Collective Multimode Vacuum Rabi Splitting

We report the experimental observation of collective multimode vacuum Rabi splitting in free space. In contrast to optical cavities, the atoms couple to a continuum of modes, and the optical thickness of the cloud provides a measure of this coupling. The splitting, also referred as normal mode splitting, is monitored through the Rabi oscillations in the scattered intensity, and the results are fully explained by a linear-dispersion theory.

Figure 1

Eigenenergies of an atomic cloud coupled to vacuum mode(s). Left: $N$ atoms coupled to a resonant single mode cavity; the mode splitting scales as $\sqrt{N}{g}_0$, with $g_0$ the single-atom coupling to the mode. Right: A cloud with resonant optical thickness $b_0$, coupled to the continuum of vacuum modes of free space; the mode splitting scales as $\sqrt{b_0-1}\mathrm{\Gamma }$, with $\mathrm{\Gamma }$ the single-atom decay rate in free space (see text for details).

Figure 2

(a) Sketch of the experiment: A monochromatic plane-wave laser suddenly illuminates a Gaussian cloud of two-level atoms, while the dynamics of the scattered intensity is measured at an angle of $\theta \approx 35°$ from the axis of the incident beam by a hybrid photon detector (HPM). (b),(c) Experimentally recorded intensity (dots) after the laser switch-on, normalized to 1 in the long-time limit, for different values of the optical thickness of the cloud and laser detunings [(b) $b_0\approx 8.1$ and $\mathrm{\Delta }=-7\mathrm{\Gamma }$; (c) $b_0\approx 20.3$ and $\mathrm{\Delta }=-4\mathrm{\Gamma }$]. The plain lines are obtained by a numerical simulation of the linear-dispersion theory (see text), the dashed lines denote the single-atom response (not normalized, for visibility purposes).

Figure 3

Frequency response of the cloud to the switch-on of the laser, at an angle $\theta =35°$. $|t_{\theta }(\omega ){|}^2$ is computed from the Fourier transform of the intensity (a) from the experiment and (b) from the linear-dispersion theory. The black lines denote, in both figures, the maxima derived from the linear-dispersion approach. $\overline{\omega }={\omega }_L-{\omega }_{\mathrm{opt}}$, i.e., it corresponds to the difference between the laser frequency (here detuned by $\mathrm{\Delta }=-12\mathrm{\Gamma }$) and the optical frequency. (c) Frequency response of the cloud for increasing optical thicknesses, for a slab and in the forward direction (dash-dotted curves), and for a spherical cloud with a Gaussian density and for light at an angle $\theta =35°$ (plain lines).

Figure 4

Splitting amplitude extracted from the experiment (full symbols), from the coupled-dipole simulations (empty symbols) and from the linear-dispersion theory (lines), for different detunings and optical thicknesses.