Time-dependent collisional approximations for two-level systems

We derive two types of perturbative approximation for the response of collisionally broadened two-level atoms to a time-dependent field. The perturbative approximations for the inversion and polarization are valid to any desired order in the derivatives of the field. Comparison with a numerical solution of the full density-matrix equations of motion demonstrates that the new approximations significantly improve upon the adiabatic-following and collisional steady-state approximations for time-dependent pulses in the appropriate damping regime. Application of the approximations to pulse propagation, sideband instability, and population pulsations is discussed. The collisional steady-state and adiabatic-following approximations find extensive use in many areas of nonlinear optics. The deficiencies of these approximations are well known: The collisional steady-state approxirnation is valid only for monochromatic (cw) radiation and the adiabatic-following approximation is valid only in the absence of damping and only if the product of the detuning and a characteristic time is much larger than one. In spite of their manifest faults, the collisional steady-state and adiabatic-following approximations are often exceptionally good approximations, in the appropriate damping regime, for the local response of a vapor of two-level atoms to a slowly varying field. However, timedependent nonlinear processes modify a propagating field zuch that it varies less slowly and the neglected portions of the atomic response become more important. ' Therefore improved approximations for the damped atomic response to a time-dependent field are desirable. In this paper we present two improved perturbative approximations for the inversion and polarization of collisionally broadened two-level atoms induced by a time-dependent field. The adiabatic-following approximation for two-level systems was derived phenomenologically by Grischkowsky, who used it to explain certain experimental results of near-resonance laser-pulse propagation. Motivated by the work of Grischkowsky, Crisp provided an analytic derivation for the adiabatic-following approximation and obtained a quadrature for the error. The essential feature of Crisp's derivation is the expansion of the kernel of the integral representation of the off-diagonal density-matrix element p2, in a Fourier series with integration proceeding term by term. Both the adiabatic-following approximation and the collisional steady-state approximation were obtained in first order. Both Crisp and Grischkowsky used the adiabatic inversion to derive a correction to the adiabatic-following approximation for the polarization. We have shown how Crisp's method can be extended to obtain corrections to the adiabatic inversion and to improve corrections to the polarization in the absence of damping. In this paper we show how the integrated Fourier series can be used to obtain two types of perturbative approximation for the inversion and polarization of collisionally broadened twolevel systems that are valid to any desired order in the derivatives of the field. The two types of perturbative approximation to the inversion are (i) a formal solution of the rate equation for the inversion which is obtained from the density-matrix equations for the populations; and (ii) an approximation obtained using the approximation (p22 — p») —I — 4p2, P, 2, valid for weak damping. The approximations for the polarization are obtained by substituting either (i) or (ii) for the inversion in the integrated-Fourier-series expansion of p2, . The approximation obtained using the first type of approximate inversion converges most rapidly for strong damping and reduces to the collisional steady-state approximation when the magnitude and phase of the field are constant. The approximation obtained using the second type of approxirnate inversion converges most rapidly for weak damping and in first order reduces to the adiabaticfollowing approximation for zero damping. Comparison with a numerical solution of the density-matrix equations demonstrates that retention of higher-order derivatives by retaining more terms in the integrated Fourier series results in a significant improvement in accuracy for a time-dependent field. However, Rabi oscillations, which can be an important component of the nonlinear response, ' cannot be obtained by finite-order approxi