Fig. 1: Illustration of the experimental protocol.

a An ensemble of cold ⁸⁷Rb atoms with an optical depth of ~5 is confined in a 1012-nm dipole trap. The counter-propagating 780 and 479 nm excitation beams are combined on a dichroic mirror (DM) and tightly focused onto the ensemble, with waists of 6 and 30 μm, respectively. Atoms are initialized in the ground state \(|{{{{{{{{{\rm{g}}}}}}}}}} \rangle\) and excited to a high-lying Rydberg state \(|{{{{{{{{{\rm{r}}}}}}}}}} \rangle\) with a single-photon detuning of Δ/2π = −200 MHz. After the excitation, a 479-nm readout light resonant with \(|{{{{{{{{{\rm{r}}}}}}}}}} \rangle \leftrightarrow|{{{{{{{{{\rm{e}}}}}}}}}} \rangle\) transition converts the single collective excitation state \(|{{{{{{{{{\rm{R}}}}}}}}}} \rangle\) into a single photon that is coupled into a single-mode fiber. The insert shows the atomic levels involved in the excitation and readout processes: ground state \(\vert {{{{{{{{{\rm{g}}}}}}}}}} \rangle=\vert 5{{{{{{{{{{\rm{S}}}}}}}}}}}_{1/2},F=2,{m}_{{{{{{{{{{\rm{F}}}}}}}}}}}=2 \rangle\), intermediate state \(\vert {{{{{{{{{\rm{e}}}}}}}}}} \rangle=\vert 5{{{{{{{{{{\rm{P}}}}}}}}}}}_{3/2},F=3,{m}_{{{{{{{{{{\rm{F}}}}}}}}}}}=3 \rangle\), and Rydberg state \(|{{{{{{{{{\rm{r}}}}}}}}}} \rangle=\vert 90{{{{{{{{{{\rm{S}}}}}}}}}}}_{1/2},J=1/2,{m}_{{{{{{{{{{\rm{J}}}}}}}}}}}=1/2 \rangle\). b Two single photons are sequentially generated with an interval of 5 μs, and their temporal wave packets are well overlapped at the interferometer using a polarization switching electro-optic modulator (EOM) and a 1-km delay fiber. Before the two-photon interference at the first PPBS, two half-wave plates (HWPs) are employed to prepare the input state of the control and target qubits. Two more PPBSs and HWPs are used after the interference to complete the CNOT gate operation. The output state is measured by a polarization-sensitive detection setup consisting of HWPs, polarization beam splitters (PBSs), and SPCMs (c_0,1 and t_0,1).