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Fr\"ohlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that far-from-equilibrium boson condensation has a topological order parameter: the winding number of a probability curl-flux network, which jumps from 0 to 1 at the pump threshold.

desk verdict The topological order parameter is an interesting idea, but it is not proven: Eq. (17) depends on an absent appendix and numerical observation, so the paper needs major revision. read the letter →

arxiv 2506.00792 v3 pith:R2UY5MDT submitted 2025-06-01 cond-mat.stat-mech

Feihong Liu , Chase Slowey , Xuanhua Wang , Dangyuan Lei , Jeremie Torres , Zhiyue Lu , Zhedong Zhang This is my paper
classification cond-mat.stat-mech MSC 82C3182C2682C70 PACS 05.70.Ln64.60.-i05.40.-a
keywords nonequilibriumphasetransitionFröhlichcondensationcurlfluxnetworkwindingnumberdriven-dissipativebosonsdetailedbalancebreakinggraphtopologypolaritoncondensate
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 claims that a driven-dissipative gas of bosons, pumped and damped at different energies, condenses into a Fröhlich-coherent state that has a graph-topological order parameter invisible to the usual U(1) symmetry-breaking picture. The order parameter is the winding number of a probability curl-flux network on a hexagonal lattice whose axes are the lowest-mode occupation $n$ and the total particle number $N$. The paper develops a reduced quantum master equation for the lowest mode, solves it in steady state, and shows that above a pump threshold the steady-state probability currents form closed loops with winding number $Q = 1$, while below threshold $Q = 0$. If right, this gives a new way to detect and classify nonequilibrium condensates, including cavity polaritons, through counting statistics and flux circulation rather than through symmetry breaking alone.

What carries the argument

The load-bearing object is the probability curl flux network built on the hexagonal grid of the reduced master equation. In the large-volume limit the occupation variables $n$ and $N$ become continuous coordinates $(X,Y)$, and the net transition currents become a flux field $J$; extending $J$ to complex variables gives $J_c(z,z^*) = J_X - i J_Y$. The phase of $J_c$ winds around a zero $z_m$, and the integral $Q = \frac{1}{2\pi} \oint A \cdot dR$ with connection $A = \nabla S$ counts that winding. The paper also derives the elementary-triangle affinity $\Phi_{\triangledown} = \frac{(R+1)S}{R(S+1)}\left(1 + \frac{1}{\bar{n}}\right)$, which is nonzero and site-independent, proving detailed-balance breaking and fixing the loop directions on the graph.

What would settle it

Compute or measure the continuous-limit flux field $J_c$ for a broad range of pump rates, mode counts, and temperature-dependent Planck factors: if the zero $z_m$ disappears or the contour integral in Eq. (17)/(19) fails to give an integer for some parameters above threshold, the winding-number order parameter is not generic. Experimentally, a delayed photon-coincidence measurement across the threshold should either show the predicted $g_2(\tau)$ oscillations or rule them out.

Watch

Extended reading notes

Core claim

The central claim is that Fröhlich condensation is a topological phase transition in the space of quantum fluctuations: when the energy pump exceeds the threshold $R_c + M S_c$, the steady-state currents on the $(n,N)$-occupation graph organize into a curl flux network around a singularity at which the complexified flux field $J_c(z,z^*)$ vanishes. Tracing a loop around that singularity rotates the phase of $J_c$ by $2\pi$, giving integer winding number $Q = 1$; below threshold the flux lines are open and $Q = 0$. This number is proposed as a new order parameter for the nonequilibrium transition, one that is not attainable from symmetry breaking, and the paper shows that the polariton condensate obeys the same reduced equation, placing it in this nonequilibrium-condensate class.

Load-bearing premise

The winding number is well defined only if, in the continuous limit, the flux field has an isolated zero with nonzero circulation, as asserted from numerical plots rather than proven for the full model; the result also leans on the uniform-thermal-occupation approximation for the environmental Planck factors.

Editorial extensions

If this is right

  • Above the pump threshold the condensation fraction jumps and the $n$–$N$ fluctuations become strongly correlated, with a ring of maximal curl flux around the peak of the occupation distribution.
  • A delayed photon-coincidence counting signal proportional to $M g_2(\tau)$ should show oscillations in the nonequilibrium-condensate phase, reflecting the cycling nature of the flux.
  • THz time-domain spectroscopy should reveal transient spectral fingerprints such as mode softening and linewidth narrowing and allow reconstruction of cyclic current patterns.
  • Polariton condensates, sharing the same reduced equation structure, should lie in the nonequilibrium-condensate regime with graph topology $Q = 1$, distinct from atomic BECs with vanishing flux.

