All-Optical Quantum Information Processing Using Rydberg Gates

In this Letter, we propose a hybrid scheme to implement a photonic controlled-$z$ (CZ) gate using photon storage in highly excited Rydberg states, which controls the effective photon-photon interaction using resonant microwave fields. Our scheme decouples the light propagation from the interaction and exploits the spatial properties of the dipole blockade phenomenon to realize a CZ gate with minimal loss and mode distortion. By excluding the coupling efficiency, fidelities exceeding 95% are achievable and are found to be mainly limited by motional dephasing and the finite lifetime of the Rydberg levels.

Figure 1

Optical layout: Control ($|C⟩$) and target ($|T⟩$) photonic qubits, in dual-rail encoding, are stored as Rydberg polaritons (dark red) in a cold atomic ensemble (yellow). The spatial modes corresponding to the qubit states $|1_C⟩$ and $|1_T⟩$ are stored in adjacent sites at a distance $d_{11}$, and the others arbitrarily further apart from the interacting ones ($d_{10}$). After storage we attempt a $2\pi$ rotation on the target qubit using the microwave field with Rabi frequency ${\mathrm{\Omega }}_{\mu }$ and an intermediate state. This succeeds except for $|1_C{1}_T⟩$, in which the intermediate state is shifted by resonant DD interactions, which have a characteristic length scale $R_b^{(3)}$. We need to ensure that the van der Waals interactions between stored states (characterized by the blockade radius $R_b^{(6)}$) are small.

Figure 2

Coupled basis of the two inner polariton sites $|1_C{1}_T⟩$. (a) The control (left) and target (right) photonic qubits are stored in the atom cloud in two different Rydberg states, $|r^{\prime }⟩$ and $|r⟩$, respectively. If the control qubit is in $|0⟩$ (states $|gX⟩$ in the pair state basis, with $X\in \{g,e,r,p\}$), we can perform a resonant $2\pi$ rotation on the $|r⟩\leftrightarrow |p⟩$ transition. (b) If the control qubit is in $|1⟩$ (states $|r^{\prime }X⟩$), it shifts the auxiliary state $|p⟩$ via a resonant DD interactions and the microwave $2\pi$ pulse is no longer resonant. This gives rise to a homogeneous, conditional phase shift in the site.

Figure 3

(Left) The characteristic length scales $R_b$ as a function of principal quantum number $n$ for different pair states in ${}^{87}\mathrm{Rb}$. The solid red line is the blockade radii for vdW interactions $R_b^{(6)}(n{S}_{1/2},(n+1)S_{1/2})$. The dash-dotted, yellow line is the standard vdW blockade radii for a same-level pair state with coefficient $C_6(n{S}_{1/2},n{S}_{1/2})$ shown for reference. The dashed blue line is the long-range resonant interaction length scale $R_b^{(3)}(n{S}_{1/2},n{P}_{1/2})$ for the $M=0$ pair state. All radii are calculated for a coupling of 1 MHz. Note the Förster resonance for the coupling $38s39s\leftrightarrow 38p_{3/2}38p_{3/2}$ [31]. (Right) The figure of merit $O$ [see (4)] for $|nS⟩$, $|(n+1)S⟩$, and $|nP⟩$ as a function of the principal quantum number $n$.

Figure 4

Estimation of the fidelity of the gate protocol, including the effects of finite lifetimes of the Rydberg states and motional dephasing, as a function of the microwave Rabi frequency ${\mathrm{\Omega }}_{\mu }$. The red, continuous line shows the case where $n=70$, $\mathrm{\Omega }=2\pi \times 10\mathrm{MHz}$, and $q=0.2$ (see the text for details), for a temperature of $T=0.1\mu \mathrm{K}$. Each plate shows the changes in the fidelity by varying one parameter: (a) EIT linewidth $\mathrm{\Omega }$; (b) temperature $T$; (c) principal quantum number $n$; and (d) different waist to blockade ratios $q=w_0/R_b^{(6)}$ used in the Gaussian averaging.