# Entangled Frequency-Tunable Microwave Photons in a Superconducting Circuit
## 1 Introduction  
A source emitting entangled microwave photons is an indispensable quantum resource for quantum information processing [1, 2]. It can be used in many quantum application technologies, for example, entangling distant superconducting qubits [3, 4] and/or other physical qubits [5], realizing quantum key distribution [6, 7, 8] and quantum teleportation [9, 10], acting as quantum repeaters and sensors [11], performing continuous-variable (CV) quantum computing [12], and accomplishing quantum illumination [13].  
In the optical domain, using a nonlinear optical crystal to generate entangled photons with tunable frequency has been sufficiently investigated [14, 15] through the spontaneous parametric down-conversion (SPDC) [16] in an optical parametric system. In the microwave domain, one promising physical candidate to construct such sources is superconducting devices [17, 18, 19, 20]. One of the previous proposals takes advantage of cooper pairs tunneling in a Josephson junction [2]. However, it is difficult to tune the frequencies of entangled two-mode photons freely. Other proposals use the nonlinearity of SQUID with suitable parametric excitation [21, 22, 23, 24, 25] or the dynamical Casimir effect [26, 27]. However, the entanglement property and frequency tunability are restricted to the nonlinearity of SQUID. A coplanar waveguide terminating with an asymmetric SQUID has also been used to construct a multi-mode superconducting crystal with its high-order nonlinearity, in which multi-mode SPDC entangled photons in the microwave domain are generated by suitably driving the SQUID [22, 23, 24]. However, the tunability of frequencies is limited by the Kerr-nonlinearity [28], and it is hard to excite the high-order modes with an external pump field [24]. Recently a Josephson ring modulator (JRM), a nonlinear device with four ports working as a three-wave mixer, has been applied to entangle two superconducting coplanar waveguide resonators (CPWRs) [29, 30, 31, 32, 33]. This kind of entanglement source uses the nonlinearity of JRM to emit entangled two-mode microwave signals under the proper externalparametric excitement. Its tunable spectrum is limited due to the harsh working conditions for nonlinearity. Almost all the superconducting entanglement sources mentioned above need the nonlinearity originally provided by the Josephson junction. This limits the tunability of frequency due to the unstablity of nonlinearity at different working points. Therefore, building an entanglement source to emit entangled microwave photons without using the nonlinearity of Josephson junctions will increase the tunable frequency range, which is promising and attractive in the quantum information process.  
In this paper, we propose a source of generating entangled frequency-tunable microwave photons with proper extra excitation to obtain a similar SPDC process in a superconducting quantum circuit, which is composed of a frequency-fixed resonator coupled to a frequency-tunable resonator. In other words, we construct a correlative microwave beam splitter that down-converts the coherent input microwave pump into two entangled microwave beams through the device. The photons are generated in pairs out of the system with a high level of entanglement characterized by the logarithmic negativity (\(\mathcal{N}\)) [34]. The frequencies of the two inseparable modes can be tuned in situ by adjusting the external flux bias applied.