In the past decades, quantum entanglement^[1,2] as a peculiar nonclassical effect of physical system and an indispensable resource of quantum information has been deeply studied and great progress has been made both theoretically^[3] and experimentally,^[4] thus greatly promotes the development of quantum computing and quantum communication. The ultimate goal of quantum information is to achieve large scale quantum computing and quantum communication. In this sense, the challenges faced by the field of quantum information in the course of development mainly come from two aspects, one is the maintenance and storage for quantum entanglement, the other is the scalability of the physical system. It is widely convinced that dissipation induced by the environment is the main obstacle in preparing and storing entanglement, for the reason that dissipation may inevitably degrade the coherence of the quantum system. From this perspective it is a negative factor for quantum information processing (QIP) and should be avoided as much as possible. However, recent theories and experiments show an interesting fact that the dissipation can be used as a positive resource for quantum computation and entanglement generation,^[5–8] so that an alternative approach for engineering dissipation to generate entanglement has emerged. Many dissipation-assisted entanglement schemes have been proposed with cavity quantum electrodynamics (QED),^[9–24] atomic ensembles,^[25–27] ion traps,^[28–30] plasmonic systems,^[31–33] and optical lattices.^[34,35] The relevant experimental demonstrations are also have been achieved in atomic ensembles^[36] and ion traps.^[30] Different from the unitary evolution schemes, these schemes use dissipation as a positive resource in the preparation process, and have the advantages in robustness against decoherence and parameter fluctuations and dose not require a specifical initial state or precise evolution time.^[28]

On the other hand, circuit QED can produce an extremely strong interaction between the superconducting qubits and the microwave photons in a transmission line resonator, which is beneficial to quantum system to resist the environmental dissipation and parameter fluctuations.^[37] In view of its high integratability, high tunability, and addressability for individual resonators, the circuit QED has been widely regarded as the most promising architectures for QIP.^[38–41] It is essential for large scale quantum computation to engineer entanglement in coupled cavity networks, from this perspective, it would be of great importance to prepare entanglement among superconducting qubits in separated resonators. In this paper, based on the circuit QED architecture, we design an alternative scheme to prepare the entangled steady-state of two superconducting transmons separately interact with two transmission line resonators in a coupled circuit QED dimer.^[42,43] The resonant transmon-resonator interaction and the photons hopping between two resonators lead to a dressed state subspace of the qutrits with different excitation numbers of photon modes. By employing two specified microwave fields to resonantly drive the system to the well-defined state in dressed state subspace and resorting to the dissipations of photon leakage and qutrit decay, a steady entanglement between two transmons can be obtained. Numerical simulations show that the entangled steady-state can be achieved with high fidelity with current experimental parameters. The scheme is independent of the initial state and robust against parameter fluctuations.

The remainder of the paper is organized as follows. In the Section 2, we describe the interaction system and the dressed state subspace by calculation. In Section 3, the steady-state entanglement of two transmons is obtained by engineering the dissipative dynamics of the system with microwave pulses. In Section 4, the fidelity of the target steady-state and Clauser–Horne–Shimony–Holt (CHSH) correlation are simulated numerically with the current available experimental parameters, by which the feasibility of the present scheme is demonstrated. A conclusion appears in Section 5.

The system under consideration is depicted in Fig. 1. It consists of two transmission line microwave resonators coupled through hopping of microwave photons with a hopping rate J that is controllable and could be determined by a coupling capacitance of the resonators. Each resonator can be modeled as a single harmonic oscillator mode of frequency , and contains a single Λ-type superconducting qutrit which possesses two ground states and and one excited state with the corresponding energies , , and , respectively. The qutrit transition is resonantly coupled to the resonator with the coupling constant g, and the transitions and are driven by two microwave fields with Rabi frequencies and , respectively. Then the total Hamiltonian of the system can be written as

Fig. 1.

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Fig. 1. Experimental schematic illustration for dissipative preparation of entangled steady state between two transmission line microwave resonators coupled through hopping of microwave photons with a hopping rate J in circuit QED. κ, γ, and are single-photon decay rate, superconducting qutrit spontaneous emission rate, and superconducting qutrit dephasing rate. Each resonator contains a single Λ-type superconducting qutrit which possesses two ground states and and one excited state . The qutrit transition is resonantly coupled to the resonator with the coupling constant g, and the transitions and are driven by two microwave fields with Rabi frequencies and , respectively.

Notice that the excitation number operator of the total system is independent of Hamiltonian H_NM and H₂, so that the excitation number is conserved under the domination of these two Hamiltonians. Nevertheless, Hamiltonian H₁ would change the excitation number of the system since it does not commute with the excitation number operator. When the Rabi frequency of the first microwave pulse satisfies and the system is initially in a state with null excitation, the probability for the system to be excited to the subspace more than a single excitation can be neglected.

Here, we obtain the relevant energy eigenstates of Hamiltonian H_NM and use them as dressed states within the specific excitation number. The energy of level is taken to be 0 as the energy reference point. In Table 1 and Table 2, we show the eigenstates and the corresponding eigenvalues in zero and single excitation subspaces with the notation

Table 1.

Table 1.

