Detecting photon-photon interactions in a superconducting circuit

A local interaction between photons can be engineered by coupling a nonlinear system to a transmission line. The required transmission line can be conveniently formed from a chain of Josephson junctions. The nonlinearity is generated by side-coupling this chain to a Cooper pair box. We propose to probe the resulting photon-photon interactions via their effect on the current-voltage characteristic of a voltage-biased Josephson junction connected to the transmission line. Considering the Cooper pair box to be in the weakly anharmonic regime, we find that the dc current through the probe junction yields features around the voltages $2eV=n\hslash {\omega }_s$, where ${\omega }_s$ is the plasma frequency of the superconducting circuit. The features at $n\ge 2$ are a direct signature of the photon-photon interaction in the system.

Figure 1

The system consists of a transmission line that is capacitively coupled (capacitor $C_c$) to a Josephson junction, shown inside the dashed box. The transmission line is realized using a chain of Josephson junctions with Josephson energy $E_J$ much larger than the charging energy $E_C$. The system is probed at node $m$ using another Josephson junction (outside the dashed box) whose current-voltage characteristic is sensitive to the properties of the system.

Figure 2

The linear regime: Current-voltage characteristic of the probe Josephson junction when placed at $m=0$. The parameters are $\mathrm{\Gamma }/{\omega }_s=0.02,E_{\mathrm{cut}\text{−}\mathrm{off}}/{\omega }_s=20$, and different $Z_0$ ($Z_0/R_Q=0.01,0.1,0.2$). The side-coupled Josephson junction causes a resonance at $2eV={\omega }_s$. In the limit $Z_0/R_Q\to 0$, the current vanishes at the resonance.

Figure 3

The resonance in the current-voltage characteristic for different values of the distance between the side-coupled Josephson junction and the probe Josephson junction, $\alpha =2m{\omega }_s/{\omega }_0$. Results in the single-photon, linear regime are plotted [Eq. (18)] with $\mathrm{\Gamma }/{\omega }_s=0.02$. Note the effect of interference on the shape of the resonance.

Figure 4

Dyson equation for the two-point Green's function. The nonlinearity results in a self-energy, $\mathrm{\Sigma }=E_J^{\mathrm{s}}{\langle {\varphi }_{\delta }^2\rangle }_{H^{\left(0\right)}}/2$.

Figure 5

The Feynman diagrams for the interaction correction to the four-point Green's functions. (a) $\delta {\mathcal{G}}^{\mathrm{int}}\left[{\varphi }_n^2\left(\tau \right){\varphi }_m^2\left(0\right)\right]$. (b) $\mathcal{G}\left[{\varphi }_n^3\left(\tau \right){\varphi }_m\left(0\right)\right]$ and $\mathcal{G}\left[{\varphi }_n^3\left(\tau \right){\varphi }_m\left(0\right)\right]$.

Figure 6

The nonlinear regime: Current-voltage characteristic of the probe Josephson junction when placed at $m=0$. The parameters are ${\omega }_s^{\prime }/E_J^{\mathrm{s}}=0.9,\mathrm{\Gamma }/{\omega }_s^{\prime }=0.02,E_{\mathrm{cut}\text{−}\mathrm{off}}/{\omega }_s^{\prime }=20$, and $Z_0/R_Q=0.2$. Photon-photon interactions lead to a second resonant feature at $2eV=2{\omega }_s^{\prime }$. A zoom on that feature with amplitude $\delta I_2/I_0\propto (Z_0/R_Q)({\omega }_s^{\prime }/E_J^{\mathrm{s}})$ is shown in the inset.

Figure 7

The second resonance in the current-voltage characteristic for different values of the distance between the side-coupled Josephson junction and the probe Josephson junction, ${\alpha }^{\prime }=2m{\omega }_s^{\prime }/{\omega }_0$. Results are plotted near $2eV=2{\omega }_s^{\prime }$ in the two-photon, nonlinear regime [Eq. (32)] with $\mathrm{\Gamma }/{\omega }_s^{\prime }=0.02$.