Colloquium: Quantum networks with trapped ions

Quantum computation and communication exploit the quantum properties of superposition and entanglement in order to perform tasks that may be impossible using classical means. In this Colloquium recent experimental and theoretical progress in the generation of entangled quantum networks based on the use of optical photons as carriers of information between fixed trapped atomic ion quantum memories are reviewed. Taken together, these quantum platforms offer a promising vision for the realization of a large-scale quantum network that could impact the future of communication and computation.

Figure 1

(Color online) Illustration of different quantum networks. (a) Illustration of a quantum repeater network to extend the communication distance. (b) Illustration of a quantum computation network with many interconnected quantum nodes to extend the computational size.

Figure 2

(Color online) Reduced energy-level diagram of ${}^{171}\mathrm{Yb}^{+}$, showing how photons are generated and entangled with internal ion qubits denoted by |0⟩ and |1⟩. (a) Due to the atomic selection rules, an ultrafast $\pi$-polarized laser pulse drives |0⟩ to $|0^{\prime }⟩$ and |1⟩ to $|1^{\prime }⟩$. The pulse bandwidth is large enough to simultaneously drive both transitions with near-unit excitation probability. (b) After excitation, the ion spontaneously decays back to ${{}^{2}S}_{1∕2}$ and emits a single photon at $369.53\mathrm{nm}$. If we consider only $\pi$-polarized photons, then the frequency of the emitted photon is entangled with the internal electronic state of the atom, with the separation of the different frequency modes equal to the sum of the ${{}^{2}S}_{1∕2}$ and ${{}^{2}P}_{1∕2}$ hyperfine splittings $({\nu }_{\text{blue}}-{\nu }_{\text{red}}=14.7\mathrm{GHz})$. Polarizing beam splitters (PBS) are used to filter out the $\sigma$-polarized light.

Figure 3

Two photon interference at a beam splitter. (a) Spatial modes $a$ and $b$ are straight paths through the beam splitter (BS), and the beam splitter couples these two modes. (b)–(e) The four possible output modes of two photons entering a beam splitter from different ports. A negative phase is acquired only upon reflection from low to high index of refraction—mode $a$ in (c) and (e).

Figure 4

Normalized experimental two-photon second-order correlation function for identical (parallel polarized) photons (diamonds) and distinguishable (perpendicularly polarized) photons (circles) (47). As expected, two identical photons incident on the beam splitter always exit by the same port, resulting in suppression of joint detections at time delay $t_d=0$.

Figure 5

(Color online) Experimental setup for the heralded entanglement and quantum gate operation between two ions. Spontaneously emitted $\pi$-polarized photons are coupled in a single-mode fiber and directed to interfere on a $50∕50$ nonpolarizing BS. Coincident detection of two photons by photon-counting photomultipliers (PMTs) announces the success of the gate between the two ions. PBSs are polarizers used to filter the photons so that only $\pi$-polarized photons are detected. The state of each ion is measured by state-dependent fluorescence, detected by a PMT on the opposite side of the vacuum chamber.

Figure 6

(Color online) State tomography of the singlet entangled state of two ions $|0⟩|1⟩-|1⟩|0⟩$. The (a) real and (b) imaginary elements of the reconstructed density matrix are shown. The density matrix was obtained with a maximum-likelihood method from 601 events measured in nine different bases. From this density matrix we calculate an entangled state fidelity of $\mathcal{F}=0.87\left(2\right)$, a concurrence of $C=0.77\left(4\right)$, and an entanglement of formation $E_F=0.69\left(6\right)$.

