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Universal dynamics of superradiant phase transition in the anisotropic quantum Rabi model

T0 review · 3 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read The anisotropic quantum Rabi model has a superradiant phase transition whose critical exponents z=2 and ν=1/4 are independent of the anisotropy ratio.

desk verdict The analytical exponents are solid, but the numerical KZ confirmation rests on an arbitrary freeze-time threshold and lacks sensitivity analysis. read the letter →

arxiv 2009.11047 v1 pith:MSUGJOER submitted 2020-09-23 quant-ph

Xunda Jiang , Bo Lu , Chengyin Han , Ruihuan Fang , Minhua Zhao , Zhu Ma , Tian Guo , Chaohong Lee This is my paper · ORCID
classification quant-ph
keywords anisotropicquantumRabimodelsuperradiantphasetransitioncriticalexponentsKibble-ZurekmechanismquenchdynamicsSchrieffer-Wolfftransformationuniversalityclassposition-momentumvariance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the anisotropic quantum Rabi model—a single oscillator mode coupled to a two-level system with different rotating and counter-rotating coupling strengths—undergoes a normal-to-superradiant phase transition whose critical exponents are the same for every anisotropy value except exactly zero. Using a Schrieffer-Wolff transformation, the authors derive effective low-energy harmonic oscillators in both phases, read off the gap and the wavefunction width near criticality, and obtain $z = 2$ and $\nu = 1/4$ for all $\lambda \neq 0$. They then simulate linear quenches across the critical point and extract the same exponents from the phase-transition delay and the diverging position or momentum variance, matching the analytic predictions. If correct, this means the universality class of the Rabi superradiant transition is stable under anisotropy, and a zero-dimensional quantum-optical model can display Kibble-Zurek scaling.

What carries the argument

The Schrieffer-Wolff transformation is the central tool: a unitary rotation that block-diagonalizes the light-matter coupling into low-energy and high-energy parts. In the normal phase it yields an effective harmonic oscillator with coupling-dependent coefficients, gap $\omega\sqrt{(1-\xi^2)(1-\xi'^2)}$, and ground-state width $\alpha_0 = \sqrt{m_0 \omega_0}$. At the critical coupling $\xi \to 1$ the gap vanishes as $|\varepsilon|^{1/2}$, giving $vz = 1/2$, while the width scales as $|\varepsilon|^{-1/4}$, giving $\nu = 1/4$ and $z = 2$. In the superradiant phase, displaced-oscillator generators and new Schrieffer-Wolff generators produce the same scalings, and for $\lambda < 0$ the roles of position and momentum are interchanged.

What would settle it

Simulate the same quench while varying the freeze-time threshold $n_c^{\text{fix}}$ (for example, 1, 5, and 20) and check whether the extracted delay exponent stays $-2/3$; if it shifts, the reported $z$ and $\nu$ are threshold-dependent. Alternatively, extend the model to a one-dimensional lattice of coupled Rabi sites and compute the actual correlation length $\xi(\varepsilon)$; the width exponent $\nu = 1/4$ would be confirmed only if $\xi \sim |\varepsilon|^{-1/4}$ with the same $z$.

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Extended reading notes

Core claim

The central claim is that the normal-to-superradiant transition of the anisotropic quantum Rabi model belongs to a single universality class for all $\lambda \neq 0$. Near the critical coupling, the excitation gap closes as $|\varepsilon|^{1/2}$ and the relevant wavefunction width diverges as $|\varepsilon|^{-1/4}$; comparing these with the Kibble-Zurek scalings gives $vz = 1/2$, hence $z = 2$ and $\nu = 1/4$. The same exponents appear in the superradiant phase, where for $\lambda > 0$ the position variance plays the role of the diverging length and for $\lambda < 0$ the momentum variance does. Numerical simulations of a linear quench extract exponents within a few percent of these values, such as $z \approx 1.99$–$2.02$ and $\nu \approx 0.248$–$0.251$ in Table I, supporting the claim that the universality is shared across anisotropies.

Load-bearing premise

The load-bearing premise is that the single-mode wavefunction width $\Delta x$ (or $\Delta p$) is a genuine diverging correlation length obeying Kibble-Zurek scaling, and that the freeze time fixed by an arbitrary phonon-number threshold does not bias the fitted exponents.

