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Dissipative Phase Transition in the Two-Photon Dicke Model

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A small amount of two-photon loss stabilizes the intrinsically unstable two-photon Dicke model and produces a superradiant phase that coexists with the normal vacuum.

desk verdict A genuine stabilization result for the two-photon Dicke model, with an uncontrolled closure that tempers the quantitative claims. read the letter →

arxiv 2412.14271 v1 pith:RCIYOPOS submitted 2024-12-18 quant-ph cond-mat.quant-gasphysics.atom-ph

Aanal Jayesh Shah , Peter Kirton , Simone Felicetti , Hadiseh Alaeian This is my paper · ORCID
classification quant-phcond-mat.quant-gasphysics.atom-ph
keywords two-photonDickemodeldissipativephasetransitionlosssuperradiantZ4symmetrysecond-ordercumulantexpansionopenquantumsystemsthermodynamiclimit
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

The paper sets out to show that the driven-dissipative two-photon Dicke model, in which each spin exchanges two cavity photons at once, has a dissipative phase transition in the thermodynamic limit. With only single-photon loss, the model never reaches a steady state above a critical coupling: the two-photon coupling acts as an effective pump, and the photon population grows without converging as the Fock-space truncation is increased. Adding even a small two-photon loss counterbalances that pumping, and a stable superradiant phase appears beyond a critical coupling, coexisting with the normal vacuum state. Using a second-order cumulant expansion for the photons, the paper derives closed thermodynamic-limit equations whose phase boundaries match exact diagonalization and quantum trajectory results for small and intermediate system sizes. The Wigner function of the cavity field shows four side lobes, revealing the breaking of the model's $Z_4$ symmetry in the superradiant phase.

What carries the argument

The load-bearing object is the second-order cumulant (cluster) expansion for the photon field, applied to the Heisenberg equations of motion. It replaces four-photon correlators such as $\langle a^{\dagger 2}a^2\rangle$ with products of one- and two-point functions, while spin-photon correlators are factorized at the mean-field level as $\langle O_{\mathrm{spin}}O_{\mathrm{photon}}\rangle = \langle O_{\mathrm{spin}}\rangle\langle O_{\mathrm{photon}}\rangle$. This closure turns the open-system dynamics into a closed set of $N$-independent equations in the thermodynamic limit, whose fixed points are the normal and superradiant phases and whose linear stability is decided by the eigenvalues of the Bogoliubov matrix. The $Z_4$ symmetry of the Hamiltonian, encoded in the generalized parity $\Pi = e^{i\pi(a^{\dagger}a/2+J_z)}$, becomes a weak symmetry of the Liouvillian when dissipation is added, and the Wigner function is used to visualize the resulting fourfold phase-space structure.

What would settle it

Perform a third- or fourth-order cumulant expansion, or exact quantum-trajectory simulations at the predicted onset for a range of system sizes, and check whether the critical coupling and the stability of the upper superradiant branch in Fig. 3(g) survive; if the onset moves or the upper branch becomes unstable, the thermodynamic-limit picture is wrong.

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Extended reading notes

Core claim

The central claim is that two-photon loss, not one-photon loss, is the dissipation that stabilizes the two-photon Dicke model. For the Hamiltonian $H = \omega_c a^{\dagger}a + \frac{\omega_a}{2}J_z + \frac{\lambda}{N}J_x(a^{\dagger 2}+a^2)$ with Lindblad losses $L_1 = \sqrt{\kappa_1}a$ and $L_2 = \sqrt{\kappa_2/N}\,a^2$, the one-photon-loss case admits only an unstable superradiant solution above the critical coupling $\lambda_c = \frac{1}{4}\sqrt{\kappa_1^2 + 4\omega_c^2}$, while exact numerics show the photon population never converging as the Fock truncation grows. Adding $\kappa_2 > 0$ restores convergence for all system sizes studied, and the second-order cumulant equations yield three steady-state branches: the normal phase, stable for all $\lambda$; a stable upper superradiant branch; and an unstable lower superradiant branch that appears only in finite systems. The stable superradiant branch emerges discontinuously and coexists with the normal vacuum, which the paper interprets as a first-order dissipative phase transition, and the four-lobed Wigner function expresses the $Z_4$ symmetry of the Liouvillian.

Load-bearing premise

The load-bearing premise is that the second-order cumulant closure, which replaces four-photon correlations by products of one- and two-photon correlations and treats spin-photon correlations as products, stays quantitatively accurate near the critical coupling, so that the predicted phase boundary and the stability of the upper superradiant branch do not depend on uncomputed higher-order correlations.

