(a) General PTAI scheme: PTAI is implemented by transferring a small fraction of the cloud from |g⟩ to |a⟩ and then imaging on the cycling transition |a⟩ to |e⟩. (b) Sodium D2 hyperfine structure : Our implementation of PTAI with 23Na uses the |3 2S1/2F=1⟩ (|g⟩), |3 2S1/2F=2⟩(|a⟩), and the |3 2P3/2F’=3⟩(|e⟩) states.
Imaging an optically thick cloud: (a) Standard absorption image of an optically thick cloud (b) The corresponding PTAI image of an almost identical cloud using a fixed ≈15% transfer fraction for the image. From this image we determine the initial maximum OD to be ≈20. (c), (d) Corresponding measured OD profiles. (e) Azimuthal column density profiles inferred from the PTAI (black line) and absorption (red line) images (angles shown in (d)). The PTAI image has been rescaled based on the known transfer fraction. Due to the severe attenuation of the probe (seen in (a), (c)), the absorption image fails to show the true OD of the cloud (see (e)) and consequently, spatial features such as the azimuthal density variation are suppressed and are disproportionately affected by shot noise (discussed in Sec. IV). In contrast, the PTAI image shows clear spatial features, particularly the density variations due to azimuthal inhomogeneities of the toroidal potential. All images are 85 μm × 85 μm. There are small (<15%) additional corrections for saturation of the transition (both images) and optical pumping (PTAI image only). These corrections have not been made, but would not affect the overall shape of plots shown in (e).
Analysis scheme: An average of photons of the incoming probe beam within cross section A are incident on the cloud and pass through the enclosed volume, which contains N atoms. The transmitted photons are ultimately incident on a single detector element (pixel) of a two-dimensional array of photosensors (CCD). For simplicity, we ignore the imaging system.
Measurement uncertainty vs fractional atom loss : We calculate the uncertainty of PTAI (from Eq. (10)) and PCI (Eq. (16)), and compare them assuming f r to be equivalent to f. (a) For an optically thick cloud, β = 100, PCI (dashed) gives a lower uncertainty than the PTAI technique (solid). A = 1.5 × 1.5 μm2. (b) For an optically thin cloud, β = 0.5, PTAI (solid) works better than PCI (dashed). To achieve atom numbers comparable to that of the optically thick cloud shown in (a), the imaging area in (b) is a factor of 200 larger (A ≈ 20 × 20 μm2). For both techniques, δβ/β decreases with increasing f for f ≪ 1, showing the trade-off between measurement uncertainty and perturbation of the sample. However, for PTAI at large β, the uncertainty reaches a minimum (as seen in (a)), before increasing with higher transfer fractions due to attenuation of the probe beam (high fβ). The PCI detuning is chosen so that Δ2 ≫ 1 and the phase-shift is modest (δϕ < π/4). In both plots, for PTAI, we use M p = 75, the approximate value for our sodium atom experiments, and we have set η = 1.
Lower sensitivity of PTAI to imaging losses: For β = 2, PCI (dashed) gives a lower uncertainty than the PTAI technique (solid) in the absence of imaging losses (η = 1, black). However, when one has high imaging losses (η = 0.3, blue), the performance of PTAI is only marginally affected and is better than PCI. A = 4 × 4 μm2, M p = 75.
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