Towards a measurement of the Debye length in very large magneto-optical traps

We propose different experimental methods to measure the analog of the Debye length in a very large magneto-optical trap, which should characterize the spatial correlations in the atomic cloud. An analytical, numerical, and experimental study of the response of the atomic cloud to an external modulation potential suggests that this Debye length, if it exists, is significantly larger than what was expected.

Figure 1

Density ${\rho }_x\left(x\right)$ obtained by fluorescence for $-\delta /\mathrm{\Gamma }=4,6$ compared with MD simulation of a trapped Coulomb gas, using $N^{\left(\mathrm{sim}\right)}=16384$ particles. The inset shows the Debye length ${\lambda }_D$ and the cloud FWHM diameter extracted from simulations. (The density plots for $-\delta /\mathrm{\Gamma }=5,8$ are not shown here.) The simulated plasma parameter ranges from ${\mathrm{\Gamma }}_p\simeq 4\times {10}^{-2}$ for $\delta /\mathrm{\Gamma }=-4$ to ${\mathrm{\Gamma }}_p\simeq 5\times {10}^{-5}$ for $\delta /\mathrm{\Gamma }=-8$. For all experiments, the number of trapped atoms is of the order of ${10}^{11}$.

Figure 2

Sketch of an incident beam ${\vec{k}}_{\text{inc}}$ diffracted on an atom in direction ${\vec{k}}_{\text{end}}$ corresponding to angles ${\theta }_k$ and ${\varphi }_k$. We define and show the vector $\vec{k}={\vec{k}}_{\text{inc}}-{\vec{k}}_{\text{end}}$.

Figure 3

MD simulations with $N^{\left(\mathrm{sim}\right)}=16384$ particles of the structure factor $S\left(k\right)$, averaged over all $\vec{k}$ such that $|\vec{k}|=k$. The horizontal axis is adimensionalized by the mean interparticle distance $a$, which is in the simulation $a/L=0.039$. For the dashed red curve ${\mathrm{\Gamma }}_p\simeq 0.043$ with the same ratio ${\lambda }_D/L$ as the black dashed fit in Fig. 1, for the dotted blue curve ${\mathrm{\Gamma }}_p\simeq 0.215$ (these values for the plasma parameter are much higher than expected in the atomic cloud; smaller, more realistic, values are difficult to reach numerically while keeping a small ${\lambda }_D/L$). The waist of the Gaussian probe beam is $w\simeq 0.76L$. The black curve corresponds to randomly distributed particles with the same average density: the two-body correlation obviously vanishes in this case, and accordingly, the characteristic dip is absent.

Figure 4

Principle of modulation experiment. (a) A sinusoidal modulation is applied by crossing two laser beams on the cloud. (b) The atoms are released from the MOT and the diffraction grating due to the atomic density modulation being probed. (c) Images of the $\pm 1$ diffracted orders vs modulation wavelength ${\lambda }_e$. The zeroth order is blocked to avoid saturation of the CCD, and the display is adjusted for each image of the figure to improve readability (see text).

Figure 5

Comparison of the total diffracted power $R\left({\lambda }_e\right)$ in the experiment (black dots) and theory (lines). The detuning is $\delta /\mathrm{\Gamma }=-4, N\sim {10}^{11}, w=2.2\mathrm{mm}$. The theoretical curves use $w=2.2\mathrm{mm}$, and $L=7.41\mathrm{mm}$, which is the value extracted from Fig. 1 for $\delta /\mathrm{\Gamma }=-4$; they are computed with Debye length ${\lambda }_D=100,300\mu \mathrm{m}$. The steepness $l$ of the step function in Eq. (A8) is chosen to be $l=1\mathrm{mm}$ (the theoretical curve only weakly depends on $l$). We also show the theoretical limit case with no interactions $B\left({\lambda }_e\right)=1$. The vertical dotted line indicates the theoretical position of the Bragg/Raman-Nath crossover ${\lambda }_e^{\left(c\right)}=136\mu \mathrm{m}$. The corresponding experimental value ${\lambda }_e^{\left(c\right),\exp }=142\mu \mathrm{m}$ is obtained at the intersection of the fitted experimental data (for $\delta /\mathrm{\Gamma }=-4$) in the Bragg $\propto {\lambda }_e^{3.35}$ and Raman-Nath region $\propto {\lambda }_e^{1.34}$. This latter exponent is not far ($1.34\simeq 1$) from the prediction of Eq. (27) in the sub-Debye Raman-Nath regime without interactions. The exponent in the Bragg regime depends on the specific details of the real experimental profile. The vertical dashed lines indicate a local maximum and a local minimum of the response in the Bragg regime; see Appendix pp2.

Figure 6

Experimental (top) and theoretical (bottom) diffracted beams for ${\lambda }_e=64.2$ and $75.68\mu \mathrm{m}$. The color scale is adjusted to improve readability, and $i,j$ are the pixel indexes.