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

arXiv:1512.08581 · Physical Review A

Parity oscillations and photon correlation functions in the Z2−U(1) Dicke model at a finite number of atoms or qubits

Yu Yi-Xiang, Jinwu Ye, Cunlin Zhang

Published Aug 17, 2016• 6 citations
arXivHTMLPDF

Abstract

Four standard quantum optics models, that is, the Rabi, Dicke, Jaynes-Cummings, and Tavis-Cummings models, were proposed by physicists many decades ago. Despite their relative simple forms and many previous theoretical works, their physics at a finite $N$, especially inside the superradiant regime, remain unknown. In this work, by using the strong-coupling expansion and exact diagonalization (ED), we study the ${Z}_{2}\text{-U}(1)$ Dicke model with independent rotating-wave coupling $g$ and counterrotating-wave coupling ${g}^{\ensuremath{'}}$ at a finite $N$. This model includes the four standard quantum optics models as its various special limits. We show that in the superradiant phase, the system's energy levels are grouped into doublets with even and odd parity. Any anisotropy $\ensuremath{\beta}={g}^{\ensuremath{'}}/g\ensuremath{\ne}1$ leads to the oscillation of parities in both the ground and excited doublets as the atom-photon coupling strength increases. The oscillations will be pushed to the infinite coupling strength in the isotropic ${Z}_{2}$ limit $\ensuremath{\beta}=1$. We find nearly perfect agreement between the strong-coupling expansion and the ED in the superradiant regime when $\ensuremath{\beta}$ is not too small. We also compute the photon correlation functions, squeezing spectrum, and number correlation functions that can be measured by various standard optical techniques.

DOI
10.1103/physreva.94.023830

Related papers

Photon Berry phases, Instantons and Schrodinger Cats with oscillating parities in cavity QED

Yu Yi-Xiang, Jinwu Ye, W.-M. Liu +1 more

The four standard quantum optics model in cavity QED such as Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model were proposed many decades ago. Despite their relative simple forms and many previous theoretical works, their solutions at a finite $ N $ inside the superradiant regime remain unknown. In view of recent remarkable experimental advances in several experimental systems such as cold atoms inside a cavity or superconducting qubits inside a microcavity to realize these models, it becomes topical and important to understand what would be the new phenomena in these models at a finite $ N $, especially inside the superradaint regime. In this work, we resolved this outstanding problem. We use three independent methods such as $ 1/J $ expansion, the strong coupling expansion and exact diagonization (ED) to study $ U(1)/Z_2 $ Dicke model at a finite $ N $ and different anisotropy parameters $ 0 \leq \beta \leq 1 $ as the atom-photon interaction strength increases. This model include the four standard quantum optics model as its special limit. We identify 4 different regimes: the normal, $ U(1) $, strong quantum tunneling (QT) and weak QT regime. We map out the evolution of energy level structures, compute all kinds of photon correlation functions and find nearly perfect agreements between the strong coupling expansion and the ED not only in the weak and strong QT regimes, but also in the $ U(1) $ regime not too close to the critical point at $ N=\infty $.

•Jun 23, 2015•0 citations

Photon Berry phases, Instantons, Schrodinger Cats with oscillating parities and crossover from $ U(1) $ to $ Z_2 $ limit in cavity QED systems

Yu Yi-Xiang, Jinwu Ye, W.-M. Liu +1 more

In this work, we study the $ U(1)/Z_2 $ Dicke model at a finite $ N $ by using the $ 1/J $ expansion and exact diagonization. This model includes the four standard quantum optics model as its various special limits. The $ 1/J $ expansions is complementary to the strong coupling expansion used by the authors in arXiv:1512.08581 to study the same model in its dual $ Z_2/U(1) $ representation. We identify 3 regimes of the system's energy levels: the normal, $ U(1) $ and quantum tunneling (QT) regime. The system's energy levels are grouped into doublets which consist of scattering states and Schrodinger Cats with even ( e ) and odd ( o ) parities in the $ U(1) $ and quantum tunneling (QT) regime respectively. In the QT regime, by the WKB method, we find the emergencies of bound states one by one as the interaction strength increases, then investigate a new class of quantum tunneling processes through the instantons between the two bound states in the compact photon phase. It is the Berry phase interference effects in the instanton tunneling event which leads to Schrodinger Cats oscillating with even and odd parities in both ground and higher energy bound states. We map out the energy level evolution from the $ U(1) $ to the QT regime and also discuss some duality relations between the energy levels in the two regimes. We also compute the photon correlation functions, squeezing spectrum, number correlation functions in both regimes which can be measured by various experimental techniques. The combinations of the results achieved here by $ 1/J $ expansion and those in arXiv:1512.08581 by strong coupling method lead to rather complete understandings of the $ U(1)/Z_2 $ Dicke model at a finite $ N $ and any anisotropy parameter $ β$.

