Collective Cooling and Self-Organization of Atoms in a Cavity

We theoretically investigate the correlated dynamics of $N$ coherently driven atoms coupled to a standing-wave cavity mode. For red detuning between the driving field and the cavity as well as the atomic resonance frequencies, we predict a light force induced self-organization of the atoms into one of two possible regular patterns, which maximize the cooperative scattering of light into the cavity field. Kinetic energy is extracted from the atoms by superradiant light scattering to reach a final kinetic energy related to the cavity linewidth. The self-organization starts only above a threshold of the pump strength and atom number. We find a quadratic dependence of the cavity mode intensity on the atom number, which demonstrates the cooperative effect.

Figure 1

Trajectories of 10 atoms exhibiting the self-organization for a weakly bound case. The system parameters are $(g,\kappa ,\gamma )=(50,10,20)\mu {\mathrm{s}}^{\mathrm{-}\mathrm{1}}$. The atomic detuning is very large, ${\Delta }_A=-500\gamma$, and the pumping constant is $\eta =1000\mu {\mathrm{s}}^{\mathrm{-}\mathrm{1}}$, resulting in a mean photon number about 7. The atomic mass number was chosen as 85 corresponding to Rb.

Figure 2

The steady-state mean number of cavity photons $|\alpha {|}^2$ as a function of (a) the number of atoms $N$ and (b) the pumping strength $\eta$ for fixed number of atoms $N=30$. In (a) the pumping strength $\eta$ is $500\mu {\mathrm{s}}^{\mathrm{-}\mathrm{1}}$ for the lower curve, and $1000\mu {\mathrm{s}}^{\mathrm{-}\mathrm{1}}$ for the upper curve. Dots are the results obtained by numerically simulating the Eqs. (1). In (a) the lines represent the solution Eq. (6) with perfect localization $\overline{x^2}=0$ (dashed lines) and with fitted values $\overline{x^2}=0.14$ and 0.06 for $\eta =500$ and $1000$ (solid lines), respectively. The system parameters are the same as in Fig. 1.

Figure 3

$X\mathrm{\text{−}}Z$ motion of 20 atoms in the cavity with parameters$(g,\kappa ,\gamma )=(30,10,20)\mu {\mathrm{s}}^{\mathrm{-}\mathrm{1}}$, ${\Delta }_A=100\gamma$, and $\eta =2500$.

Figure 4

Time evolution of the photon number $|\alpha {|}^2$ and the longitudinal kinetic energy $E_x$ corresponding to the self-organization process shown in Fig. 3.