Three-dimensional simulations of spatiotemporal instabilities in a magneto-optical trap

Large clouds of atoms in a magneto-optical trap (MOT) are known to exhibit spatiotemporal instabilities when the frequency of the trapping lasers comes close to the atomic resonance. Such instabilities possess similarities with stars and confined plasmas, where corresponding nonlinearities may give rise to spontaneous oscillations. In this paper, we describe the kinetic model that has recently been employed in three-dimensional (3D) simulations of spatiotemporal instabilities in a MOT, yielding qualitative agreements with experimentally observed instability thresholds and regimes. Details surrounding its implementation are included, and the impact of its physical effects on the instabilities is investigated to improve the understanding of the complex mechanism at work.

Figure 1

(a) Sketch of a MOT, displaying the considered conventions for the MOT magnetic field and beams in the derivation of the kinetic model employed in our 3D simulations of instabilities. The coils produce a magnetic field (curved arrows) that has a positive gradient along the $\mathsf{z}$ axis. Accordingly, the beams have the following circular helicities (circular arrows): Right-handed for the $\mathsf{z}$-axis beams, and left-handed for the $\mathsf{x}$- and $\mathsf{y}$-axis beams. (b) Simplification of the Zeeman sublevel structure of the hyperfine transition $F=0\to F^{\prime }=1$, in the employed kinetic model. Each of the three Zeeman transitions $m=0\to m^{\prime }=-1,0,+1$ between $F=0$ and $F^{\prime }=1$ is treated as an independent 2-level system (right picture). They are induced by, respectively, only ${\sigma }^{-}, \pi , {\sigma }^{+}$ polarized light. The MOT magnetic field leads to the Zeeman splitting of the hyperfine levels depending on the atom position [$\mu B\left(\mathbf{r}\right)$].

Figure 2

(a) Depiction of the mechanism behind the shadow force. As the oppositely directed beams travel through the cloud, their respective intensities $I_{\alpha }^{+}$ and $I_{\alpha }^{-}$ ($\alpha =\mathsf{x},\mathsf{y},\mathsf{z}$) become attenuated, giving rise to an intensity imbalance in the cloud. This imbalance yields an additional compression, i.e., the shadow force. (b) Depiction of the mechanism behind the rescattering force with two 2-level atoms. Atom 1 first scatters the laser light of intensity $I_L$, and atom 2 then rescatters the scattered light, thus experiencing a repulsion. The rescattered light has the intensity $I_R$ determined by the inverse-square law of propagation, such that this repulsion is Coulomb-like. The scattering cross section ${\sigma }_L$ and the rescattering cross section ${\sigma }_R$ determine respectively the power scattered and rescattered.

Figure 3

Illustration of the rescattering in the $F=0\to F^{\prime }=1$ model. Each of the ${\sigma }^{-}, \pi , {\sigma }^{+}$ transitions of atom 1 produces a characteristic radiation pattern of light scattered from the total laser field (respectively, ${\eta }_{-}, {\eta }_0, {\eta }_{+}$). The scattered light polarization seen by atom 2 depends on the quantization-axis orientations of the two atoms, such that each transition of atom 2 rescatters light in specific fractions (respectively, $b_{-}^{-}, b_0^{-}, b_{+}^{-}$, when considering light from the ${\sigma }^{-}$ transition of atom 1).

Figure 4

Display of 3D images (upper row) and corresponding 2D images (lower row) of simulated clouds belonging to different instability regimes discussed in Ref. [24]. Each image is a single-shot image. In the 3D images, the individual dots are superparticles. The 2D images are the cloud densities integrated along the $\mathsf{z}$ axis and are Gaussian filtered; the diagonals correspond to the directions of two pairs of MOT beams; the field of view is $10\times 10{\mathrm{cm}}^2$ for all these images. The $\mathsf{x}$ and $\mathsf{y}$ axes have been rotated by ${45}^{\circ }$ between the 3D and 2D images. A video version of this figure is available as online Supplemental Material [38].

Figure 5

(a) First and (b) second part of a 2D illustration of the numerical method called the tube method. (a) The upper two drawings display the same cloud composed of superparticles (small dots), and it is imagined that each beam is segmented into rectangular tubes parallel to the beam's propagation direction. During the cloud's evolution, the tubes remain at fixed positions, with their width $W$ being of a set size. The lower drawing displays the same cloud, with the added large dots indicating the positions of the points where the intensity of each beam is calculated first. These dots are placed at grid-point positions located through the center of the tubes. (b) In the calculation of, e.g., the $+\widehat{\mathsf{z}}$ directed beam's intensity at the position of a given large dot, the superparticles that contribute are those inside the tube of this dot and positioned before it. Once the intensities of each beam at the positions of the large dots are calculated, the intensities at the positions of the superparticles are found by means of interpolation.

Figure 6

Investigation on how the threshold detuning ${\mathrm{\Delta }}_{\mathrm{thr}}$ for the magnetic field gradient $B^{\prime }=3$ G/cm is affected as the stochastic force ${\mathbf{F}}_{\mathcal{D}}$ [Eq. (28)] is scaled by a constant factor $d$. The open stars are the test results, and the solid star is the result of the ordinary simulation (from Fig. 6 in Ref. [23]).

Figure 7

Investigation on how the threshold detuning ${\mathrm{\Delta }}_{\mathrm{thr}}$ for the magnetic field gradient $B^{\prime }=3$ G/cm is affected after removing either ${\sigma }_{\mathrm{el},q}$ or ${\sigma }_{\mathrm{inel},q}$, being the parts of rescattering cross section that result from the contribution of, respectively, the elastically and inelastically scattered spectra. The small asterisks are the test results, and the large asterisk is the result of the ordinary simulation (from Fig. 6 in Ref. [23]).