Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Excited-state quantum phase transition in light-matter systems with discrete and continuous symmetries

Gaoke Hu, Wen‐Long You, Maoxin Liu, Hai‐Qing Lin

Published Sep 21, 2023• 5 citations
Publisher

Abstract

We systematically investigate the excited-state quantum phase transition (ESQPT) in the anisotropic quantum Rabi model, which interpolates between the quantum Rabi model with ${\mathbb{Z}}_{2}$ symmetry and the Jaynes-Cummings model with $\mathbb{U}(1)$ symmetry. We calculate the model energy spectra and density of states (DOS) with the cumulants in both analytical and numerical ways to describe the ESQPT by the singularities. In the Jaynes-Cummings limit, its continuous $\mathbb{U}(1)$ symmetry presents different nonanalytic behaviors from the quantum Rabi model with the discrete ${\mathbb{Z}}_{2}$ symmetry, and there exists a finite discontinuous jump at the critical energy. For the general anisotropy case with ${\mathbb{Z}}_{2}$ symmetry, there are two types of ESQPTs characterized by the finite discontinuous jump and the logarithmic divergence in the DOS, respectively. Different from the ground-state quantum phase transition, the ESQPT of the anisotropic quantum Rabi model strongly depends on the anisotropy which leads to a discontinuity even though the ${\mathbb{Z}}_{2}$ symmetry is still preserved.

DOI
10.1103/physreva.108.033710

Related papers

Excited-state quantum phase transition in the Rabi model

Ricardo Puebla, Myung-Joong Hwang, Martin Bodo Plenio

The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.

arXiv:1607.00283•Physical Review A•Aug 18, 2016•73 citations

Excited-state quantum phase transitions in Kerr nonlinear oscillators

Qing-Wei Wang, Sha Wu

Excited-state quantum phase transitions (ESQPTs) generalize the concept of quantum phase transitions to excited levels. Here we investigate how such ESQPTs can emerge in Kerr nonlinear oscillators, where a bosonic mode with a Kerr nonlinearity is subject to both coherent (one-photon) and parametric (two-photon) driving. We determine the structure and properties of the complete phase diagram by defining proper dimensionless parameters. Rich phases and phase transitions exist in the phase diagram, including phases with two or four critical energies above the ground-state level and first-order quantum phase transitions. We characterize the ESQPTs by singularities in the density of states, which show universal logarithmic divergence or discontinuity at the critical energies. Furthermore, we find that observables such as the photon number also exhibit singular behaviors at the critical energies and obey scaling laws. These results may extend our understanding of the driven Kerr oscillator and ESQPTs in finite-component systems.

Physical Review A•Dec 24, 2020•21 citations

From quantum Rabi model to Jaynes-Cummings model: symmetry-breaking quantum phase transitions, topological phase transitions and multicriticalities

Zu‐Jian Ying

We study the ground state (GS) and excitation gap of anisotropic quantum Rabi model (QRM) which connects the fundamental QRM and the Jaynes-Cummings model (JCM). While the GS has a second-order quantum phase transition (QPT) in the low frequency limit, turning on finite frequencies we shed a novel light on the phase diagram to illuminate a fine structure of first-order transition series. We find the QPT is accompanied with a hidden symmetry breaking, whereas the emerging series transitions are topological transitions without symmetry breaking. The topological structure of the wave function provides a novel universality classification in bridging the QRM and the JCM. We show that the conventionally established triple point is actually a quintuple or sextuple point and following the penta-/hexa-criticality emerge a series of tetra-criticalities.

arXiv:2010.15113•Oct 28, 2020•0 citations

Universal Scaling and Critical Exponents of the Anisotropic Quantum Rabi Model

Maoxin Liu, Stefano Chesi, Zu‐Jian Ying +3 more

We investigate the quantum phase transition of the anisotropic quantum Rabi model, in which the rotating and counterrotating terms are allowed to have different coupling strengths. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches, we extract the phase diagram, scaling functions, and critical exponents, which determine the universality class at finite anisotropy (identical to the isotropic limit). We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are extended to the few-body quantum phase transitions with N>1 spins, where we expose the same effective parameters, scaling properties, and phase diagram. Thus, a stronger form of universality is established, valid from N=1 up to the thermodynamic limit.

arXiv:1702.06641•Physical Review Letters•Nov 28, 2017•166 citations

Some remarks on ‘superradiant’ phase transitions in light-matter systems

Jonas Larson, E. K. Twyeffort Irish

Abstract In this paper we analyze properties of the phase transition that appears in a set of quantum optical models; Dicke, Tavis–Cummings, quantum Rabi, and finally the Jaynes–Cummings model. As the light-matter coupling is increased into the deep strong coupling regime, the ground state turns from vacuum to become a superradiant state characterized by both atomic and photonic excitations. It is pointed out that all four transitions are of the mean-field type, that quantum fluctuations are negligible, and hence these fluctuations cannot be responsible for the corresponding vacuum instability. In this respect, these are not quantum phase transitions. In the case of the Tavis–Cummings and Jaynes–Cummings models, the continuous symmetry of these models implies that quantum fluctuations are not only negligible, but strictly zero. However, all models possess a non-analyticity in the ground state in agreement with a continuous quantum phase transition. As such, it is a matter of taste whether the transitions should be termed quantum or not. In addition, we also consider the modifications of the transitions when photon losses are present. For the Dicke and Rabi models these non-equilibrium steady states remain critical, while the criticality for the open Tavis–Cummings and Jaynes–Cummings models is completely lost, i.e. in realistic settings one cannot expect a true critical behaviour for the two last models.

