Fault-tolerant quantum repeater with atomic ensembles and linear optics

We present a detailed analysis of a robust quantum repeater architecture building on the original Duan-Lukin-Cirac-Zoller (DLCZ) protocol [L.M. Duan et al. Nature (London) 414, 413 (2001)]. The architecture is based on two-photon Hong-Ou-Mandel-type interference which relaxes the long-distance interferometric stability requirements by about seven orders of magnitude, from subwavelength for the single photon interference required by DLCZ to the coherence length of the photons, thereby removing the weakest point in the DLCZ scheme. Our proposal provides an exciting possibility for robust and realistic long-distance quantum communication.

Figure 1

(Color online) Setups for entanglement generation and entanglement swapping in the DLCZ protocol. (a) Forward-scattered Stokes photons, generated by an off-resonant write laser pulse via spontaneous Raman transition, are directed to the beam splitter (BS) at the middle point. Entanglement is generated between atomic ensembles at sites $a$ and $b$, once there is a click on either of the detectors. The inset shows the atomic level structure, with the ground state $\mid g⟩$, metastable state $\mid s⟩$, and excited state $\mid e⟩$. (b) Entanglement has been generated between atomic ensembles $(a,b^L)$ and $(b^R,c)$. The atomic ensembles at site $b$ are illuminated by near resonant read laser pulses, and the retrieved anti-Stokes photons are subject to BS at the middle point. A click on either of the detectors will prepare the atomic ensembles at $a$ and $c$ into an entangled state.

Figure 2

(Color online) In the DLCZ protocol, two entangled pairs are generated in parallel. The relative phase between the two entangled states has to be stabilized during the entanglement generation process.

Figure 3

(Color online) Elementary entangled pairs are created locally. Entanglement swapping is performed remotely to connect atomic ensembles between adjacent nodes $a$ and $b$.

Figure 4

Entangled pairs are generated between neighboring communication nodes as shown in Fig. 3. The entangled pairs are connected by performing further entanglement swapping to construct entanglement between remote communication sites $A$ and $B$. The entanglement connection process, as well as the accumulated phase, is shown step by step.

Figure 5

(Color online) Setup for entanglement generation between sites $A$ and $B$. Forward-scattered Stokes photons, generated by an off-resonant write laser pulse via spontaneous Raman transition, are subject to BSM-i at the middle point. The Stokes photons generated at the same site are assumed to have different polarization, i.e., $\mid H⟩$ and $\mid V⟩$. PBS $\left({\mathrm{PBS}}_{\pm }\right)$ reflects photons with polarization $\mid V⟩$ $\left(\mid -⟩\right)$ and transmits photons with polarization $\mid H⟩$ $\left(\mid +⟩\right)$, where $\mid \pm ⟩=\frac{1}{\sqrt{2}}(\mid H⟩\pm \mid V⟩)$. After passing through the ${\mathrm{PBS}}_{\pm }$ and PBS successively, the Stokes photons are detected by single photon detectors. A coincidence count between single photon detectors ${\mathrm{D}}_1$ and ${\mathrm{D}}_4$ (${\mathrm{D}}_1$ and ${\mathrm{D}}_3$) or ${\mathrm{D}}_2$ and ${\mathrm{D}}_3$ (${\mathrm{D}}_2$ and ${\mathrm{D}}_4$) will project the four atomic ensembles into the complex entangled state ${\mid \psi ⟩}_{AB}$ up to a local unitary transformation.

Figure 6

(Color online) Setup for entanglement connection between sites $A$ and $C$ via entanglement swapping. Complex entangled states have been prepared in the memory qubits between sites $(A,B_L)$ and $(B_R,C)$. The memory qubits at site $B$ are illuminated by near resonant read laser pulses, and the retrieved anti-Stokes photons are subject to BSM-ii at the middle point. The anti-Stokes photons at the same site have different polarizations $\mid H⟩$ and $\mid V⟩$. After passing through PBS and ${\mathrm{PBS}}_{\pm }$ successively, the anti-Stokes photons are detected by single photon detectors. Coincidence counts between ${\mathrm{D}}_1$ and ${\mathrm{D}}_4$ (${\mathrm{D}}_1$ and ${\mathrm{D}}_3$) or ${\mathrm{D}}_2$ and ${\mathrm{D}}_3$ (${\mathrm{D}}_2$ and ${\mathrm{D}}_4$) are registered. The memory qubits will be projected into an effectively maximally entangled state ${\rho }_{AC}$ up to a local unitary transformation. Note that the sequence of PBSs in BSM-II is different from BSM-I. This helps to eliminate the spurious contributions from second-order excitations.