arXiv:1506.07030•Jun 23, 2015•0 citations

Universal fluctuations and squeezing in a generalized Dicke model near the superradiant phase transition

D. S. Shapiro, W. V. Pogosov, Yu. E. Lozovik

In view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) nonstationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields, and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of a generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete ${\mathbb{Z}}_{2}$ symmetry), and that of anti-Tavis-Cummings and Tavis-Cummings $U(1)$ models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of a photon number take universal values. We also find a temperature scale below which one approaches a zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss the Bethe ansatz solution for integrable $U(1)$ limits.

arXiv:1909.10252•Physical Review A•Aug 7, 2020•29 citations

Superradiance, Berry phase, photon phase diffusion, and number squeezed state in the U(1) Dicke (Tavis-Cummings) model

Jinwu Ye, Cunlin Zhang

Recently, strong-coupling regimes of superconducting qubits or quantum dots inside a microwave circuit cavity and BEC atoms inside an optical cavity were achieved experimentally. The strong-coupling regimes in these systems were described by the Dicke model. Here, we solve the Dicke model by a $1/N$ expansion. In the normal state, we find a $\sqrt{N}$ behavior of the collective Rabi splitting. In the superradiant phase, we identify an important Berry phase term that has dramatic effects on both the ground state and the excitation spectra of the strongly interacting system. The single photon excitation spectrum has a low-energy quantum phase diffusion mode in imaginary time with a large spectral weight and also a high-energy optical mode with a low spectral weight. The photons are in a number squeezed state that may have wide applications in high sensitive measurements and quantum-information processing. Comparisons with exact diagonalization studies are made. Possible experimental schemes to realize the superradiant phase are briefly discussed.

arXiv:0912.1982•Physical Review A•Aug 25, 2011•25 citations

Modulation-based superradiant phase transition in the strong-coupling regime

Jin‐Feng Huang, Lin Tian

The Dicke model can exhibit quantum phase transition between the normal and the superradiant phases when the strength of the light-matter coupling exceeds the ultrastrong-coupling regime. However, it is challenging to observe this phase transition in practical systems due to limited coupling strength or finite two-photon ${A}^{2}$ terms. Here we show that by applying a periodic modulation to the frequency of the two-level systems in a standard Dicke model in the strong-coupling regime, an anisotropic Dicke model with tunable rotating and counter-rotating terms in the ultrastrong-coupling regime can be achieved. We calculate the ground state and the excitation spectrum of this model in terms of the modulation parameters. Our result shows that the superradiant phases can be observed in cavity- or circuit-quantum electrodynamics systems with strong coupling.

