Detection of small atom numbers through image processing

We demonstrate improved detection of small trapped atomic ensembles through advanced postprocessing and optimal analysis of absorption images. A fringe-removal algorithm reduces imaging noise to the fundamental photon-shot-noise level and proves beneficial even in the absence of fringes. A maximum-likelihood estimator is then derived for optimal atom-number estimation in well-localized ensembles and is applied to real experimental data to measure the population differences and intrinsic atom shot noise between spatially separated ensembles each comprising between $10$ and $2000$ atoms. The combined techniques improve our signal-to-noise ratio by a factor of $3$, to a minimum resolvable population difference of $17$ atoms, close to our ultimate detection limit.

Figure 1

Typical absorption image of the atomic distribution ($90\times 71$ pixels) for a hold time of $1.6$ s. Four independent atomic ensembles are visible (horizontally distributed) with two mirror images (vertically separated). The two centermost clouds each contain $\approx 400$ atoms ($\approx 100$ in outermost wells). Dashed boxes indicate the background pixels for the fringe-removal algorithm, and the signal regions are indicated by solid lines.

Figure 2

Imaging noise for various probe intensities with and without fringe removal, $\mathrm{var}(A_x/Q_x)$ (circles) and $\mathrm{var}(A_x/R_x)$ (squares), respectively. The predicted photon shot noise plus camera readout noise is shown as a solid line. The dashed line shows the ultimate photon-shot-noise limit.

Figure 3

Relative variance as a function of total atom number measured by maximum-likelihood estimation (squares) or straight integration (circles). The dash-dotted line shows the expected fluctuations combining atom shot noise (dashed line) and the minimum detection-noise contribution given by the CRB (dotted line). Solid lines indicate fits for the detection-noise contributions. The inset shows the covariance between $N^{-}/N^{+}$ in the upper and lower mirror images (squares) along with the predicted atom shot noise (dashed line).