Heralded Entanglement of Two Ions in an Optical Cavity

We demonstrate precise control of the coupling of each of two trapped ions to the mode of an optical resonator. When both ions are coupled with near-maximum strength, we generate ion-ion entanglement heralded by the detection of two orthogonally polarized cavity photons. The entanglement fidelity with respect to the Bell state ${\Psi }^{+}$ reaches $F\ge (91.9\pm 2.5)\%$. This result represents an important step toward distributed quantum computing with cavities linking remote atom-based registers.

Figure 1

Two ions are trapped in a linear Paul trap within an optical cavity. The cavity axis is defined as $\widehat{x}$. (a) There is a 4° angle between the trap axis and $\widehat{z}$. The separation $d$ between the ions can be tuned. (b) A piezostage translates the cavity along ${\widehat{y}}^{\prime }$. As ${\widehat{y}}^{\prime }$ is tilted at an angle $\varphi \sim 5°$ with respect to $\widehat{y}$, the coupling of each ion to the ${\mathrm{TEM}}_{00}$ mode is sinusoidally modulated. The projection of $d$ in the $xy$ plane is $d^{\prime }$. (c) For a projected ion-ion separation of $d^{\prime }=670\mathrm{nm}$, as the cavity is translated along ${\widehat{y}}^{\prime }$, the ions couple to the cavity with phase difference $0.9\pi$. The cavity standing wave at 866 nm is used to repump the ions, and fluorescence of the ions at 397 nm is measured on an EM-CCD. (d) For $d^{\prime }=370\mathrm{nm}$, the relative phase difference is $0.2\pi$.

Figure 2

(a) Both ions are prepared in the electronic state $|S⟩$. A Raman process, driven by a bichromatic field at 393 nm detuned from the excited state $|P⟩$, generates two cavity photons at 854 nm. The photons’ polarization, either horizontal ($H$) or vertical ($V$), is entangled with the states $|D⟩$ and $|D^{\prime }⟩$ of both ions. An optical $\pi$ pulse at 729 nm then coherently transfers population in $|D^{\prime }⟩$ to $|S⟩$. (b) A polarizing beam splitter (PBS) at the cavity output enables measurement of polarization components using avalanche photodiodes (APDs). Measurement of one $H$- and one $V$-polarized photon projects the ions into an entangled state.

Figure 3

Three measurements are used to bound the fidelity of the entangled state. First, the sum of population terms ${\rho }_{SD,SD}$ and ${\rho }_{DS,DS}$ is determined directly after entanglement. This measurement is independent of phase and is indicated by a red line whose width represents the uncertainty. Second, after two $\pi /2$ rotations on the $|S⟩\leftrightarrow |D⟩$ optical transition, the parity is measured. The parity oscillates as a function of the relative phase of the rotations (blue circles). A sinusoidal fit is indicated by a dashed line. Third, the parity is measured after only one $\pi /2$ pulse as a function of that pulse’s phase (green triangles). Each data point represents about 50 entanglement events. Error bars represent 1 standard deviation, where the sources of error are projection noise and the determination of $p_k$ from fluorescence data (Supplemental Material [26]).

Figure 4

Lower bound of the fidelity of the entangled state with respect to ${\Psi }^{+}$ as a function of the detection interval $T$ between photons. As scattering within this interval removes the indistinguishability of the ions, the fidelity decreases with increasing $T$. The fidelity drops below 50% for large $T$ values due to the possibility of coherent evolution after a scattering event, which may result in the Bell state ${\Psi }^{-}$. This process is reproduced by a Monte Carlo simulation of the fidelity (red line). Error bars correspond to 1 standard deviation.