Abstract We propose a method to generate a new type of quantum entangled states, i.e., “double” NOONstates (|NN00⟩ + |00NN⟩)=√2, with a setup of two superconducting flux qutrits and five circuitcavities. This scheme operates essentially by employing a two-photon process, i.e., two photonsare simulaneously and separatedly emitten into two cavities when each coupler qutrit is initiallyin a higher-energy excited state. As a consequence, the “double” NOON-state creation needs onlyN+2 steps (including two basic steps for initial preparation of a two-qutrit Bell state).PACS numbers: 03.67.Lx, 42.50.DvI. INTRODUCTIONQuantum entanglement plays an important role in quantum communication, quantummeasurement, and quantum computing. 48089

Recently, the NOON states, i.e., the entangledstates (|N0⟩ + |0N⟩)/√2, have attracted considerable attention because of their significantapplications in quantum optical lithography [1], quantum metrology [2], precision measure-ment of transmons [3], and quantum information processing [4]. For the generation of theNOON states with photons in two cavities or resonators, there have been some proposalspresented for the past few years. Ref. [5] employs a circuit of two superconducting resonatorsplus a tunable qubit and is based on alternative resonant interactions of the coupler qubitwith two cavity modes and a classical pulse. Ref. [6] employs a relatively complicated cir-cuit of three superconducting resonators and two qutrits and is based on qutrit-cavity/pulseresonant interactions and requires N + 1 operational steps only. The scheme presented inRef. [6] was later implemented in experiment for creating the NOON states with N ≤ 3 [7].Ref. [8] employs a setup of one superconducting transmon qutrit and two resonators andis based on qutrit-cavity/pulse resonant interaction. As discussed in [8], the NOON statepreparation is faster when compared with the previous proposal [5] and employs a simplersetup than [2,6], but requires 2N steps of operation. Recently, an improved scheme hasbeen proposed [9], which also employs a simple setup of one four-level superconducting fluxdevice and two resonators, and when compared with [8] the number of operational stepsis reduced by half (i.e., only requiring N + 1 operational steps). These previous works areimportant and useful because they provide different ways for generating the NOON statesof photons in two cavities or resonators.Different from the previous works, we here focus on how to generate the NOON states withphotons in four cavities or resonators. In the following, we consider a setup of three qutritsand five cavities (Fig.1). By applying the idea of tranferring the Bell state of the couplerqutrits to the cavities [6,7], we present an efficient scheme for generating the entangledNOON states (|NN00⟩+|00NN⟩)/√2 (hereafter, called “double” NOON states). As shownbelow, the major advantage of this proposal is that only N +2 operational steps are neededto generate the “double” NOON states, and thus the operation is significantly simplified.As shown below, our scheme operates essentially by employing a two-photon process,i.e., two photons are simulaneously and separatedly emitten into two left (right) cavitiesFIG. 1: (Color online) (a) Setup for two superconducting flux qutrits and five cavities. A commoncavity (i.e., the central one) is used to initially prepare the two qutrits in a Bell state, while the otherfour cavities are utilized to create the double NOON state. Here, each cavity is a one-dimensionaltransmission line resonator. (b) Illustration of each qutrit dispersively interacting with its twocavities. Here, g1 is the coupling strength between cavity 1 (3) with the |g⟩ ↔ |e⟩ while g2 is thecoupling strength between cavity 2 (4) with the |e⟩ ↔ |f⟩ transition.when the left (right) coupler qutrit connecting to the two left (right) cavities is initially inthe excited state |f⟩ . Thus, we stress that this work is quite different from the previousworks [6,7], because the physical mechanism used for transferring the Bell state of the twoqutrits onto the four cavities in our work is different from that in [6,7] where a single-photonprocess was employed. We believe that this work is of interest because our work is thefirst to demonstrate that the NOON states with photons in more than two cavities can begenerated in cavity/circuit QED with only N + 2 operational steps.This paper is arranged as follows. In Sec. II, we derive an effective Hamiltonian fora subsystem composed of two cavities and a coupler qutrit, and then discuss the timeevolution of the states of the subsystem under the effective Hamiltonian. In Sec. III, wediscuss preparation of the Bell state for the two coupler qutrits. In Sec. IV, we introducethe generation of the “double” NOON states with two superconducting flux qutrits and fiveresonators, which requires N +2 steps of operation only In Sec. V, we give a brief discussionof the possible experimental implementation of this proposal. Generating double NOON states of photons in circuit QED:http://www.751com.cn/yingyu/lunwen_50429.html