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

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.

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 and 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 $90/10$ beam splitter [which allows for the vertical magneto-optical trap (MOT) beam to enter the chamber], the light is then reimaged onto an electron-multiplying CCD (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 reimaging 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 $1\mathrm{s}$ 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. (6) 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. (6). Different focal plane shifts $\mathrm{\Delta }z$ 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.

Figure 6

Visualization of the simulated field emanating from a single atom without aberrations for a NA of $0.6$ (a) in the SLM plane and (b) in the image plane at $z=0$, after modification by the SLM into a DH-PSF. The SLM plane intensity is normalized to its maximum and increases towards the edges due to the apodization of the objective lens. The pupil radius corresponding to the NA of $0.6$ is $a=230$ SLM pixels and we use the same Laguerre-Gaussian waist $w_0$ as in the experiment such that $a/w_0=2.97$. Only a central section of the calculated output matrix is shown, so that the PSF is clearly visible. Here, the axes are given in object/atom plane coordinates. The image plane intensity is normalized to the maximal intensity of the standard PSF.

Figure 7

Computed rotation angles and corresponding Laguerre-Gaussian model fits of the DH-PSF for three different NAs. The PSF was calculated at each of 81 axial positions in the interval from $-4$ to $4d_{\mathrm{L}}$.

Figure 8

Angle of rotation of the DH-PSF as a function of axial emitter position in the presence of aberrations for an NA of 0.6. Each subplot shows the effect of a different Zernike polynomial wavefront aberration (up to Noll index 13, omitting the trivial piston, horizontal, and vertical tilt) for varying Zernike coefficients or aberration strengths $W$. The rotation angle $\theta$ is determined from the computed DH-PSF using the Radon transform for every integer position within the axial range of $-10-10d_{\mathrm{L}}$ (see text for details). The rotation angle of our DH-PSF with $V=2$ is only unique within an angle interval of length $\pi$. Since our angle determination is restricted to $\theta \in [-\pi /2,+\pi /2)$ by construction, the computed angles jump by $\pm \pi$ at the edges of this interval.