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

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 ͑1981͔͒ 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.

FDTD Simulation of the Nonlinear Gain Dynamics in Active Optical Waveguides and Semiconductor Microcavities

“…This ability has been demonstrated by the authors in [32], [33] for two-dimensional geometries and a degenerate three-level quantum system. Further steps in the development of the model are currently under investigation.…”

“…These investigations definitely show that new physical effects become important in this regime which require treatments beyond the usual approximations [10], [11]. We have demonstrated the new features present in the new full-wave Maxwell-Bloch solution in tackling the self-induced transparancy (SIT) problem in one-(1-D) [18] and two-dimensions (2-D), [32], [33] that are absent in the standard approximations. The time derivatives of the electric field of the SIT pulses play an essential role in the nonlinear evolution of the quantum system.…”

“…With this discretization, we have found that the iterative scheme implemented converges and within 25 iterations gives differences between the new and the old values that are smaller than 1 10 . The chosen predictor-corrector scheme has been thoroughly investigated in terms of its ability to satisfy the lossless-case constraint [18] and proved to be very efficient even applied to the much more computationally demanding 2-D problem [32], [33].…”

Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers

“…Moreover, when studying quantum information systems that are dominated by radiative decay, it is critical to preserve the coherent radiative contributions that a MB analysis provides [41], without recourse to adding in phenomenological damping constants. Otherwise, if nonradiative processes dominate, a more straightforward MB formalism may be used, in which phenomenological damping terms like pure dephasing are implemented [42,43]. Direct MB simulations have been successfully used for a number of years.…”

Coherent control of extreme nonlinear optical properties via few-cycle laser pulses with transverse distribution

“…Based on the coupled Maxwell-Bloch equations with the slowly varying envelope approximation (SVEA) and rotating-wave approximation (RWA), the influences of the transverse effects on selffocusing or induced focusing of coherent optical pulses have been investigated extensively [6,7,8]. When the laser pulse duration approaches the optical cycle, the limitations of SVEA and RWA has been demonstrated, and several new features arise in the full Maxwell-Bloch equations without SVEA and RWA [3,9,10]. Recently, we have demonstrated an ultrafast transient population grating by using a two-dimensional (2D) ultrashort dipole soliton via solving the full-wave Maxwell-Bloch equations without any approximations [11].…”