Atom counting in ultracold gases using photoionization and ion detection

We analyze photoionization and ion detection as a means of accurately counting ultracold atoms. We show that it is possible to count clouds containing many thousands of atoms with accuracies better than $N^{-1∕2}$ with current technology. This allows the direct probing of sub-Poissonian number statistics of atomic samples. The scheme can also be used for efficient single-atom detection with high spatiotemporal resolution. All aspects of a realistic detection scheme are considered, and we discuss experimental situations in which such a scheme could be implemented.

Figure 1

(a) In the proposed atom-counting scheme atoms are ionized from an optical dipole trap via a three-photon process. The resulting ions are accelerated into an ion detector. (b) The ionization scheme showing the relevant energy levels of ${}^{87}\mathrm{Rb}$ (not to scale). The scheme consists of $2\times 778\mathrm{nm}$ photon excitation to the $5D_{5∕2}$ state, followed by ionization with a Nd:YAG laser, which also forms the dipole trap.

Figure 2

Contours of the normalized count uncertainty $\kappa$ for an atom-counting system with ion detection efficiency ${\eta }_i\pm {\sigma }_{{\eta }_i}$, for ion number $N_I={10}^3$. The shaded area shows the region of parameter space expected for common commercial ion detectors calibrated as detailed in Sec. 4.

Figure 3

Histogram of the inferred values of the ion detection system efficiency ${\eta }_{\mathit{det}}$ for 1000 runs of the calibration simulation with ${\eta }_{\mathit{det}}=0.9$, electron detection system efficiency ${\eta }_{\mathit{det},e}=0.9$, ionization rate ${\dot{N}}_I={10}^4{\mathrm{s}}^{-1}$, calibration time ${\tau }_{\mathit{cal}}=10\mathrm{s}$, pulse resolution time ${\tau }_r=10\mathrm{ns}$, and background electron and ion detection rates ${\dot{N}}_{ib}={\dot{N}}_{eb}={10}^2{\mathrm{s}}^{-1}$, respectively.

Figure 4

Contour plot of the normalized count uncertainty $\kappa$, showing how $\kappa$ increases as the ionization time ${\tau }_I$ decreases, for various combinations of initial values of the overall ion detection efficiency ${\eta }_i$ and its uncertainty ${\sigma }_{{\eta }_i}$. Crosses, circles, and triangles correspond to ${\tau }_I\to \infty$, ${\tau }_I=2\mathrm{ms}$ and ${\tau }_I=1\mathrm{ms}$, respectively.

Figure 5

The optimal $1∕e$ time required to ionize the cloud is shown as a function of the atom number and the loss rate from the trap.

Figure 6

The minimum achievable value of the relative count uncertainty $\kappa$ expressed relative to ${\kappa }_{\mathit{min}}$ (the value for zero loss) is shown as a function of atom number and trap lifetime. For the given values of the overall ion detection efficiency ${\eta }_i$ and its uncertainty ${\sigma }_{{\eta }_i}$, ${\kappa }_{\mathit{min}}\approx 0.11$.