Editorial extensions

If this is right

  • Above the critical coupling, the open two-photon Dicke model has a stable superradiant steady state and a stable normal steady state at the same parameters, so the dissipative phase transition should be first-order, with coexistence rather than a continuous transition.
  • Two-photon loss acts as a control parameter: tuning $\kappa_2$ from zero to a small value switches the model from runaway dynamics to a well-defined steady-state phase diagram, even for large systems.
  • In the thermodynamic limit, only one of the two superradiant solutions remains stable; finite-size systems show both, so the upper superradiant branch is the one that persists as $N\to\infty$.
  • The four-lobed Wigner function at strong coupling directly visualizes the spontaneous breaking of the $Z_4$ symmetry, giving a phase-space signature that cavity experiments could look for.
  • The same stabilization mechanism could be used to engineer nonclassical steady states, including those relevant for cat-qubit encoding and critical quantum sensing, which the paper lists as future directions.

Reading between the lines

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

  • The precise location of the critical coupling near the transition remains somewhat open because convergence slows there; a higher-order cumulant closure or a larger-scale trajectory calculation could shift $\lambda_c$ and would be the natural next test.
  • Because normal and superradiant states coexist, ramping the coupling up and down should produce hysteresis in the photon number; the paper's steady-state analysis implies this but does not demonstrate the dynamical switch.
  • The two-photon-loss-only sector has a degenerate Liouvillian kernel with even- and odd-Fock parity sectors, so initialization in one parity sector may matter for experiments aiming at cat-qubit or sensing applications.
  • The mechanism may extend to other nonlinear loss channels: any sufficiently strong multiphoton loss that truncates high Fock occupancy could stabilize models whose two-photon coupling otherwise causes spectral collapse.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the open two-photon Dicke model (Eq. (1)) with one- and two-photon cavity losses (Eq. (5)). The authors report that single-photon loss alone cannot stabilize the model: above a critical coupling the exact (small-N) photon distributions fail to converge as the Fock-space truncation increases. Adding two-photon loss (with a 1/N scaling) restores convergence and gives rise to superradiant steady states that coexist with the normal vacuum. Using a second-order cumulant expansion for photon correlations, the authors derive a thermodynamic-limit description and a stability analysis, yielding a phase diagram (Fig. 3(g)) with one always-stable normal phase, one stable superradiant branch, and one unstable superradiant branch. They compare this analytical result against exact diagonalization (N=5) and quantum-trajectory simulations (N=13,15), find agreement, and present Wigner functions that show the expected Z4 symmetry.

Significance. If the analytical phase boundary is quantitatively reliable, the paper provides a clear and nontrivial result: in a two-photon Dicke model, one-photon dissipation cannot prevent instability, while even weak two-photon dissipation stabilizes a superradiant phase whose upper branch remains stable in the thermodynamic limit. This goes beyond previous mean-field studies of the dissipative two-photon Dicke model and is backed by a parameter-free analytical model (no fitted constants enter the equations of motion) plus independent ED and QT validations. The Wigner-function analysis and the coexistence picture for the superradiant and vacuum states are also valuable. The main open risk is the uncontrolled second-order cumulant closure used to locate the critical point and assess branch stability; this is the central issue that must be addressed before the quantitative phase boundary can be fully trusted.

major comments (3)
  1. [Introduction; SM Eqs. (35)-(42) and (43)-(50)] The thermodynamic-limit phase boundary in Fig. 3(g) is derived from the second-order cumulant equations (SM Eqs. (35)-(42) and (43)-(50)), which factorize four-photon correlators such as <a†²a²> into products of lower-order moments and treat spin-photon correlators at mean-field level (Eq. (7) of the SM). The paper itself notes in the Introduction that 'only around the critical point do higher-order correlations play a significant role' (refs. [34-36]). Because the critical coupling and the stability of the upper branch are extracted from precisely this closure, the quantitative accuracy of the closure near the phase boundary is load-bearing. The manuscript does not provide a systematic error estimate, a comparison with a higher-order cumulant, or a controlled finite-size extrapolation that would validate the closure. I ask the authors to quantify the closure error near λ_c, either by computing a third-order cumulant correction or by performing a finite-size scaling analysis of the exact data in Fig. 3(g). Without this, the predicted location of λ_c and the stability assignments for the superradiant branches remain quantitatively unsecured.
  2. [SM 'Equations of motion for one- and two-photon losses' (after Eq. (50))] The text states that 'These equations can be solved analytically to determine the steady states' for the two-photon-loss case, but the explicit steady-state solutions, including the critical coupling λ_c as a function of κ1, κ2, ω_c, and ω_a, are not written out. In contrast, the one-photon-only case has a transparent closed-form solution in the SM (Eqs. (17)-(22) and Eq. (14) for λ_c). For the two-photon case, the reader cannot verify the phase boundary in Fig. 3(g), the coexistence window, or the stability assignments from the text. Please provide the explicit steady-state expressions and the resulting λ_c formula, or state clearly if the equations have no closed-form solution and must be solved numerically.
  3. [Fig. 3(g), Fig. 7, SM 'Quantum trajectory'] The quantitative comparison with exact numerics is presented without statistical error bars or convergence metrics. The average photon numbers for N=5,13,15 are extracted from Gaussian fits to P(n) (Fig. 3(a)-(f)), but the fit uncertainties are not reported. The quantum-trajectory data in Fig. 7 show only N=15 for NT=500 and 3000, and the residual disagreement with the analytical curve near the transition is attributed to 'computational limitations' without a quantitative criterion. To support the claim of excellent agreement and the identification of the phase-boundary onset, the authors should report statistical errors from trajectory sampling and histogram fitting, and show a systematic convergence test (e.g., in trajectory number and system size) especially in the critical region where the deviations are largest.
minor comments (4)
  1. [SM Eq. (52)] The Bogoliubov matrix in Eq. (52) is typeset in a way that is effectively unreadable, with missing delimiters and incorrectly placed brackets. Please reformat this matrix so that its entries can be checked.
  2. [SM and main text] There are several spelling and terminology issues: 'Bougolibov' should be 'Bogoliubov' (Figs. 5 and 6 captions), 'stablility' should be 'stability', and 'hashed regions' should be 'hatched regions' (Fig. 6 caption).
  3. [Fig. 2(a)-(b) caption] The caption refers to 'black, blue, and red dashed lines', but the figure panels (a) and (b) appear to show a single dashed unstable branch; please clarify the color coding and how the different colors correspond to the plot.
  4. [Fig. 3 caption] The sentence 'The light blue curve in (d) and (c) is the Gaussian fit...' is ambiguous because (c) and (d) correspond to different system sizes and different couplings; please specify precisely which panels contain which fits.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the phase boundaries and stability assignments are computed from explicitly stated equations of motion with externally fixed parameters, and the finite-size simulations are independent validation rather than fitted inputs.

