Optical coherent transients in cold atoms: From free-induction decay to optical precursors

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.

Figure 1

Schematic of an EIT $\Lambda$ system. ${\omega }_p$ and ${\omega }_c$ are the optical angular frequencies of the probe beam and coupling beam, respectively. ${\Gamma }_{31}$ and ${\Gamma }_{32}$ denote the atomic population decay rate of the transitions $|3\rangle \to |1\rangle$ and $|3\rangle \to |2\rangle$, respectively. $\Delta {\omega }_p$ is the detuning of the probe beam from the transition $|1\rangle \to |3\rangle$.

Figure 2

Transmission spectrum of the probe laser in (a) a two-level system (${\Omega }_c=0$) and (b) an EIT system at different optical depths. The red dashed lines denote the spectrum of an input step pulse.

Figure 3

Experimental configuration for measuring optical transients in cold ${}^{85}\mathrm{Rb}$ atoms. Both coupling and probe laser beams are directed from a single laser to eliminate EIT two-photon phase-frequency fluctuation. (Inset) EIT energy level diagram.

Figure 4

Observation of FID signals at the rising and falling edges with a weak probe pulse (${\Omega }_p=0.07{\gamma }_{13}$). (a) The input pulse passing through the vacuum (${\alpha }_{0}L=0$). (b) ${\alpha }_{0}L=2.2$ and $\Delta {\omega }_p=0$. (c) ${\alpha }_{0}L=2.2$ and $\Delta {\omega }_p=2\pi \times 20$ MHz. Panel (c3) is an enlarged view of (c1). The black circles are experimental data. The blue solid lines are theoretical curves obtained from Eq. (17) using FFT or solving the coupled Eqs. (2) and (4).

Figure 5

Power pumping and saturation effects in FID for the on-resonance case ($\Delta {\omega }_p=0$). (a) ${\Omega }_p=0.07{\gamma }_{13}$; (b) ${\Omega }_p=0.45{\gamma }_{13}$. ${\alpha }_{0}L=3$.

Figure 6

Power pumping and saturation effects in FID for the off-resonance case ($\Delta {\omega }_p=2\pi \times 8$ MHz). (a) ${\Omega }_p=0.07{\gamma }_{13}$; (b) ${\Omega }_p=0.81{\gamma }_{13}$. ${\alpha }_{0}L=4$.

Figure 7

Observation of optical precursors from a weak square-modulated pulse propagating through a two-level system. ${\Omega }_c=0$; ${\alpha }_{0}L=30$. The black circles are experimental data. The blue solid lines are numerical simulation from Eq. (17) using FFT or solving the coupled Eqs. (2) and (4) with the input pulse the same as in Fig. 4a. The red dash lines, overlapping with the blue solid lines, are calculated with hybrid-asymptotic analysis by taking into account the finite rise and fall time of 7 ns.

Figure 8

Observation of optical precursors from a weak square-modulated pulse propagating through an EIT system. ${\Omega }_c=4{\gamma }_{13}$; ${\alpha }_{0}L=30$. The black circles are experimental data. The blue solid lines are numerical simulations from Eq. (17) using FFT or solving the coupled Eqs. (2) and (4). The red dashed lines are calculated with hybrid-asymptotic analysis by taking into account the finite rise and fall time of 7 ns.

Figure 9

Off-resonance optical precursors measured at the two-level and EIT systems with ${\alpha }_{0}L=27$. Rows (a)–(c) correspond to $\Delta {\omega }_p/(2\pi )=0$, 13, and 300 MHz, respectively. Columns 1 and 2 represent the two-level system, and columns 3 and 4 represent the EIT system.

Figure 10

The transmission of the first transient peak after the falling edge as a function of optical depth in the two-level system with the finite fall time of 1, 3, and 7 ns. The probe laser is on resonance $\Delta {\omega }_p=0$. The circular and square markers are experimental data. The solid and dashed lines are theoretical curves.