Transformation of subradiant states to superradiant states in a thick resonant medium

The propagation of a step pulse through a thick resonant absorber with a homogeneously broadened absorption line is considered. It is shown that a specific subradiant state is naturally developed in the absorber due to the formation of the spatial domains of the atomic coherence with opposite phases. It is proposed to divide the absorber into slices in accordance with these domains and place the phase shifters in front of the first slice and between the other slices. If the phase shifters are switched on simultaneously at a particular moment of time, elapsed from the beginning of the step pulse, a strong sharp pulse is generated at the output of the last slice of the absorber. The effect is explained by the phasing of the atomic coherence along all slices of the absorber, which transforms the subradiant state of the atom-field system to a superradiant state.

Figure 1

Spatial dependencies of the imaginary part of the atomic coherence, $-i{\sigma }_{\mathrm{eg}}(z,t_p)$, (solid line in red) and the radiation field, $\Omega (z,t_p)$, (dotted black line) along the absorber at $t=t_p$. Both are normalized to 1 (see the text for details). Vertical bold blue lines divide the plot into domains (I, II, and III), where the imaginary part of the atomic coherence has one sign (plus or minus). Thin horizontal lines show the values of the two first extrema of the field amplitude. The excitation scheme of the absorber, cut into slices, and phase shifters (PSs) placed between them, are shown in the bottom.

Figure 2

The field intensity and the shape of the pulse, generated at the output of the sample by the phase shift of the radiation field, applied at the input at time $t_p$ satisfying the condition $b_l{t}_p=3.67$. $b_l$ is the superradiant parameter of the sample. The field intensity is normalized to $|{\Omega }_0{|}^2$. The decay rates of the atomic coherence are $\gamma =0.003b_l$ (solid line in red) and $\gamma =0.3b_l$ (dotted line in blue).

Figure 3

The field intensity at the output of the composite absorber consisting of two slices. The $\pi$-phase shift of the fields at the input of the first and second slices of the composite absorber are applied at time $t_p$ satisfying the condition $b_1{t}_p=3.67$, where $b_1$ is the superradiant rate of the first slice. The time scale is normalized to this rate. The field intensity is normalized to $|{\Omega }_0{|}^2$. The decay rates of the atomic coherence are $\gamma =0.01b_1$ (solid line in red) and $\gamma =0.1b_1$ (dotted line in blue).

Figure 4

The field intensity at the output of the composite absorber consisting of three slices. The $\pi$-phase shift of the fields at the input of all slices of the composite absorber are applied at time $t_p$ satisfying the condition $b_1{t}_p=3.67$, where $b_1$ is the superradiant rate of the first slice. The time scale is normalized to this rate. The field intensity is normalized to $|{\Omega }_0{|}^2$. The decay rates of the atomic coherence are $\gamma =0.01b_1$ (a) and $\gamma =0.1b_1$ (b). The dependence of the normalized maximum of the pulse intensity $I_n$, induced by the phase shifts, on the number of slices $n$ is shown in the inset to (a) for the case $\gamma t_p\ll 1$.

Figure 5

Time dependence of the phase variation in time (bold solid line). The parameters are ${\tau }_{ph}=8$ ps and $t_p=25$ ps. Thin horizontal lines show the borders of the phase change within the time interval 50 ps.

Figure 6

Time dependence of intensity of the field $I_n\left(t\right)$ at the exit of the $n$th slice $\left(n=1,2,3\right)$. $I_0$ is the intensity of the field incident to the composite absorber. The solid line (in red) shows the intensity time dependence for the instantaneous phase shifts. Dots (in blue) show the intensity if the phases change smoothly. Parameters are defined in the text.

Figure 7

Normalized maximum intensity of the second superradiant pulse $I_{2\max }$ versus $t_2-t_1$, bold solid line. Thin horizontal line shows maximum intensity for $t_2=t_1$.

Figure 8

Time dependence of the pulses induced by nonsimultaneous phase shifts of the fields in front of two slices of the composite absorber. The solid line (in red) corresponds to the instantaneous phase shifts and the dotted line (in blue) corresponds to the case of smooth phase shifts (during 50 ps). The time delay between phase shifts is 1.8 ns. The moments of time of the phase shifts are $t_1=16.5$ ns, $t_2=18.3$ ns in (a), and $t_1=18.3$ ns, $t_2=16.5$ ns in (b).

Figure 9

Time dependence of the fields at the exit of the first slice. They are produced by the input step pulse (solid line in red) and pulse whose front grows slowly according to Eq. (52) (dots in blue). Details are in the text.