# Entangled Frequency-Tunable Microwave Photons in a Superconducting Circuit
## 2 The Circuit Model
By moving the system into the rotating frame by defining the unitary operator as \(\hat{U}=\hat{U}_{1}\times\hat{U}_{2}\), in which  
\[\hat{U}_{1}=\exp[-i(\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b }\hat{b}^{\dagger}\hat{b})t],\] \[\hat{U}_{2}=\exp[ia_{p}\text{cos}(\omega_{p}t+\varphi_{p0})\hat{b }^{\dagger}\hat{b}], \tag{4}\]  
with a dimensionless parameter \(\alpha_{p}=\beta/\omega_{p}\), we can obtain the transformed Hamiltonian of the system  
\[\hat{H}_{\rm trans} =\hat{U}^{\dagger}\hat{H}\hat{U}-i\hat{U}^{\dagger}\frac{\partial \hat{U}}{\partial t}\] \[=\hat{U}_{2}^{\dagger}\hat{U}_{1}^{\dagger}[-g(\hat{a}-\hat{a}^{ \dagger})(\hat{b}-\hat{b}^{\dagger})]\hat{U}_{1}\hat{U}_{2}\] \[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]0 \[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]1 \[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]2  
where \(\Delta=\omega_{a}-\omega_{b}\), \(\Omega=\omega_{a}+\omega_{b}\). Then, the Jacobi-Anger identity with \(m\)-th order Bessel function of the first kind \(J_{m}(\alpha_{p})\) is used to expand the cosine oscillation of exponential term,  
\[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]3  
\[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]4 **(a)** Equivalent circuit of our proposed scheme to emit entangled frequency-tunable microwave photons. A \(\lambda/2\) frequency-fixed TLR (orange) coupled to a \(\lambda/2\) frequency-tunable TLR (green) via a coupling capacitor \(C_{S}\). The tunable resonator is inserted with a SQUID that is composed of two identical Josephson junctions. The SQUID is driven by an external direct current (dc) flux bias \(\Phi_{\rm dc}\) and a continuous microwave tone (ac) \(\Phi_{\rm ac}\). The output signals are processed by a detector. **(b)** (Upper panel) A proposed sketch of a setup for detecting the correlations of output photons. The second-order coherence function can be calculated by recording the intensities of modes such as \(\hat{D}_{a}(t)\), \(\hat{D}_{a}(t+\tau)\), \(\hat{D}_{b}(t)\), and \(\hat{D}_{b}(t+\tau)\). (Lower panel) A proposed schematic to observe the in-phase and quadrature voltages of output signals from the circuit. The emitted microwave signals are demodulated via a local oscillator (LO) and into the detector for calculating the quadrature components, which can be used to reconstruct the quantum properties of output signals using \(\hat{I}_{a}\), \(\hat{Q}_{a}\), \(\hat{I}_{b}\), and \(\hat{Q}_{b}\) for modes \(a\) and \(b\). **(c)** Model of generating the entangled microwave photons. The green and orange energy levels represent the harmonic behavior of modes \(a\) and \(b\). \(\kappa_{a}\) and \(\kappa_{b}\) are the rates of decay into the environment of the two modes \(a\) and \(b\), respectively. The curves with double arrows indicate the coupling between two oscillators. The red microwave signal denotes the extra pump field. The dark gray oval is the output envelope consisting of the entangled modes \(a\) and \(b\). Now, we truncate Equation (6) into the order \(-1\), \(0\), and \(1\) and set the initial phase \(\varphi_{p0}\) to zero for convenience. Then, Equation (5) is reduced to  
\[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]5 \[=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger} \hat{b}-g(\hat{a}-\hat{a}^{\dagger})(\hat{b}-\hat{b}^{\dagger})\]6  
The frequency of the pump field that we implement is a sum of resonant frequencies of the two modes, i.e., \(\omega_{p}=\omega_{a}+\omega_{b}\).