Collective excitation dynamics of a cold atom cloud

We study experimentally and theoretically the time-dependent response of a cold atom cloud illuminated by a laser beam immediately after the light is switched on. We show that cooperative effects, which have been previously investigated in the decay dynamics after the laser is switched off, also give rise to characteristic features in this configuration. In particular, we show that collective Rabi oscillations exhibit a superradiant damping. We first consider an experiment that is performed in the linear-optics regime and is well described by a linear coupled-dipole theory. We then show that this linear-optics model breaks down when increasing the saturation parameter, and that the experimental results are then well described by a nonlinear mean-field theory.

Figure 1

(a) Sketch of the experiment: A probe beam (detuned by $\mathrm{\Delta }={\omega }_L-{\omega }_0$ from the atomic transition) suddenly illuminates a Gaussian cloud of two-level atoms (transition linewidth $\mathrm{\Gamma }$, optical depth $b_0$). The time-dependent scattered light intensity is measured at an angle of $\theta ={35}^{\circ }$ from the axis of the incident beam. (b)–(d) Experimental time-dependent intensity response for various cloud and laser parameters showing damped Rabi oscillations. The damping depends on the optical depth $b_0$ and $\mathrm{\Delta }$. Lines denote a full numerical simulation of linear-optics equations of motion (see text) and reproduce the experiment very well due to the low saturation parameter ($s\approx 0.02$).

Figure 2

Examples of an effective mode fitting [Eq. (6)] to signals from the experiment (points: data; solid lines: fit) and to CD simulations (triangles: simulation; dashed lines: fit) in various regimes: In the underdamped case (a) the fit to the CD theory is nearly perfect. In the more damped cases (b),(c) small deviations from the effective mode (also in CD simulations) are visible.

Figure 3

CD simulations for the parameters from Fig. 2. Besides the realization averaged curves (orange lines), we also show 100 single realization results (thin grey lines). (a)–(c) $N_{\mathrm{eff}}=5000$, (d) $N_{\mathrm{eff}}=2000$. For small $\left|\mathrm{\Delta }\right|$ and large $b_0$ we observe large fluctuations that are independent of $N_{\mathrm{eff}}$.

Figure 4

Collective decay rate ${\mathrm{\Gamma }}_N$ showing superradiant behavior, reminiscent of previous switch-off experiments [5]. (a) Fits to CD simulations. (b) Fits to experimental data. We only kept fits for which the fitting function Eq. (6) works well with a value of $R^2>0.85$. For small $b\left(\mathrm{\Delta }\right)$ the CD theory agrees with the single-mode prediction [lines from Eq. (18)]. Generally, the experiment exhibits a more damped behavior.

Figure 5

Comparison between exact simulations of the master equation (E), Eq. (23), mean-field simulations (MF), Eqs. (29) and (30), and the coupled-dipole model (CD), Eq. (1). The intensity is not normalized to the long-time value. We use a small cloud with uniform density ($N_{\mathrm{eff}}=6$). Here, $\mathrm{\Delta }=-4$ and we use two sphere radii $kR=1$ (a,b) and $kR=5$ (c). We tune the Rabi frequency to obtain small and large saturation parameters, $s=0.01$ (a) and $s=2$ (b),(c), respectively. MF is superior to CD in reproducing the full quantum results for high saturation (c) and only fails for dense clouds and high saturation (b). Exclusion distance $k{d}_{\min }=0.1$, average over 21 realizations.

Figure 6

Comparison between the normalized intensity evolution from the experiment with MF and CD simulations in a large saturation regime ($s\gtrsim 0.1$) and with $b_0\sim 60$: (a) $s=0.13$, (b) $s=0.34$, (c) $s=0.63$. Here, $\mathrm{\Delta }=-4\mathrm{\Gamma }$. For the simulations $N_{\mathrm{eff}}=5000$ and results for 78 random atom positions have been averaged.