Passive CPHASE Gate via Cross-Kerr Nonlinearities

A fundamental and open question is whether cross-Kerr nonlinearities can be used to construct a controlled-phase (cphase) gate. Here we propose a gate constructed from a discrete set of atom-mediated cross-Kerr interaction sites with counterpropagating photons. We show that the average gate fidelity $F$ between a cphase and our proposed gate increases as the number of interaction sites increases and the spectral width of the photon decreases; e.g., with 12 sites we find $F>99\%$.

Figure 1

(a) The physical system inside our unit cell. It consists of two coupled two-level atoms, with internal energies $\mathrm{\Delta }$, and which interact via $H=\chi (\mathbb{1}-A_z)(\mathbb{1}-B_z)=\chi |1,1⟩⟨1,1|$. The input-output fields couple to the atoms via the relation $a_{\text{out}}=\sqrt{\gamma }{A}_{-}+a_{\text{in}}$, and similarly for mode $b$. It was shown in Ref. [28] that this system gives rise to the same $S$ matrices for single- and two-photon scattering as a pair of crossed cavities with cross-Kerr interaction between them. In the limit $\chi \to \infty$, this reduces to a three-level atom in a “V” configuration, such as considered in Ref. [19]. (b) Our main proposal using $N$ discrete interaction sites with counterpropagating photons.

Figure 2

In the top row, solid lines represent the average gate fidelity with respect to the cphase gate, i.e., $F(\varphi =\pi )$, while dashed lines denote the fidelity $F(\varphi )$ maximized with respect to some $\varphi$. The bottom row plots the corresponding phase shift ${\varphi }_{\text{opt}}={\text{argmax}}_{\varphi }F(\varphi )$. We compare three cases of interest: (i) a single-site Kerr interaction (circles), (ii) a two-site interaction with copropagating photons (squares), (iii) the two-site interaction with counterpropagation (stars). In the left column we have chosen $({\omega }_0,\gamma ,\chi )=(0,10,10000)$, which maximizes $F$ for the counterpropagating case, resulting in $F_{\text{counter}}=0.8628$ (only for this case, whenever $F\gtrsim 0.6$, the dashed and solid lines coincide). In the middle column we chose $({\omega }_0,\gamma ,\chi )=(0,6,2.67)$ to maximize $F$ for the copropagating case, obtaining $F_{\text{co}\text{−}\text{prop}.}=0.7326$, and in the right column we chose $({\omega }_0,\gamma ,\chi )=(1.1,4.5,5)$ to optimize the single-site $F$, obtaining $F_{\text{single}}=0.7810$.

Figure 3

(a) Average gate fidelity between our proposal and the cphase as a function of frequency bandwidth $\sigma$ for increasing number of interaction sites $N$. (b) We take the maximum of the gate fidelity in (a), and plot the infidelity ($1-F$) and the corresponding maximizing ${\sigma }_{\max }$ as functions of the number of interaction sites $N$. Small red dots correspond to $1-F$ for photons that have undergone two rounds of single-photon deformation. The dashed lines correspond to the fits $1-F({\sigma }_{\max },N)=0.537N^{-1.61}$ and ${\sigma }_{\max }=0.350N^{-0.81}$, where we fit to $N\in [4,20]$.

Figure 4

Saturation of the average gate infidelity for fixed $\sigma$ as the number of interaction sites $N$ increases. For illustration, we choose the $\sigma \mathrm{s}$ that maximize the fidelity for specific numbers of interaction sites $n$, i.e., ${\sigma }_{\max }^n={\text{argmax}}_{\sigma }F(\sigma ,n)$ where $n=\{2,4,6,8,10\}$. The lines then correspond to the fidelities at ${\sigma }_{\max }^n$ for increasing $N$, i.e., $F({\sigma }_{\max }^n,N)$. Finally, the black dots in each curve correspond to $N=1/{\sigma }_{\max }^n$, where we predict the fidelity to saturate.