full rationale

The paper's derivation chain is self-contained: the Hamiltonian and Lindblad dissipators (Eqs. 1, 3, 5) fix all parameters, the scaled equations of motion (SM Eqs. 8-13 and 43-50) are written out explicitly, and the steady states and Bogoliubov stability matrix determine the phase diagram in Fig. 3(g) without any parameter being fitted to the subsequent exact-diagonalization or quantum-trajectory results. The ED/QT data are overlaid as independent checks, and the Gaussian fits are used only to extract simulated average photon numbers, not to adjust the analytical curves. The second-order cumulant closure is an approximation whose accuracy near the critical point is not quantified, but that is a validity assumption explicitly stated in the paper rather than a circular reimportation of the target result; hard rule 5 therefore places it under correctness risk, not circularity. The only mild self-reference is the citation of [71] (which includes a co-author) as one of several references for the cumulant expansion, but the method is also supported by independent references [36,69,70] and is re-derived in the SM, so no load-bearing self-citation is present.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The analytical phase diagram is not fitted: all rates and frequencies are inputs, and the numerical ED and QT data are used for validation. The central claim nonetheless rests on unproved modeling choices: Gaussian closure of the photon hierarchy and the 1/N scaling of two-photon loss. Gaussian fits to the numerical distributions are used only for data extraction.

free parameters (2)
  • Two-photon loss rate kappa2 = Not tabulated for main figures; SM example uses 0.05
    Two-photon loss rate is an input parameter, but its value in the main phase-diagram figures (Fig. 3 and Fig. 4) is not stated in the main text, which is needed for reproduction.
  • Gaussian fit mean and width for extracting average photon number = Not reported; one fit per (lambda, N)
    Used in Fig. 3 to convert P(n) into average photon numbers for comparison with the analytical model; these fits are data-extraction tools and do not enter the analytical phase diagram.
assumptions (5)
  • domain assumption Markovian Lindblad master equation with only photon loss channels L1 and L2; no spin decay or dephasing is included.
    Invoked in Eq. (3)-(5). The authors defer robustness to spin decay and dephasing to future work, so the phase diagram is specific to this dissipation set.
  • domain assumption Second-order cumulant (Gaussian) factorization closes the photon operator hierarchy, and spin-photon correlators are factorized at mean-field level.
    Used in SM Eqs. (35)-(42) and Eq. (7). No systematic error estimate or higher-order closure check is provided.
  • ad hoc to paper Scaling of operators J/N and a/sqrt(N), with two-photon loss written as sqrt(kappa2/N) a^2, yields a non-trivial N-independent thermodynamic limit.
    Introduced in Eq. (2) and Eq. (5). This is a modeling choice: a constant physical two-photon loss rate would scale differently in the thermodynamic limit and could change the phase diagram.
  • standard math Total spin J^2 is a conserved quantity because the jump operators act only on the photon sector.
    Follows from [H, J^2]=0 and the form of L1 and L2; it limits the spin Hilbert space to N+1 states.
  • domain assumption Fock-truncated exact diagonalization and quantum trajectory simulations converge to the true NESS for the truncation M and trajectory count NT used.
    Relied on for Figs. 2-4 and SM Fig. 7. Convergence is checked only for selected parameter points and no error bars are shown.