Editorial extensions

If this is right

  • For any anisotropy $\lambda \neq 0$, tuning the model across its critical coupling produces the same universal exponents $z = 2$ and $\nu = 1/4$; $\lambda$ changes only the critical coupling and nonuniversal prefactors.
  • A linear quench should show a phase-transition delay scaling as $\tau_Q^{-2/3}$ and a diverging width scaling as $\tau_Q^{1/6}$, so two independent observables give the same universality test.
  • The $\lambda = 0$ Jaynes-Cummings-like case belongs to a different universality class with $vz = 1$, marking exact isotropy as a special boundary.
  • Single-mode quantum-optical systems, not just extended lattices, can serve as quantitative testbeds for Kibble-Zurek scaling, provided the wavefunction-width-as-length analogy is accepted.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I would expect the width-versus-correlation-length analogy to be the main target of further scrutiny: a lattice extension of the anisotropic Rabi model would let one compute a true two-point correlation length and check whether its $\nu$ equals $1/4$; if not, the width exponent is an effective rather than fundamental quantity.
  • The predicted $\lambda$-independence could be tested directly in circuit-QED or trapped-ion simulators by sweeping the coupling at several values of $\lambda$ and comparing the prefactor $f(\lambda)$ of the scaling laws, not just the exponents.
  • The same Schrieffer-Wolff-based derivation might extend to other few-mode light-matter models with different rotating and counter-rotating imbalances, predicting whether their critical exponents collapse onto the same family or branch into new ones.
  • A systematic check of the freeze-time threshold $n_c^{\text{fix}}$ would reveal whether the reported scalings are sensitive to the arbitrary value 5; if they are, the dynamical exponent extraction would need a threshold-independent method.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the superradiant phase transition in the anisotropic quantum Rabi model, in which the rotating and counter-rotating couplings have different strengths λ. Using position and momentum operators and a Schrieffer-Wolff transformation, the authors derive low-energy effective Hamiltonians for the normal and superradiant phases, obtaining ground states and excitation gaps (Sec. II). From the gap and the variance of the position or momentum operator, they extract static and dynamic critical exponents and claim that they are independent of the anisotropy ratio for λ ≠ 0, namely z = 2 and ν = 1/4 (Sec. III.A). They then simulate the real-time dynamics of the full Hamiltonian with a linear quench of the coupling through the critical point, and from the phase-transition delay and the diverging variance they numerically extract exponents that are reported to be consistent with the analytical values (Sec. III.B and Table I). The central claim is that the anisotropic Rabi model exhibits universal Kibble-Zurek scaling for all λ ≠ 0.

Significance. If the result holds, the paper extends the known quantum-Rabi-model universality to the anisotropic case and provides a concrete single-mode setting in which Kibble-Zurek scalings can be tested. The analytical part is a strength: the Schrieffer-Wolff derivations are standard, the gap formulas (6), (17), and (27) have the correct behavior at the critical point, and the analytical exponent combinations vz = 1/2 and ν = 1/4 are derived explicitly rather than assumed. The numerical simulations use the full Hamiltonian rather than only the effective model, which is a genuine check. The main weaknesses are that the numerical extraction of exponents relies on an arbitrary freeze-time threshold and that no statistical uncertainties or sensitivity tests are reported; therefore the agreement in Table I is not yet fully established. The conceptual mapping of the single-mode wavefunction width to the diverging correlation length also needs justification.

major comments (3)
  1. [§IV.B, Eq. (50) and the paragraph defining n_c^fix] The freeze time t̂ is defined as the time when the phonon number reaches the fixed threshold n_c^fix = 5, and both the phase-transition delay b_d and the variances Δx, Δp are evaluated at this time. Since the KZM only fixes the scaling of the freeze-out time up to a constant, an absolute threshold can bias the fitted exponents if the post-freeze growth rate of n_c depends on τ_Q. The paper reports no sensitivity analysis, no number of quench times, and no error bars for the fits in Table I. Please repeat the extraction for several thresholds (for example n_c^fix = 1, 2, 5, 10, 20), report the resulting spread in the fitted exponents, and show the fit ranges in Fig. 4 and Fig. 5. Without this, the stated consistency with z = 2 and ν = 1/4 cannot be distinguished from a threshold artifact.
  2. [§III.A, Eqs. (29)-(33) and Eqs. (37)-(41)] The paper identifies the single-mode position variance Δx (or momentum variance Δp) with the diverging correlation length ζ in the Kibble-Zurek scaling relation. For a zero-dimensional model with no spatial degrees of freedom, this identification is not automatic; the exponent relation ν/(1+νz) in Eq. (33) presumes a spatial correlation length. The manuscript should either justify the mapping to a true correlation length (for example by relating the variance to the correlation length of the corresponding lattice or many-cavity model) or explicitly state that the KZM is applied by analogy to the known Rabi-model literature. As written, the extracted exponent ν is not demonstrated to be a true correlation-length exponent.
  3. [§IV.B, Table I and Fig. 4(b)] Table I lists separate numerical values for z and ν, but the text does not state how the two exponents are obtained independently from the fitted slopes. A fit of b_d versus τ_Q gives the combination 1/(1+νz), while a fit of Δx versus τ_Q gives ν/(1+νz); the two slopes do jointly determine ν and z, but the extraction should be stated explicitly. Please report the fitted slopes, the number of τ_Q values, the fitting range, and the uncertainties, so that the reader can verify that the reported exponents are not obtained by fixing one exponent to its analytical value.
minor comments (4)
  1. [Throughout] There are several typographical errors that should be corrected: “well-konwn” in Sec. I, “Ads. Phys.” in Ref. [23], “Syace Quantization” in Ref. [41], “Nat. Commum.” in Refs. [31,58,59], “TABLE. I” in Sec. III.B, and “Combing” in Sec. IV.
  2. [Fig. 2] The insets of Fig. 2 are very small and the axes are not labeled; adding labels such as log(ω₀) versus log|ε| would make the claimed exponent 1/2 visible to the reader.
  3. [Sec. IV.B] The simulation truncation parameter L = 96 appears only in the figure captions; the main text should state the Hilbert-space truncation and check that the results are converged for the largest τ_Q used.
  4. [Sec. II] The notation ξ = g̃(1+λ)/2 and ξ′ = g̃(1−λ)/2 is convenient, but the dependence of the critical point on the sign of λ is only stated piecewise; a single expression g̃_c = 2/(1+|λ|) in the main text would clarify the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: analytical exponents are derived from perturbation theory and the numerical scalings come from independent time-dependent Schrödinger simulations.

