Accurate Atom Counting in Mesoscopic Ensembles

Many cold atom experiments rely on precise atom number detection, especially in the context of quantum-enhanced metrology where effects at the single particle level are important. Here, we investigate the limits of atom number counting via resonant fluorescence detection for mesoscopic samples of trapped atoms. We characterize the precision of these fluorescence measurements beginning from the single-atom level up to more than one thousand. By investigating the primary noise sources, we obtain single-atom resolution for atom numbers as high as 1200. This capability is an essential prerequisite for future experiments with highly entangled states of mesoscopic atomic ensembles.

Figure 1

(a) Histogram of collected fluorescence signal (detection time $t=100\mathrm{ms}$) and Gaussian fits to the resulting distributions (red lines). (b) Example time traces for different fluorescence levels. The upper and center insets show the signal of a single atom and 100 atoms, respectively.

Figure 2

Atom number variance as a function of exposure time. For atom numbers in the range $N=100$ to 200, the measured variance (circles) reaches a minimum near $t=100\mathrm{ms}$. For short integration times ${\sigma }_N^2$ is limited by photon shot noise and additional fluorescence noise, which average in time (dash-dot line). For long integration times, the finite lifetime is the dominant noise contribution (dashed line), where the main loss processes, depicted in the diagram, are collisions with background gas and light-assisted collisions. Higher atom numbers ($N=1000$ to 1100 in the inset) exhibit a higher overall variance, but the optimal detection time is found to be similar. The error bars represent 1-$\sigma$ statistical uncertainties and the solid lines are fits based on the model described in the text.

Figure 3

Single-atom resolution for mesoscopic atom numbers. For the near-optimal exposure time of 100 ms the variance ${\sigma }_N^2$ reaches the limit of single-atom resolution at $N=1080$. A fit to the data based on a noise model incorporating fluorescence noise and atom loss (upper green line) allow us to estimate the relative contribution of these noise sources (shaded regions) and compensate for the mean atom loss (red diamonds) as described in the text. A logarithmic plot (inset) shows how the scaling of the variance with respect to atom number transitions from linear to quadratic as the additional fluorescence noise begins to dominate the shot noise.

Figure 4

Efficiency of state-selective detection via radiation pressure. (a) Level scheme of the ${}^{87}\mathrm{Rb}$ $D_2$ transition. The push beam accelerates atoms in $|F=2⟩$ out of the capture volume while the MOT beams and repumper are off. The atoms can be prepared in $|F=1⟩$ via optical pumping. (b) Three example measurements of the $|F=2⟩$ push efficiency, detecting the atom number before and after the push pulse (dashed line). (c) The measured error rate without push pulse (diamonds) is consistent with isotropic thermal expansion (green line). The errors for $|F=2⟩$ atoms (squares) can be qualitatively understood based on an analysis of the rate of depumping from $|F=2⟩$ to $|F=1⟩$ including the effects of imperfect push beam polarization, and imperfect state preparation (blue line). The error rate for atoms prepared in $|F=1⟩$ (circles) includes both the effects of thermal expansion and off-resonant optical pumping (red line). All error bars represent 1-$\sigma$ statistical uncertainties.