Theory of Microwave Parametric Down-Conversion and Squeezing Using Circuit QED

We study theoretically the parametric down-conversion and squeezing of microwaves using cavity quantum electrodynamics of a superconducting Cooper-pair box (CPB) qubit located inside a transmission line resonator. The nonlinear susceptibility ${\chi }_2$ describing three-wave mixing can be tuned by dc gate voltage applied to the CPB and vanishes by symmetry at the charge degeneracy point. We show that the coherent coupling of different cavity modes through the qubit can generate a squeezed state. Based on parameters realized in recent successful circuit QED experiments, squeezing of $95\%\sim 13\mathrm{dB}$ below the vacuum noise level should be readily achievable.

Figure 1

The schematic diagram for the circuit QED coupled to a Cooper-pair box located at one edge of the cavity. The red and blue lines represent the fundamental and the second harmonic cavity modes, respectively. The change of CPB position from the edge will vary the coupling strengths $g_1$ and $g_2$.

Figure 2

The quadrature variance $V(\omega )$ for $X_1$ is plotted as a function of $\omega$ for $N=10$, $\Delta =2.6\times {10}^4$, $\Gamma =5$, $g_2|{\alpha }_p|=2827$ in units of ${\kappa }_1$. The maximum squeezing is obtained at $\omega =0$.

Figure 3

The quadrature variance $V(\omega =0)$ for $X_1$ is plotted as a function of $\Gamma$ for several different values of $N=10,12,15,20$. The minimum value of $V(\omega =0)$ decreases with the increase of the maximum photon number $N$. The critical value of $\Gamma$ is about 8.2 in units of ${\kappa }_1$. In the inset, the maximum output squeezing for several values of $N$ is plotted with respect to $1/N$. By extrapolation to $N=\infty$, we obtain the $V_{\min }(\omega =0)=0.05$, that is, $-13\mathrm{dB}$.

Figure 4

The critical value ${\Gamma }^{*}$ is plotted as a function of $\Delta$. The dotted line represents the result from the analytical formula and the filled circles from the numerical simulation. Here ${\Gamma }^{*}$ and $\Delta$ are in units of ${\kappa }_1$.