arXiv:1612.00336•Journal of Physics A Mathematical and Theoretical•Mar 10, 2017•116 citations

Dissecting the superradiant phase transition in the anisotropic Rabi model: Pattern competition and cavity-QED simulation

Yuan Qiu, Ke-Xiong Yan, Jun-Hao Lin +3 more

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.

quant-pharXiv:2511.03207•Nov 6, 2025•0 citations
We systematically investigate the excited-state quantum phase transition (ESQPT) in the anisotropic quantum Rabi model, which interpolates between the quantum Rabi model with Z2{\mathbb{Z}}_{2}Z2​ symmetry and the Jaynes-Cummings model with U(1)\mathbb{U}(1)U(1) symmetry. We calculate the model energy spectra and density of states (DOS) with the cumulants in both analytical and numerical ways to describe the ESQPT by the singularities. In the Jaynes-Cummings limit, its continuous U(1)\mathbb{U}(1)U(1) symmetry presents different nonanalytic behaviors from the quantum Rabi model with the discrete Z2{\mathbb{Z}}_{2}Z2​ symmetry, and there exists a finite discontinuous jump at the critical energy. For the general anisotropy case with Z2{\mathbb{Z}}_{2}Z2​ symmetry, there are two types of ESQPTs characterized by the finite discontinuous jump and the logarithmic divergence in the DOS, respectively. Different from the ground-state quantum phase transition, the ESQPT of the anisotropic quantum Rabi model strongly depends on the anisotropy which leads to a discontinuity even though the Z2{\mathbb{Z}}_{2}Z2​ symmetry is still preserved.
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.
Excited-state quantum phase transitions (ESQPTs) generalize the concept of quantum phase transitions to excited levels. Here we investigate how such ESQPTs can emerge in Kerr nonlinear oscillators, where a bosonic mode with a Kerr nonlinearity is subject to both coherent (one-photon) and parametric (two-photon) driving. We determine the structure and properties of the complete phase diagram by defining proper dimensionless parameters. Rich phases and phase transitions exist in the phase diagram, including phases with two or four critical energies above the ground-state level and first-order quantum phase transitions. We characterize the ESQPTs by singularities in the density of states, which show universal logarithmic divergence or discontinuity at the critical energies. Furthermore, we find that observables such as the photon number also exhibit singular behaviors at the critical energies and obey scaling laws. These results may extend our understanding of the driven Kerr oscillator and ESQPTs in finite-component systems.
We study the ground state (GS) and excitation gap of anisotropic quantum Rabi model (QRM) which connects the fundamental QRM and the Jaynes-Cummings model (JCM). While the GS has a second-order quantum phase transition (QPT) in the low frequency limit, turning on finite frequencies we shed a novel light on the phase diagram to illuminate a fine structure of first-order transition series. We find the QPT is accompanied with a hidden symmetry breaking, whereas the emerging series transitions are topological transitions without symmetry breaking. The topological structure of the wave function provides a novel universality classification in bridging the QRM and the JCM. We show that the conventionally established triple point is actually a quintuple or sextuple point and following the penta-/hexa-criticality emerge a series of tetra-criticalities.
We investigate the quantum phase transition of the anisotropic quantum Rabi model, in which the rotating and counterrotating terms are allowed to have different coupling strengths. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches, we extract the phase diagram, scaling functions, and critical exponents, which determine the universality class at finite anisotropy (identical to the isotropic limit). We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are extended to the few-body quantum phase transitions with N>1 spins, where we expose the same effective parameters, scaling properties, and phase diagram. Thus, a stronger form of universality is established, valid from N=1 up to the thermodynamic limit.
Abstract In this paper we analyze properties of the phase transition that appears in a set of quantum optical models; Dicke, Tavis–Cummings, quantum Rabi, and finally the Jaynes–Cummings model. As the light-matter coupling is increased into the deep strong coupling regime, the ground state turns from vacuum to become a superradiant state characterized by both atomic and photonic excitations. It is pointed out that all four transitions are of the mean-field type, that quantum fluctuations are negligible, and hence these fluctuations cannot be responsible for the corresponding vacuum instability. In this respect, these are not quantum phase transitions. In the case of the Tavis–Cummings and Jaynes–Cummings models, the continuous symmetry of these models implies that quantum fluctuations are not only negligible, but strictly zero. However, all models possess a non-analyticity in the ground state in agreement with a continuous quantum phase transition. As such, it is a matter of taste whether the transitions should be termed quantum or not. In addition, we also consider the modifications of the transitions when photon losses are present. For the Dicke and Rabi models these non-equilibrium steady states remain critical, while the criticality for the open Tavis–Cummings and Jaynes–Cummings models is completely lost, i.e. in realistic settings one cannot expect a true critical behaviour for the two last 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.