Bimodal momentum distribution of laser-cooled atoms in optical lattices

We study, numerically and experimentally, the momentum distribution of atoms cooled in optical lattices. Using semiclassical simulations, we show that this distribution is bimodal, made up of a central feature corresponding to “cold,” trapped atoms, with tails of “hot,” untrapped atoms, and that this holds true also for very shallow potentials. Careful analysis of the distribution of high-momentum untrapped atoms, both from simulations and experiments, shows that the tails of the distribution do not follow a normal law, hinting at a power-law distribution and nonergodic behavior. We also revisit the phenomenon leading to the existence of an optimal cooling point, i.e., a potential depth below which the temperature of the atoms starts increasing.

Figure 1

Simulated momentum distribution of atoms cooled on a $1/2\leftrightarrow 3/2$ transition in an optical lattice, for $\mathrm{\Delta }=-10\mathrm{\Gamma }$ and $\left|{\mathrm{\Delta }}^{\prime }\right|=50E_{\mathrm{rec}}/\hslash$. (a) Total distribution (full black line), trapped (blue $+$), and untrapped (red $\times$) atoms. (b) Same as (a), but in log scale. (c) Trapped atoms ($+$) and fit to a Gaussian curve (full line).

Figure 2

Fits to the entire simulated momentum distribution of atoms cooled on a $1/2\leftrightarrow 3/2$ transition in an optical lattice, for $\mathrm{\Delta }=-10\mathrm{\Gamma }$ and $\left|{\mathrm{\Delta }}^{\prime }\right|=50E_{\mathrm{rec}}/\hslash$. (b) A zoom-in of the data shown in (a). In both panels, the simulated data are indicated by crosses, with fits to a single Gaussian (dotted green line), a Tsallis (dashed blue line), and a double Gaussian (full red line) function. ${\chi }^2$ values for the fits are $7.1\times {10}^{-4}, 1.0\times {10}^{-4}$, and $2.7\times {10}^{-5}$, respectively.

Figure 3

Simulated momentum distribution of atoms cooled on a $4\leftrightarrow 5$ transition in an optical lattice, for $\mathrm{\Delta }=-10\mathrm{\Gamma }$ and $\left|{\mathrm{\Delta }}^{\prime }\right|=50E_{\mathrm{rec}}/\hslash$, shown on a logarithmic scale. Total distribution (full black line), trapped (blue $+$), and untrapped (red $\times$) atoms; including (a) a single ($F_{\mathrm{e}}=5$) excited state; (b) two ($F_{\mathrm{e}}=4,5$) excited states.

Figure 4

Simulated tails ($\times$) of the momentum distribution shown in Fig. 1, for the $1/2\to 3/2$ transition ($\mathrm{\Delta }=-10\mathrm{\Gamma }, \left|{\mathrm{\Delta }}^{\prime }\right|=50E_{\mathrm{rec}}/\hslash$), with fits to Eq. (13) (full line) and to a Gaussian (dashed line).

Figure 5

Configuration of the three-dimensional optical lattice. A red-detuned beam is split into four beams. These are aligned, and their polarizations are chosen, as shown in the figure. This provides a three-dimensional generalization of the one-dimensional $\mathrm{lin}\perp \mathrm{lin}$ laser cooling configuration.

Figure 6

Two-dimensional projection of the calculated lowest adiabatic potential (experimentally verified in Ref. [67]) of the optical lattice. Along the $\widehat{z}$ direction (in the figure, this corresponds to the diagonal from top left to bottom right), the modulation is purely sinusoidal.

Figure 7

Velocity distribution recorded for a shallow optical lattice ($U=106E_{\mathrm{rec}}$), close to the critical potential depth (average of two measurements). In the full figure, the dotted green line is a Gaussian function fitted to the data. Fits to a Tsallis function, or to a double Gaussian, are visually indistinguishable from the experimental data at this scale, and are therefore not included. In the inset is a zoom-in of the wing of the distribution. The dotted green line is still the single Gaussian, whereas the dashed blue line is a fit to a Tsallis function, and the full red line one to a double Gaussian.

Figure 8

The same data as in Fig. 7, but shown in a logarithmic scale. The mismatches of the fits, as quantified by their ${\chi }^2$ values are ${\chi }^2=0.16$ for a single Gaussian (dotted green line), ${\chi }^2=0.0068$ for a Tsallis function (dashed blue line), and ${\chi }^2=0.011$ for a double Gaussian (full red line). For all fits, the $y$ intercept is a fitting parameter, which explains why the fits flatten out. The influence of this on the least-squares fit is negligible, since it concerns data three decades smaller than the center of the distribution.

Figure 9

Root-mean-square momentum for all atoms (filled symbols), and for trapped (blue open symbols, lower points) and untrapped (red open symbols, upper points) only. (a) $1/2\leftrightarrow 3/2$ transition (simulations done for 50 000 atoms); (b) $4\leftrightarrow 5$ transition (simulations done for 5000 atoms).

Figure 10

Fraction of trapped atoms for $1/2\leftrightarrow 3/2$ (filled symbols) and $4\leftrightarrow 5$ (open symbols) transitions (simulations done for 50 000 and 5000 atoms, respectively).