Reading between the lines

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

  • One might test whether the same winding-number criterion classifies other driven-dissipative many-body phases whose probability currents wind around a fixed point, not just bosonic condensation.
  • The uniform-thermal-occupation approximation for the environmental Planck factors likely sets the robustness range of $Q$; testing a frequency-dependent Planck factor could show whether the topological transition survives realistic phonon dispersions.
  • Time-resolved counting at pump rates just below $R_c$ could reveal the flux network's loop structure as a precursor to full condensation, since the summit-crater landscape of $|J_c|$ appears before the sharp jump in the condensation fraction.
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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

5 major / 5 minor

Summary. The paper studies a driven-dissipative boson model with M excited modes and a lowest mode, modeled by a Lindblad master equation. Restricting to diagonal reduced density matrix elements P_{n,N}, the authors derive a rate equation on a 2D lattice, identify steady-state probability currents, and show detailed balance breaking through nonzero loop affinities. In the large-volume limit they obtain a Fokker-Planck equation and a probability curl flux J_c; they claim J_c has an isolated zero with nonzero circulation in the condensed phase, leading to winding number Q=1 as a new order parameter. They classify NCB, thermal, and BEC phases according to ODLRO, symmetry, homotopy, and winding number, and propose experimental signatures in delayed photon-coincidence counting and THz spectroscopy.

Significance. If the central claims hold, the winding number of the flux network would be a genuinely new, nonlocal order parameter for Fröhlich-type nonequilibrium condensation, distinct from U(1) symmetry breaking and potentially observable through coherent oscillations in coincidence counting. The paper contains several useful explicit results: the diagonal rate equation (2), the closed-form affinity (8), the mean-field threshold estimate R_c + M S_c = C, and numerical steady-state distributions. The graph-theoretic cycle decomposition and relation to Schnakenberg network theory are appropriate tools. However, the topological order parameter is not established: Eq. (17) is deferred to an absent appendix, and the ODLRO calculation is explicitly deferred; as written the manuscript is a promising but incomplete derivation.

major comments (5)
  1. [§VII, Eq. (17)] The existence of an isolated zero z_m of J_c with nonzero circulation is the load-bearing input for Q in Eq. (19), but it is asserted rather than proven: the derivation is relegated to Appendix E, which is not present in this version. A divergence-free steady-state vector field need not have any isolated zeros, and when it does the zero can be a saddle (index -1) or a center (index +1); Fig. 2 for a single parameter set cannot rule out parameter-dependent changes. Please provide the proof or, if only numerical evidence is available, state the property as a conjecture and test it systematically with a parameter sweep, a finite-graph analogue of Q, and a sensitivity check of the uniform-thermal approximation in Eq. (3).
  2. [§VI, Eq. (10) and footnote 52] The paper identifies the condensed phase with ODLRO and contrasts it with symmetry-breaking BEC, but the ODLRO value Eq. (10) is stated without derivation, and footnote 52 says the off-diagonal equations of motion will be presented elsewhere. Since ODLRO is central to the claim that NCB is a coherent phase, the derivation of the off-diagonal reduced matrix elements, or an explicit statement that Eq. (10) is an assumption, must be included before the phase classification in Table I can be accepted.
  3. [§V–§VII, Appendices C–E] The main technical steps—the derivation of the rate-equation coefficients in Eq. (3), the graph-current decomposition Eq. (6), the affinity computation Eq. (8), the continuous limit Eqs. (12)–(14), and Theorem 1 in Eq. (20)—are all relegated to appendices that are absent from the submitted text. These steps connect the microscopic model to the flux network and to Eq. (17), so they cannot be checked; at minimum the appendices must be included or the steps reproduced in the main text.
  4. [§VII, Fig. 2] The claim that Q changes from 0 to 1 exactly at the condensation threshold is not demonstrated. Fig. 2 presents one above-threshold and one below-threshold parameter set, which shows a difference but not that the topological quantum number tracks the phase boundary, nor that it is independent of the arbitrary contour in Eq. (19). A sweep of R (or S) with Q computed from the discrete currents of Eq. (6) and from the continuum J_c would be needed to support the 'order parameter' terminology.
  5. [§VII, Eq. (19)] The definitions of A and F_{\mu\nu} are inconsistent: if A = \nabla S, then F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu vanishes identically, and the integral (1/4\pi)\int F would be zero, contradicting the claimed Q=1. The standard winding number uses the phase S itself and does not require a curvature tensor; please replace Eq. (19) with a correct definition or explain the intended meaning.
minor comments (5)
  1. [§IX, last paragraph] There is a typo: 'conisderable' should be 'considerable'.
  2. [Table I] The entries in the 'Homotopy' column, such as 'Z±1' and '×0', are not defined; please specify the fundamental group or winding-number value for each phase explicitly.
  3. [§VII, Eq. (11)] The transformation from (x,y) to (X,Y) appears to be non-orthogonal; please clarify the metric used in the vector calculus operations in Eqs. (13)–(15).
  4. [Reference [32]] The author list of Ref. [32] contains a malformed name ('C. Bennenh, ei'); please correct it.
  5. [Throughout] The notation π1[e] and π1[∅] in Eq. (18) is nonstandard; please define the topological spaces e and ∅ and state what 'winding number 0' means for them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the winding-number order parameter is computed from the model's steady-state flux, not fitted or defined in terms of itself.

