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Physical Review A

Quasiadiabatic approximation for a dense collection of two-level atoms

Michael E. Crenshaw

Published Oct 1, 1996• 18 citations
Publisher

Abstract

In dense media, the Lorentz local-field condition results in a generalization of the usual semiclassical Maxwell-Bloch formulation for two-level atoms in which the optical Bloch equations are recast in terms of macroscopic spatially averaged atomic variables. We derive an adiabaticlike, or stationary, approximation for the generalized Bloch equations in the ultratransient limit and establish validity criteria. The significance of this quasiadiabatic approximation is that a point response solution of the generalized Bloch equations is obtained in terms of the field strength, thereby allowing the atomic response to be expressed as a nonlinear susceptibility that can be used to predict and analyze propagation effects in dense media. The quasiadiabatic approximation is quite general, allowing for a combination of detuning, chirping, and a variety of local-field and mean-field effects. It spans the range from cases for which local-field effects can be neglected, as in the usual adiabatic approximation, to cases in which local-field effects cause a large inversion-dependent frequency renormalization. The approximation is used to interpret previous numerical results for the resonant interaction of ultrashort pulses with dense media and it is shown that dense media can exhibit behavior analogous to adiabatic following and adiabatic inversion. The nonlinear index of refraction in the quasiadiabatic limit is purely real for large detunings. However, at and near resonance, the nonlinear index is purely imaginary. The imaginary index relates to coherent reflectivity arising from the reaction field of the cooperating atoms, rather than absorption, and transmission and reflection coefficients for thin films are derived. In the intermediate range of detunings, the intensity-dependent index of refraction can undergo a phase transition due solely to the variation of the field intensity during excitation by an ultrashort pulse. Propagation effects are investigated using a finite-difference time-domain method to integrate the generalized Bloch-Maxwell equations and the results of the numerical simulations are analyzed in the context of the nonlinear index of refraction obtained using the quasiadiabatic approximation.

DOI
10.1103/physreva.54.3559

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In dense media, the Lorentz local-field condition results in a generalization of the usual semiclassical Maxwell-Bloch formulation for two-level atoms in which the optical Bloch equations are recast in terms of macroscopic spatially averaged atomic variables. We derive an adiabaticlike, or stationary, approximation for the generalized Bloch equations in the ultratransient limit and establish validity criteria. The significance of this quasiadiabatic approximation is that a point response solution of the generalized Bloch equations is obtained in terms of the field strength, thereby allowing the atomic response to be expressed as a nonlinear susceptibility that can be used to predict and analyze propagation effects in dense media. The quasiadiabatic approximation is quite general, allowing for a combination of detuning, chirping, and a variety of local-field and mean-field effects. It spans the range from cases for which local-field effects can be neglected, as in the usual adiabatic approximation, to cases in which local-field effects cause a large inversion-dependent frequency renormalization. The approximation is used to interpret previous numerical results for the resonant interaction of ultrashort pulses with dense media and it is shown that dense media can exhibit behavior analogous to adiabatic following and adiabatic inversion. The nonlinear index of refraction in the quasiadiabatic limit is purely real for large detunings. However, at and near resonance, the nonlinear index is purely imaginary. The imaginary index relates to coherent reflectivity arising from the reaction field of the cooperating atoms, rather than absorption, and transmission and reflection coefficients for thin films are derived. In the intermediate range of detunings, the intensity-dependent index of refraction can undergo a phase transition due solely to the variation of the field intensity during excitation by an ultrashort pulse. Propagation effects are investigated using a finite-difference time-domain method to integrate the generalized Bloch-Maxwell equations and the results of the numerical simulations are analyzed in the context of the nonlinear index of refraction obtained using the quasiadiabatic approximation.
We investigate ultrashort pulse propagation in a dense medium of two-level atoms. We numerically solve the coupled nonlinear Maxwell-Bloch equations by assuming that the fields are slowly varying in time only. We find that while intrinsic optical switching of the atomic inversion persists in extended media, self-phase-modulation effects, coherent energy transfer, and reflections can be large even for films that are only a small fraction of a wavelength thick. If the propagation distance is sufficiently large, a physical boundary that separates fully excited atoms from atoms in the ground state can be created within the medium. In turn, this leads to a rapid spatial modulation of the nonlinear polarization, and a deterioration of the intrinsic optical switching mechanism. We then show that, although the slowly varying envelope approximation in space may be preempted by the fast longitudinal variation of the field and atomic variables, the mean-field approximation remains valid in the regime of ultrafast optical switching, provided the film thickness is sufficiently small such that the medium is nearly uniformly excited. These results are generated by utilization of our calculational method that fully accounts for the longitudinal dynamics of the fields, including reflections.
Transient processes are analyzed based on modified Bloch equations describing the interaction of optical radiation with dense resonant media with allowance for dipole-dipole interaction between atoms (the local-field effect). An analytical solution to the equation for the population difference of resonant atomic levels, taking into account the up-conversion processes, is obtained for a rectangular pulse in the quasi-stationary approximation. Computer simulation is performed for the kinetics of the population difference of resonant atomic levels in the field of a long sine-shaped pulse. It is shown that the time dynamics of the population difference of resonant atomic levels manifests characteristic features inherent in the intrinsic optical bistability effect under adiabatically slow variation of the acting field.
Transient processes are analyzed based on modified Bloch equations describing the interaction of optical radiation with dense resonant media with allowance for dipole-dipole interaction between atoms (the local-field effect). An analytical solution to the equation for the population difference of resonant atomic levels, taking into account the up-conversion processes, is obtained for a rectangular pulse in the quasi-stationary approximation. Computer simulation is performed for the kinetics of the population difference of resonant atomic levels in the field of a long sine-shaped pulse. It is shown that the time dynamics of the population difference of resonant atomic levels manifests characteristic features inherent in the intrinsic optical bistability effect under an adiabatically slow variation of the acting field.
Summary form only given. The dynamics of a medium consisting of a dense collection of two-level atoms interacting via the electromagnetic field has been derived from first principles. In dense media and condensed matter, the microscopic local field that couples with a single atom is related to the macroscopic field and volume propagation by the Lorentz-Lorenz relation written for an isotropic homogeneous medium. The dynamical equations for the medium, coupled self-consistently with the Maxwell inhomogeneous wave equation, are given in the rotating wave and dipole coupling approximations. The microscopic development is used within the framework of full quantization, and the explicit effects of spontaneous relaxation for the host as well as the atoms are treated, together with corresponding fluctuations. Particular effects upon intrinsic adiabatic inversion, self-induced transparency, and lasing without inversion, are presented and discussed. In particular, the equilibrium, stationary, and dynamical effects of the host medium, and medium and field fluctuations upon local field effects are discussed.