Self-organization of atoms in a cavity field: Threshold, bistability, and scaling laws

We present a detailed study of the spatial self-organization of laser-driven atoms in an optical cavity, an effect predicted on the basis of numerical simulations [P. Domokos and H. Ritsch, Phys. Rev. Lett. 89, 253003 (2002)] and observed experimentally [A. T. Black et al., Phys. Rev. Lett. 91, 203001 (2003)]. Above a threshold in the driving laser intensity, from a uniform distribution the atoms evolve into one of two stable patterns that produce superradiant scattering into the cavity. We derive analytic formulas for the threshold and critical exponent of this phase transition from a mean-field approach. Numerical simulations of the microscopic dynamics reveal that, on a laboratory time scale, a hysteresis masks the mean-field behavior. Simple physical arguments explain this phenomenon and provide analytical expressions for the observable threshold. Above a certain density of the atoms a limited number of “defects” appear in the organized phase and influence the statistical properties of the system. The scaling of the cavity cooling mechanism and the phase-space density with the atom number is also studied.

Figure 1

(Color online) The setup: transversely driven atoms moving in a standing-wave cavity. The laser intensity is given by the maximum free-space Rabi frequency (pumping strength) $\eta$. The atoms couple to the cavity mode with one-photon Rabi frequency $g$. The loss channels are spontaneous emission (rate $2\gamma$) and cavity loss (rate $2\kappa$).

Figure 2

(Color online) Two-dimensional trajectories of 40 rubidium atoms in a cavity, during the initial $50\mu \mathrm{s}$. A checkerboard pattern of trapped atoms emerges; untrapped atoms move mainly along the cavity axis. Parameters: $\gamma =20∕\mu \mathrm{s}$, $(g,\kappa )=(2.5,0.5)\gamma$, atomic detuning ${\Delta }_A=-500\gamma$, cavity detuning ${\Delta }_C=-\kappa +N{U}_0$, and the pumping strength $\eta =50\gamma$.

Figure 3

(Color online) The time evolution of the photon number in the cavity on a long time scale, for $N=40$ and $N=160$ atoms (note the different vertical scaling). The parameters are the same as in Fig. 2

Figure 4

(Color online) Evolution of the phase-space volume, along the cavity axis (up) and along the transverse pump (bottom), on a long time scale for various settings, the same parameters as in Fig. 3. The number of atoms is $N=40$ and $N=160$ (black line) and $N=40$ but pumped below threshold, $\eta =10\gamma$, for the dashed line.

Figure 5

(Color online) Numerical iterations of the mean-field approximation. The percentage of atoms near odd sites after 10 (solid circles) and 100 (open diamonds) iterations is shown varying the pumping strength $\eta$. The vertical line is at ${\eta }^{*}$ of Eq. (19). The two insets show two steady-state position distribution functions at two different pumping strengths. Parameters: $\kappa =\gamma ∕2$, ${\Delta }_A=-500\gamma$, $N{g}^2=200{\gamma }^2$, ${\Delta }_C=-\kappa -N{g}^2∕\mid {\Delta }_A\mid$, and $k_{B}T=\hslash \kappa$.

Figure 6

(Color online) Ratio of defect atoms against pumping strength $\eta$, $4\mathrm{ms}$ after the loading of the trap with a uniform (“up”) or organized (“down”) gas of atoms. The different curves show the approach towards the thermodynamic limit. The parameters are the same as in Fig. 5.

Figure 7

(Color online) Ratio of defect atoms in the ensemble (measure of the spatial order) as a function of the control parameter $\eta$ (given in units of $\gamma$). There is a clear threshold for self-organization which is independent of the atom number provided $N{g}^4$ is kept constant. Parameters are the same as in Figs. 6, 5.

Figure 8

(Color online) Defect atom ratio $5\mathrm{ms}$ after loading the trap as a function of the atom number. Each point is an average over 25 runs of the simulation. The inset shows the average number defects. The physical parameters are $\kappa =\gamma ∕2$, $g=2.5\gamma$, ${\Delta }_A=-500\gamma$, and $\eta =50\gamma$.

Figure 9

(Color online) The normalized phase-space volume of the system. Dots represent the values taken at individual trajectories at $5\mathrm{ms}$; the thick solid line is their average. The thin lines correspond to the average over the trajectories at 2, 3, and $4\mathrm{ms}$. The physical parameters are the same as in Fig. 8.

Figure 10

(Color online) Photon number in the cavity $5\mathrm{ms}$ after loading. At high atom numbers $(N>20)$ the simulation results (dots) are fitted well by a quadratic function (black line), indicating that the atoms scatter cooperatively. The physical parameters are the same as in Fig. 8.

Figure 11

(Color online) The localization parameter ${\mathcal{D}}_z={\sum }_i{(k{z}_i∕\pi )}^2∕N$ measured by the simulation (dots) and averaged over runs (solid black line) shows an overall $1∕N$ dependence as a function of the atom number. The dashed line is the approximation of Eq. (31). The physical parameters are the same as in Fig. 8.