Assessing the number of atoms in a Rydberg-blockaded mesoscopic ensemble

The dipole blockade of multiple Rydberg excitations in mesoscopic atomic ensembles allows the implementation of various quantum information tasks using collective states of cold, trapped atoms. Precise coherent manipulations of the collective ground and single Rydberg excitation states of an atomic ensemble requires the knowledge of the number of atoms with small uncertainty. We present an efficient method to acquire such information by interrogating the atomic ensemble with resonant pulses while monitoring the resulting Rydberg excitations. We show that after several such steps accompanied by feedback the number of atoms in an ensemble can be assessed with high accuracy. This will facilitate the realization of high-fidelity quantum gates, long-term storage of quantum information, and deterministic sources of single photons with Rydberg-blockaded atomic ensembles.

Figure 1

(a) Single realization of the algorithm for the initial mean $\langle N\rangle =175$ and actual $N^{\left(\mathrm{a}\right)}=200$ number of atoms: Main graph shows the probability distribution $P_i\left(N\right)$ and the deduced atom number $N_i^{\left(\mathrm{p}\right)}$ after $i$ steps ($N^{\left(\mathrm{a}\right)}$ decreases by 1 after each Rydberg detection event—steps $i=1,2$), and the inset shows the corresponding infidelity $1-F$ of the swap gate [cf. Eq. (2)]. (b) Many (${10}^3$) independent realizations of the algorithm for the same initial conditions: Main graph shows the mean value $\langle N^{\left(\mathrm{a}\right)}-N^{\left(\mathrm{p}\right)}\rangle$ of the difference between the actual and deduced atom numbers after $i$ steps, and the inset shows the corresponding mean infidelity $1-\langle F\rangle$ (error bars are one standard deviation).

Figure 2

Many independent realizations of the algorithm for the initial mean number of atoms $\langle N\rangle =250$ after (a) $i=20$ and (b) $i=30$ steps, with the Rydberg excitation detection efficiency $\eta =1$ (black, solid symbols) and $\eta =0.7$ (blue, open symbols). The mean values $\langle N^{\left(\mathrm{a}\right)}-N^{\left(\mathrm{p}\right)}\rangle$ of the difference between the actual $N^{\left(\mathrm{a}\right)}$ and deduced $N^{\left(\mathrm{p}\right)}$ atom numbers (bottom) and the corresponding infidelity $1-\langle F\rangle$ of the swap gate (top) are plotted versus the initial $N^{\left(\mathrm{a}\right)}$ (error bars are one standard deviation). Red dashed line is the initial infidelity [cf. Eq. (2) with $N^{\left(\mathrm{p}\right)}=\langle N\rangle$].

Figure 3

Average infidelity $1-\overline{F}$ of the swap gate versus the detector efficiency $\eta$ and the number of steps. The isolines at $(1.5;3;6)\times {10}^{-4}$ are approximated by function $f\left(\eta \right)=\frac{a}{\eta +b}$ with $a,b=(17.45,-0.378;8.056,-0.341;3.84;-0.295)$, respectively (blue dotted; dashed; dot-dashed lines).