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
The response of a two-level atom to a smoothly varying near-resonant driving pulse can be described by the usual adiabatic-following approximation if the conditions $\frac{{T}_{1}^{\ensuremath{-}1},{T}_{2}^{\ensuremath{-}1}\ensuremath{\ll}{t}_{p}^{\ensuremath{-}1}\ensuremath{\ll}\ensuremath{\Omega}(t)}{2\ensuremath{\pi}}$ are satisfied throughout the pulse. Here, $\ensuremath{\Omega}(t)={[{\ensuremath{\Delta}}^{2}(t)+\frac{{p}^{2}{\mathcal{E}}^{2}(t)}{{\ensuremath{\hbar}}^{2}}]}^{\frac{1}{2}}$ is the atomic precession frequency, $\ensuremath{\Delta}(t)$ the detuning,
E(t) the field envelope,
tp the effective pulse width, and
T1 and
T2 the atomic level and phase relaxation times, respectively. From Bloch's equations, we have developed a generalized version of this approximation applicable to cases where
T1 and
T2 can be comparable to or less than
tp. It allows the condition ${T}_{1}^{\ensuremath{-}1}, {T}_{2}^{\ensuremath{-}1}\ensuremath{\ll}{t}_{p}^{\ensuremath{-}1}$ to be replaced by the weaker requirement ${T}_{1}^{\ensuremath{-}1}$,$\frac{{T}_{2}^{\ensuremath{-}1}{p}^{2}{\mathcal{E}}^{2}(t)}{{\ensuremath{\hbar}}^{2}{\ensuremath{\Omega}}^{2}(t)\ensuremath{\ll}\ensuremath{\Omega}(t)}$. If ${T}_{1}\ensuremath{\gg}{t}_{p}$, the atomic Bloch vector remains aligned nearly parallel to the effective driving field
E(t), 0, $\frac{\ensuremath{\hbar}\ensuremath{\Delta}(t)}{p}$) in the rotating reference frame (as in the pure adiabatic following case); however, it decays in length if
T2 is comparable to
tp. The approximation bridges the gap between pure adiabatic following behavior for ${t}_{p}^{\ensuremath{-}1}\ensuremath{\gg}{T}_{1}^{\ensuremath{-}1}, {T}_{2}^{\ensuremath{-}1}$ and rate-equation behavior for ${t}_{p}^{\ensuremath{-}1}\ensuremath{\ll}{T}_{2}^{\ensuremath{-}1}\ensuremath{\approx}\ensuremath{\Omega}(t)$, thereby allowing a more complete description of phenomena such as adiabatic inversion and resonantly enhanced self-focusing and defocusing.
A perturbation scheme based on the slow adiabatic variation of the radiation-field characteristics is applied to the Maxwell-Bloch equations. The two leading orders in the perturbation expansion yield a set of partial differential equations for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$, $\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$, and $a(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ (the local wave vector and frequency and the field amplitude). All the coefficients of these equations are derivable from a single dispersion function $H(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},\ensuremath{\omega},a)$, this feature is used to further reduce the equations to a set of Hamilton-Jacobi-type ordinary differential equations (ray equations). Unlike the linear geometric optics case the properties of the "local lens" seen by a ray depend also on the amplitude of the field propagating along neighboring rays. A method to derive additional equations for the evolution of the local lens along a single ray and to decouple the equations for different rays is described, the resultant enlarged set of ray equations is derived. The limitations and advantages of this approximation scheme are discussed in comparison with other approaches.
A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.
A more accurate description of the 2 pi pulse of self-induced transparency is obtained. This is an exact solution of an approximate form of the Maxwell-Bloch equations in which only backscattering is neglected; the equations are valid at densities less than about 10 18 atoms cm -3 . At such densities the 2 pi sech pulse of McCall and Hahn (1967, 1969) remains a good approximate well into the picosecond range. The more accurate solution is chirped by a factor proportional to the square of the ratio of the spectral width of the pulse to its carrier wave frequency. The stability of this pulse solution and other N soliton solutions of the approximate Maxwell-Bloch equations is demonstrated.
Short pulses of narrow-line low-intensity dye-laser light nearly resonant with the Zeeman-split
2P21 resonance line (7948 \AA{}) of rubidium were observed to propagate through dilute rubidium vapor as slowly as
(141)c. These slow pulse velocities showed that most of the energy in the propagating wave was contained by the vapor as coherent atomic excitation. The observed pulse velocities
vp are in good agreement with the equation ${v}_{p}=\frac{d\ensuremath{\omega}}{\mathrm{dk}}$ for the group velocity obtained from linear-dispersion theory. Also, the experimental results are quantitatively explained by adiabatic following, in which the pseudomoments of the atoms remain aligned along the effective field of the laser light. The adiabatic-following model allows for a direct comparison of our results with the work on self-induced transparency. For high-intensity light, adiabatic following predicts a nonlinear pulse velocity and the possibility of observing self-steepening.