# Entangled Frequency-Tunable Microwave Photons in a Superconducting Circuit
## 2 The Circuit Model  
The circuit in our proposal is illustrated in Figure 1a. A frequency-fixed transmission-line resonator (TLR) (orange) is capacitively coupled to a frequency-tunable TLR (green), which is inserted with a SQUID in the center that behaves as a tunable inductance modulated by the flux bias through its loop. In the standard quantized manner, assuming that \(\hbar=1\), the Hamiltonian of the circuit is  
\[\hat{H} =\hat{H}_{a}+\hat{H}_{b}+\hat{H}_{ab}\] \[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\] \[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}+g\{(\hat{a}\hat{b}^{\dagger}+\hat{a}^{\dagger}\hat{b})-(\hat{a}\hat{ b}+\hat{a}^{\dagger}\hat{b}^{\dagger})\}, \tag{1}\]  
where \(\hat{a}\) (\(\hat{a}^{\dagger}\)) and \(\hat{b}\) (\(\hat{b}^{\dagger}\)) are the annihilation (creation) operator of modes \(a\) and \(b\), respectively, and \(\omega_{a}\) and \(\omega_{b}\) are the corresponding resonant angular frequencies of the two modes of our system, respectively. \(g\) is the capacitive coupling strength between the two modes. (\(\hat{a}\hat{b}^{\dagger}+\hat{a}^{\dagger}\hat{b}\)) is the rotating term, and (\(\hat{a}\hat{b}+\hat{a}^{\dagger}\hat{b}^{\dagger}\)) is the counterrotating (CR) term. The two modes of the system come from the coupled resonators. The eigenfrequency of the frequency-fixed resonator (orange) is \(\omega_{f}\). The resonant frequency of the frequency-tunable resonator (green) is \(\omega_{t}=\omega_{t}^{0}/(1+\gamma_{0}\text{cos}(\pi\Phi_{\text{dc}}/\Phi_{ 0}))\) with geometric frequency \(\omega_{t}^{0}\), i.e., the resonant frequency without a SQUID. The participation ratio \(\gamma_{0}\) is the ratio between the equivalent inductance of SQUID (at zero dc flux bias), and the inductance of the tunable resonator. \(\Phi_{0}=h/2e\) is the flux quantum. Then, the two different modes [35; 36] of the superconducting circuit are  
\[\omega_{a}=\frac{1}{2}(\omega_{f}+\omega_{t})-\sqrt{g^{2}+( \frac{\Delta\omega}{2})^{2}},\] \[\omega_{b}=\frac{1}{2}(\omega_{f}+\omega_{t})+\sqrt{g^{2}+(\frac {\Delta\omega}{2})^{2}}, \tag{2}\]  
where \(\Delta\omega=\omega_{t}-\omega_{f}\), and the coupling strength \(g=C_{g}Z_{0}(2\omega_{f}^{2}+\omega_{t}^{2})/2\pi\) with impedance \(Z_{0}\) of resonators. Here, we have set \(\omega_{f}=5.416\) GHz, \(\omega_{t}^{0}=5.844\) GHz, \(\gamma_{0}=4.8\%\), \(C_{g}=5.1\) fF and \(Z_{0}=50\) \(\Omega\). The rates of decay into the environment of the two modes are \(\kappa_{a}/2\pi=0.02\) GHz and \(\kappa_{b}=2\kappa_{a}\), respectively. Equation (1) describes the quantum states of two coupled modes. The entangled photons between the two modes hardly escape from the circuit without external energy excitation due to the usual neglect of the CR term in the rotating-wave approximation [35; 36]. In our proposal, the high-frequency oscillation terms of the last part in Equation (1) are excited with the help of a strong external pump through the SQUID [37; 38; 39]. In this case, the resonant frequency of SQUID is modulated by \(\Phi_{e}=\Phi_{\rm dc}+\Phi_{\rm ac}\) periodically as  
\[\omega_{b}(t)=\omega_{b}(\Phi_{\rm dc})+\beta\text{sin}(\omega_{p}t+\varphi_{p0}), \tag{3}\]  
where \(\Phi_{\rm dc}\) is the dc flux bias and \(\Phi_{\rm ac}=\beta\text{sin}(\omega_{p}t+\varphi_{p0})\) is a microwave pump. \(\beta\), \(\omega_{p}\), and \(\varphi_{p0}\) are the amplitude, frequency, and initial phase of the pump field, respectively.