arXiv:2208.12524•Physical Review A•Jun 28, 2023•14 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
Four standard quantum optics models, that is, the Rabi, Dicke, Jaynes-Cummings, and Tavis-Cummings models, were proposed by physicists many decades ago. Despite their relative simple forms and many previous theoretical works, their physics at a finite NNN, especially inside the superradiant regime, remain unknown. In this work, by using the strong-coupling expansion and exact diagonalization (ED), we study the Z2-U(1){Z}_{2}\text{-U}(1)Z2​-U(1) Dicke model with independent rotating-wave coupling ggg and counterrotating-wave coupling g\ensuremath′{g}^{\ensuremath{'}}g\ensuremath′ at a finite NNN. This model includes the four standard quantum optics models as its various special limits. We show that in the superradiant phase, the system's energy levels are grouped into doublets with even and odd parity. Any anisotropy \ensuremathβ=g\ensuremath′/g\ensuremath≠1\ensuremath{\beta}={g}^{\ensuremath{'}}/g\ensuremath{\ne}1\ensuremathβ=g\ensuremath′/g\ensuremath=1 leads to the oscillation of parities in both the ground and excited doublets as the atom-photon coupling strength increases. The oscillations will be pushed to the infinite coupling strength in the isotropic Z2{Z}_{2}Z2​ limit \ensuremathβ=1\ensuremath{\beta}=1\ensuremathβ=1. We find nearly perfect agreement between the strong-coupling expansion and the ED in the superradiant regime when \ensuremathβ\ensuremath{\beta}\ensuremathβ is not too small. We also compute the photon correlation functions, squeezing spectrum, and number correlation functions that can be measured by various standard optical techniques.
The four standard quantum optics model in cavity QED such as Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model were proposed many decades ago. Despite their relative simple forms and many previous theoretical works, their solutions at a finite $ N $ inside the superradiant regime remain unknown. In view of recent remarkable experimental advances in several experimental systems such as cold atoms inside a cavity or superconducting qubits inside a microcavity to realize these models, it becomes topical and important to understand what would be the new phenomena in these models at a finite $ N $, especially inside the superradaint regime. In this work, we resolved this outstanding problem. We use three independent methods such as $ 1/J $ expansion, the strong coupling expansion and exact diagonization (ED) to study $ U(1)/Z_2 $ Dicke model at a finite $ N $ and different anisotropy parameters $ 0 \leq \beta \leq 1 $ as the atom-photon interaction strength increases. This model include the four standard quantum optics model as its special limit. We identify 4 different regimes: the normal, $ U(1) $, strong quantum tunneling (QT) and weak QT regime. We map out the evolution of energy level structures, compute all kinds of photon correlation functions and find nearly perfect agreements between the strong coupling expansion and the ED not only in the weak and strong QT regimes, but also in the $ U(1) $ regime not too close to the critical point at $ N=\infty $.
In this work, we study the $ U(1)/Z_2 $ Dicke model at a finite $ N $ by using the $ 1/J $ expansion and exact diagonization. This model includes the four standard quantum optics model as its various special limits. The $ 1/J $ expansions is complementary to the strong coupling expansion used by the authors in arXiv:1512.08581 to study the same model in its dual $ Z_2/U(1) $ representation. We identify 3 regimes of the system's energy levels: the normal, $ U(1) $ and quantum tunneling (QT) regime. The system's energy levels are grouped into doublets which consist of scattering states and Schrodinger Cats with even ( e ) and odd ( o ) parities in the $ U(1) $ and quantum tunneling (QT) regime respectively. In the QT regime, by the WKB method, we find the emergencies of bound states one by one as the interaction strength increases, then investigate a new class of quantum tunneling processes through the instantons between the two bound states in the compact photon phase. It is the Berry phase interference effects in the instanton tunneling event which leads to Schrodinger Cats oscillating with even and odd parities in both ground and higher energy bound states. We map out the energy level evolution from the $ U(1) $ to the QT regime and also discuss some duality relations between the energy levels in the two regimes. We also compute the photon correlation functions, squeezing spectrum, number correlation functions in both regimes which can be measured by various experimental techniques. The combinations of the results achieved here by $ 1/J $ expansion and those in arXiv:1512.08581 by strong coupling method lead to rather complete understandings of the $ U(1)/Z_2 $ Dicke model at a finite $ N $ and any anisotropy parameter $ β$.
In view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) nonstationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields, and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of a generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete Z2{\mathbb{Z}}_{2}Z2​ symmetry), and that of anti-Tavis-Cummings and Tavis-Cummings U(1)U(1)U(1) models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of a photon number take universal values. We also find a temperature scale below which one approaches a zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss the Bethe ansatz solution for integrable U(1)U(1)U(1) limits.
Recently, strong-coupling regimes of superconducting qubits or quantum dots inside a microwave circuit cavity and BEC atoms inside an optical cavity were achieved experimentally. The strong-coupling regimes in these systems were described by the Dicke model. Here, we solve the Dicke model by a 1/N1/N1/N expansion. In the normal state, we find a N\sqrt{N}N​ behavior of the collective Rabi splitting. In the superradiant phase, we identify an important Berry phase term that has dramatic effects on both the ground state and the excitation spectra of the strongly interacting system. The single photon excitation spectrum has a low-energy quantum phase diffusion mode in imaginary time with a large spectral weight and also a high-energy optical mode with a low spectral weight. The photons are in a number squeezed state that may have wide applications in high sensitive measurements and quantum-information processing. Comparisons with exact diagonalization studies are made. Possible experimental schemes to realize the superradiant phase are briefly discussed.
The Dicke model can exhibit quantum phase transition between the normal and the superradiant phases when the strength of the light-matter coupling exceeds the ultrastrong-coupling regime. However, it is challenging to observe this phase transition in practical systems due to limited coupling strength or finite two-photon A2{A}^{2}A2 terms. Here we show that by applying a periodic modulation to the frequency of the two-level systems in a standard Dicke model in the strong-coupling regime, an anisotropic Dicke model with tunable rotating and counter-rotating terms in the ultrastrong-coupling regime can be achieved. We calculate the ground state and the excitation spectrum of this model in terms of the modulation parameters. Our result shows that the superradiant phases can be observed in cavity- or circuit-quantum electrodynamics systems with strong coupling.
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.