Simulation of three-level Dicke quantum phase transitions using solid-state spins synergistically coupled to acoustics and microwave

Abstract A theoretical scheme is proposed to simulate the quantum phase transitions (QPTs) of the three-level Dicke model. In this designed hybrid system, an ensemble of solid-state spins can simultaneously couple 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> parity symmetry and therefore potentially supports a rich phase diagram. We obtain analytic expressions for the critical couplings of the three phases: the normal, superradiant I and superradiant II phases. We obtain analytic expressions for the critical couplings of three phases: the normal, superradiant I, and superradiant II phases. Among these, we can deterministically identify both first-order (discontinuous) and second-order (continuous) quantum phase transitions with distinct boundary characteristics. Additionally, rigorous diagonalization of the quadratic fluctuation Hamiltonian yields four excitation energies, whose behaviors further clarify the order and continuity of each phase transition. This attempt opens an interesting avenue for simulating multilevel QPTs by jointly using different coupling transducers, including acoustics, mechanics, magnons, and electromagnetics.