The Doppler-broadened steady state and transient response of a thermal ensemble of anharmonic oscillators to incident laser radiation is investigated. Non-linear polarization of this classical system is obtained by analytically solving the dynamical equations up to third order in the electric field. Except for the sign of the third-order transient polarization, the results are in striking contrast to those obtained in semi-classical theory. In particular, the classical response at resonance is unsaturated up to this order. The Doppler-integrated third-order free induction decay (FID) signal vanishes in the usual approximation of the homogeneous width being smaller than the Doppler width. The physical conclusion of the paper is negative: resonant non-linear and coherent effects such as saturation and nutation typical of the quantum mechanical two-level atoms do not arise for classical anharmonic oscillators.
The Doppler-broadened steady state and transient response of a thermal ensemble of anharmonic oscillators to incident laser radiation is investigated. Non-linear polarization of this classical system is obtained by analytically solving the dynamical equations up to third order in the electric field. Except for the sign of the third-order transient polarization, the results are in striking contrast to those obtained in semi-classical theory. In particular, the classical response at resonance is unsaturated up to this order. The Doppler-integrated third-order free induction decay (FID) signal vanishes in the usual approximation of the homogeneous width being smaller than the Doppler width. The physical conclusion of the paper is negative: resonant non-linear and coherent effects such as saturation and nutation typical of the quantum mechanical two-level atoms do not arise for classical anharmonic oscillators.
The propagation effects of linearly polarized near-resonant laser radiation through a thermal ensemble of Doppler-broadened anharmonic oscillators are considered. The dynamical equations of motion are analytically solved by a perturbation approach to obtain the nonlinear polarization. Subsequent self-consistent analysis of the Maxwell wave equation then leads to a classical area theorem, which is in qualitative agreement up to third order with the corresponding quantum mechanical result. There is, however, a discrepancy between the respective coefficients of the third-order terms.
We present a theoretical treatment of the coherent three-level beat signals occurring in the transient response of a Doppler-broadened molecular system with coupled closely spaced levels resonating with an intense monochromatic cw laser field. When the degenerate levels of the molecular system are suddenly split by means of an external Stark or Zeeman field, the transmitted laser radiation exhibits decaying beats at frequencies related to the level splitting or Raman frequency. This effect has been the subject of earlier theoretical analyses, but some relevant aspects such as the velocity integration, were not taken into account. A perturbation treatment, complete to third order in laser field strength, is used to calculate the velocity-integrated induced polarization and the corresponding reradiated field. The calculation takes into account both Raman-type processes, which lead to transient oscillations at the Raman frequency ${\ensuremath{\omega}}_{21}$, and population-saturation effects. In particular, population-induced transient oscillations at half the Raman frequency, $\frac{{\ensuremath{\omega}}_{21}}{2}$, are predicted. The relative importance of the various oscillations is analyzed as a function of the laser detuning and other parameters, including velocity-changing collisions. The relationship between the three-level beat signals and the well-known quantum-beat effect studied in fluorescence is explored.
A class of molecular two-photon processes, occurring under transient or steady-state conditions, is analyzed exactly for certain cases using a density-matrix formalism. Our results are relevant to the recently observed phenomena of transient coherent Raman beats and steady-state two-photon absorption of oppositely directed laser beams. Lowest-order perturbation theories have shown that these processes are insensitive to the Doppler effect, and consequently the linewidth is unaffected by Doppler broadening and by elastic collisions that change molecular velocity. The solutions presented here are of the same form for the Raman and two-photon problems and reveal significant power-dependent frequency shifts and line broadening that correct the ideal Doppler-free solutions. For the transient case, we assume a molecular three-level quantum structure that can be switched in or out of optical resonance with cw laser radiation by means of a pulsed dc Stark field, but the solutions also apply when resonant optical pulses are introduced. During the resonant condition when the Stark pulse is on, the three levels are prepared in coherent superposition, and during the nonresonant condition following the Stark pulse, the laser field(s) stimulates the coherently prepared sample in a transient two-photon process. One of these processes, the Ramanbeat effect, has been observed in forward scattering, but the second one, transient two-photon emission, should be observable in backward scattering. Bloch-like equations are derived for this three-level problem that facilitate an exact pulse solution for state preparation. Following the pulse, transient decay is well-approximated by a perturbation calculation. For the steady-state case, an exact solution is also obtained and is of interest for continuous-wave spectroscopy or for transient experiments requiring an initial preparation of quantum states, prior to Stark switching. These solutions exhibit a power-dependent line broadening and a frequency shift of magnitude $\ensuremath{\sim}\frac{\ensuremath{\Delta}({\ensuremath{\alpha}}^{2}\ensuremath{-}{\ensuremath{\beta}}^{2})}{({\ensuremath{\Delta}}^{2}+\frac{1}{{T}_{2}^{2}})}$, in agreement with our earlier estimate, where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are the Rabi flopping frequencies for the two intermediate transitions and $\ensuremath{\Delta}$ is the off-resonance tuning behavior for one of them.
The transient response of a three-level optical maser is calculated for a density of homogeneously broadened atoms coupled to one homogeneously broadened cavity mode. The atoms are treated in terms of the density matrix and the cavity mode is treated classically. Solutions are found analytically for the steady state and for the linearized form of two simplifications of the general equations. These special solutions have been previously obtained by Bloom and Statz and de Mars. Numerical solutions are found for a certain set of parameters. The response linewidth is found assuming one cavity mode coupled to a density of inhomogeneously broadened atoms.
The transient dynamics of electromagnetically induced transparency (EIT) are fundamental to understanding coherent light–atom interactions and the advancement of quantum technologies such as optical switching and quantum memory. However, in room-temperature atomic vapors, Doppler broadening significantly alters these dynamics, yet a comprehensive understanding of its impact on the transient EIT response remains lacking. In this study, we combine analytical and numerical methods to investigate the absorption dynamics of a weak probe field in a three-level Λ-type system driven by a strong coupling field, based on the optical Bloch equations and Laplace transform techniques. Our results show that the transient response is highly sensitive to both the atomic spontaneous emission rate and the Rabi frequency of the coupling field. Increasing the coupling field intensity not only accelerates the approach to steady state but also induces oscillatory dynamics and negative absorption. Under Doppler broadening, the time required to reach steady state increases by approximately three orders of magnitude compared to the Doppler-free case—an effect that is surprisingly insensitive to temperature variations across the 100–400 K range. Moreover, restoring a short steady-state time under broadened conditions necessitates increasing the coupling laser intensity by two orders of magnitude. These findings provide key insights into the influence of Doppler broadening on coherent transient processes and offer practical guidelines for the design of room-temperature atomic devices, including quantum memories and optical modulators.
We report theoretical and experimental studies of the propagation of a square-modulated laser pulse through a laser-cooled atomic ensemble of two-level absorbers, as well as through a three-level system with electromagnetically induced transparency. We find that the transmission characteristics over a wide range of optical depth can be fully accounted for as optical coherent transients excited by the wide spectral content of the square-modulated optical pulse. We show that both time-domain atom-field coupled equations and frequency-domain linear dispersion theory give precisely the same description of the optical transients at a weak power limit. At low optical depth with moderate absorption, resonant excitation dominates and free-induction decay (FID) contributes mostly to the transient field. At high optical depth when absorption and propagation effects become significant, lossless optical precursors start to dominate the transient response. By varying the optical depth from 0 to 45, we observe that optical transients evolve gradually from FID to optical precursors. We thus show that FID and optical precursors, which have been considered as two different optical transients for many decades, can be unified within a single theoretical frame and they are the manifestations of the same physical process in two different regimes.