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Physical Review A

Coupled Maxwell-pseudospin equations for investigation of self-induced transparency effects in a degenerate three-level quantum system in two dimensions: Finite-difference time-domain study

Gabriela M. Slavcheva, John M. Arnold, Iain D. Wallace, Richard W. Ziolkowski

Published Dec 31, 2002• 60 citations
Publisher

Abstract

We extend to more than one spatial dimension the semiclassical full-wave vector Maxwell-Bloch equations for the purpose of achieving an adequate and rigorous description of ultrashort pulse propagation in optical waveguides containing resonant nonlinearities. Our considerations are based on the generalized pseudospin formalism introduced by Hioe and Eberly [Phys. Rev. Lett. 47, 838 (1981)] for treatment of the resonant coherent interactions of ultrashort light pulses with discrete-multilevel systems. A self-consistent set of coupled curl Maxwell-pseudospin equations in two spatial dimensions and time for the special case of a degenerate three-level system of quantum absorbers is originally derived. Maxwell's curl equations are considered to be coupled via macroscopic medium polarization to the three-level atom model for the resonant medium. Two distinct sets of pseudospin equations are obtained corresponding to the TE- and TM-polarized optical waves. For the case of TM polarization, the electromagnetic wave is polarized in a general direction in the plane of incidence inducing two dipole transitions in a degenerate three-level system by each E-field component along the propagation axis and in transverse direction. 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. The relationship between the induced polarization and the state vector components that describe the evolution of the discrete-level system 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 by introducing nonuniform decay times corresponding to the various dipole transitions occurring in a three-level system. The system has been discretized using finite differences on a Yee grid and solved numerically by an iterative predictor-corrector finite-difference time-domain method. Self-induced transparency soliton propagation through a degenerate three-level quantum system of absorbers in two spatial dimensions and time is demonstrated in planar parallel-mirror waveguide geometries.

DOI
10.1103/physreva.66.063418

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We extend to more than one spatial dimension the semiclassical full-wave vector Maxwell-Bloch equations for the purpose of achieving an adequate and rigorous description of ultrashort pulse propagation in optical waveguides containing resonant nonlinearities. Our considerations are based on the generalized pseudospin formalism introduced by Hioe and Eberly [Phys. Rev. Lett. 47, 838 (1981)] for treatment of the resonant coherent interactions of ultrashort light pulses with discrete-multilevel systems. A self-consistent set of coupled curl Maxwell-pseudospin equations in two spatial dimensions and time for the special case of a degenerate three-level system of quantum absorbers is originally derived. Maxwell's curl equations are considered to be coupled via macroscopic medium polarization to the three-level atom model for the resonant medium. Two distinct sets of pseudospin equations are obtained corresponding to the TE- and TM-polarized optical waves. For the case of TM polarization, the electromagnetic wave is polarized in a general direction in the plane of incidence inducing two dipole transitions in a degenerate three-level system by each E-field component along the propagation axis and in transverse direction. 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. The relationship between the induced polarization and the state vector components that describe the evolution of the discrete-level system 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 by introducing nonuniform decay times corresponding to the various dipole transitions occurring in a three-level system. The system has been discretized using finite differences on a Yee grid and solved numerically by an iterative predictor-corrector finite-difference time-domain method. Self-induced transparency soliton propagation through a degenerate three-level quantum system of absorbers in two spatial dimensions and time is demonstrated in planar parallel-mirror waveguide geometries.
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.
We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.
The authors develop and apply a novel group-theoretical approach for studying the coherent dynamics of ultrashort pulse propagation in nonlinear optical waveguides and passive semiconductor microresonators. The resonant nonlinearity is modeled by a degenerate three-level system of saturable absorbers in order to allow for a two-dimensional medium polarization. The resulting Maxwell-pseudospin equations are solved in the time domain using the finite-difference time-domain method. Conditions of onset of the self-induced transparency (SIT) regime of propagation are investigated. Numerical evidence of multidimensional solitons localized both in space and in time is given for the planar optical waveguides. Pattern formation and cavity SIT-soliton formation are demonstrated for a passive semiconductor microcavity filled with saturable absorbers.
The coherent propagation of two optical short pulses through a resonant medium consisting of two three-level atoms in the configuration is investigated. A self-consisting analytical solution without steady state or adiabatic approximations is presented. The electric field in the model studied consists of two co-propagating plane waves, each of which is in near resonance with a transition in the absorber. The density matrix of two three level atoms is studied. Also, the reduced density matrices are stated. The present approach for the semiclassical treatment of resonant coherent interactions in three-level atoms represents a generalization of the Maxwell- Bloch equations for a two-level system. The Maxwell-Bloch equations reduced to the non-linear pendulum equation. The solution shows that the two pulses can propagate simultaneously without loss under some conditions about the pulses and the medium. The propagation of the two pulses through the medium without any constrains approves the self -induced transparency (SIT) phenomenon.
A study is made of steady-state ultrashort electromagnetic radiation pulses of two different wavelengths propagating at the same velocity in a three-level medium under exact resonance conditions. It is shown that such pulses (simultons) correspond to the soliton solutions of the corresponding generalized reduced Maxwell–Bloch equations which thus represent a new example of a system of evolution equations that can be solved exactly by the inverse scattering method. A full characterization of a simulton requires the knowledge of its duration and phase and also of a two-dimensional vector (called the internal polarization vector) with the components equal to the relative amplitudes of the pulses forming a simulton. Collision of two different simultons generally changes their polarizations. An expression is obtained for the angle of rotation of the polarization vectors as a function of the simulton amplitudes and their initial polarizations.
A new model for rigorous theoretical description of circularly (elliptically) polarized ultrashort optical pulse interactions with singly charged quantum dots embedded in semiconductor optical waveguides and microcavities is proposed. The method is based on self-consistent solution in the time domain of the vector Maxwell equations coupled via microscopic polarization to the coherent time-evolution equations of a discrete N-level quantum system in terms of the real pseudospin (coherence) vector. The model is applied to a 4-level system describing the optical dipole transitions of the trion state in a singly charged quantum dot. Selective optical excitation of specific spin states by predefined helicity of the optical field and an onset of self-induced transparency and polarized soliton formation in the nonlinear regime are numerically demonstrated