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arXiv:2212.05186 · Chinese Physics Letters

Dissecting Superradiant Phase Transition in the Quantum Rabi Model

Yun-Tong Yang, Hong‐Gang Luo

Published Nov 19, 2024• 4 citations
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Abstract

Abstract Phase transitions are both thermodynamically and quantum-mechanically ubiquitous in nature and laboratories, and their understanding remains one of the most active issues in modern physics and related disciplines. The Landau theory provides a general framework to describe phenomenologically phase transitions by introducing order parameters and associated symmetry breaking. This theory is also taken as a starting point to explore critical phenomena in connection with phase transitions in the renormalization group, which provides a complete theoretical description of behaviors close to critical points. In this context, the microscopic mechanism of phase transitions remains unclear. In this study, we explore the microscopic mechanism of the superradiant phase transition in the quantum Rabi model (QRM). First, we perform a diagonalization operation in an operator space to obtain three fundamental patterns, denoted as λ 1 , λ 2 , and λ 3 , involved in the QRM. Then, we explicitly analyze the energy evolutions of these patterns with increasing coupling strength. The observed characteristic behaviors reveal the microscopic mechanism of the superradiant phase transition as a consequence of competition between patterns due to different phase relations. In other words, with increasing coupling strength, the pattern λ 1 drives the phase transition, the pattern λ 2 exhibits a similar response speed but less energy compensation than the pattern λ 1 , and the pattern λ 3 exhibits a slow response speed but plays a key role in the balance between it and the pattern λ 1 , which stabilizes the new phase. This type of dissecting mechanism explains why and how the superradiant phase transition occurs in the QRM and paves the way for exploring the microscopic mechanism of phase transitions that occur frequently in nature.

DOI
10.1088/0256-307x/41/12/120501

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Abstract Phase transitions are both thermodynamically and quantum-mechanically ubiquitous in nature and laboratories, and their understanding remains one of the most active issues in modern physics and related disciplines. The Landau theory provides a general framework to describe phenomenologically phase transitions by introducing order parameters and associated symmetry breaking. This theory is also taken as a starting point to explore critical phenomena in connection with phase transitions in the renormalization group, which provides a complete theoretical description of behaviors close to critical points. In this context, the microscopic mechanism of phase transitions remains unclear. In this study, we explore the microscopic mechanism of the superradiant phase transition in the quantum Rabi model (QRM). First, we perform a diagonalization operation in an operator space to obtain three fundamental patterns, denoted as λ 1 , λ 2 , and λ 3 , involved in the QRM. Then, we explicitly analyze the energy evolutions of these patterns with increasing coupling strength. The observed characteristic behaviors reveal the microscopic mechanism of the superradiant phase transition as a consequence of competition between patterns due to different phase relations. In other words, with increasing coupling strength, the pattern λ 1 drives the phase transition, the pattern λ 2 exhibits a similar response speed but less energy compensation than the pattern λ 1 , and the pattern λ 3 exhibits a slow response speed but plays a key role in the balance between it and the pattern λ 1 , which stabilizes the new phase. This type of dissecting mechanism explains why and how the superradiant phase transition occurs in the QRM and paves the way for exploring the microscopic mechanism of phase transitions that occur frequently in nature.
Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalizations to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus, one is able to see how the phase transition occurs and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition takes place and further exhibits the statistics of Gaussian unitary ensemble with increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.
In this manuscript, we analyze the mechanism of the superradiant phase transition in the anisotropic Rabi model under the classical oscillator limit using the pattern picture. By expanding the anisotropic Rabi model Hamiltonian in operator space, we obtained three patterns, and we find that the phase transition arises from the competition between patterns. The difficulty in achieving the classical oscillator limit motivates our investigation into the quantum phase transition within a parametrically-driven Jaynes-Cummings model. This parametrically-driven Jaynes-Cummings model can reproduce the dynamics of an ultrastrong-coupling anisotropic Rabi model in a squeezed-light frame. According to the eigenenergies and eigenstates of the normal and superradiant phases of this equivalent anisotropic Rabi model, we find that the excitation energy of the normal phase and the superradiant phase vanishes at the critical point. The photon number becomes infinite beyond the critical point. These results indicate that the system undergoes a superradiant phase transition at the critical point.
Although the superradiant phase transition (SRPT) is prohibited in the paradigmatic quantum Rabi model due to the no-go theorem caused by the A2{\mathbf{A}}^{2}A2 term, we demonstrate two distinct types of SRPTs emerging from the normal phase in the anisotropic quantum Rabi model. A discontinuous phase transition between the two types of superradiant phases also emerges in the presence of a strong A2{\mathbf{A}}^{2}A2 term. Additionally, a rich phase diagram featuring a triple point, which connects first- and second-order phase transitions, is derived analytically and confirmed through numerical diagonalization at large effective system sizes. Finally, distinct critical behavior at the triple point is revealed and contrasted with that of a single continuous SRPT. This work may open a new avenue for observing SRPTs in their intrinsic form without altering the A2{\mathbf{A}}^{2}A2 term, while also offering a practical platform for exploring rich quantum phenomena.
Superradiant phase transitions play a fundamental role in understanding the mechanism of collective light–matter interaction at the quantum level. Here we investigate multiple superradiant phases and phase transitions with different symmetry-breaking patterns in a two-mode V-type Dicke model. Interestingly, we show that there exists a quadruple point where one normal phase, one global symmetry-breaking superradiant phase, and two local symmetry-breaking superradiant phases meet. Such a global phase results from the phase competition between two local superradiant phases and cannot occur in the standard Λ- and Ξ-type three-level configurations in quantum optics. Moreover, we exhibit a sequential first-order quantum phase transition from one local to the global again to the other local superradiant phase. Our study opens up a perspective of exploring multilevel quantum critical phenomena with global symmetry breaking.
Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well established through the renormalization-group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between particle/spins drive the system across distinct phases as parameters vary, and how do these particle/spins respond to parameter variations during this transition process? In this work, we investigate these questions through the paradigmatic example of quantum phase transitions (QPT) in the one-dimensional transverse Ising model (TIM). We first introduce 2L collective structures, referred to as patterns, for the TIM with L ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analog of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified at the interaction strength J_{c}=1. We systematically explore this process for small lattice sizes of L=6,8,10,12, whose ground-state energies are identical to those obtained by directly numerical exact diagonalization. Increasing the system size up to L=128, the actual QPT point located at J_{c}=1 in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.
Quantum Rabi model (QRM) is fascinating not only because of its broad relevance but also due to its few-body quantum phase transition. In practice, both the bias and the nonlinear coupling in QRM are important controlling parameters in experimental setups. We study the interplay of the bias and the nonlinear interaction with the linear coupling in the ground state which exhibits various patterns of symmetry breaking and different orders of transitions. Several situations of tricriticalities are unveiled in the low-frequency limit and at finite frequencies. We find that the full quantum-mechanical effect leads to transitions, tricriticalities, quadruple points, and a fine structure of spontaneous symmetry breaking, which are much beyond the semiclassical picture. We clarify the underlying mechanisms by analyzing the energy competitions and the essential changeovers of the quantum states, which enables us to extract most analytic phase boundaries.