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A concluding summary ispresented in Sec. VI. II. EFFECTIVE HAMILTONIAN AND TIME EVOLUTIONConsider a setup consisting of two superconducting flux qutrits and five cavities [Fig.1(a)]. The left-side qutrit is called qutrit L while the right-side qutrit is called qutrit R.The two qutrits are coupled to a common cavity and each qutrit is coupled to two cavities[Fig. 1(a)]. The three levels of each qutrit are denoted as |g⟩ , |e⟩ , and |f⟩ . In the following,we will present an effective Hamiltonian for a subsystem composed of the left two cavitiesand the left qutrit and then give a discussion on the time evolution of the states of thesubsystem under the effective Hamiltonina. It is noted that the Hamiltonian and the statetime evolution obtained below also apply to the subsystem composed of the right two cavitiesand the right qutrit.Suppose that cavity 1 is coupled to the |g⟩ ↔ |e⟩ transition while cavity 2 is coupled tothe |e⟩ ↔ |f⟩ transition of the coupler qutrit L. In the interaction picture, the Hamiltoniandescribing the interaction of the coupler qutrit L with the two cavities 1 and 2 is given byHI(a1, a2) = ~g1(eiδta1σ+eg,L + h.c.) + ~g2(e iδta2σ+fe,L + h.c.), (1)where δ = ωeg,L −ωc1 = ωc2 −ωfe,L < 0, σ+eg,L = |e⟩L⟨g|, σ+fe,L = |f⟩L⟨e|, ωeg is the |g⟩ ↔ |e⟩transition frequency, ωfe is the |e⟩ ↔ |f⟩ transition frequency, ωc1 (ωc2) is the frequencyof cavity 1 (2), g1 (g2) is the coupling strength between cavity 1 (2) and the |g⟩ ↔ |e⟩(|e⟩ ↔ |f⟩) transition, a1 (a2) is the photon annihilation operator of cavity 1 (2).According to the derivation in Ref. [10], we find that under the large-detuning condition|δ| >> g , g , the Hamiltonian (1) becomes the following effective Hamiltonian