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Pith. "Pith review of Dissipative Phase Transition in the Two-Photon Dicke Model." pith.science (2026). https://pith.science/paper/RCIYOPOS

@misc{pith2026241214271,
  author       = {Pith},
  title        = {Pith review of: Dissipative Phase Transition in the Two-Photon Dicke Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCIYOPOS}},
  note         = {Machine review of arXiv:2412.14271}
}
read the original abstract

We explore the dissipative phase transition of the two-photon Dicke model, a topic that has garnered significant attention recently. Our analysis reveals that while single-photon loss does not stabilize the intrinsic instability in the model, the inclusion of two-photon loss restores stability, leading to the emergence of superradiant states which coexist with the normal vacuum states. Using a second-order cumulant expansion for the photons, we derive an analytical description of the system in the thermodynamic limit which agrees well with the exact calculation results. Additionally, we present the Wigner function for the system, shedding light on the breaking of the Z4-symmetry inherent in the model. These findings offer valuable insights into stabilization mechanisms in open quantum systems and pave the way for exploring complex nonlinear dynamics in two-photon Dicke models.

Figures

Figures reproduced from arXiv: 2412.14271 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic diagram of an ensemble of identical [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Instability of the open two-photon Dicke model with [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The stabilized open two-photon Dicke model with [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The photonic Wigner functions distribution of the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Real part of the Bougolibov matrix spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Real part of the Bougolibov matrix spectrum vs. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Convergence of the quantum trajectory approach for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) The first 8 eigenvalues of the Liouvillian for [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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Works this paper leans on

84 extracted references · 58 canonical work pages · cited by 1 Pith paper

  1. [1]

    R. H. Dicke, Coherence in spontaneous radiation pro- cesses, Phys. Rev. 93, 99 (1954)

    1954

  2. [2]

    Gross and S

    M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Physics Reports 93, 301 (1982)

    1982

  3. [3]

    B. M. Garraway, The dicke model in quantum op- tics: Dicke model revisited, Philosophical Transactions: Mathematical, Physical and Engineering Sciences 369, 1137 (2011)

    2011

  4. [4]

    Kirton, M

    P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the dicke model: From equilibrium to nonequilibrium, and vice versa, Advanced Quantum Technologies 2, 10.1002/qute.201800043 (2019)

    doi:10.1002/qute.201800043 2019

  5. [5]

    S. J. Masson and A. Asenjo-Garcia, Universality of dicke superradiance in arrays of quantum emitters, Na- ture Communications 13, 10.1038/s41467-022-29805-4 (2022)

    doi:10.1038/s41467-022-29805-4 2022

  6. [6]

    Villase˜ nor, S

    D. Villase˜ nor, S. Pilatowsky-Cameo, J. Ch´ avez-Carlos, M. A. Bastarrachea-Magnani, S. Lerma-Hern´ andez, L. F. Santos, and J. G. Hirsch, Classical and quantum proper- ties of the spin-boson dicke model: Chaos, localization, and scarring (2024), arXiv:2405.20381 [quant-ph]

    arXiv 2024

  7. [7]

    Forn-D ´ ıaz, L

    P. Forn-D ´ ıaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys. 91, 025005 (2019)

    2019

  8. [8]

    F. J. Garcia-Vidal, C. Ciuti, and T. W. Ebbesen, Ma- nipulating matter by strong coupling to vacuum fields, Science 373, eabd0336 (2021)

    2021

Show all 84 references
  1. [9]

    Frisk Kockum, A

    A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nature Reviews Physics 1, 19 (2019)

    2019

  2. [10]

    Le Boit´ e, Theoretical methods for ultrastrong light– matter interactions, Advanced Quantum Technologies 3, 1900140 (2020)

    A. Le Boit´ e, Theoretical methods for ultrastrong light– matter interactions, Advanced Quantum Technologies 3, 1900140 (2020)

    2020

  3. [11]

    Rza˙ zewski, K

    K. Rza˙ zewski, K. W´ odkiewicz, and W.˙Zakowicz, Phase transitions, two-level atoms, and the A2 term, Phys. Rev. Lett. 35, 432 (1975)

    1975

  4. [12]

    De Bernardis, P

    D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, and P. Rabl, Breakdown of gauge invariance in ultrastrong- coupling cavity qed, Phys. Rev. A 98, 053819 (2018)

    2018

  5. [13]

    Di Stefano, A

    O. Di Stefano, A. Settineri, V. Macr ` ı, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum elec- trodynamics, Nature Physics 15, 803 (2019)

    2019

  6. [14]

    Stokes and A

    A. Stokes and A. Nazir, Gauge ambiguities imply jaynes- cummings physics remains valid in ultrastrong coupling qed, Nature communications 10, 499 (2019)

    2019

  7. [15]