full rationale

The paper's central claim (critical exponents z=2 and ν=1/4 independent of the anisotropy ratio) is built in two independent ways. First, the analytic exponents are extracted from explicit Schrieffer-Wolff transformed effective Hamiltonians, Eqs. (5), (16), and (26), yielding gaps ϕ0,ϕ1,ϕ2 and variances Δx,Δp, whose critical scalings are written out in Eqs. (34), (39), (41), (42), (45), and (48). These are not fitted inputs; they are direct consequences of the perturbation expansion about the bare Hamiltonian. Second, the numerical verification solves the real-time dynamics of the original Hamiltonian (2) and extracts the freeze-time delay b_d and diverging length Δx (or Δp) as functions of the quench time τ_Q, using the standard KZM relations b_d~τ_Q^{-1/(1+vz)} and Δx~τ_Q^{v/(1+vz)}. The numerical fits in Table I are compared with, not forced to equal, the analytic values (z=2,ν=1/4). No parameter of the simulation is set by the analytic exponents, and the freeze-time criterion n_c^fix=5 is a fixed simulation-analysis threshold, not a quantity tuned to reproduce the target slopes. The citations to prior work on the anisotropic Rabi model [55] and on the isotropic Rabi model [51] are external and are used to contextualize or recover known limits; neither citation is load-bearing in the sense of supplying the universality claim without independent derivation. A possible robustness limitation of the numerical freeze-time threshold is a validity concern, not a circularity, because the threshold is not varied or used as a fitting parameter. Therefore no step in the derivation chain reduces by construction to its inputs, and no self-citation chain forces the result.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard perturbation theory and the KZM analogy. No new entities are introduced. The main analysis choices are the freeze-time threshold and the parameter regime Ω/ω=1000.

free parameters (1)
  • Freeze-time phonon number threshold n_c^fix = 5
    Chosen by hand in Sec. IV.B to define the freeze time when the phonon number reaches this value. The extracted scaling exponents could in principle depend on this threshold, and no sensitivity analysis is provided.
assumptions (3)
  • domain assumption The thermodynamic limit Ω/ω ≫ 1 is taken, so that the low-energy subspace is projected onto ⟨σx⟩ ≈ -1.
    The phase transition in the quantum Rabi model is only well-defined in this limit. Used throughout Sec. II to derive the effective Hamiltonians and gaps.
  • domain assumption The Schrieffer-Wolff transformation is truncated at second order in the coupling.
    The effective Hamiltonians in Eqs. (5), (16), and (26) are second-order approximations. The validity requires g̃ ≪ sqrt(Ω/ω) or similar, which is satisfied deep in the phases but may be questionable near the critical point.
  • domain assumption The Kibble-Zurek mechanism applies to a single-mode model with no spatial degrees of freedom.
    The paper uses the wavefunction width Δx as the diverging length scale ζ in Eqs. (29)-(33). This is an analogy with extended systems and is not rigorously justified for zero-dimensional systems.

