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Physica Scripta

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

Hao Jiang, Zeyun Shi, Bo Li, Long Wang, Xiu-Juan Dong, Ming Ma, Kai Chen, Cheng Chen, Chengrui Wu, Dong‐Yan Lü, Yuan Zhou

Published Sep 1, 2025• 2 citations
Publisher

Abstract

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 Z 2 × Z 2 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.

DOI
10.1088/1402-4896/ae01f8

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Results in Physics•Jun 18, 2021•2 citations
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 Z 2 × Z 2 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.
Abstract We theoretically investigate the quantum phase transitions of a hybrid quantum system consisting of an ensemble of nitrogen-vacancy (NV) centers coupled to two microwave cavity resonators. By leveraging the Zeeman effect with a static magnetic field, we achieve a tunable switch between different three-level Dicke model configurations. We map out the global ground-state phase diagram and identify three distinct operating regimes: (i) a V type regime characterized by the competition between two superradiant phases, (ii) an effective two-level regime emerging near the level-crossing point due to channel decoupling and (iii) a cascade Ξ type regime induced by level inversion. Particularly in the cascade regime, we analyze a mechanism of ‘sequential phase transition’, where the onset of the second superradiant phase is strictly contingent upon the condensation of the first channel. This mechanism leads to a nested structure in the phase diagram, which is fundamentally distinct from the phase competition observed in V type systems. Furthermore, by calculating the excitation spectra, we observe the emergence of soft modes at continuous critical points, establishing the second-order quantum nature of the relevant transitions. Our proposal provides a feasible solid-state platform for simulating multicritical phenomena and realizing controllable light–matter interactions.
We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
We consider an ensemble of three-level particles in Lambda configuration interacting with two bosonic modes. The Hamiltonian has the form of a generalized Dicke model. We show that in the thermodynamic limit this model supports a superradiant quantum phase transition. Remarkably, this can be both a first- and a second-order phase transition. A connection of the phase diagram to the symmetries of the Hamiltonian is also given. In addition, we show that this model can describe atoms interacting with an electromagnetic field in which the microscopic Hamiltonian includes a diamagnetic contribution. Even though the parameters of the atomic system respect the Thomas-Reiche-Kuhn sum rule, the system still shows a superradiant phase transition.
Abstract The Dicke model describes the cooperative interaction of an ensemble of two-level atoms with a single-mode photonic field and exhibits a quantum phase transition as a function of light–matter coupling strength. Extending this model by incorporating short-range atom–atom interactions makes the problem intractable but is expected to produce new physical phenomena and phases. Here, we simulate such an extended Dicke model using a crystal of ErFeO 3 , where the role of atoms (photons) is played by Er 3+ spins (Fe 3+ magnons). Through terahertz spectroscopy and magnetocaloric effect measurements as a function of temperature and magnetic field, we demonstrated the existence of a novel atomically ordered phase in addition to the superradiant and normal phases that are expected from the standard Dicke model. Further, we elucidated the nature of the phase boundaries in the temperature–magnetic-field phase diagram, identifying both first-order and second-order phase transitions. These results lay the foundation for studying multiatomic quantum optics models using well-characterized many-body solid-state systems.
We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio η of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer–Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete Z 2 symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1) symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu–Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.
In this article, we obtain the multistable macroscopic quantum states and quantum tricritical behavior in generalized Dicke model (DM) analytically by means of spin-coherent-state (SCS) variational method for a finite atom number N. The quantum phase transition (QPT) is characterized by the ground state from the normal phase (NP) to the superradiant phases (SP). The second-order QPT turns to the first-order one at the tricritical point of phase boundary. The tricritical behavior of QPT emerges when the coupling strength of the symmetry breaking mechanism increases to a certain extent, which leads to an additional dimension of phase diagram. Beyond the ground state the symmetry-breaking coupling can manipulate the pseudospin flip of higher-level macroscopic states. The critical property of QPT is also demonstrated from the average photon-number, the atomic population as well as Berry phase.