Energy levels and transitions of the system for Δ ϕ = ϕ₂ ∓ ϕ₁ = 2nπ. Atom–cavity system can either be excited via driving cavity [red, see panel (a)] or driving atoms [blue, see panel (b)]. The transition strength between states is described by the thick colored lines (red and blue). The symbols ω_c, η, Δ, and g correspond to cavity frequency, driving strength, detuning, and coupling strength, respectively. The mean photon number 〈a†a〉 for driving the cavity (red η_c/κ = 1.0) or driving the atoms (blue, η_a/κ = 1.0) is shown in panel (c). The normalized parameters, γ/κ = 1.0 and g₁/κ = g₂/κ = g/κ = 15 are taken in panel (c).