    Stokes and A

    A. Stokes and A. Nazir, Implications of gauge freedom for nonrelativistic quantum electrodynamics, Rev. Mod. Phys. 94, 045003 (2022)

    2022

  8. [16]

    M. J. Bhaseen, J. Mayoh, B. D. Simons, and J. Keeling, Dynamics of nonequilibrium dicke models, Phys. Rev. A 85, 013817 (2012)

    2012

  9. [17]

    V. M. Bastidas, C. Emary, B. Regler, and T. Brandes, Nonequilibrium quantum phase transitions in the dicke model, Phys. Rev. Lett. 108, 043003 (2012)

    2012

  10. [18]

    Kirton and J

    P. Kirton and J. Keeling, Suppressing and restoring the dicke superradiance transition by dephasing and decay, Phys. Rev. Lett. 118, 123602 (2017)

    2017

  11. [19]

    Kirton and J

    P. Kirton and J. Keeling, Superradiant and lasing states in driven-dissipative dicke models, New Journal of Physics 20, 015009 (2018)

    2018

  12. [20]

    S. J. Masson and S. Parkins, Extreme spin squeezing in the steady state of a generalized dicke model, Phys. Rev. A 99, 023822 (2019)

    2019

  13. [21]

    J. T. Reilly, S. B. J¨ ager, J. D. Wilson, J. Cooper, S. Eg- gert, and M. J. Holland, Speeding up squeezing with a pe- riodically driven dicke model, Phys. Rev. Res. 6, 033090 (2024)

    2024

  14. [22]

    Baumann, C

    K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature 464, 1301 (2010)

    2010

  15. [23]

    Hamner, C

    C. Hamner, C. Qu, Y. Zhang, J. Chang, M. Gong, C. Zhang, and P. Engels, Dicke-type phase transition in a spin-orbit-coupled bose–einstein condensate, Nature Communications 5, 10.1038/ncomms5023 (2014)

    2014 doi

  16. [24]

    Klinder, H

    J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hem- merich, Dynamical phase transition in the open dicke model, Proceedings of the National Academy of Sciences 112, 3290 (2015)

    2015

  17. [25]

    Brookes, G

    P. Brookes, G. Tancredi, A. D. Patterson, J. Rahamim, M. Esposito, T. K. Mavrogordatos, P. J. Leek, E. Gi- nossar, and M. H. Szymanska, Critical slowing down in circuit quantum electrodynamics, Sci. Adv. 7, eabe9492 (2021)

    2021

  18. [26]

    Q.-M. Chen, M. Fischer, Y. Nojiri, M. Renger, E. Xie, M. Partanen, S. Pogorzalek, K. G. Fedorov, A. Marx, F. Deppe, et al., Quantum behavior of the duffing oscil- lator at the dissipative phase transition, Nat. Commun. 14, 2896 (2023)

    2023

  19. [27]

    J. M. Fink, A. Dombi, A. Vukics, A. Wallraff, and P. Domokos, Observation of the photon-blockade break- down phase transition, Phys. Rev. X 7, 011012 (2017)

    2017

  20. [28]

    Beaulieu, F

    G. Beaulieu, F. Minganti, S. Frasca, V. Savona, S. Fe- licetti, R. Di Candia, and P. Scarlino, Observation of first-and second-order dissipative phase transitions in a two-photon driven Kerr resonator, arXiv preprint arXiv:2310.13636 (2023)

    2023

  21. [29]

    Cai, Z.-D

    M.-L. Cai, Z.-D. Liu, W.-D. Zhao, Y.-K. Wu, Q.-X. Mei, Y. Jiang, L. He, X. Zhang, Z.-C. Zhou, and L.-M. Duan, Observation of a quantum phase transition in the quan- tum rabi model with a single trapped ion, Nature Com- munications 12, 10.1038/s41467-021-21425-8 (2021)

    2021 doi

  22. [30]

    S. R. K. Rodriguez, W. Casteels, F. Storme, N. Car- lon Zambon, I. Sagnes, L. Le Gratiet, E. Galopin, A. Lema ˆ ıtre, A. Amo, C. Ciuti, and J. Bloch, Probing a dissipative phase transition via dynamical optical hys- teresis, Phys. Rev. Lett. 118, 247402 (2017)

    2017

  23. [31]

    T. Fink, A. Schade, S. H¨ ofling, C. Schneider, and A. Imamoˇ glu, Signatures of a dissipative phase transi- tion in photon correlation measurements, Nat. Phys. 14, 365 (2018)

    2018

  24. [32]

    Z. Li, F. Claude, T. Boulier, E. Giacobino, Q. Glo- rieux, A. Bramati, and C. Ciuti, Dissipative phase transi- tion with driving-controlled spatial dimension and diffu- sive boundary conditions, Phys. Rev. Lett. 128, 093601 (2022)

    2022

  25. [33]

