The effect of uniform dephasing and relaxation on the dynamical evolution of the population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the extreme excited state by an externally applied (
c-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field, which is treated quantum mechanically. The authors examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration ${\ensuremath{\tau}}_{p}$. It is found that in the homogeneously broadened regime, in the mean-field limit, for uniform pumping, and under conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective relaxation time ${\ensuremath{\tau}}_{R}$ and
T1 and
T2 satisfy the inequalities ${\ensuremath{\omega}}_{R}{\ensuremath{\tau}}_{R}$, ${\ensuremath{\omega}}_{R}{T}_{1}$, ${\ensuremath{\omega}}_{R}{T}_{2}\ensuremath{\gg}1$, collective transverse polarization will evolve if $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}>1$, ${\ensuremath{\omega}}_{R}{\ensuremath{\tau}}_{p}\ensuremath{\approx}\ensuremath{\pi}$ ($\ensuremath{\gamma}$ is the uniform dephasing rate, i.e., $\ensuremath{\gamma}={T}_{2}^{\ensuremath{-}1}$). If, on the other hand, the above conditions are met, but $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}<1$, no collective transverse polarization will evolve during the pump-pulse duration ${\ensuremath{\tau}}_{p}$. In the first case super-radiant pulse evolution (classical collective evolution) may evolve after pump-pulse termination. In the latter case it is possible for collective spontaneous relaxation (superfluorescence) to take place if other conditions are satisfied. Further, a convenient algorithm is established whereby results for the super-radiance or superfluorescence model in the mean-field, small-tipping-angle limit are shown to be formally identical to corresponding results for the lethargic, asymptotic regime of swept-gain super-radiance. In the latter case, the equations of motion are with respect to the retarded time frame and the field equations are replaced by Maxwell's equations. It is shown that the mean-field approximation for the case of super-radiance or superfluorescence is formally equivalent to the asymptotic-limit approximation for swept-gain super-radiance. It is found that a solitary pulse will evolve if the gain-to-loss ratio $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}>1$; if $\frac{{\ensuremath{\gamma}}_{R}}{4\ensuremath{\gamma}}<1$, no solitary pulse will evolve.
The density matrix equations have been solved in conjunction with the time-dependent field amplitude equations for a three level optically pumped gas. The solutions are valid in the limit in which the pumping pulse is longer than the dephasing time of the system and true coherence effects may be neglected. In order to obtain closed form solutions, it is assumed that the FIR field is smaller than the pump field and saturation effects associated with this field are absent. When the pump pulse is resonant with the absorbing FIR transition, particularly simple solutions for the time-dependent FIR field amplification are obtained. These solutions exhibit the specific dependence of the FIR output pulse on the shape of the input pulse and display certain pulse narrowing features.
It is shown that the density matrix equations for a three-level quantum system interacting with external radiation fields can lead to negative populations if arbitrary dephasing rates and shifts are included in these equations. To guarantee non-negative populations, the equations themselves impose certain restrictions on the dephasing widths and shifts. The constraints on the widths are shown to be identical to those that can be derived from a model of Markovian dephasing events, independent of any atom-field interaction.
For the interaction of coherent radiation with a nondegenerate, homogeneously broadened multilevel system the nonlinear susceptibilities and excitations and their time dependence are evaluated using a perturbation approach to the density matrix up to fourth order. This is done for the "off-resonance" case, ${(\ensuremath{\Delta}{\ensuremath{\omega}}^{2}+{\ensuremath{\Gamma}}^{2})}^{\frac{1}{2}}\ensuremath{\gg}\frac{1}{\ensuremath{\tau}}$, as well as for the "on-resonance" case, ${(\ensuremath{\Delta}{\ensuremath{\omega}}^{2}+{\ensuremath{\Gamma}}^{2})}^{\frac{1}{2}}\ensuremath{\ll}\frac{1}{\ensuremath{\tau}}$, with $\ensuremath{\tau}$ being the pulse duration, by taking into account the temporal change of the electric field amplitude and the phase and energy relaxation times,
T2 and
T1. It is shown that the transient one-photon excitation known as adiabatic following, is not only important for a pulse duration $\ensuremath{\tau}<{T}_{2}$, but may also be important for ${T}_{2}\ensuremath{\lesssim}\ensuremath{\tau}<{T}_{1}$. In addition, expressions for the transient two-photon excitation in a multilevel system are derived. Finally, the two-level limit is evaluated to all orders of the perturbation including damping effects, which gives results valid also for large incident light intensities. For the "off-resonance" case an expression is obtained which describes not only the transient adiabatic following process, but also the usual one-photon absorption including saturation effects giving rise to power broadening.
Starting with the complete molecular Liouville equation and making use of the Mori–Zwanzig projection operator formalism we derive generalized rate equations which describe the collisionless dynamics of level populations in the quasicontinuum of molecular multiphoton processes. We define precisely the complete molecular information required for the description of these processes which is expressed in the form of a hierarchy of intramolecular dipole correlation functions. We then discuss the general conditions under which the equations attain the form of simple rate equations and show how the complete molecular information reduces considerably in this case to three relevant quantities per transition: an integrated Rabi frequency, a dephasing time and ratios of statistical weights of the levels.
The rate of absorption and emission of radiation is calculated for two laser fields interacting with a homogeneously broadened, three-level molecular system. The time-dependent Schrödinger equation is used in the density matrix formalism and the laser fields are treated classically. The laser fields, at frequencies ωpand ωs, are off resonance by δpand δs, respectively, such that|\delta_{p}|, |\delta_{s}| \ll \omega_{p}, \omega_{s}. Analytic expressions are derived for the imaginary part of the electric susceptibility,\chi"(\omega_{p})and\chi"(\omega_{s}). The present work differs from previous calculations in that the laser fields may have arbitrary values of field strength and each field may be on or off resonance. A discussion is presented of the properties of the solutions for\chi"(\omega_{p})and\chi"(\omega_{s}), for both strong and weak laser fields, based on computer evaluation of the general analytic expressions. It is shown that the solutions may be divided into single-photon and two-photon, or Raman, contributions. Conditions for optimizingdI(\omega_{s})/dzand the photon conversion efficiency are derived. These results may be applied to analyzing the pulsed, optically pumped submillimeter laser and to studying two-photon absorption in a gas.