full rationale

The central object, the winding number Q of Eq. (19), is computed from the steady-state probability flux J_c(z,z*) defined in Eq. (15), which in turn follows from the reduced master equation (2) through the continuous-limit expansions (12). No parameter is fitted to a subset of data and then 'predicted' in a closely related quantity; the condensation threshold is estimated from the mean equations (5), and Q is a derived diagnostic of the same model. The paper's self-citations [61,62] introduce the curl-flux/network framework, but the relevant flux-current expressions (Eqs. (6)-(8) and (20)-(21)) are re-derived here from Eq. (2), so the argument does not reduce to those citations. The main weakness—Eq. (17)'s assertion that J_c has an isolated zero with nonzero circulation, deferred to an apparently absent Appendix E and illustrated in Fig. 2—is a verification gap or a correctness risk, not circularity: it is a claim about the computed J_c, not an input assumed to force the conclusion. No equation defines the order parameter in terms of itself, and no known result is merely renamed as a prediction. Therefore no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The model treats R, S, alpha, nbar, and M as physical inputs, not fitted values, though several are chosen by hand for plots and estimates. No new particles, forces, or dimensions are introduced; the winding number Q is a derived quantity. The audit is incomplete because appendices containing key derivations are missing.

free parameters (4)
  • Pump rate ratio R/S = R=3S in Fig. 1
    Chosen for representative plots; no fitting to data. The threshold condition Rc+MSc=C depends on R and S.
  • Internal conversion rate alpha = alpha=0.05 in experimental estimate
    Assumed for THz vibrations; used only for an order-of-magnitude pump power estimate.
  • Mode count M = M=100 in experimental estimate
    Assumed for the power estimate; not measured for a specific system.
  • Thermal occupation nbar = nbar~10 at ambient temperature
    Assumed uniform for all modes in Eq. (3); central to the simplified coefficients K and H.
assumptions (4)
  • domain assumption Lindblad-form master equation Eq. (1) with linear and nonlinear superoperators captures the driven-dissipative boson dynamics.
    Assumes Markovian environment and the specific nonlinear internal-conversion form, stated in Section II.
  • domain assumption Uniform thermal occupation nbar_j0 ~ nbar and simplified coefficients A, B, K, H in Eq. (3).
    Introduced in Section III to close Eq. (2); not derived from a microscopic model for general environments.
  • standard math Large-volume continuum limit with exponential shift operators in Eq. (12) and smooth steady-state P(X,Y).
    Standard asymptotic expansion, but requires smoothness of P; used to derive Eq. (13)-(15).
  • ad hoc to paper Neglect of off-diagonal reduced density matrix elements; ODLRO deferred to future work (footnote 52).
    The paper states the off-diagonal equation of motion will be presented elsewhere, so the diagonal dynamics Eq. (2) is treated as closed.

