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The quantum ground-state and excitation properties of two independent chains of pseudospins (two-level systems) interacting with the same bosonic field are theoretically investigated. Each chain is coupled to a different quadrature of the field, leading to two independent symmetry breakings for increasing values of the two collective spin-boson interaction constants ΩC and ΩI. A phase diagram is provided in the plane (ΩC,ΩI), with four phases that can be characterized by the complex bosonic coherence of the ground states and can be manipulated via geometric Berry effects. In particular, when ΩC and ΩI are both larger than two critical values, the fundamental subspace has a fourfold degeneracy. Finite-size properties are shown for both critical and ultrastrong values of ΩC and ΩI. Possible implementations in superconducting quantum circuits are detailed.
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