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Pith. "Pith review of Universal dynamics of superradiant phase transition in the anisotropic quantum Rabi model." pith.science (2026). https://pith.science/paper/MSUGJOER

@misc{pith2026200911047,
  author       = {Pith},
  title        = {Pith review of: Universal dynamics of superradiant phase transition in the anisotropic quantum Rabi model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSUGJOER}},
  note         = {Machine review of arXiv:2009.11047}
}
read the original abstract

We investigate the universally non-equilibrium dynamics of superradiant phase transition in the anisotropic quantum Rabi model. By introducing position and momentum operators, we obtain the ground states and their excitation gaps for both normal and superradiant phases via perturbation theory. We analytically extract the critical exponents from the excitation gap and the diverging length scale near the critical point, and find that the critical exponents are independent upon the anisotropy ratio. Moreover, by simulating the real-time dynamics across the critical point, we numerically extract the critical exponents from the phase transition delay and the diverging length scale, which are well consistent with the analytical ones. Our study provides a dynamic way to explore universal critical behaviors in the quantum Rabi model.

Figures

Figures reproduced from arXiv: 2009.11047 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) Density distributions of the ground [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) The excitation gap obtained from the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) The time-evolution of density distri [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) (a) The time evolution of the phonon [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) (a,c) Universal scalings of the diver [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [55]

    Quantum Critical- ity and State Engineering in the Simulated Anisotropic Quantum Rabi Model,

    Y. M. Wang, W. L. You, M. X. Liu, Y. L. Dong, H. G. Luo, G. Romero, and J. Q. You, “Quantum Critical- ity and State Engineering in the Simulated Anisotropic Quantum Rabi Model,” New J. Phys. 20, 053061 (2018)

    work page 2018

  2. [1]

    02 and 1

    5, 1 . 02 and 1 . 5, respectively. In which, ˜ g/ ˜gc = 1 is the critical point. In the weak coupling region, ˜g <˜gc, the ground state is dominated by the oscillator term in the normal phase as shown in Fig. 1(b), which is the vacuum of the phonon field and atom in the low energy space. However, as the field-matter coupling is increased to the deep strong ...

    work page

  3. [2]

    Sachdev, Quantum Phase Transitions 1st edn (Cam- bridge University Press, 2011)

    S. Sachdev, Quantum Phase Transitions 1st edn (Cam- bridge University Press, 2011)

    work page 2011

  4. [3]

    Cosmological Inflation as a Quantum Phase Transition,

    M. Morikawa, “Cosmological Inflation as a Quantum Phase Transition,” Progr. Theoret. Phys. 93, 685 (1995)

    work page 1995

  5. [4]

    Some Implications of a Cosmological Phase Transition,

    T. W. B. Kibble, “Some Implications of a Cosmological Phase Transition,” Phys. Rep. 67, 183 (1980)

    work page 1980

  6. [5]

    Topology of Cosmic Domains and Strings,

    T. W. B. Kibble, “Topology of Cosmic Domains and Strings,” J. Phys. A: Math. Gen. 9, 1387 (1976)

    work page 1976

  7. [6]

    Universality and Anomalous Mean-Field Break- down of Symmetry-Breaking Transitions in a Coupled Two-Component Bose-Einstein Condensate,

    C. Lee, “Universality and Anomalous Mean-Field Break- down of Symmetry-Breaking Transitions in a Coupled Two-Component Bose-Einstein Condensate,” Phys. Rev. Lett. 102, 070401 (2009)

    work page 2009

  8. [7]

    Phase Sep- aration and Pattern Formation in a Binary Bose-Einstein Condensate,

    J. Sabbatini, W. H. Zurek, and M. J. Davis, “Phase Sep- aration and Pattern Formation in a Binary Bose-Einstein Condensate,” Phys. Rev. Lett. 107, 230402 (2011)

    work page 2011

Show all 67 references
  1. [8]

    Causality and Defect Formation in the Dynamics of an Engineered Quantum Phase Transition in a Coupled Binary Bose- Einstein Condensate,

    J. Sabbatini, W. H. Zurek, and M. J. Davis, “Causality and Defect Formation in the Dynamics of an Engineered Quantum Phase Transition in a Coupled Binary Bose- Einstein Condensate,” New. J. Phys. 14, 095030 (2012)

    2012

  2. [9]

    Double Universality of a Quantum Phase Transition in Spinor Condensates: Modification of the Kibble-Zurek Mechanism by a Conservation Law,

    T. ´Swislocki, E. Witkowska, J. Dziarmaga, and M. Ma- tuszewski, “Double Universality of a Quantum Phase Transition in Spinor Condensates: Modification of the Kibble-Zurek Mechanism by a Conservation Law,” Phys. Rev. Lett. 110, 045303 (2013)

    2013

  3. [10]

    Coarsening Dynamics of Binary Bose Condensates,

    J. Hofmann, S. S. Natu, and S. D. Sarma, “Coarsening Dynamics of Binary Bose Condensates,” Phys. Rev. Lett. 113, 095702 (2014)

    2014

  4. [11]

    Kibble- Zurek Scalings of Continuous Magnetic Phase Transi- tions in Spin-1 Spin-Orbit-Coupled Bose-Einstein Con- densates,