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Cite this review

Pith. "Pith review of Fr\"ohlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties." pith.science (2026). https://pith.science/paper/R2UY5MDT

@misc{pith2026250600792,
  author       = {Pith},
  title        = {Pith review of: Fr\"ohlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R2UY5MDT}},
  note         = {Machine review of arXiv:2506.00792}
}
read the original abstract

Nonequilibrium condensates of bosons subject to energy pump and dissipation are investigated, manifesting the Fr\"ohlich coherence proposed in 1968. A quantum theory is developed to capture such a nonequilibrium nature, yielding a certain graphic structure arising from the detailed-balance breaking. The results show a network of probability curl fluxes that reveals a graph topology. The winding number associated with the flux network is thus identified as a new order parameter for the phase transition towards the Fr\"ohlich condensation (FC), not attainable by the symmetry breaking. Our work demonstrates a global property of the FCs, in significant conjunction with the coherence of cavity polaritons that may exhibit robust cooperative phases driven far from equilibrium.

Figures

Figures reproduced from arXiv: 2506.00792 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic illustration of phonon dispersion [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Curl flux (white arrows) J [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of the loop affinity Φ. (a) The affinities [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Works this paper leans on

70 extracted references · 69 canonical work pages

  1. [1]

    Perez-Martin, T

    E. Perez-Martin, T. Beranger, L. Bonnet, F. Teppe, A. Lisauskas,et al.,Unveiling long-range forces in light- harvesting proteins: Pivotal roles of temperature and light, Sci. Adv.11, eadv0346 (2025)

    work page 2025

  2. [2]

    A. V. Zasedatelev, A. V. Baranikov, D. Urbonas, F. Scafirimuto, U. Scherf, T. St¨ oferle, R. F. Mahrt, and P. G. Lagoudakis,A room-temperature organic polariton transistor, Nat. Photonics13, 378 (2019)

    work page 2019

  3. [3]

    W. Xu, A. A. Bagrov, F. T. Chowdhury, L. D. Smith, D. R. Kattnig, H. J. Kappen, and M. I. Katsnelson, Fr¨ ohlich versus Bose-Einstein condensation in pumped bosonic systems, Phys. Rev. Research7, 023111 (2025)

    work page 2025

  4. [4]

    A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valleet al.,Collective fluid dynamics of a polariton condensate in a semiconductor microcavity, Nature457, 291 (2009)

    work page 2009

  5. [5]

    Xiong,Molecular vibrational polariton dynamics: What can polaritons do?, Acc

    W. Xiong,Molecular vibrational polariton dynamics: What can polaritons do?, Acc. Chem. Res.56, 776 (2023)

    work page 2023

  6. [6]

    Liang, P

    S. Liang, P. De Los Rios, and D. M. Busiello, Thermodynamic bounds on symmetry breaking in linear and catalytic biochemical systems, Phys. Rev. Lett.132, 228402 (2024)

    work page 2024

  7. [7]

    Sheng, J.-W

    J. Sheng, J.-W. Mei, L. Wang, X. Xu, W. Jiang,et al., Bose-Einstein condensation of a two-magnon bound state in a spin-1 triangular lattice, Nat. Mater.24, 544 (2025)

    work page 2025

  8. [8]

    Fr¨ ohlich,Long-range coherence and energy storage in biological systems, Int

    H. Fr¨ ohlich,Long-range coherence and energy storage in biological systems, Int. J. Quantum Chem.2, 641 (1968)

    work page 1968

Show all 70 references
  1. [9]

    Preto,Semi-classical statistical description of Fr¨ ohlich condensation, J

    J. Preto,Semi-classical statistical description of Fr¨ ohlich condensation, J. Bio. Phys.43, 167 (2017)

    2017

  2. [10]

    T. M. Wu and S. Austin,Bose-Einstein condensation in biological systems, J. Theor. Bio.71, 209 (1978)

    1978

  3. [11]

    I. V. Lundholm, H. Rodilla, W. Y. Wahlgren, A. Duelli, G. Bourenkov,et al.,Terahertz radiation induces non-thermal structural changes associated with Fr¨ ohlich condensation in a protein crystal, Struct. Dyn.2, 054702 (2015)

    2015

  4. [12]

    Prigogine and R

    I. Prigogine and R. Lefever,Theory of dissipative structures, Synergetics, pp. 124-135, eds. H. Haken, (Vieweg+Teubner Verlag, Wiesbaden, 1973)

    1973

  5. [13]

    F. A. Popp, W. Nagl, K. H. Li, W. Scholz, O. Weing¨ artner, and R. Wolf,Biophoton emission: New 7 evidence for coherence and DNA as source, Cell Biophys. 6, 33 (1984)

