Slowing 80-ns light pulses by four-wave mixing in potassium vapor

We experimentally and theoretically study propagation of 80-ns Gaussian-like probe pulses in hot potassium vapor under conditions of four-wave mixing (FWM). The atomic scheme for FWM is off-resonant, double-$\mathrm{\Lambda }$ atomic scheme, with pump and probe photons, mediated in the K vapor, generating new probe and conjugate photons. We define the subset of FWM parameters, one-photon pump detuning, two-photon pump-probe Raman detuning, vapor density, the pump Rabi frequencies, when slowed pulses exit the vapor are also Gaussian-like. When Gaussian-like pulses exit the cell we are able to compare theoretical and experimental results for fractional delays and broadening for the probe and conjugate. We have obtained fractional delays above 1. Results of the model are compared with the experiment, with and without the model of Doppler averaging, when the atom velocity distribution is divided into different number of groups. We analyze possible causes for pulse broadening and distortion of slowed probe pulses and show that they are the result of quite different behavior of the probe pulse in the FWM vapor. Besides presenting the first results of slowing 80-ns probe pulses, this work is a useful test of the numerical model and values of parameters taken in the model that are not known in experiments.

Figure 1

(a) Experimental setup. (a) Double-$\mathrm{\Lambda }$ scheme for $D_1$ line in potassium. Pump beam–thick line (blue for online version), probe beam–thin line (red for online version), conjugate beam–dashed line (green for online version).

Figure 2

Experimental observations of slow light, probe (thick black), conjugate (thin black), and reference 80-ns incoming probe beam (green) waveforms for (a) $\delta =-4$ MHz, (b) $\delta =-8$ MHz, and (c) $\delta =-12$ MHz. In all three cases (a), (b), and (c), $\mathrm{\Delta }=0.7$ GHz, $T=120{}^{\circ }\mathrm{C}$, pump power $P_d=220\mathrm{mW}$, probe power $P_p=20\mu \mathrm{W}$.

Figure 3

Probe (solid squares, black for online version) and conjugate (solid triangles, red for online version) (a) gains, (b) fractional broadenings, and (c) fractional delays vs $\delta$, for $\mathrm{\Delta }=0.7\mathrm{GHz},T=120{}^{\circ }\mathrm{C},P_d=220\text{mW},P_p=20\mu \mathrm{W}$.

Figure 4

Calculated waveforms of probe (thick black line) and conjugate (thin black line) for 80-ns incoming probe pulse (green line). (a) $\delta =-4$ MHz, (b) $\delta =-8$ MHz, (c) $\delta =-12$ MHz. Values of other parameters were kept constant, ${\mathrm{\Omega }}_d=1.38$ GHz, ${\mathrm{\Omega }}_p=18.9$ MHz, $\gamma =0.5\times {10}^7$ Hz, ${\gamma }_{\mathrm{deph}}=1.5\times {10}^7$ Hz, $\mathrm{\Delta }=0.7$ MHz, $N_c=3\times {10}^{12}{\mathrm{cm}}^{-3}$ ($T=120{}^{\circ }\mathrm{C}$), $\theta =3$ mrad.

Figure 5

Calculated probe (solid squares, black for online version) and conjugate (solid triangles, red for online version) (a) gains (note that probe gain is multiplied by 10), (b) fractional broadenings, and (c) fractional delays vs $\delta$ of 80-ns input Gaussian pulses in K vapor at $3\times {10}^{12}{\mathrm{cm}}^{-3}$ ($T=120{}^{\circ }\mathrm{C}$), ${\mathrm{\Omega }}_d=1.38$ GHz, ${\mathrm{\Omega }}_p=18.9$ MHz, $\mathrm{\Delta }=0.7$ GHz.

Figure 6

Numerical simulations of probe (thick black) and conjugate (thin red) waveforms. a) without Doppler averaging, b) with Doppler averaging using 3 velocity groups, and c) using 5 velocity groups. Results are for 80 ns probe inbound pulse, $\mathrm{\Delta }=1\mathrm{GHz}$, and $\delta =4\mathrm{MHz}$. Other parameters for simulations: ${\mathrm{\Omega }}_d=2.31\mathrm{GHz}, {\mathrm{\Omega }}_p=18.9$ Mhz, $N_c=3\times {10}^{12}{\mathrm{cm}}^{-3}, \gamma =5\times {10}^7$ Hz, ${\gamma }_{\mathrm{deph}}=1.5\times {10}^7$ Hz.

Figure 7

Dynamics of 80-ns probe pulse propagation through K vapor cell of 4 cm, example of pulse destruction and revival. (a)–(f) Pulse waveforms at 0, 8, 16, 24, 52, and $100\%$ of the total cell length, respectively. Small perturbations at the top of the probe pulse are followed in several figures representing pulse waveform at a later time (distance). Parameters for the simulations: $\mathrm{\Delta }=0.7\mathrm{GHz}, \delta =-2$ MHz, ${\mathrm{\Omega }}_d=3.08\mathrm{GHz}, {\mathrm{\Omega }}_p=18.9$ MHz, $N_c=3\times {10}^{12}{\mathrm{cm}}^{-3}, \gamma =0.5\times {10}^7$ Hz, ${\gamma }_{\mathrm{deph}}=1.5\times {10}^7$ Hz.

Figure 8

Dynamics of 80-ns probe pulse propagation through K vapor cell of 4 cm, example of pulse broadening. (a)–(d) Pulse waveforms at different percentages of cell length, 0%, 20%, 60%, $100\%$, respectively. ${\mathrm{\Omega }}_d=1.72\mathrm{GHz}, N_c=1\times {10}^{12}{\mathrm{cm}}^{-3}$ ($T=110{}^{\circ }\mathrm{C}$), $\mathrm{\Delta }=0.7\mathrm{GHz}$, and $\delta =0$ MHz. $\gamma$ and ${\gamma }_{\mathrm{deph}}$ are the same as in Fig. 7.