Cavity Carving of Atomic Bell States

We demonstrate entanglement generation of two neutral atoms trapped inside an optical cavity. Entanglement is created from initially separable two-atom states through carving with weak photon pulses reflected from the cavity. A polarization rotation of the photons heralds the entanglement. We show the successful implementation of two different protocols and the generation of all four Bell states with a maximum fidelity of $(90\pm 2)\%$. The protocol works for any distance between cavity-coupled atoms, and no individual addressing is required. Our result constitutes an important step towards applications in quantum networks, e.g., for entanglement swapping in a quantum repeater.

Figure 1

Illustration of quantum state carving via the Husimi $Q$ distribution. The Husimi $Q$ distribution of a two-particle density matrix $\rho$ is defined as $Q(\rho ;\theta ,\varphi )=(3/4\pi )⟨\theta ,\varphi |\rho |\theta ,\varphi ⟩$, where $|\theta ,\varphi ⟩={\otimes }_{j=1}^2[\cos (\theta /2){|\uparrow ⟩}_j-e^{i\varphi }\sin (\theta /2){|\downarrow ⟩}_j]$ are coherent spin states defined by the spherical coordinates $\theta$ and $\varphi$ on the generalized Bloch sphere. The initial distribution (the left sphere) is rotated around the $y$ axis (the middle sphere) before the $|\downarrow \downarrow ⟩$ and $|\uparrow \uparrow ⟩$ components are carved out. After the carving step, the poles are not populated anymore and the maximally entangled Bell state $|{\mathrm{\Psi }}^{+}⟩$ is prepared (the right sphere). The color code is normalized on each sphere and $Q$ increases from dark to bright.

Figure 2

Schematic representation of the setup. Two ${}^{87}\mathrm{Rb}$ atoms are trapped near the cavity center. The internal state of the atoms is controlled with a Raman laser pair impinging from the side of the cavity, colinear with a state-detection beam (blue, from above). State detection of the atomic qubits can additionally be performed with a laser probing the cavity transmission (green, from the left). Coherent light pulses impinge on the cavity’s outcoupling mirror through a nonpolarizing beam splitter (NPBS) and are reflected from the cavity. Light from the cavity is detected by a polarization-resolving detection setup consisting of a half- and a quarter-wave plate ($\lambda /2$ and $\lambda /4$), a polarizing beam splitter (PBS), and two single-photon detectors (SPDs). (Left inset) Simplified level scheme of a single atom with the two ground states $|\uparrow ⟩$ and $|\downarrow ⟩$ and the excited state $|e⟩$. $|R⟩$ photons couple $|\uparrow ⟩$ and $|e⟩$. (Right inset) Fluorescence image of two trapped atoms.

Figure 3

Quantum circuit diagrams. (a) Double-carving scheme. If the initial atom-atom state is chosen as ① $|\downarrow \downarrow ⟩$, the symmetric state $|{\mathrm{\Psi }}^{+}⟩$ is prepared via the reflection of two antidiagonally ($|A⟩$) polarized photons, each followed by postselection, and interleaved by a $\pi$ pulse. A subsequent, optional $\pi /2$ rotation (the dashed box) generates $|{\mathrm{\Phi }}^{\pm }⟩$, depending on the phase of the last rotation pulse. Starting with atoms in opposite states ② $|\uparrow \downarrow ⟩$ or $|\downarrow \uparrow ⟩$ or a statistical mixture of them, the singlet state $|{\mathrm{\Psi }}^{-}⟩$ can be prepared with the same protocol. (b) Single-carving scheme following weak initial excitation. Starting from $|\downarrow \downarrow ⟩$, the scheme creates $|{\mathrm{\Psi }}^{+}⟩$, which can afterwards be transformed into $|{\mathrm{\Phi }}^{\pm }⟩$ by an optional $\pi /2$ pulse (the dashed box) of the appropriate phase. The state $|{\mathrm{\Psi }}^{-}⟩$ cannot be created with this scheme.

Figure 4

Parity oscillations of the four Bell states and the Husimi $Q$ distribution of $|{\mathrm{\Phi }}^{-}⟩$. (a) Shown are experimental data for $\mathrm{\Pi }(\varphi )$ from the double-carving entangling scheme with standard errors of each mean value, and a fit of $\mathrm{\Pi }(\varphi )$ with $\mathrm{Re}({\rho }_{\uparrow \downarrow ,\downarrow \uparrow })$, $\mathrm{Im}({\rho }_{\uparrow \uparrow ,\downarrow \downarrow })$, and $\mathrm{Re}({\rho }_{\uparrow \uparrow ,\downarrow \downarrow })$ as free parameters. (b) Comparison between the theoretically expected Husimi distribution and experimental data for the $|{\mathrm{\Phi }}^{-}⟩$ state. The Mollweide projection of the respective spheres is shown.

Figure 5

(a) Entangled state fidelity $F$ vs average incident photon number $\overline{n}$ in each coherent pulse in the double-carving scheme. The dashed line shows the simple theoretical model of exponential decoherence through photons scattered by the atoms. The predicted decay rate for our cavity parameters is applied without any free fitting parameters. The model plotted as a solid line additionally considers dark counts, which become dominant for low $\overline{n}$, and shows a fitted dark count rate of 0.011 per pulse. (b) Fidelity vs rotation angle $\alpha$ in the single-carving scheme, showing measured data and a fitted model including dark counts. All error bars are standard deviations of the mean.