    1984

  6. [14]

    T. M. Wu and S. J. Austin,Fr¨ ohlich’s model of Bose condensation in biological systems, J. Bio. Phys.9, 97 (1981)

    1981

  7. [15]

    J. A. Tuszynski, R. Paul, R. Chatterjee, and S. R. Sreenivasan,Relationship between Fr¨ ohlich and Davydov models of biological order, Phys. Rev. A30, 2666 (1984)

    1984

  8. [16]

    Chandran, T

    A. Chandran, T. Iadecola, V. Khemani, and R. Moessner, Quantum many-body scars: A quasiparticle perspective, Ann. Rev. Condens. Matter Phys.14, 443 (2022)

    2022

  9. [17]

    Hameroff,Quantum computation in brain micro- tubules? The Penrose-Hameroff ‘Orch OR’ model of consciousness, Phil

    S. Hameroff,Quantum computation in brain micro- tubules? The Penrose-Hameroff ‘Orch OR’ model of consciousness, Phil. Trans. R. Soc. A356, 1869 (1998)

    1998

  10. [18]

    J. Song, S. Ghosh, X. Deng, C. Li, Q. Shang,et al.,Room-temperature continuous-wave pumped exciton polariton condensation in a perovskite microcavity, Sci. Adv.11, eadr1652 (2025)

    2025

  11. [19]

    Schneider, A

    C. Schneider, A. Rahimi-Iman, N. Y. Kim, J. Fischer, I. G. Savenko,et al.,An electrically pumped polariton laser, Nature497, 348 (2013)

    2013

  12. [20]

    Galego, F

    J. Galego, F. J. Garcia-Vidal, and J. Feist,Suppressing photochemical reactions with quantized light fields, Nat. Commun.7, 13841 (2016)

    2016

  13. [21]

    Hagan, S

    S. Hagan, S. R. Hameroff, and J. A. Tuszynski,Quantum computation in brain microtubules: Decoherence and biological feasibility, Phys. Rev. E65, 061901 (2002)

    2002

  14. [22]

    J. R. Reimers, L. K. McKemmish, R. H. McKenzie, A. E. Mark, and N. S. Hush,Weak, strong, and coherent regimes of Fr¨ ohlich condensation and their applications to terahertz medicine and quantum consciousness, PNAS 106, 4219 (2009)

    2009

  15. [23]

    Z. D. Zhang, G. S. Agarwal, and M. O. Scully,Quantum fluctuations in the Fr¨ ohlich condensate of molecular vibrations driven far from equilibrium, Phys. Rev. Lett. 122, 158101 (2019)

    2019

  16. [24]

    Kasprzak, M

    J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun,et al.,Bose-Einstein condensation of exciton polaritons, Nature443, 409 (2006)

    2006

  17. [25]

    R. Su, A. Fieramosca, Q. Zhang, H. S. Nguyen, E. Deleporte, Z. Chen, D. Sanvitto, T. C. H. Liew, and Q. Xiong,Perovskite semiconductors for room-temperature exciton-polaritonics, Nat. Mater.20, 1315 (2021)

    2021

  18. [26]

    Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L. N. Pfeiffer, K. West, D. W. Snoke, and K. A. Nelson, Bose-Einstein condensation of long-lifetime polaritons in thermal equilibrium, Phys. Rev. Lett.118, 016602 (2017)

    2017

  19. [27]

    Nardecchia, J

    I. Nardecchia, J. Torres, M. Lechelon, V. Giliberti, M. Ortolani,et al.,Out-of-equilibrium collective oscillation as phonon condensation in a model protein, Phys. Rev. X8, 031061 (2018)

    2018

  20. [28]

    J. Miao, P. Charalambous, J. Kirz, and D. Sayre, Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens, Nature400, 342 (1999)

    1999

  21. [29]

    D. A. Turton, H. M. Senn, T. Harwood, A. J. Lapthorn, E. M. Ellis, and K. Wynne,Terahertz underdamped vibrational motion governs protein-ligand binding in solution, Nat. Commun.5, 3999 (2014)

    2014

  22. [30]

    Lechelon, Y

    M. Lechelon, Y. Meriguet, M. Gori, S. Ruffenach, I. Nardecchia,et al.,Experimental evidence for long- distance electrodynamic intermolecular forces, Sci. Adv. 8, eabl5855 (2022)

