content_image/326114.jpg
D1 and D2 lines). 

To study quantitatively pulse propagation in our system, we simulate numerically the propagation of an incident pulse with a field envelope E (0 , t ) and intensity full width at half maximum (FWHM)) T 0 . After traveling a distance z in the vapor, the field envelope of the pulse is 

<img src='content_image/326114.jpg'>

where F represents the Fourier transform and H ( ω ) = exp ( in ( ω ) ω/c ) is the transfer function of the vapor. The fractional delay and broadening are calculated, respectively, as f$_{D}$ = t$_{D}$/T 0 and f$_{B}$ = ( T − T 0 ) /T 0 , where t$_{D}$ is the time delay of the pulse peak and T is the FWHM of the pulse when it exits the vapor. 

## 3. Experimental Details 

In this paper, we carry out two distinct but related experiments in order to fully characterize the optical response of the Rubidium vapor. In the first experiment, we use a monochromatic source to measure directly the frequency-dependent susceptibility of the vapor (see Fig. 1 a)). We use an external cavity diode laser (Toptica, DL Pro) to generate continuous wave light with a narrow linewidth ( < 1 MHz) that we tune across the D1 and D2 resonances (from 772- 796 nm). We monitor the frequency of the light via a wavemeter with 2 MHz resolution (WS Ultimate Precision, High Finesse). In order to modulate the amplitude of the optical signal, we couple the light into a Mach-Zehnder modulator (MZM) via a free-space-to-fiber port, and operate the MZM in the linear regime. We note that the MZM introduces dispersion, but we explicitly account for this in our analysis. An RF signal generator (SG, HP 70340A) drives the MZM at frequencies of up to 7 GHz (limited by the SG). We then couple the modulated optical signal back to free space where it interacts with a heated 7 cm vapor cell containing natural Rb in a multi-pass geometry. After three passes through the cell ( L = 21 cm), we couple the light into an optical fiber, detect the signal, and record the resulting waveforms with a high-speed oscilloscope (Tektronix 11801B). 

