Polarization-Sensitive Imaging in Magnetic Environments

TL;DR

The authors address polarization-sensitive imaging of ultracold spinor gases in magnetic environments, where field-induced birefringence distorts σ^+ and σ^- images and biases spin-density reconstructions. They develop a 3D framework that couples tensor polarizability with local quantization-axis rotations to predict the accumulated phase shifts for arbitrary field geometries, and validate it using spin-dependent off-axis holography in a sodium Ioffe–Pritchard trap. The model reproduces observed distortions across temperatures, with a small residual offset likely due to optical asymmetries; for off-axis holography, a Fourier-space phase mask enables post-processing removal of magnetically induced aberrations. This approach provides universal corrections for polarization-dependent aberrations, enabling reliable spin-resolved imaging in complex magnetic landscapes and informing future precision measurements in quantum gases.

Abstract

Nondestructive spin-resolved imaging of ultracold atomic gases requires calculating the differences of the refractive indices seen by two circular probe polarizations. Perfect overlap of the two images, corresponding to two different polarizations, is required well below the feature size of interest. In this paper, we demonstrate that the birefringence of atoms in magnetic field gradients results in polarization-dependent aberrations in the image, which deteriorates the overlap. To that end, we develop a model that couples atomic tensor polarizability with position-dependent spin orientation and yields aberration predictions for accumulated phase shifts in arbitrary field geometries. Applied to data from an ultra-cold atomic cloud trapped in a Ioffe-Pritchard trap, the model quantitatively reproduces the observed distortion across a range of temperatures. A residual offset of $\sim1\;μ\mathrm{m}$ remains even under uniform field conditions, likely due to optical asymmetries. For images obtained through off-axis holography, the full complex field of the probe enables post-processing removal of all magnetically induced aberrations through a single numerically calculated Fourier-space phase mask.

Figures (5)

• Figure 1: Analysis of the relative shift between the two images corresponding to the opposite circular polarization components of the probe beam. The left-most images show the accumulated phase of a)$\sigma_+$ polarized light (red) and b)$\sigma_-$ polarized light (blue) at the beginning of the cooling sequence. The shaded regions indicate the integration window used to compute vertical linecuts of the phase profile; image within this region is averaged along the axial direction to yield one-dimensional cross-sections shown in c). The differences between the profiles at various stages of cooling are plotted in d); each line is vertically offset for clarity, and the color encodes the magnitude of the relative shift. The trend clearly indicates a relative shift between two distributions, which diminishes as the temperature decreases, resembling a zipping behavior. Additionally, each distribution is fitted and their relative displacement of the centers is plotted in e), indicating the same pattern. For reference, the corresponding temperatures extracted from the fits to the data are shown in f).
• Figure 2: Coupling strengths of the various transitions from the $F=1$ ground state of sodium with right-handed circular polarized light. All strengths have to be normalized to the strongest transitions, which has a strength of 60. Note that the total coupling from the different magnetic substates to all excited states are $5/6$, $4/6$, and $3/6$ for $m =$ -1, 0, and +1, respectively. Values taken from Steck2001
• Figure 3: Orientation of the magnetic field characterized by the angle $\beta$ relative to the $y$-direction in the $xz$-plane, at $y = 0$ (color). The dashed ellipse represents the spatial extent of a thermal atomic cloud at $T = 30 \ \mu\rm{K}$ in in-situ conditions. Here, $\sigma_z =\sqrt{k_BT/m\omega_z^2} \approx 1.1$ mm and $\sigma_r = \sqrt{k_BT/m\omega_\rho^2} \approx 0.12$ mm, with $m$ in this case denoting the mass of sodium atoms, and $\omega_{z}$, $\omega_{\rho}$ the trapping frequency in axial and radial direction, respectively. The arrows in the right-hand plot indicate the direction of the quantization axis relative to the $y$-direction - the direction of the propagation of light - along the center of the trap ($y=0, \ z=0$).
• Figure 4: Simulated phase delay contours for a)$\sigma_{+}$ and b)$\sigma_{-}$ polarizations showing an apparent displacement of the phase centroid in a thermal atomic cloud of $3\times10^8$ atoms at $30 \ \rm{\mu K}$. The dashed white line indicates its Gaussian widths. c) Vertical and d) horizontal line cuts through the center of the clouds highlight an apparent radial shift of about $\pm 40 \ \mu\text{m}$ for $\sigma^{\pm}$-polarized light, relative to the center of the ground truth (GT) phase delay for $\beta = \pi$. Note the different x-axis scales in the plots in c) and d) due to the cigar-shaped geometry of the cloud. The vertical cut is scaled for better visualization of the displacement.
• Figure 5: Displacement between the centers of the two circular polarization components as a function of temperature. For each image taken during cooling, the temperature is extracted from fits to the data, and the corresponding displacement is computed using the model described above. At high temperatures, the model predicts the behavior well, but it seems not to fully account for the shift at lower temperatures. The solid black dots show the data; empty dots with a dashed red guide show the simulation.