High-efficiency cluster-state generation with atomic ensembles via the dipole-blockade mechanism

We demonstrate theoretically a scheme for cluster-state generation, based on atomic ensembles and the dipole-blockade mechanism. In the protocol, atomic ensembles serve as single-qubit systems. Therefore, we review single-qubit operations on qubit defined as collective states of atomic ensemble. Our entangling protocol requires nearly identical single-photon sources, one ultracold ensemble per physical qubit, and regular photodetectors. The general entangling procedure is presented, as well as a procedure that generates in a single step $Q$-qubit GHZ states with success probability $p_{\text{success}}\sim {\eta }^{Q∕2}$, where $\eta$ is the combined detection and source efficiency. This is significantly more efficient than any known robust probabilistic entangling operation. GHZ states form the basic building block for universal cluster states, a resource for the one-way quantum computer.

Figure 1

The dipole-blockade mechanism. A laser pulse couples ground state $\mid g⟩$ and Rydberg state $\mid r⟩$. (a) One of the atoms is excited to the Rydberg state $\mid r⟩$. (b) Presence of a single atom in the Rydberg state $\mid r⟩$ shifts energy levels and blocks further excitations.

Figure 2

(Color online) Relevant atomic level structure with allowed atomic transitions. States $\mid g⟩$, $\mid e⟩$, and $\mid s⟩$ can be realized by the electronic low-lying, long-lived states of alkali-metal atoms. The transition between states $\mid g⟩$ and $\mid s⟩$ is always dipole forbidden. The state $\mid g⟩$ is coupled to the state $\mid e⟩$ through a classical field ${\Omega }_g$. A second classical field ${\Omega }_s$ is applied to the transition between the highly excited Rydberg level $\mid r⟩$ and the state $\mid s⟩$ (it may possibly be a two-photon process). States $\mid e⟩$ and $\mid r⟩$ are coupled via a quantum field. In general, $\Delta$ is a small detuning.

Figure 3

(Color online) The bit-flip operation $\left(X\right)$ and the Hadamard gate $\left(H\right)$. (a) Rotation ${\mid 0⟩}_L\to {\mid 1⟩}_L$ or ${\mid 0⟩}_L\to 1∕\sqrt{2}({\mid 0⟩}_L+{\mid 1⟩}_L)$ (b) rotation ${\mid 1⟩}_L\to {\mid 0⟩}_L$ or ${\mid 1⟩}_L\to 1∕\sqrt{2}({\mid 0⟩}_L-{\mid 1⟩}_L)$. See the text for an explanation.

Figure 4

(Color online) The collective states of a mesoscopic atomic ensemble. $\mid e⟩$ is the collective low-lying state, $\mid r⟩$ is the singly excited state, $\mid rr⟩$ is the doubly excited state, and $\overline{\Delta }$ is the mean dipole shift.

Figure 5

(Color online) A cluster state. Nodes represent physical qubits which are connected via the CZ operations. A horizontal strings of physical qubits constitute a logical qubit. The vertical links between logical qubits represent 2-qubit CZ gates.

Figure 6

(Color online) Diagram of the protocol. We send an entangled pair of photons in the state $\mid {\varphi }_{\text{light}}⟩=\frac{i}{\sqrt{2}}(\mid 02⟩+\mid 20⟩)$ into the arms of a Mach-Zehnder interferometer. The photons interact with atomic vapors: One and only one alkali-metal atom in the ensemble is excited by one of the photons to the Rydberg state $\mid r⟩$. Absorption of the second photon is prohibited by the dipole-blockade mechanism. After ${\mathrm{BS}}_2$, the state of ensemble-light system has the following form $\mid {\varphi }_{\text{out}}⟩=\frac{i}{\sqrt{2}}(\mid {\psi }^{+}⟩\mid 01⟩+\mid {\psi }^{-}⟩\mid 10⟩)$, where $\mid {\psi }^{\pm }⟩=\frac{1}{\sqrt{2}}(\mid re⟩\pm i\mid er⟩)$. Detection of a single photon will leave the atomic ensembles entangled.

Figure 7

(Color online) The scheme for creating the 4-qubit GHZ state. The four ensembles $A$, $B$, $C$, and $D$ are prepared in the state $\mid {\varphi }_{ABCD}⟩=\mid eeee⟩$. Four indistinguishable photons are sent into the beam splitters. The interaction of photons with the atomic vapors is followed by the beam splitters and four photodetectors. Conditional on photodetector clicks at the photodetector $(D_1,D_2)$, $(D_1,D_3)$, $(D_4,D_2)$ or $(D_4,D_3)$, a state of the four qubits is projected onto the 4-qubit GHZ state (up to phase correcting operations) with success probability $p_{\text{success}}={\eta }^2∕2$.

Figure 8

(Color online) Example of an experimental implementation of the protocol. The source of a single-photon pair consists of the type I nonlinear crystal. Generated single-photon pulses are entangled in the momentum (path) degree of freedom.