Figure 7

Space-time schematic of the teleportation protocol. Each ion is first initialized to |0⟩ by optical pumping. At time $t_1$ the state to be teleported $\alpha |0⟩+\beta |1⟩$ is written to ion $A$ and the known state $|0⟩+|1⟩$ is written to ion $B$ through appropriate rotation operators $R(\theta ,\varphi )$ on the Bloch sphere, with $R(\theta ,0)\equiv R_y\left(\theta \right)$ and $R(\theta ,\pi ∕2)\equiv R_x\left(\theta \right)$. At time $t_2$, synchronized laser pulses excite each atom, and the frequencies of the emitted photons are entangled with the respective ion qubits following Fig. 2. The two photons interfere at a beam splitter, and coincident detection of the emerging photons at time $t_3$ heralds the production of the entangled state of Eq. (3) with $\delta =\gamma =1∕\sqrt{2}$. Following successful entanglement, ion $A$ is coherently rotated at time $t_4$ through the operator $R_y(\pi ∕2)$ and measured in the $z$-basis at time $t_5$. Finally, depending on the measurement result, rotation operators conditioned upon the measurement result are applied to ion $B$ at time $t_6$ to complete the teleportation of the original quantum state from $A$ to $B$.

Figure 8

Tomography of the teleported quantum states. The reconstructed density matrices $\rho$ for the six unbiased basis states teleported from ion $A$ to ion $B$: (a) ${|\psi ⟩}_{\text{ideal}}=1∕\sqrt{2}(|0⟩+|1⟩)$ teleported with fidelity $\mathcal{F}=0.91\left(3\right)$, (b) ${|\psi ⟩}_{\text{ideal}}=1∕\sqrt{2}(|0⟩-|1⟩)$ teleported with fidelity $\mathcal{F}=0.88\left(4\right)$, (c) ${|\psi ⟩}_{\text{ideal}}=1∕\sqrt{2}(|0⟩+i|1⟩)$ teleported with fidelity $\mathcal{F}=0.92\left(4\right)$, (d) ${|\psi ⟩}_{\text{ideal}}=1∕\sqrt{2}(|0⟩-i|1⟩)$ teleported with fidelity $\mathcal{F}=0.91\left(4\right)$, (e) ${|\psi ⟩}_{\text{ideal}}=|0⟩$ teleported with fidelity $\mathcal{F}=0.93\left(4\right)$, and (f) ${|\psi ⟩}_{\text{ideal}}=|1⟩$ teleported with fidelity $\mathcal{F}=0.88\left(4\right)$. These measurements yield an average teleportation fidelity $\overline{\mathcal{F}}=0.90\left(2\right)$. The data shown comprise a total of 1285 events in $253\mathrm{h}$.

Figure 9

(Color online) Illustration of construction of a quantum repeater network through probabilistic entangling operations.

Figure 10

Illustration of the three properties of the cluster states which are important for our construction of such states with the probabilistic entangling gates: (a) extend cluster states with $ZZ$ measurement gates, (b) recover cluster states by removing bad qubits, and (c) shrink cluster states for more complicated links [see 26].

Figure 11

Illustration of the steps for construction of the two-dimensional square lattice cluster states from a set of cluster chains. (a) Construction of the basic $+$ shape states from cluster chains by applying a $ZZ$ gate to connect the two middle qubits and an $X$ measurement on one middle qubit to remove it. (b) and (c) Construction of the square lattice cluster state from the $+$ shape states through probabilistic $ZZ$ measurement gates along the legs and $X$ measurements to remove the remaining redundant qubits. See 26 for a similar construction with the controlled phase flip gates.

Figure 12

(Color online) Schematic of a quantum computational model based on a hybrid approach. The ancilla ions in different traps are entangled through the probabilistic entangling protocol. Deterministic gates on remote ions are constructed from the local Coulomb gates and the probabilistic remote entanglement.

Figure 13

(Color online) Schematic realization of quantum repeaters with trapped ions based on a combination of the probabilistic detection-induced remote entanglement and the local Coulomb gates.

Figure 14

(Color online) Hierarchical trapped ion quantum computer. $N$ quantum registers (left) each consists of a many-qubit crystal of trapped ions that can be entangled through deterministic Coulomb gates. One ion within each register is coupled optically to an $N\times N$ optical cross-connect switch in order to propagate quantum information between registers and create large-scale entangled states among all ions. Such a hierarchical approach to scaling the ion-photon network may allow the scaling to millions or more trapped ion qubits.