    S.-Y. Wu, Y.-G. Ke, J.-H. Huang, and C Lee, “Kibble- Zurek Scalings of Continuous Magnetic Phase Transi- tions in Spin-1 Spin-Orbit-Coupled Bose-Einstein Con- densates,” Phys. Rev. A. 95, 063606 (2017)

    2017

  5. [12]

    Kibble-Zurek Dynamics in an Array of Coupled Binary Bose Condensates,

    J. Xu, S.-Y. Wu, X.-Z. Qin, J.-H. Huang, Y.-G. Ke, H.- H. Zhong, and C. Lee, “Kibble-Zurek Dynamics in an Array of Coupled Binary Bose Condensates,” Europhys. Lett. 113, 50003 (2016)

    2016

  6. [13]

    Critical Dynamics of Spontaneous Symmetry Breaking in a Homogeneous Bose Gas,

    N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, “Critical Dynamics of Spontaneous Symmetry Breaking in a Homogeneous Bose Gas,” Science 347, 167 (2015)

    2015

  7. [14]

    Univer- sal Dynamics of Zero-Momentum to Plane-Wave Transi- 8 tion in Spin-Orbit Coupled Bose-Einstein Condensates,

    Q. Z. Ye, S. Y. Wu, X. D. Jiang, and C. H. Lee, “Univer- sal Dynamics of Zero-Momentum to Plane-Wave Transi- 8 tion in Spin-Orbit Coupled Bose-Einstein Condensates,” J. Stat. Mech. , 053110 (2018)

    2018

  8. [15]

    Univer- sality of Miscible-Immiscible Phase Separation Dynamics in Two-Component Bose-Einstein Condensates,

    X. D. Jiang, S. Y. Wu, Q. Z. Ye, and C. H. Lee, “Univer- sality of Miscible-Immiscible Phase Separation Dynamics in Two-Component Bose-Einstein Condensates,” New J. Phys. 21, 023014 (2019)

    2019

  9. [16]

    Spontaneous Vortices in the Formation of Bose-Einstein Condensates,

    C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis, and B. P. Anderson, “Spontaneous Vortices in the Formation of Bose-Einstein Condensates,” Nature 455, 948 (2008)

    2008

  10. [17]

    Kibble-Zurek Scaling and its Breakdown for Sponta- neous Generation of Josephson Vortices in Bose-Einstein Condensates,

    S. W. Su, S. C. Gou, A. Bradley, O. Fialko, and J. Brand, “Kibble-Zurek Scaling and its Breakdown for Sponta- neous Generation of Josephson Vortices in Bose-Einstein Condensates,” Phys. Rev. Lett. 110, 215302 (2013)

    2013

  11. [18]

    Universal Spatiotemporal Dynamics of Spontaneous Superfluidity Breakdown in the Presence of Synthetic Gauge Fields,

    S.-Y Wu, X.-Z. Qin, J. Xu, and C Lee, “Universal Spatiotemporal Dynamics of Spontaneous Superfluidity Breakdown in the Presence of Synthetic Gauge Fields,” Phys. Rev. A. 94, 043606 (2016)

    2016

  12. [19]

    Soliton Creation During a Bose-Einstein Condensation,

    B. Damski and W. H. Zurek, “Soliton Creation During a Bose-Einstein Condensation,” Phys. Rev. Lett. 104, 160404 (2010)

    2010

  13. [20]

    Solitons as the Early Stage of Quasicondensate Forma- tion during Evaporative Cooling,

    E. Witkowska, P. Deuar, M. Gajda, and K. Rza˙ zewski, “Solitons as the Early Stage of Quasicondensate Forma- tion during Evaporative Cooling,” Phys. Rev. Lett. 106, 135301 (2011)

    2011

  14. [21]

    Causality in Condensates: Gray Solitons as Relics of BEC Formation,

    W. H. Zurek, “Causality in Condensates: Gray Solitons as Relics of BEC Formation,” Phys. Rev. Lett. 102, 105702 (2009)

    2009

  15. [22]

    Cosmological Experiments in Superfluid Helium?

    W. H. Zurek, “Cosmological Experiments in Superfluid Helium?” Nature 317, 505 (1985)

    1985

  16. [23]

    Cosmological Experiments in Condensed Matter Systems,

    W. H. Zurek, “Cosmological Experiments in Condensed Matter Systems,” Phys. Rep. 276, 177 (1996)

    1996

  17. [24]

    Dynamics of a Quantum Phase Transi- tion and Relaxation to a Steady State,

    J. Dziarmaga, “Dynamics of a Quantum Phase Transi- tion and Relaxation to a Steady State,” Ads. Phys. 59, 1063–1189 (2000)

    2000

  18. [25]