    2022

  23. [31]

    H. Deng, G. Weihs, C. Santori, J. Bloch, and Y. Yamamoto,Condensation of semiconductor microcavity exciton polaritons, Science298, 199 (2002)

    2002

  24. [32]

    Struve, C

    M. Struve, C. Bennenh, ei, H. P. Adl, K. W. Song, H. Shan,et al.,Room-temperature polariton condensate in a two-dimensional hybrid perovskite, arXiv:2408.13677

    arXiv

  25. [33]

    J. D. Plumhof, T. St¨ oferle, L. Mai, U. Scherf, and R. F. Mahrt,Room-temperature Bose-Einstein condensation of cavity exciton-polaritons in a polymer, Nat. Mater.13, 247 (2014)

    2014

  26. [34]

    H. Deng, H. Haug, and Y. Yamamoto,Exciton-polariton Bose-Einstein condensation, Rev. Mod. Phys.82, 1489 (2010)

    2010

  27. [35]

    Pannir-Sivajothi, J

    S. Pannir-Sivajothi, J. A. Campos-Gonzalez-Angulo, L. A. Mart ´ ınez-Mart ´ ınez, S. Sinha, and J. Yuen-Zhou, Driving chemical reactions with polariton condensates, Nat. Commun.13, 1645 (2022)

    2022

  28. [36]

    Schwendimann and A

    P. Schwendimann and A. Quattropani,Statistics of the polariton condensate, Phys. Rev. B77, 085317 (2008)

    2008

  29. [37]

    Z. D. Zhang, S. Zhao, and D. Lei,Quantum statistical theory for an exciton-polariton condensate: Fluctuations and coherence, Phys. Rev. B106, L220306 (2022)

    2022

  30. [38]

    Wang and J

    X. Wang and J. Wang,Full quantum theory of nonequi- librium phonon condensation and phase transition, Phys. Rev. B106, L220103 (2022)

    2022

  31. [39]

    Klaas, E

    M. Klaas, E. Schlottmann, H. Flayac, F. P. Laussy, F. Gericke,et al.,Photon-number-resolved measurement of an exciton-polariton condensate, Phys. Rev. Lett.121, 047401 (2018)

    2018

  32. [40]

    F. P. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald,Spontaneous coherence buildup in a polariton laser, Phys. Rev. Lett.93, 016402 (2004)

    2004

  33. [41]

    Sethi, M

    G. Sethi, M. Cuma, and F. Liu,Excitonic condensate in flat valence and conduction bands of opposite chirality, Phys. Rev. Lett.130, 186401 (2023)

    2023

  34. [42]

    Pieczarka, E

    M. Pieczarka, E. Estrecho, M. Boozarjmehr, O. Bleu, M. Steger,et al.,Observation of quantum depletion in a nonequilibrium exciton-polariton condensate, Nat. Commun.11, 429 (2020)

    2020

  35. [43]

    Byrnes, N

    T. Byrnes, N. Y. Kim, and Y. Yamamoto,Exciton- polariton condensates, Nat. Phys.10, 803 (2014)

    2014

  36. [44]

    L. V. Butov,A polariton laser, Nature447, 540 (2007)

    2007

  37. [45]

    L. V. Butov and A. V. Kavokin,The behaviour of exciton- polaritons, Nat. Photonics6, 2 (2012)

    2012

  38. [46]

    Deveaud-Pledran,The behaviour of exciton- polaritons, Nat

    B. Deveaud-Pledran,The behaviour of exciton- polaritons, Nat. Photonics6, 205 (2012)

    2012

  39. [47]

    Estrecho, T

    E. Estrecho, T. Gao, N. Bobrovska, M. D. Fraser, M. Steger,et al.,Single-shot condensation of exciton polaritons and the hole burning effect, Nat. Commun.9, 2944 (2018)

    2018

  40. [48]

    C. N. Yang,Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors , Rev. Mod. Phys.34, 694 (1962)

    1962

  41. [49]

    A. J. Leggett,Bose-Einstein condensation in the alkali gases: Some fundamental concepts, Rev. Mod. Phys.73, 307 (2001)

    2001

  42. [50]

    V. V. Albert, and L. Jiang,Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014)

    2014

  43. [51]

    L. M. Sieberer, M. Buchhold, and S. Diehl,Keldysh field theory for driven open quantum systems, Rep. Prog. Phys.79, 096001 (2016)

