing of the atomic polarization and nonradiative decay caused by damped phonons. Our self-consistent theory captures stimulated emission and coherent feedback effects of the atomic Mollow sidebands, neglected in earlier treatments. This leads to remarkable high-contrast QD-population switching with relatively modest (factor of 10) jump discontinuities in the electromagnetic LDOS. Switching is demonstrated in three separate models of QD's placed (i) in the vicinity of a band edge of a 1D PC, (ii) near a cutoff frequency in a bimodal waveguide channel of a 2D PC, and (iii) in the vicinity of a localized defect mode side coupled to a single-mode waveguide channel in a 2D PC.
The two-band effective semiconductor optical Bloch equations for a quantum well have been generalized to include multisubband effects. The ultrafast transient adiabatic-following solutions are obtained from the equations for interband polarizations, intersubband polarizations, and subband-carrier distribution functions of a ${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Alx{\mathrm{Al}}_{\mathit{x}}Alx​As/GaAs quantum well, excited by an ultrafast optical pump field. Two conservation laws for the multisubband case are derived. The steady-state response and the \ensuremath{\pi}-pulse response are studied. It is found that these multisubband equations lead to quantitatively different results from the conventional two-band approximation, particularly for wide quantum wells and asymmetric quantum wells.
The simultaneous influence of pulse propagation and Coulomb many-body interaction in Four Wave Mixing experiments is theoretically studied using the Semiconductor-Maxwell-Bloch equations. In this framework, the propagation of the optical fields and the microscopic interaction of the light with the semiconductor are treated consistently. The Maxwell equation is used as a reduced wave equation and the many-body Coulomb interaction are treated at the level of a screened Hartree-Fock approximation including phenomenological decay terms. We have analyzed the interplay of the propagation and Coulomb effects by calculating the Four Wave Mixing signal (FWMS) for different sample lengths and various excitation conditions, i.e, for pulses with different intensities at the exciton and in the continuum.