This section describes how we evaluate our fluorescence images and reconstruct the atom number distribution using a deconvolution algorithm.
Determination of the lattice angles and spacing
To characterize our imaging system and to determine the lattice structure, we used a fluorescence image of a dilute thermal could [Fig.4.1(c)]. The lattice axes are oriented at approximately ±45◦ with respect to the image coordinates. A precise determina- tion of these angles and of the lattice spacing is needed so that the deconvolution al- gorithm works with high fidelity. We first determined the center positions of isolated atoms from this image by a simple fitting algorithm. The histogram of the mutual
4.4 Image evaluation and deconvolution 45.5 45.6 45.7 45.8 45.9 46.0 46.1 46.2 0.4 0.5 0.6 0.7 0.8 0.9
Coordinate rotation angle (°)
Width in histogram (pixel)
0 10 20 30 40 0 100 200 Distance (pixel) Frequency 0 10 20 30 40 0 50 100 Distance (pixel) Frequency
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b
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Figure 4.4: Determination of the lattice angles. (a), (b) Histogram of the distances between the center positions of individual atoms for different angles θ of the coordinate system. The
red line is a fit to a sum of equidistant Gaussians. The atom positions are taken from Gaussian fits to isolated atoms in a dilute cloud as in Fig.4.1(c). (c) The width of the fitted Gaussians shows a clear minimum versusθ. The red line is a parabolic fit and yields a minimum at a
rotation angle ofθ =45.85(5)◦.
distances between these center positions along the axes of a coordinate system ro- tated by an angleθclearly shows the periodicity of the lattice [see Fig.4.4(a),(b)]. The visibility of the pattern depends very sensitively onθ, because the periodicity is only visible if we project the distances along the right direction.
For a quantitative analysis, we fit a sum of equidistant Gaussians to the histogram. The width of the Gaussians for different values of θ [Fig.4.4(c)] shows a clear min- imum at θ = 45.85(5)◦. We obtained a similar graph for the other lattice axis and found an angle of−45.55(5)◦. The distance of the Gaussians is 4.269(4) pixel which corresponds to the lattice period of 532 nm. Thus, our magnification factor is 128.4(5) and one pixel of the CCD camera corresponds to 125 nm in the object plane. The an- gles and the lattice spacing determined by this method are used as fixed parameters for our deconvolution algorithm. We also found that the phases of the two lattice axes slightly drift from shot to shot. They are determined for each image by fitting the center positions of single atoms in the outer part of the images.
Reconstruction of the atom number distribution
We developed a deconvolution algorithm to reconstruct the atom number distribu- tion from a fluorescence image [Fig.4.5(a)]. The algorithm tries different model-confi- gurations for each lattice site and its nearest neighbors in order to minimize the differ- ence of the original image with the reconstructed one [Fig.4.5(b)]. This reconstructed image is obtained by convoluting the atom number distribution with the atomic PSF [Fig. 4.1(d)]. The algorithm allows for a variance of the fluorescence level of each atom of±20% of the mean photon count. These varying fluorescence levels partially arise from the inhomogeneous intensity of the molasses light.
We additionally found an increased fluorescence level of about 5%-10% in the cen- ter of very dense n = 1 shells of a Mott insulator, compared to the isolated atoms in the outer part of the images. This effect might arise from rescattering, which effec- tively blocks a significant part of the solid angle, redirecting the photons out of the plane. The cross section for rescattering a photon in a stimulated Rayleigh transition is of the order of the resonant cross section [112] which is of the same size as the distance between the atoms.
We have evaluated the fidelity of the reconstruction algorithm by creating simu- lated images of a known atom distribution using the PSF of our imaging system, the poissonian and superpoissonian noise contributions of the light hitting the EMCCD camera (including the amplification process), and the site-to-site fluorescence fluc- tuations of ±20%. Running the reconstruction algorithm over several hundred of such randomly generated images of Mott insulators at finite temperatures, we find a reconstruction fidelity of∼ 99.5%. In our experiment, the main limitations of the fi- delity are atom losses during the detection process due to collisions with background gas atoms (Sec.4.5). The imaging time of 900 ms is thus a compromise between the acquired signal and the described atom losses.
A more sophisticated algorithm is presently under development and will be de- scribed in the thesis of Peter Schauß.
Etaloning of the camera
We correct for an etaloning effect of the CCD camera, which causes a spatially de- pendent signal strength (see Fig.4.6). We determined the effect by summing over 90 images with two slices of a large thermal cloud [Fig.4.6(a) and (b)]. To eliminate the shape of the cloud, we fitted the data with an 8th order polynomial and divided the data by this fit, obtaining the etalon data containing the fringes [Fig.4.6(c)]. The outer regions with little signal were set to one. All pictures are divided by this etalon data to correct for the effect.
We correct for stray light by subtracting a second picture without the atoms but with the same molasses parameters.
4.4 Image evaluation and deconvolution
Original picture Reconstructed Reconstructed * PSF
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b
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Figure 4.5: Illustration of the deconvolution algorithm. (a) Original picture. The white dots mark the lattice sites as determined from the position of isolated atoms and the lattice angles and spacing. The white circles indicate, where the algorithm found an atom. (b) Recon- structed atom distribution on the lattice. (c) Convolution of the reconstructed image with the PSF of the imaging system. This data is subtracted from the original picture, and the re- construction is changed to minimize the residuum. The algorithm also allows for spatially varying signal levels.
Figure 4.6:Etalonning of the camera chip. (a) Picture of a thermal cloud (two slices). (b) Sum over 90 such pictures. Fringes originating from interference in the camera chip are clearly visible. (c) The picture in (b) divided by a polynomial fit of 8th order. In order to correct for the etalon effect, we divide each picture by this data.