    Nonequilibrium Dynamics of Closed Interact- ing Quantum Systems,

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Ven- galattore, “Nonequilibrium Dynamics of Closed Interact- ing Quantum Systems,” Rev. Mod. Phys. 83, 863 (2011)

    2011

  19. [26]

    Many-Body Physics with Ultracold Gases,

    I. Bloch, J. Dalibard, and W. Zwerger, “Many-Body Physics with Ultracold Gases,” Rev. Mod. Phys. 80, 885 (2008)

    2008

  20. [27]

    Vortex Formation in Neutron-Irradiated Super- fluid 3He as an Analogue of Cosmological Defect Forma- tion,

    V. M. H. Ruutu, V. B. Eltsov, A. J. Gill, and T. W. B. Kibble, “Vortex Formation in Neutron-Irradiated Super- fluid 3He as an Analogue of Cosmological Defect Forma- tion,” Nature 382, 334 (1996)

    1996

  21. [28]

    Laboratory Simulation of Cosmic String Formation in the Early Uni- verse Using Superfluid 3He,

    C. B¨ auerle, Y. M. Bunkov, and S. N. Fisher, “Laboratory Simulation of Cosmic String Formation in the Early Uni- verse Using Superfluid 3He,” Nature 382, 1332 (1996)

    1996

  22. [29]

    Spontaneous Fluxoid Formation in Superconducting Loops,

    R. Monaco, J. Mygind, R. J. Rivers, and V. P. Koshelets, “Spontaneous Fluxoid Formation in Superconducting Loops,” Phys. Rev. B. 80, 180501 (2009)

    2009

  23. [30]

    Structural Defects in Ion Chains by Quenching the External Potential: The Inhomogeneous Kibble-Zurek Mechanism,

    A. del Campo, G. De. Chiara, G. Morigi, M. B. Plenio, and A. Retzker, “Structural Defects in Ion Chains by Quenching the External Potential: The Inhomogeneous Kibble-Zurek Mechanism,” Phys. Rev. Lett. 105, 075701 (2010)

    2010

  24. [31]

    Observation of the Kibble-Zurek Scaling Law for Defect Formation in Ion Crystals,

    S. Ulm et al, “Observation of the Kibble-Zurek Scaling Law for Defect Formation in Ion Crystals,” Nat. Com- mum. 4, 2290 (2013)

    2013

  25. [32]

    Topological Defect Formation and Spon- taneous Symmetry Breaking in Ion Coulomb Crystals,

    K. Pyka et al, “Topological Defect Formation and Spon- taneous Symmetry Breaking in Ion Coulomb Crystals,” Nat. Commum. 4, 2291 (2013)

    2013

  26. [33]

    Spontaneous Nucle- ation and Dynamics of Kink Defects in Zigzag Arrays of Trapped Ions,

    S. Ejtemaee and P. C. Haljan, “Spontaneous Nucle- ation and Dynamics of Kink Defects in Zigzag Arrays of Trapped Ions,” Phys. Rev. A. 87, 051401 (2013)

    2013

  27. [34]

    Quantum Simulation of the Quantum Rabi Model in a Trapped Ion,

    D. S. Lv, S. M. An, Z. Y. Liu, J. N. Zhang, J. S. Ped- ernales, L. Lamata, E. Solano, and K. Kim, “Quantum Simulation of the Quantum Rabi Model in a Trapped Ion,” Phys. Rev. X 8, 021027 (2018)

    2018

  28. [35]

    Spontaneous Creation of Kibble-Zurek Soli- tons in a Bose-Einstein Condensate,

    G. Lamporesi, S. Donadello, S. Serafini, F. Dalfovo, and G. Ferrari, “Spontaneous Creation of Kibble-Zurek Soli- tons in a Bose-Einstein Condensate,” Nat. Phys. 9, 656 (2013)

    2013

  29. [36]

    Quantum Kibble-Zurek Mechanism in a Spin-1 Bose-Einstein Con- densate,

    M. Anquez, B. A. Robbins, H. M Bharath, M. Bogus- lawski, T. M. Hoang, and M. S. Chapman, “Quantum Kibble-Zurek Mechanism in a Spin-1 Bose-Einstein Con- densate,” Phys. Rev. Lett. 116, 155301 (2016)

    2016

  30. [37]

    Universal Space- Time Scaling Symmetry in the Dynamics of Bosons Across a Quantum Phase Transition,

    L. W. Clark, L. Feng, and C. Chin, “Universal Space- Time Scaling Symmetry in the Dynamics of Bosons Across a Quantum Phase Transition,” Science 354, 606 (2016)

    2016

  31. [38]

    Coher- ent Inflationary Dynamics for Bose-Einstein Condensates Crossing a Quantum Critical Point,

    L. Feng, L. W. Clark, A. Gaj, and C. Chin, “Coher- ent Inflationary Dynamics for Bose-Einstein Condensates Crossing a Quantum Critical Point,” Nat. Phys. 14, 269 (2018)

