Deterministic photon-photon $\sqrt{\text{SWAP}}$gate using a $\Lambda$ system

We theoretically present a method to realize a deterministic photon-photon $\sqrt{\text{SWAP}}$ gate using a three-level $\Lambda$ system interacting with single photons in reflection geometry. The $\Lambda$ system is used completely passively as a temporary memory for a photonic qubit; the initial state of the $\Lambda$ system may be arbitrary, and active control by auxiliary fields is unnecessary throughout the gate operations. These distinct merits make this entangling gate suitable for deterministic and scalable quantum computation.

Figure 1

Interaction between a $\Lambda$ system and a single photon propagating in one dimension. (a) Initial state. The photonic and atomic qubits may be in arbitrary states. (b) Final state. The $\Lambda$ system is completely deexcited through radiative decay. The photonic and atomic qubits can be completely swapped under appropriate conditions.

Figure 2

Shapes of the output wave packets, $g_1$ (thin dotted line), $g_2$ (solid line), and $g_3$ (dashed line) for the case of the atom-photon swap gate (${\Gamma }_H={\Gamma }_V$ and $\omega =0$). The natural phase factor $e^{i\Omega (r-t)}$ is removed. The pulse length is (a) $l=2.5{\Gamma }_H^{-1}$ and (b) $l=10{\Gamma }_H^{-1}$.

Figure 3

Contour plots of the average gate fidelities for the atom-photon (a) swap ($\omega =0$) and (b) $\sqrt{{\text{SWAP}}}$ ($\omega =-{\Gamma }_H$) gates, as functions of the pulse length $l$ and ${\Gamma }_V/{\Gamma }_H$.

Figure 4

Illustration of the photon-photon $\sqrt{{\text{SWAP}}}$ gate. ${\mathrm{P}}_1$ and ${\mathrm{P}}_3$ are in resonance with the atom ($\omega =0$), whereas ${\mathrm{P}}_2$ is detuned ($\omega =\pm {\Gamma }_H$). The input qubits are the polarization states of ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$, whereas the output qubits are those of ${\mathrm{P}}_3$ and ${\mathrm{P}}_2$.

Figure 5

Generation of a three-atom Greenberger-Horne-Zeilinger (GHZ) state. (PBS) polarization beam splitter; (HWP) half-wave plate; (BS) beam splitter. Initially, all atoms are in the $|0\rangle$ state. The polarization state of the input photon is $2^{-1/2}(|H\rangle +|V\rangle )$.