Three-dimensional imaging of single atoms in an optical lattice via helical point-spread-function engineering

TL;DR

This work tackles three-dimensional imaging of single atoms in a quantum gas microscope by engineering a rotating double-helix PSF (DH-PSF) with a phase-only SLM to encode axial position in the PSF rotation. The authors derive a simple model, $\theta = V \arctan(z/z_R) + \alpha$, and demonstrate experimentally that, with calibrated aberrations and a holographic focal shift, they can localize atoms to within a single lattice spacing using a single exposure. Simulations reveal how aberrations and NA affect the rotation and localization fidelity, and show that defocus and vertical astigmatism or spherical aberration shift the in-focus angle, informing calibration strategies. The method markedly extends the depth of field and enables true 3D reconstruction of atomic distributions in lattices, with potential applicability to ions or atoms in optical tweezers and to future aberration-aware PSF engineering.

Abstract

We demonstrate a method for determining the three-dimensional location of single atoms in a quantum gas microscopy system using a phase-only spatial light modulator to modify the point-spread function of the high-resolution imaging system. Here, the typical diffracted spot generated by a single atom as a point source is modified to a double spot that rotates as a function of the atom's distance from the focal plane of the imaging system. We present and numerically validate a simple model linking the rotation angle of the point-spread function with the distance to the focal plane. We show that, when aberrations in the system are carefully calibrated and compensated for, this method can be used to determine an atom's position to within a single lattice site in a single experimental image, extending quantum simulation with microscopy systems further into the regime of three dimensions.

Figures (8)

• Figure 1: (a) Intensity and phase profile of the DH-PSF at different axial positions within the Rayleigh range. The upper end of the intensity color scale corresponds to the maximum intensity of the unmodified PSF. (b)-(c) Fisher information $\mathcal{I}_\eta$ as a function of axial position $z/z_\mathrm{R}$ of the DH-PSF (blue) and the standard PSF (Std., orange) with respect to the (b) axial z-dimension, (c) x-dimension (solid) and y-dimension (dashed). Note the different units of the ordinates, as well as the fact that $\mathcal{I}_z$ = 0 at $z = 0$ for the standard PSF.
• Figure 2: A schematic of the fluorescence imaging system. The atoms are trapped in a three-dimensional lattice below a high-NA in vacuo microscope objective. After passing through a $90/10$ beamsplitter (which allows for the vertical MOT beam to enter the chamber), the light is then re-imaged onto an EMCCD camera with a magnification of $M = \frac{f_1}{f_\mathrm{obj}}\frac{f_\mathrm{tube}}{f_3}$. In the Fourier plane of this re-imaging system, the phase of the atoms' fluorescence is modified by an SLM, which gives rise to the helically-diffracting PSF that encodes information on the atoms' $z$ position along the optical axis.
• Figure 3: Example of an image section taken with the standard PSF (left) and the DH-PSF (right). The (background corrected) images at $\mathrm{NA}=0.6$ with an exposure time of $1s$ have been consecutively taken from the same ensemble of atoms. Note that there is some atom hopping and loss between images. Atom A appears relatively in focus, while atom B is clearly out-of-focus, resulting in a different angle of the DH-PSF relative to atom A. Atom C also appears moderately defocused, but based on its DH-PSF angle it can be concluded that, unlike atom B, it is located above the focal plane. Atom D shows an example of an atom that is lost between images.
• Figure 4: Axial calibration of the DH-PSF giving the effective Rayleigh range $z_\mathrm{R}$, the angle offset $\alpha$ and the mean squared error (MSE) of the nonlinear fit of Eq. \ref{['eq:calibration']} to the experimental data. The angle measured in the image with shifted focal plane $\theta_s$ is related to the angle measured in the unshifted images $\theta$ by Eq. \ref{['eq:calibration']}. Different focal plane shifts $\Delta z$ are shown in different colors. The lines corresponding to odd integer focal shifts, in units of the vertical lattice constant $d_\mathrm{L}$, are marked in gray dashed lines.
• Figure 5: (a) Histograms of the calculated $z$ position of the atoms for a given focus setting, calculated from their measured angle $\theta$. (b) Fourier transform of the histogram in (a), showing a peak around $\xi_z = 1/d_\mathrm{L}$, as expected due to the discrete nature of the lattice structure. This peak is, however, obscured due to between-planes hopping and inhomogeneous aberrations in the imaging system, as we describe in the main text.