Imaging Trapped Ion Structures via Fluorescence Cross-Correlation Detection

Cross-correlation signals are recorded from fluorescence photons scattered in free space off a trapped ion structure. The analysis of the signal allows for unambiguously revealing the spatial frequency, thus the distance, as well as the spatial alignment of the ions. For the case of two ions we obtain from the cross-correlations a spatial frequency $f_{\text{spatial}}=1490\pm 2_{\mathrm{stat}}\pm 8_{\mathrm{syst}}{\mathrm{rad}}^{-1}$, where the statistical uncertainty improves with the integrated number of correlation events as $N^{-0.51\pm 0.06}$. We independently determine the spatial frequency to be $1494\pm 11{\mathrm{rad}}^{-1}$, proving excellent agreement. Expanding our method to the case of three ions, we demonstrate its functionality for two-dimensional arrays of emitters of indistinguishable photons, serving as a model system to yield structural information where direct imaging techniques fail.

Figure 1

The spatial arrangement of SPEs in a fluorescing object (blue) can be determined by measuring the spatial photon cross-correlations. Coincident two-photon events (pairs of stars of equal color) are recorded by an ultrafast camera in the far field. For low coincident rates, the 2D distribution of the recorded data can be binned along a line (dash-dotted) to a 1D distribution. The line of binning can be rotated by an angle $\varphi$.

Figure 2

(a)–(c) One-dimensional cross-correlation signal (histogram of coincidence events) and corresponding cosine fit according to Eq. (1) (with appropriate scaling factor) for different rotation angles $\varphi =0°$ (a), $\varphi =10.5°$ (b), and $\varphi =15°$ (c); plotting the distribution of the number of bins (#) with a particular $G^{(2)}$ value in (d)–(f) yields different standard deviations (orange bars). Maximizing the standard deviation allows to find the optimal rotation angle (for details see text). (g) Standard deviation as a function of rotation angle $\varphi$. The maximum at $\varphi =0.86°\pm 0.31°$ determines the absolute orientation of the structure. (h) Uncertainty of the spatial frequency $f_{\text{spatial}}$ as a function of the number of coincidences $N$; the fit (solid line) follows $N^{-0.51}$, and for comparison $N^{-0.5}$ (dashed line); the $1\sigma$ uncertainty of the fit of 0.06 is indicated (shaded area).

Figure 3

(a) Two ions at a distance $\mathit{d}$ are used as model SPE system. A pair of fluorescence photons is scattered off the ions into directions ${\mathit{n}}_1$ and ${\mathit{n}}_2$ and recorded coincidently at detector positions ${\mathit{x}}_1$ and ${\mathit{x}}_2$. (b) Experimental setup: two ${{}^{40}\mathrm{Ca}}^{+}$ ions are trapped in a linear Paul trap and cooled by laser light, red detuned to the $S_{1/2}\text{−}{P}_{1/2}$ transition. Coincident photon events are recorded by a HBT setup (for details see text).

Figure 4

Simulating the imaging of a planar three-SPE array: (a) Analyzing the cross-correlation signal as in Fig. 2 results in different standard deviations of the one-dimensional cross-correlation signal as a function of $\varphi$. (b)–(d) For specific ${\varphi }_i$, we find a single spatial frequency $f_{\text{spatial}}^{(i)}$, $i=1$, 2, 3, suited for extracting the full structural information.