Entanglement of trapped-ion clock states

A Mølmer-Sørensen entangling gate is realized for pairs of trapped ${}^{111}\mathrm{Cd}^{+}$ ions using magnetic-field insensitive “clock” states and an implementation offering reduced sensitivity to optical phase drifts. The gate is used to generate the complete set of four entangled states, which are reconstructed and evaluated with quantum-state tomography. An average target-state fidelity of 0.79 is achieved, limited by available laser power and technical noise. The tomographic reconstruction of entangled states demonstrates universal quantum control of two ion qubits, which through multiplexing can provide a route to scalable architectures for trapped-ion quantum computing.

Figure 1

Two views of the Mølmer-Sørensen ${\widehat{\sigma }}_x\otimes {\widehat{\sigma }}_x$ entangling gate for two ions in (a) energy space [9] and (b) motional phase space [14] for the gate-diagonal spin basis. The quantum number $n$ and phase-space coordinates describe a given collective motional mode. Red and blue Raman sideband couplings are labeled by $r$ and $b$ and have detuning ${\delta }_b=\delta =-{\delta }_r$. For a closed phase-space trajectory, the phase $\Phi$ depends only on the area enclosed.

Figure 2

Average brightness $S_{\mathit{av}}$ (see text) versus MS gate detuning $\delta$. Applied gate time $\left(75\mu \mathrm{s}\right)$ is within 10% of the ideal. Dotted line indicates expected signal modified to include an initial temperature ${\overline{n}}_s=0.3$ [15]. Solid line is a fit including offset and contrast factors to account for imperfections such as spontaneous emission. The fit gives a sideband Rabi frequency $\eta \Omega ∕2\pi =6.3\mathrm{kHz}$ and initial stretch mode temperature ${\overline{n}}_s=0.3$. Vertical line shows ideal gate operation point $\delta =2\eta \Omega$, roughly at $S_{\mathit{av}}=1$. Each point is the average of 150 PMT measurements.

Figure 3

Time scan of MS gate showing (a) average brightess $S_{\mathit{av}}$ and (b) parity $\Pi$. Ideal gate evolution shown as dotted lines with best fit including exponential damping shown with solid line. The fit gives a sideband Rabi frequency $\eta \Omega ∕2\pi =6.6\mathrm{kHz}$ and detuning $\delta ∕2\pi =12.8\mathrm{kHz}\approx 2\eta \Omega ∕2\pi$ with other parameters the same as in Fig. 2. Vertical line shows gate operation time $\tau =2\pi ∕\delta \approx 80\mu \mathrm{s}$.

Figure 4

Parity versus phase of analysis $\pi ∕2$ pulse applied to the ${\Psi }_1$ state. The solid line is a sinusoidal fit yielding an amplitude 0.79(2). The fidelity of the state shown is 0.83(2). Each point is an average over 50 PMT measurements and other parameters are as in text.

Figure 5

Tomographically reconstructed density matrices (a)–(d) for the four Bell-like entangled states ${\Psi }_1$ through ${\Psi }_4$ as per Eq. (2). To allow direct comparison of diagonal and off-diagonal elements, the reconstructed matrices were rotated into the real coordinate using fit parameter ${\varphi }_e=-1.1\mathrm{rad}$ for (a) and (b), and ${\varphi }_o=0.43\mathrm{rad}$ for (c) and (d). Each state reconstruction uses 27 independent projective camera measurements averaged over 200 runs.