Table 1. The eigenstates and eigenenergies of the Hamiltonian HNM within the null-excitation subspace. Here the notation represents that atom 1 (2) is in the state ( ) and there are photons in cavity 1 (2). . Eigenstate Eigenenergy 0

Table 1. The eigenstates and eigenenergies of the Hamiltonian HNM within the null-excitation subspace. Here the notation represents that atom 1 (2) is in the state ( ) and there are photons in cavity 1 (2). .

Table 2.

Table 2.

Table 2. The eigenstates and eigenenergies of the Hamiltonian HNM within the single-excitation subspace. . Eigenstate Eigenenergy , , ,

Table 2. The eigenstates and eigenenergies of the Hamiltonian HNM within the single-excitation subspace. .

The different energy gaps between these dressed states provide the possibility to drive the specific transitions of the system with extra microwave fields.

We let the second microwave pulses applied to two transmons to be opposite in phase, i.e., in the Hamiltonian H₂. In order to see clearly the roles of H₁ and H₂, we reexpress them in the dressed state subspace and move to the interaction picture with respect to the Hamiltonian H_NM. Then the Hamiltonian H₁ can be rewritten as

The dynamics of open dissipative system in Lindblad form is described by the master equation

The processes for producing and stabilizing the Bell state is shown in Fig. 2 which presents four preparation channels. By choosing different frequency of the first microwave field, the target state can be prepared in different ways. This is due to the fact that the microwave field with specified frequency can drive a certain resonant couplings between the states in zero subspace and one excitation subspace, but the transitions from to the states in the one excitation subspace are non-resonant. In Fig. 2(a), by choosing = , the state would couple resonantly to and while the other terms in Eq. (5) undergo non-resonant transitions with different detunings. Affected by the dissipative factors mentioned in Eq. (8), the and would be converted to and , which is a superposition of the states and . As has been shown that the state can be driven to ground state by microwave field , which would be redriven to the states and , while the state is far off-resonant with microwave field . This means that the population of will accumulate with time increasing. Similarly, this target state also can be achieved through the other three channels in Figs. 2(b)–2(d) by choosing = , , and , respectively.

Fig. 2.

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Fig. 2. Level configuration and transitions in the dressed state picture, the microwave field causes resonant transitions among states , , and in null excitation subspace by choosing the frequency suitably, while microwave field causes resonant transitions forming the following four preparation channels: (a) with , (b) with , (c) with , and (d) with , respectively, but the transitions from to the states in the one excitation subspace is non-resonant. The notations and in the figure are defined as = - and =- .

To demonstrate the feasibility of our Bell-state stabilization mechanism, we assess the performance of our scheme by numerically solving the Lindblad master equation. In Fig. 3, we plot the fidelity of the states in null excitation with optimized parameters corresponding to the four preparation channels in Fig. 2. The results show that the target state could be prepared for different choice of the with fidelity 91.28%, 97.09%, 93.15%, and 87.93%, respectively. In addition, in order to verify the quantum coherence degree of the target state, we also consider the Clauser–Horne–Shimony–Holt (CHSH) correlation S(t) which is defined as

Fig. 3.

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	Fig. 3. Populations of , , , corresponding to four channels in Fig. 2 with the initial state . The optimized parameters are chosen as (a) , , , , , and . (b) , , J = 1.1g, , , and . (c) , , J = 1.5g, κ = 0.008g, , , and . (d) , , , , , , and .

Fig. 4.

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	Fig. 4. The CHSH correlation of the target steady state as a function of the time. We choose the experimental parameters: , J = 1.1g, κ = 0.035g, γ = (1/3600)g, and .

Fig. 5.

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Fig. 5. (a) The fidelity of the target steady state as a function of the Rabi frequencies and with the initial state at time . (b) The fidelity of the target steady state versus the variance of the single-photon decay κ and qubit decay γ at time . (c) The fidelity of the target steady state versus photon hopping rate J and evolution time. The rest of the parameters are the same as those in Fig. 4.

Since charge noise may causes random fluctuations of the qutrit frequency in circuit QED, resulting in dephasing,^[45] different from the qutrit decay and photon decay of the resonator, qutrit dephasing is a detrimental effect for the current scheme. In Fig. 6, the influence of dephasing is considered and the fidelity of the target state is plotted when the system is in stabilization (for the evolution time ). The fidelity can be observed to monotonously increase with . The fidelity can be higher than 90% when the ratio appears about . This result occurs because qutrit dephasing results in a population transfer between the singlet states and , decreasing the fidelity of the target steady-state.

Fig. 6.

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	Fig. 6. The fidelity of the target steady-state, plotted versus different ratios . The rest of the parameters are the same as those in Fig. 4.

In conclusion, we have proposed a dissipative scheme for creating an entangled steady-state between two transmons separately interacts with two coupled transmission line resonators in circuit QED. The combination of coherent driving fields and dissipative processes would drive the system into a steady-state. By choosing different frequencies of the microwave field, the target steady-state can be generated with different preparation channels. The numerical simulations and analysis show that the present scheme is robust against the qutrit decay and the cavity decay due to they are used as the positive resources in the scheme. Since the transmon qubits have been demonstrated in experiments as well as its scalability in circuit QED, the current scheme can be used to physical realization of quantum computer and quantum communication.