Measuring the group velocity dispersion in near resonant hot atomic vapors

TL;DR

Measuring group-velocity dispersion (GVD) in near-resonant hot atomic vapors is challenging due to absorption, distortion, and nonlinearities that degrade conventional methods. The authors propose a simple technique using a weakly phase-modulated continuous-wave laser and slow photodetection, where two weak sidebands co-propagate and accumulate dispersive phases; the resulting transmitted modulation contrast exhibits minima that map directly to the GVD, with the relation $D_0 = \frac{2\pi}{\delta\omega_{\min}^2 L}$. Applied to hot rubidium D2 transitions, the method reveals a strong detuning- and temperature-dependent GVD and enables reconstruction of the dispersion via parabolic fits to multiple minima, yielding the GVD coefficient $D_0$. In the far-detuned regime, theory predicts $\delta\omega_{\min} \propto |\Delta|^{3/2}$, which is confirmed experimentally, while near resonance higher-order susceptibilities cause deviations. Overall, the technique provides a robust, accessible tool for characterizing dispersive media and exploring paraxial fluids of light under strongly dispersive conditions.

Abstract

Group velocity dispersion (GVD) in near-resonant hot atomic vapors is difficult to measure with standard pulse broadening or interferometric techniques, as absorption, pulse distortion and nonlinearities strongly affect the probe and reduce the signal-to-noise ratio. We introduce a simpler method using a continuous-wave laser with weak phase modulation and a slow photodetector, directly inspired by Bragg-like spectroscopy in fluids of light. During propagation, the red and blue-detuned sidebands accumulate different dispersive phase shifts, leading to oscillations in the transmitted modulation contrast as the modulation frequency is scanned. Vanishing contrast at well-defined frequencies directly yields the GVD. We apply this technique to hot rubidium vapors and observe the strong frequency dependence of the GVD across a broad detuning range of the D2 line at different temperatures.

Figures (4)

• Figure 1: (a) Experimental setup (b) Probe spectrum showing carrier and modulation sidebands in a rubidium Doppler-broaden absorption line. (c) Simulated electric field intensity for medium of 9 cm with a modulation of 4 GHz. The time is represented in the comoving frame $t'$. (d) Interference inside the cell with examples of destructive/constructive phases.
• Figure 2: (a) Measured contrast at a cell temperature of $120^{\circ}\mathrm{C}$ for a detuning $\Delta=-3.70\;\mathrm{GHz}$ and an input power of 15 mW. Thin blue line: directly measured spectrum; thick blue line: reconstructed contrast after Hilbert-phase analysis. Blue dots: experimental minima (b) Same measurement as in Fig.(a) but for a cell temperature of $99^{\circ}\mathrm{C}$ (orange curves). (c) Dispersion relation $\omega(k_t)$ extracted from the minima of the contrast for the $120^{\circ}\mathrm{C}$ data. Blue dots: experimental points $\omega(k_t)=p\pi/L$ for successive minima; thin blue line: continuous dispersion retrieved from the reconstructed contrast; thick blue line: parabolic fit. (d) Same analysis as in Fig.(c) for the $99^{\circ}\mathrm{C}$ data (orange). These panels illustrate how both the contrast spectra and the derived dispersions depend on the vapor temperature.
• Figure 3: (a) GVD $D_0$ as a function of the detuning $\Delta$. The blue and orange curves represent the results obtained at two temperatures, respectively $T = 99^{\circ}\text{C}$ (blue dots) and $T = 120^{\circ}\text{C}$ (orange dots). The experimental conditions are the same as for the corresponding curves in Fig. 2, except that here the detuning is scanned over the range $-4$ to $-2~\text{GHz}$ (x-axe in log scale). (b) Positions of the experimentally measured contrast minima $\delta\omega_{\min}^{(n)}$ as a function of the detuning $\Delta$ for several orders $n$, for a cell temperature of $120^{\circ}\mathrm{C}$, illustrating their systematic shift with detuning.
• Figure 4: Positions of the first minima $\delta\omega_{\min}$ (blue dots) as a function of detuning at $120^\circ$C. For detunings closer to resonance (>-4 GHz), $\delta\omega_{\min}$ are extracted from the parabolic fit of the reconstructed dispersion curve, while for detunings far from resonance (<-4 GHz) they correspond to the measured contrast minima. The solid blue line shows the power-law fit $\delta\omega_{\min} \propto |\Delta|^{3/2}$ (x- and y-axes in log scale).