    2018

  32. [39]

    Ultrastrong Coupling Regimes of Light- Matter Interaction,

    P. Forn-Diaz, L. Lamata, E. Rico, J. Kono, and E. Solano, “Ultrastrong Coupling Regimes of Light- Matter Interaction,” Rev. Mod. Phys. 91, 025005 (2019)

    2019

  33. [40]

    Ultrastrong Coupling Between Light and Matter,

    A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong Coupling Between Light and Matter,” Nat. Rev. Phys. 1, 19 (2019)

    2019

  34. [41]

    On the Process of Space Quantization,

    I. I. Rabi, “On the Process of Space Quantization,” Phys . Rev. 49, 324 (1936)

    1936

  35. [42]

    Syace Quantization in a Gyrating Magnetic Field,

    I. I. Rabi, “Syace Quantization in a Gyrating Magnetic Field,” Phys. Rev. 51, 652 (1937)

    1937

  36. [43]

    Analytical Eigenstates for the Quantum Rabi Model,

    H. H. Zhong, Q. T. Xie, M. T. Batchelor, and C. H. Lee, “Analytical Eigenstates for the Quantum Rabi Model,” J. Phys. A: Math. Theor. 46, 415302 (2013)

    2013

  37. [44]

    Analytical Energy Spec- trum for Hybrid Mechanical Systems,

    H. H. Zhong, Q. T. Xie, X. W. Guan, M. T. Batchelor, K. L. Gao, and C. H. Lee, “Analytical Energy Spec- trum for Hybrid Mechanical Systems,” J. Phys. A: Math. Theor. 47, 045301 (2014)

    2014

  38. [45]

    The Quantum Rabi Model: Solution and Dynamics,

    Q. T. Xie, H. H. Zhong, M. T. Batchelor, and C. H. Lee, “The Quantum Rabi Model: Solution and Dynamics,” J. Phys. A: Math. Theor. 50, 113001 (2017)

    2017

  39. [46]

    Superradiance Transition in a System with a Single Qubit and a Single Oscillator,

    S. Ashhab, “Superradiance Transition in a System with a Single Qubit and a Single Oscillator,” Phys. Rev. A. 87, 013826 (2013)

    2013

  40. [47]

    Application of the Coupled Cluster Method to the Jaynes-Cummings Model Without the Rotating-Wave Approximation,

    R. F. Bishop and N. J. Davidson, “Application of the Coupled Cluster Method to the Jaynes-Cummings Model Without the Rotating-Wave Approximation,” Phys. Rev. A. 54, 4657 (1996)

    1996

  41. [48]

    Some Remarks on ’Super- radiant’ Phase Transitions in Light-Matter Systems,

    J. Larson and E. K. Irish, “Some Remarks on ’Super- radiant’ Phase Transitions in Light-Matter Systems,” J. Phys. A: Math. Theor. 50, 174002 (2017)

    2017

  42. [49]

    Variational Study of a Two-Level System Coupled to a Harmonic Oscillator in an Ultrastrong-Coupling Regime,

    M. J. Hwang and M. S. Choi, “Variational Study of a Two-Level System Coupled to a Harmonic Oscillator in an Ultrastrong-Coupling Regime,” Phys. Rev. A. 82, 025802 (2010)

    2010

  43. [50]

    Dissipative Phase Transition in the Open Quantum Rabi Model,

    M. J. Hwang, P. Rabl, and M. B. Plenio, “Dissipative Phase Transition in the Open Quantum Rabi Model,” Phys. Rev. A. 97, 013825 (2018)

    2018

  44. [51]

    Probing the Dynamics of a Superradiant Quantum Phase Transition with a Single Trapped Ion,

    R. Puebla, M. J. Hwang, J. Casanova, and M. B. Ple- nio, “Probing the Dynamics of a Superradiant Quantum Phase Transition with a Single Trapped Ion,” Phys. Rev. Lett. 1198, 073001 (2017)

    2017

  45. [52]

    Quantum Phase Transition and Universal Dynamics in the Rabi 9 Model,

    M. J. Hwang, R. Puebla, and M. B. Plenio, “Quantum Phase Transition and Universal Dynamics in the Rabi 9 Model,” Phys. Rev. Lett. 115, 180404 (2015)

    2015

  46. [53]

    Anisotropic Rabi Model,

    Q. T. Xie, S. Cui, J. P. Cao, L. G. Amico, and H. Fan, “Anisotropic Rabi Model,” Phys. Rev. X. 4, 021046 (2014)

    2014

  47. [54]

    Analytical Solutions by Squeezing to the Anisotropic Rabi Model in the Nonper- turbative Deep-Strong-Coupling Regime,