    Pallmann, K

    M. Pallmann, K. K¨ oster, Y. Zhang, J. Heupel, T. Eich- horn, C. Popov, K. Mølmer, and D. Hunger, Cavity- mediated collective emission from few emitters in a di- 7 amond membrane (2023), arXiv:2311.12723 [quant-ph]

    2023 arXiv

  26. [34]

    Emary and T

    C. Emary and T. Brandes, Chaos and the quantum phase transition in the dicke model, Phys. Rev. E 67, 066203 (2003)

    2003

  27. [35]

    D. Nagy, G. K´ onya, G. Szirmai, and P. Domokos, Dicke- model phase transition in the quantum motion of a bose- einstein condensate in an optical cavity, Phys. Rev. Lett. 104, 130401 (2010)

    2010

  28. [36]

    Reiter, T

    F. Reiter, T. L. Nguyen, J. P. Home, and S. F. Yelin, Cooperative breakdown of the oscillator blockade in the dicke model, Phys. Rev. Lett. 125, 233602 (2020)

    2020

  29. [37]

    Felicetti, J

    S. Felicetti, J. S. Pedernales, I. L. Egusquiza, G. Romero, L. Lamata, D. Braak, and E. Solano, Spectral collapse via two-phonon interactions in trapped ions, Physical Review A 92, 033817 (2015)

    2015

  30. [38]

    Puebla, M.-J

    R. Puebla, M.-J. Hwang, J. Casanova, and M. B. Plenio, Protected ultrastrong coupling regime of the two-photon quantum Rabi model with trapped ions, Physical Review A 95, 063844 (2017)

    2017

  31. [39]

    Casanova, R

    J. Casanova, R. Puebla, H. Moya-Cessa, and M. B. Plenio, Connecting nth order generalised quantum Rabi models: Emergence of nonlinear spin-boson coupling via spin rotations, npj Quantum Information 4, 47 (2018)

    2018

  32. [40]

    Felicetti, D

    S. Felicetti, D. Z. Rossatto, E. Rico, E. Solano, and P. Forn-D ´ ıaz, Two-photon quantum rabi model with su- perconducting circuits, Phys. Rev. A 97, 013851 (2018)

    2018

  33. [41]

    Felicetti, M.-J

    S. Felicetti, M.-J. Hwang, and A. Le Boit´ e, Ultrastrong- coupling regime of nondipolar light-matter interactions, Phys. Rev. A 98, 053859 (2018)

    2018

  34. [42]

    Gautier, A

    R. Gautier, A. Sarlette, and M. Mirrahimi, Combined Dissipative and Hamiltonian Confinement of Cat Qubits, PRX Quantum 3, 020339 (2022)

    2022

  35. [43]

    Ayyash, X

    M. Ayyash, X. Xu, S. Ashhab, and M. Mariantoni, Driven multiphoton qubit-resonator interactions, Phys. Rev. A 110, 053711 (2024)

    2024

  36. [44]

    X. Wang, A. Miranowicz, H.-R. Li, and F. Nori, Method for observing robust and tunable phonon blockade in a nanomechanical resonator coupled to a charge qubit, Physical Review A 93, 063861 (2016), publisher: Amer- ican Physical Society

    2016

  37. [45]

    S´ anchez Mu˜ noz, A

    C. S´ anchez Mu˜ noz, A. Lara, J. Puebla, and F. Nori, Hy- brid Systems for the Generation of Nonclassical Mechan- ical States via Quadratic Interactions, Physical Review Letters 121, 123604 (2018), publisher: American Physi- cal Society

    2018

  38. [46]

    Menotti, B

    M. Menotti, B. Morrison, K. Tan, Z. Vernon, J. Sipe, and M. Liscidini, Nonlinear Coupling of Linearly Uncoupled Resonators, Physical Review Letters 122, 013904 (2019)

    2019

  39. [47]

    Zatti, J

    L. Zatti, J. E. Sipe, and M. Liscidini, Generation of pho- ton pairs by spontaneous four-wave mixing in linearly uncoupled resonators, Physical Review A 107, 013514 (2023), arXiv:2301.08603 [physics, physics:quant-ph]

    2023 arXiv

  40. [48]

    Travˇ enec, Solvability of the two-photon Rabi Hamilto- nian, Physical Review A 85, 043805 (2012)

    I. Travˇ enec, Solvability of the two-photon Rabi Hamilto- nian, Physical Review A 85, 043805 (2012)

    2012

  41. [49]

    Duan, Y.-F

    L. Duan, Y.-F. Xie, D. Braak, and Q.-H. Chen, Two- photon Rabi model: analytic solutions and spectral col- lapse, Journal of Physics A: Mathematical and Theoret- ical 49, 464002 (2016)

    2016

  42. [50]

    Cong, X.-M

    L. Cong, X.-M. Sun, M. Liu, Z.-J. Ying, and H.-G. Luo, Polaron picture of the two-photon quantum Rabi model, Physical Review A 99, 013815 (2019)

