## 4 Discussion
### Comparison of THz Data with Superradiant Theory  
The full width of half maximum (FWHM) of the pulse plotted in Figure 4b is estimated to be \(\sim\)0.8 ps, much shorter than the depopulation time T\({}_{1}\)\(\approx\) 60 ns. In addition, the modeling of Ref. [6] predicts that the FWHM is less than the dephasing time of the polarization T\({}_{2}\) (on the order of \(\sim\)1 ps) because of the radiation field-QD coupling [6]. Figure 7: (a) GaAs-ErAs band offsets and the most probable band-to-band transition between a \(\Gamma\)-pt S state (quantum numbers \(l\) = 0, n = 1) and a X-pt P state (\(l\) = 1, n = 2, m = \(-\)1, 0 or 1). (b) Spontaneous emission rate for transition in (a). The ringing in Figure 4b cannot be explained by the time-of-flight of the traveling wave to the contact pads, which is estimated to be \(\sim\)\(2\times 20=40\) ps (arm length = 2200 \(\upmu\)m, effective refraction index n\({}_{\text{eff}}\) = 2.65). It can't be explained by the reflection of the traveling wave from each corner of the square spiral as the length difference of each turn is only \(\sim\)14 \(\upmu\)m (\(\sim\)0.12 ps) either. Instead, we propose that the ringing is a telltale sign of superradiance, which has been observed experimentally in several material systems, and studied theoretically. It originates from the feedback between a radiation field and a collective system of multiple optical dipoles. The radiation field swings both positive and negative, displaying a displacement current-like alternation, which cannot be explained as normal photoconductive relaxation. And the fact that the same device pumped at 780 nm displayed no ringing rules supports this proposal. To explain the experimental ringing behavior quantitatively, we applied the sine-Gordon equation, which is an approximation of the famous Maxwell-Bloch set of equation that has been used successfully for description of superradiance of two-energy level systems [2; 3; 12; 13]. To solve the sine-Gordon equation for the sample used for Figure 4b, we need just a few parameters. The first one is the total number of QDs under the direct illumination of the pump pulses, which we estimate by N \(\approx\) nQDW\({}^{2}\)L = 1.62 \(\times\) 10\({}^{8}\) where nQD = 1 \(\times\) 10\({}^{18}\) cm\({}^{-3}\) is the quantum dot density, W = 9 \(\upmu\)m is the gap width (W \(\times\) W = active area), and L is the quantum-dot layer thickness (2 \(\upmu\)m). The instantaneous polarization is the summation over all available dipoles, and **P(t)** = \(\sum_{i}^{N}\mathbf{p}_{i}\approx\) Np when the dipoles are initially aligned by the strong ultrafast laser field. The initial tipping angle of the Bloch vector **P** is approximated by \(\theta_{0}\approx 1/\sqrt{N}=7.9\times 10^{-5}\). Assuming the dipoles radiate coherently, we solved the sine-Gordon equation with this initial value and obtained the \"ringing\" electric field plotted in Figure 8a. Only two other quantities were required for the solution, the superradiance time constant T\({}_{\text{R}}\) of \(\approx\)201 fs, and the individual quantum-dot spontaneous lifetime T\({}_{\text{SP}}\approx\) 60 ns. The fact that, T\({}_{\text{R}}\)\(<<\) T\({}_{\text{SP}}\) is an essential aspect of superradiant systems. The comparison between experiment and theory, Figure 4b vs. Figure 8a, is good in the ringing behavior beyond the first peak. Namely, the polarization field oscillates about zero, and is damped over about four periods. As a telltale sign of sine-Gordon behavior, the period increases with time as the ringing evolves, which is why the waveform in Figure 8a is a \"quasi-sinusoid\" rather than a pure sinusoid. The other essential aspect of the superradiant ringing is coupling to an antenna.