    Y. Y. Zhang and X. Y. Chen, “Analytical Solutions by Squeezing to the Anisotropic Rabi Model in the Nonper- turbative Deep-Strong-Coupling Regime,” Phys. Rev. A. 96, 063821 (2017)

    2017

  48. [56]

    Universal Scaling and Critical Exponents of the Anisotropic Quantum Rabi Model,

    M.-X. Liu, S. Chesi, Z.-J. Ying, X.-S. Chen, H.-G. Luo, and H.-Q. Lin, “Universal Scaling and Critical Exponents of the Anisotropic Quantum Rabi Model,” Phys. Rev. Lett. 119, 220601 (2017)

    2017

  49. [57]

    Ground-State Phase Diagram of the Quantum Rabi Model,

    Z.-J. Ying, M.-X. Liu, H.-G. Luo, H.-Q. Lin, and J. Q. You, “Ground-State Phase Diagram of the Quantum Rabi Model,” Phys. Rev. A. 92, 053823 (2015)

    2015

  50. [58]

    Quantum Phase Tran- sition in the Finite Jaynes-Cummings Lattice Systems,

    M. J. Hwang and M. B. Plenio, “Quantum Phase Tran- sition in the Finite Jaynes-Cummings Lattice Systems,” Phys. Rev. Lett. 117, 123602 (2016)

    2016

  51. [59]

    Experimentally Simulating the Dynamics of Quantum Light and Matter at Deep-Strong Coupling,

    N.K. Langford, R. Sagastizabal, M. Kounalakis, C. Dickel, A. Bruno, F. Luthi, D.J. Thoen, A. Endo, and L. DiCarlo, “Experimentally Simulating the Dynamics of Quantum Light and Matter at Deep-Strong Coupling,” Nat. Commum. 8, 1715 (2017)

    2017

  52. [60]

    Analog Quantum Simulation of the Rabi Model in the Ultra- Strong Coupling Regime,

    J. Braum¨uller, M. Marthaler, A. Schneider, A. Stehli, H. Rotzinger M. Weides, and A. V. Ustinov, “Analog Quantum Simulation of the Rabi Model in the Ultra- Strong Coupling Regime,” Nat. Commum. 8, 779 (2017)

    2017

  53. [61]

    Enhanc- ing Cavity Quantum Electrodynamics via Antisqueezing: Synthetic Ultrastrong Coupling,

    C. Leroux, L. C. G. Govia, and A. A. Clerk, “Enhanc- ing Cavity Quantum Electrodynamics via Antisqueezing: Synthetic Ultrastrong Coupling,” Phy. Rev. Lett. 120, 093602 (2018)

    2018

  54. [62]

    Quan- tum Rabi Model in a Superfluid Bose-Einstein Conden- sate,

    S. Felicetti, G. Romero, E. Solano, and C. Sab ´ ın, “Quan- tum Rabi Model in a Superfluid Bose-Einstein Conden- sate,” Phys. Rev. A 96, 033839 (2017)

    2017

  55. [63]

    Quantum Rabi Model with Trapped Ions,

    J. S. Pedernales, I. Lizuain, S. Felicetti, G. Romero, L. Lamata, and E. Solano, “Quantum Rabi Model with Trapped Ions,” Sci. Rep. 5, 15472 (2015)

    2015

  56. [64]

    Variational Study of the ν = 1 Quantum Hall Ferromagnet in the Presence of Spin-Orbit interaction,

    J. Schliemann, J. C. Egues, and D. Loss, “Variational Study of the ν = 1 Quantum Hall Ferromagnet in the Presence of Spin-Orbit interaction,” Phys. Rev. B 67, 085302 (2003)

    2003

  57. [65]

    The Energy-Level Crossing Behavior and Quantum Fisher In- formation in a Quantum Well with Spin-Orbit Coupling,

    Z. H. Wang, Q. Zheng, X. G. Wang, and Y. Li, “The Energy-Level Crossing Behavior and Quantum Fisher In- formation in a Quantum Well with Spin-Orbit Coupling,” Sci. Rep. 6, 22347 (2016)

    2016

  58. [66]

    Controlling Discrete and Con- tinuous Symmetries in “Superradiant

    A. Baksic and C. Ciuti, “Controlling Discrete and Con- tinuous Symmetries in “Superradiant” Phase Transi- tions with Circuit QED Systems,” Phys. Rev. Lett. 112, 173601 (2014)

    2014

  59. [67]

    Ultrastrong-Coupling Quantum-Phase-Transition Phenomena in a Few-Qubit Circuit QED System,

    W. J. Yang and X. B. Wang, “Ultrastrong-Coupling Quantum-Phase-Transition Phenomena in a Few-Qubit Circuit QED System,” Phys. Rev. A 95, 043823 (2017)

    2017

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