Fig. 1. (a) Waveguide dispersion as a function of the wavelength, calculated for a waveguide height of 300 nm and different waveguide widths. (b) SEM image of the resonator. (c) Normalized transmission spectrum of the resonator. (d) Measured quality factor as a function of the wavelength.

Fig. 2. Measured FSR as a function of wavelength and optical frequency around 1540 nm.

Fig. 3. (a) Setup for measurement of the joint spectral intensity. BP, bandpass filter; NF, notch filter; PF, programmable filter; PC, polarization controller; and SNSPDs, superconducting single-photon detectors. (b) Anti-diagonal elements of the JSI measurement for every accessible signal-idler pair from $n=3$ to $n=83$.

Fig. 4. Photon pair generation and heralded single-photon characterization of the SOI MR. (a) Setup for measuring single counts, two-photon coincidences, and three-photon coincidences. NF, notch filter; PF, programmable filter; and TT, time tagger. (b) Single counts, (c) coincidences, (d) generated number of pairs, (e) heralded $g^{(2)}(0)$, and (f) CAR, each as a function of input power (refer to text).

Fig. 5. Setup for the quantum state tomography. PF, programmable filter; EOM, electro-optic phase modulator. Insets are the action of the PFs on the frequency modes. PF1 is used both as an amplitude filter to select the four modes of the two qubits, and as a phase gate implementing a phase ${\varphi }_i$ and ${\varphi }_s$ on the frequency modes $I_n$ and $S_n$. The boxed devices implement identity or Hadamard gate on the qubits. All the projections required for the tomography are accessible with these two gates. PF3 selects two modes $I_p$ and $S_q$, where $p,q\in \{n,n+1\}$.

Fig. 6. Numerical reconstruction of the experimental density matrix of a two-qubit frequency-bin entangled state generated by the SOI resonator + PF1. (a) Real part and (b) imaginary part.

Fig. 7. Fidelity to a maximally entangled state for several frequency-bin entangled photon pairs. The $x$-axis corresponds to the number of resonances between the frequency qubit and the pump mode.

Fig. 8. (a) Raw coincidences (bars) and QBER (dots) between two users and (b) sifted key rate, calculated using the method in Ref. 34 as a function of $n$, spectral detuning from the pump. Each link between two users is a two-qubit frequency-bin entangled pair and is color-coded. For example, the bottom users of the network in the inset of (b) receive, respectively, the two frequency channels $|I_{10}⟩$, $|I_{11}⟩$ and $|S_{10}⟩$, $|S_{11}⟩$, which connect them in the network.

Fig. 9. Classical characterization of the quantum gate. (a) Principle of a frequency-domain operation. (b) Principle of a frequency-domain quantum gate. (c) Phase pattern applied by the PF. (d) Measured tunability of the quantum gate operation.

Fig. 10. (a) Light coupling from mode $|{\omega }_0⟩$ (black) or $|{\omega }_1⟩$ (red) into nine neighboring modes. The gray shaded areas represent the guard modes that are not used for parallelization. (b) Transmission measurement for nine frequency modes when a Hadamard gate is applied to two qubits separated by two guard modes.

Table 1. Comparison of internal brightness, FSR, and frequency channels accessible with commercial EOMs and other SOI implementations. We measured the brightness B on the signal-idler pair |I15,S15⟩. *: Clementi et al.⁹ and Borghi et al.¹⁰ used several coupled rings to achieve, respectively, an effective 18 GHz and a 15 GHz mode spacing.

Table 2. Coincidences for the two-photon projections Ca,b integrated for 125 s, in a coincidence window of 1 ns.