    2019

  43. [51]

    R. J. A. Rico, F. H. Maldonado-Villamizar, and B. M. Rodriguez-Lara, Spectral collapse in the two-photon quantum rabi model, Phys. Rev. A 101, 063825 (2020)

    2020

  44. [52]

    L. Cong, S. Felicetti, J. Casanova, L. Lamata, E. Solano, and I. Arrazola, Selective interactions in the quantum Rabi model, Physical Review A 101, 032350 (2020)

    2020

  45. [53]

    Zou, X.-Y

    F. Zou, X.-Y. Zhang, X.-W. Xu, J.-F. Huang, and J.-Q. Liao, Multiphoton blockade in the two-photon Jaynes- Cummings model, Physical Review A 102, 053710 (2020)

    2020

  46. [54]

    Shen, J.-W

    L.-T. Shen, J.-W. Yang, Z.-R. Zhong, Z.-B. Yang, and S.-B. Zheng, Quantum phase transition and quench dy- namics in the two-mode Rabi model, Physical Review A 104, 063703 (2021)

    2021

  47. [55]

    Piccione, S

    N. Piccione, S. Felicetti, and B. Bellomo, Two-photon- interaction effects in the bad-cavity limit, Phys. Rev. A 105, L011702 (2022)

    2022

  48. [56]

    Banerjee, D

    P. Banerjee, D. Sharma, and A. B. Bhattacherjee, En- hanced photon squeezing in two-photon dicke model, Physics Letters A 446, 128287 (2022)

    2022

  49. [57]

    C. J. Villas-Boas and D. Z. Rossatto, Multiphoton Jaynes-Cummings Model: Arbitrary Rotations in Fock Space and Quantum Filters, Physical Review Letters 122, 123604 (2019)

    2019

  50. [58]

    P. Laha, P. A. A. Yasir, and P. Van Loock, Genuine non- Gaussian entanglement of light and quantum coherence for an atom from noisy multiphoton spin-boson interac- tions, Physical Review Research 6, 033302 (2024)

    2024

  51. [59]

    Crescente, M

    A. Crescente, M. Carrega, M. Sassetti, and D. Ferraro, Ultrafast charging in a two-photon dicke quantum bat- tery, Phys. Rev. B 102, 245407 (2020)

    2020

  52. [60]

    Delmonte, A

    A. Delmonte, A. Crescente, M. Carrega, D. Ferraro, and M. Sassetti, Characterization of a Two-Photon Quantum Battery: Initial Conditions, Stability and Work Extrac- tion, Entropy 23, 612 (2021)

    2021

  53. [61]

    Alushi, T

    U. Alushi, T. Ramos, J. J. Garc ´ ıa-Ripoll, R. Di Can- dia, and S. Felicetti, Waveguide qed with quadratic light- matter interactions, PRX Quantum 4, 030326 (2023)

    2023

  54. [62]

    J.-C. Tang, J. Zhao, H. Yang, J. Tian, P. Tang, S.- P. Wang, L. Lamata, and J. Peng, Deterministic two- photon C-Z gate with the two-photon quantum Rabi model (2024), arXiv:2411.03762 [quant-ph]

    2024 arXiv

  55. [63]

    Z.-J. Ying, S. Felicetti, G. Liu, and D. Braak, Critical Quantum Metrology in the Non-Linear Quantum Rabi Model, Entropy 24, 1015 (2022)

    2022

  56. [64]

    Garbe, I

    L. Garbe, I. L. Egusquiza, E. Solano, C. Ciuti, T. Coudreau, P. Milman, and S. Felicetti, Superradiant phase transition in the ultrastrong-coupling regime of the two-photon dicke model, Phys. Rev. A95, 053854 (2017)

    2017

  57. [65]

    Chen and Y.-Y

    X.-Y. Chen and Y.-Y. Zhang, Finite-size scaling analysis in the two-photon Dicke model, Physical Review A 97, 053821 (2018)

    2018

  58. [66]

    Garbe, P

    L. Garbe, P. Wade, F. Minganti, N. Shammah, S. Fe- licetti, and F. Nori, Dissipation-induced bistability in the two-photon dicke model, Scientific Reports 10, 10.1038/s41598-020-69704-6 (2020)

    2020 doi

  59. [67]

    J. Li, R. Fazio, and S. Chesi, Nonlinear dynamics of the dissipative anisotropic two-photon Dicke model, New Journal of Physics 24, 083039 (2022)

    2022

  60. [68]

    Li and S

    J. Li and S. Chesi, Routes to chaos in the balanced two- photon Dicke model with qubit dissipation, Physical Re- view A 109, 053702 (2024)

    2024

  61. [69]

    Kira and S

    M. Kira and S. W. Koch, Cluster-expansion representa- tion in quantum optics, Phys. Rev. A 78, 022102 (2008)

    2008

  62. [70]

