Transient effects of dephasing and relaxation in cooperative evolution among three-level systems: II

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 ${\tau }_p$. 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 ${\omega }_R$, the characteristic collective relaxation time ${\tau }_R$ and $T_1$ and $T_2$ satisfy the inequalities ${\omega }_R{\tau }_R$, ${\omega }_R{T}_1$, ${\omega }_R{T}_2\gg 1$, collective transverse polarization will evolve if $\frac{{\gamma }_R}{4\gamma }>1\mathrm{}$, ${\omega }_R{\tau }_p\approx \pi$ ($\gamma$ is the uniform dephasing rate, i.e., $\gamma =T_2^{-1\mathrm{}}$). If, on the other hand, the above conditions are met, but $\frac{{\gamma }_R}{4\gamma }<1\mathrm{}$, no collective transverse polarization will evolve during the pump-pulse duration ${\tau }_p$. 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 $\frac{{\gamma }_R}{4\gamma }>1\mathrm{}$; if $\frac{{\gamma }_R}{4\gamma }<1\mathrm{}$, no solitary pulse will evolve.