An integrated photonic circuit for color qubit preparation by third-order nonlinear interactions

Abstract

This work presents a feasible design of an integrated photonic circuit performing as a device for single-qubit preparation and rotations through the third-order nonlinear process of difference frequency generation (DFG) and defined in the temporal mode basis. The first stage of our circuit includes the generation of heralded single photons by spontaneous four-wave mixing in a micro-ring cavity engineered for delivering a single-photon state in a unique temporal mode. The second stage comprises the implementation of DFG in a spiral waveguide with controlled dispersion properties for reaching color qubit preparation fidelity close to unity. We present the generalized rotation operator related to the DFG process, a methodology for the device design, and qubit preparation fidelity results as a function of user-accessible parameters.

Figure 1

Design of the proposed integrated photonic circuit for color qubit preparation and rotations. All components in the device are based on silicon nitride (Si${}_3$N${}_4$) waveguides above a substrate of silicon dioxide (SiO${}_2$) on silicon (Si). The top scheme describes the circuit operation, comprising the stage for generating a heralded single-photon state, which acts as input to the quantum gate waveguide (the second stage), where the DFG process occurs. The bottom-right scheme depicts the DFG process that leads to a generalized rotation operator for single-qubits defined in the temporal mode basis. Note that $|0⟩=|{\varphi }_j⟩$ and $|1⟩=|{\psi }_j⟩$ are written on a computational basis in pairs of temporal modes. The bottom-left image is a representation of the waveguide’s geometry.

Figure 2

Spectral properties of the proposed SFWM and DFG sources. (a) The normalized joint spectral intensity of the SFWM source. (b) and (c) The three first temporal-mode pairs of the JSA (${\phi }_m({\omega }_s)$, ${\mathrm{\Omega }}_m({\omega }_i)$), plotted as dashed lines. (d) The normalized DFG mapping function. (b) and (e) The three first temporal-mode pairs of the MF (${\varphi }_j({\omega }_s)$, ${\psi }_j({\omega }_i)$), plotted as solid lines. Note that the eigenfunctions have been normalized to the maximum value of the fundamental mode, while the joint spectral functions were normalized to their corresponding maximum value.The insets of panels (a) and (d) show the Schmidt coefficients of the JSA and MF, respectively.

Figure 3

Color qubit preparation fidelity as a function of (a) the pump wavelengths, (b) the pump bandwidths, and (c) variations in the SFWM and DFG waveguide widths, while the remaining parameters of the SFWM and DFG sources are fixed to the values in Table 1.

Figure 4

Method for simultaneous SFWM and DFG phasematching. (a) The minimum value of the objective function in terms of the core height h, when the widths of the two waveguides are allowed to vary. (b) Value of the objective function for $h=0.7\mu$m in terms of the two waveguide widths. (c) Phasematching contours for SFWM (red) and DFG (blue), obtained with parameters $h=0.7\mu$m, $w_{\mathit{sfwm}}=0.953\mu$m and $w_{\mathit{dfg}}=1.617\mu$m.

Figure 5

Methodology for increasing the qubit preparation fidelity. Single-qubit preparation fidelity in terms of the length of the DFG waveguide (L), the bandwidth of pump 1 (${\sigma }_1$ ), and the length ($l_c$ ) and reflectivity ($R_i$) of the SFWM cavity, for ${\sigma }_2=0.5$ THz and ${\sigma }_f=4$ THz. The black circle marker indicates the combination of parameters we choose for our example.

Figure 6

Methodology for increasing the qubit preparation fidelity. (a) Fidelity as a function of ${\sigma }_2$ and ${\sigma }_f$, while fixing the remaining parameters at the values corresponding to the black circle marker in Fig. 5. (b) The photon-pair emission rate per pump-power squared in terms of ${\sigma }_f$. (c) Minimum product of the average powers of the two pump fields required for a complete frequency conversion in terms of ${\sigma }_2$. The black circle markers correspond to the choice for our design.