Seeding patterns for self-organization of photons and atoms

When atoms scatter photons from a transverse laser into a high-finesse optical cavity, they form crystalline structures which maximize the intracavity light field and trap the atoms in the ordered array. Stable organization occurs when the laser field amplitude exceeds a certain threshold. For planar single-mode cavities there exist two equivalent possible atomic patterns, which determine the phase of the intracavity light field. Under these premises, we show that the effect of an additional laser pumping the cavity makes one pattern more favorable than the other and that it can dynamically force the system into a predetermined configuration. This is an instance of pattern formation and seeding in a nonlinear quantum-optical system.

Figure 1

Sketch of the system. Atoms are confined inside a cavity with decay rate $\kappa$ and illuminated by a transverse laser beam of strength $\eta$. When $\left|\eta \right|$ exceeds a certain threshold value the atoms self-organize in stable Bragg gratings which maximize scattering into the cavity mode. Here, we study how these dynamics are modified when the cavity mode is additionally driven by a longitudinal laser of strength ${\eta }_{\mathrm{p}}$.

Figure 2

Simulation of self-organization of atoms driven by a laser and coupled to a high-finesse cavity for ${\eta }_{\mathrm{p}}=0$, i.e., without additional longitudinal pump. (a) Spatial distribution at $t=0$ and (b) after reaching the steady state at $t={10}^4{\omega }_{\mathrm{R}}^{-1}$. The time evolution of the order parameter $\Theta$ is displayed in (c). The curves have been obtained by numerically integrating the SDEs (4). Parameters: $N=1000$, $\sqrt{N}\eta =500{\omega }_{\mathrm{R}}$, $N{U}_0=-100{\omega }_{\mathrm{R}}$, $\kappa =100{\omega }_{\mathrm{R}}$, and ${\Delta }_{\mathrm{c}}=N{U}_0/2-\kappa$. Ensemble average over $50\times 10$ trajectories, i.e., 50 initial conditions and 10 realizations of the white noise process. The critical pump strength for the considered parameters and initial gas temperature $k_{\mathrm{B}}T=2\hslash \kappa$ is $\sqrt{N}{\eta }_{\mathrm{crit}}=200{\omega }_{\mathrm{R}}$.

Figure 3

Spatial distribution of the atoms after reaching the steady state at $t={10}^4{\omega }_{\mathrm{R}}^{-1}$ (a) when the transverse laser is switched off ($\eta =0$) and the cavity is pumped by a laser of intensity ${\eta }_{\mathrm{p}}=500{\omega }_{\mathrm{R}}$, corresponding to a potential depth $V_0\sim 2E_{\mathrm{R}}$. Subplots (b) and (c) show the spatial distribution in the presence of both longitudinal and transverse pump, with ${\eta }_{\mathrm{p}}=-500{\omega }_{\mathrm{R}}$ and ${\eta }_{\mathrm{p}}=500{\omega }_{\mathrm{R}}$, respectively. Ensemble average over $20\times 10$ trajectories. The other parameters are the same as in Fig. 2.

Figure 4

Order parameter $\Theta$ (for all trajectories) as a function of time for the parameters corresponding to the three subplots of Fig. 3, respectively.

Figure 5

Probability that the atoms organize in an odd pattern as a function of the longitudinal pump strength ${\eta }_{\mathrm{p}}\ge 0$. The second pattern occurs with a small, but finite probability which decreases as ${\eta }_{\mathrm{p}}$ is increased. This figure was obtained by integrating the SDEs (4) for a short time ($t=1{\omega }_{\mathrm{R}}^{-1}$) and computing the ratio of the number of trajectories for which $\Theta <0$ and the ensemble size. Ensemble average over $5000\times 5$ trajectories. The other parameters are the same as in Fig. 2.

Figure 6

Order parameter $\Theta$ evaluated in steady state (at $t=20N{\omega }_{\mathrm{R}}^{-1}$) for ${\eta }_{\mathrm{p}}=0$ (red dots), ${\eta }_{\mathrm{p}}=-500{\omega }_{\mathrm{R}}$ (blue squares), and ${\eta }_{\mathrm{p}}=-5000{\omega }_{\mathrm{R}}$ (blue diamonds). For ${\eta }_{\mathrm{p}}<0$ the order parameter is always positive when driving the cavity. The black-dotted lines are the asymptotic steady-state predictions from kinetic theory in the weak-coupling limit [7]. Inset: corresponding bunching parameter $\mathcal{B}$. Ensemble average of $5\times 5$ trajectories. The other parameters are the same as in Fig. 2.

Figure 7

Order parameter $\Theta$ for $5\times 5$ trajectories simulated using Eqs. (4) as a function of time for ${\eta }_{\mathrm{p}}=0$ (red), ${\eta }_{\mathrm{p}}=-500{\omega }_{\mathrm{R}}$ (blue), and ${\eta }_{\mathrm{p}}=-5000{\omega }_{\mathrm{R}}$ (green). From top to bottom (left to right) the value of $\eta$ is $\sqrt{N}\eta =(0,90,120,180){\omega }_{\mathrm{R}}$. The other parameters are the same as in Fig. 2.

Figure 8

Example of dynamically flipping patterns with a longitudinal pump. The system behavior is shown as a function of time for a sequential change of the cavity drive by plotting (a) the order parameter, (b) the intracavity photon number, and the real (c) and imaginary (d) part of the field amplitude. First, ${\eta }_{\mathrm{p}}=0$ and $\sqrt{N}\eta =500{\omega }_{\mathrm{R}}$. At $t=2000{\omega }_{\mathrm{R}}^{-1}$ the longitudinal laser is switched on with ${\eta }_{\mathrm{p}}=2i\times {10}^4{\omega }_{\mathrm{R}}$ in order to compensate for the scattered field for a short time. Finally, at $t=2100{\omega }_{\mathrm{R}}^{-1}$ the external laser is reduced to ${\eta }_{\mathrm{p}}=500{\omega }_{\mathrm{R}}$. The other parameters are the same as in Fig. 2.