    2016

  44. [52]

    The equation of motion for the off-diagonal elements will be presented elsewhere. 8

  45. [53]

    ¯nj0 = [e(ωj −ω0)/T −1] −1 is the Planck factor

    The coefficientsA n,N = PM j=1⟨nj + 1⟩ n,N ,B n,N =PM j=1⟨nj⟩n,N ,K n,N = PM j=1(¯nj0 + 1)⟨nj⟩n,N ,H n,N =PM j=1 ¯nj0⟨nj + 1⟩n,N where⟨· · · ⟩n,N denotes the mean, givenn, N. ¯nj0 = [e(ωj −ω0)/T −1] −1 is the Planck factor

  46. [54]

    T. L. Hill and O. Kedem,Studies in irreversible thermodynamics. 3. Models for steady state and active transport across membranes, J. Theor. Biol.10, 399 (1966)

    1966

  47. [55]

    T. L. Hill and Y.-D. Chen,Stochastics of cycle completions (fluxes) in biochemical kinetic diagrams, PNAS72, 1291 (1975)

    1975

  48. [56]

    Schnakenberg,Network theory of microscopic and macroscopic behavior of master equation systems, Rev

    J. Schnakenberg,Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys.48, 571 (1976)

    1976

  49. [57]

    Wang,Landscape and flux theory of nonequilibrium dynamical systems with application to biology, Adv

    J. Wang,Landscape and flux theory of nonequilibrium dynamical systems with application to biology, Adv. Phys. 64, 1 (2015)

    2015

  50. [58]

    J. Wang, L. Xu, and E. Wang,Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations, PNAS105, 12271 (2008)

    2008

  51. [59]

    Qian,Phosphorylation energy hypothesis: Open chemical systems and their biological functions, Ann

    H. Qian,Phosphorylation energy hypothesis: Open chemical systems and their biological functions, Ann. Rev. Phys. Chem.58, 113 (2007)

    2007

  52. [60]

    T. L. Hill and R. V. Chamberlin,Extension of the thermodynamics of small systems to open metastable states: An example, PNAS95, 12779 (1998)

    1998

  53. [61]

    Z. D. Zhang and J. Wang,Curl flux, coherence, and pop- ulation landscape of molecular systems: Nonequilibrium quantum steady state, energy (charge) transport, and thermodynamics, J. Chem. Phys.140, 245101 (2014)

    2014

  54. [62]

    Z. D. Zhang and J. Wang,Landscape, kinetics, paths and statistics of curl flux, coherence, entanglement and energy transfer in nonequilibrium quantum systems, New J. Phys.17, 043053 (2015)

    2015

  55. [63]

    L. Wang, Z. Wang, C. Wang, and J. Ren,Cycle flux ranking of network analysis in quantum thermal devices, Phys. Rev. Lett.128, 067701 (2022)

    2022

  56. [64]

    M. O. Scully,Condensation ofNbosons and the laser phase transition analogy, Phys. Rev. Lett.82, 3927 (1999)

    1999

  57. [65]

    The space of circles is isomorphic to the Abelian group, as seen from the product operationϕ 2 ·ϕ 1 that is commutative.e inθ forms an irreducible representation of the Abelian group where the integerncounts the number of rounds for a periodic boundary condition

  58. [66]

    V. Y. Shishkov, E. S. Andrianov, A. V. Zasedatelev, P. G. Lagoudakis, and Y. E. Lozovik,Exact analytical solution for the density matrix of a nonequilibrium polariton Bose-Einstein condensate, Phys. Rev. Lett.128, 065301 (2022)

    2022

  59. [67]

    T. L. Hill,Thermodynamics for Chemists and Biologists (Addison-Wesley, Boston, 1968)

    1968

  60. [68]

    Polettini and M

    M. Polettini and M. Esposito,Transient fluctuation the- orems for the currents and initial equilibrium ensembles, J. Stat. Mech. (2014) P10033

    2014

  61. [69]

    X. L. Qi and S.-C. Zhang,Topological insulators and superconductors, Rev. Mod. Phys.83, 1057 (2011)

    2011

  62. [70]

    E. Tang, J. Agudo-Canalejo, and R. Golestanian, Topology protects chiral edge currents in stochastic systems, Phys. Rev. X11, 031015 (2021)

    2021

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