Simulation of Three-Level Dicke Quantum Phase Transitions Using Solid-State Spins Synergistically Coupled to Acoustics and Microwave

Abstract We believe that a novel theoretical scheme has been proposed to simulate the quantum phase transitions (QPTs) of the three-level Dicke model. In this designed hybrid system, an ensemble of solid-state spin can simultaneously couples to a superconducting microwave (MW) resonator and a surface-acoustic-wave (SAW) cavity, and then ensure the simulation of the Λ type Dicke model with the acoustic-MW dual-channel transitions. Using a generalized Holstein-Primakoff (HP) mapping in the thermodynamic limit, we derive an effective Hamiltonian that preserves a Z2 ×Z2 parity symmetry, and it can therefore supports a rich phase diagram potentially. We obtain analytic expressions for the critical couplings of the three phases, e.g., the normal, superradiant I and superradiant II three different phases. Among which, we can deterministically acquire the first-order (discontinuous) and second-order (continuous) quantum phase transitions with the relevant boundaries obviously. In addition, a rigorous diagonalization of the quadratic fluctuation Hamiltonian yields four excitation energies, whose behaviors can further clarify the order and continuity of each phase transition. This attempt open an interesting avenue for simulating the multilevel QPTs by jointly using the different coupling transducers, including such as the acoustics, mechanics, magnon and electromagnetics.