    S´ anchez-Barquilla and R

    M. S´ anchez-Barquilla and R. Silva, Cumulant expansion for the treatment of light–matter interactions in arbitrary material structures, The Journal of chemical physics 152 8 (2020)

    2020

  63. [71]

    Fowler-Wright, K

    P. Fowler-Wright, K. B. Arnard´ ottir, P. Kirton, B. W. Lovett, and J. Keeling, Determining the validity of cu- mulant expansions for central spin models, Phys. Rev. Res. 5, 033148 (2023)

    2023

  64. [72]

    Johansson, P

    R. Johansson, P. Nation, Anubhav Vardhan, A. Pitch- ford, C. Granade, A. L. Grimsmo, M. Baden, P. Migda l, K. Fischer, D. Vasilyev, B. Criger, J. Feist, Alexbrc, Shahnawaz Ahmed, F. Henneke, D. Meiser, F. Ziem, N. Lambert, S. Krastanov, R. Brierley, R. Heeres, J. H¨ orsch, A. ...

    2017

  65. [73]

    Buˇ ca, Unified theory of local quantum many-body dy- namics: Eigenoperator thermalization theorems, Phys

    B. Buˇ ca, Unified theory of local quantum many-body dy- namics: Eigenoperator thermalization theorems, Phys. Rev. X 13, 031013 (2023)

    2023

  66. [74]

    Lo, Demystifying the spectral collapse in two-photon rabi model, Scientific Reports 10 (2020)

    C. Lo, Demystifying the spectral collapse in two-photon rabi model, Scientific Reports 10 (2020)

    2020

  67. [75]

    Roberts and A

    D. Roberts and A. A. Clerk, Driven-dissipative quantum kerr resonators: New exact solutions, photon blockade and quantum bistability, Phys. Rev. X10, 021022 (2020)

    2020

  68. [76]

    R. Rota, F. Minganti, C. Ciuti, and V. Savona, Quan- tum critical regime in a quadratically driven nonlinear photonic lattice, Phys. Rev. Lett. 122, 110405 (2019)

    2019

  69. [77]

    Shammah, S

    N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, Open quantum systems with local and col- lective incoherent processes: Efficient numerical simula- tions using permutational invariance, Phys. Rev. A 98, 063815 (2018)

    2018

  70. [78]

    S. Wang, S. Chen, and J. Jing, Effect of system energy on quantum signatures of chaos in the two-photon dicke model, Phys. Rev. E 100, 022207 (2019)

    2019

  71. [79]

    H.-G. Xu, J. Jin, G. D. M. Neto, and N. G. de Almeida, Universal quantum otto heat machine based on the dicke model (2023), arXiv:2308.06664 [quant-ph]

    2023 arXiv

  72. [80]

    Ding, Z.-K

    D.-S. Ding, Z.-K. Liu, B.-S. Shi, G.-C. Guo, K. Mølmer, and C. S. Adams, Enhanced metrology at the critical point of a many-body Rydberg atomic system, Nat. Phys. 18, 1447 (2022)

    2022

  73. [81]

    Beaulieu, F

    G. Beaulieu, F. Minganti, S. Frasca, M. Scigliuzzo, S. Felicetti, R. Di Candia, and P. Scarlino, Criticality- enhanced quantum sensing with a parametric super- conducting resonator, arXiv preprint arXiv:2409.19968 (2024)

    2024

  74. [82]

    Gravina, F

    L. Gravina, F. Minganti, and V. Savona, Critical schr¨ odinger cat qubit, PRX Quantum4, 020337 (2023). 9 EQUA TION OF MOTION FOR ONE-PHOTON LOSS For the two-photon Dicke model subject to the one- photon loss, the Heisenberg equations of motion for the cavity field and collecti...

    2023

  75. [83]

    ⟨X⟩ss = ⟨Y ⟩ss = ⟨n⟩ss = ⟨Jx⟩ss = ⟨Jy⟩ss = 0 and ⟨Jz⟩ss = −1

    The normal phase for λ ≤ λc, where all spins are in the ground state and there is no macroscopic occupation of the cavity. ⟨X⟩ss = ⟨Y ⟩ss = ⟨n⟩ss = ⟨Jx⟩ss = ⟨Jy⟩ss = 0 and ⟨Jz⟩ss = −1

  76. [84]

    ⟨X⟩ss = ⟨Y ⟩ss = ⟨n⟩ss = ⟨Jx⟩ss = ⟨Jz⟩ss ̸= 0 and ⟨Jy⟩ss = 0

    The superradiant phase for λ ≥ λc, where the spins are polarized, and there is a non-zero cavity occupation. ⟨X⟩ss = ⟨Y ⟩ss = ⟨n⟩ss = ⟨Jx⟩ss = ⟨Jz⟩ss ̸= 0 and ⟨Jy⟩ss = 0. Since there is no spin-related jump operators, the total spin is a conserved quantity, i.e. the strong sym...

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