Figure 7

(Color online) Elementary entangled pairs are first locally generated via the standard DLCZ protocol. The anti-Stokes photons are subject to BSM-i to connect neighboring communication nodes. We also assume the anti-Stokes photons retrieved from atomic ensembles at the same site have different polarization. Note that BSM-i also helps to eliminate the spurious contributions from higher order excitations.

Figure 8

(Color online) Quantum memory for photonic polarization qubits. Two ensembles are driven by a classical control field. Classical and quantized light fields are fed into the first PBS and will leave at two different outputs of the second PBS. As each atomic cell works as a quantum memory for single photons with polarization $\mid H⟩$ or $\mid V⟩$ via the adiabatic transfer method, the whole setup is then a quantum memory of any single-photon polarization states. The inset shows the relevant level structure of the atoms. The $\mid e⟩-\mid s⟩$ transition is coherently driven by the classical control field of Rabi frequency ${\Omega }_c$, and the $\mid g⟩-\mid e⟩$ transition is coupled to a quantized light field.

Figure 9

(a) Entanglement swapping between adjacent communications nodes $A$ and $B$. Two pairs of entangled memory qubits are first generated by storing the event-ready entanglement of two photons at each node. Then the two photons stored in the $U$ memory at the two nodes are simultaneously retrieved and subject to a two-photon BSM at the middle point. This entanglement swapping process will in an event-ready way entangle the two distant $D$ memory qubits. (b) Entanglement connection to extend the communication length. Two well-entangled pairs of memory qubits, one across nodes $(A,B_L)$ and $(B_R,C)$ are prepared in parallel. The BSM on the two photons released simultaneously from the two memories at node $B$ results in, with a probability of $1∕2$, well-entangled quantum memories across nodes $A$ and $C$ in a definite Bell state.

Figure 10

(Color online) Setup for quantum entanglement purification. Entangled states have been prepared in the memory qubits between two distant nodes $I$ and $J$. The memory qubits at the two sites are illuminated by a near resonant read laser pulse, and the retrieved entangled photon pairs are directed to two PBS, respectively. The photons in mode $b_1$ and $b_2$ are detected in the $\mid \pm ⟩=\frac{1}{\sqrt{2}}(\mid H⟩\pm \mid V⟩)$ basis and the remaining photons in mode $a_1$ and $a_2$ are restored in the memory qubits at the two sites, respectively.

Figure 11

(Color online) Deterministic single-photon polarization entangler. PBS (${\mathrm{PBS}}_{\pm }$; ${\mathrm{PBS}}_{R∕L}$) reflects photons with vertical polarization $\mid V⟩\left(\mid -⟩\right);\mid L⟩$ and transmits photons with horizontal-polarization $\mid H⟩$ $(\mid +⟩;\mid R⟩)$. Here $\mid \pm ⟩=\frac{1}{\sqrt{2}}(\mid H⟩+\mid V⟩);\mid R∕L⟩=\frac{1}{\sqrt{2}}(\mid H⟩\pm i\mid V⟩)$. The four single photons are prepared on demand in an initial state ${\mid -⟩}_1{\mid V⟩}_2{\mid +⟩}_{1^{\prime }}{\mid H⟩}_{2^{\prime }}$. After passing through the first PBS and ${\mathrm{PBS}}_{\pm }$, one selects the “four-mode” case where there is one and only one photon in each of the four output modes. Then the BSM will collapse photons in modes $a$ and $b$ into a Bell state conditioned on the result of the BSM. In our case, a coincidence count between single-photon detectors D1 and D4 (D1 and D3) or between D2 and D3 (D2 and D4) leaving photons along paths $a$ and $b$ deterministically entangled in ${\mid {\psi }^{+}⟩}_{ab}\left({\mid {\varphi }^{-}⟩}_{ab}\right)$.