pith. sign in

arxiv: 2512.20459 · v2 · submitted 2025-12-23 · ❄️ cond-mat.quant-gas

Opening a gap in the dispersion of the collective excitations of a driven-dissipative condensate subject to an external coherent drive

E. Stazzu , G. A. P. Sacchetto , I. Carusotto This is my paper

Pith reviewed 2026-05-16 20:12 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas
keywords driven-dissipative condensatephase lockingcollective excitationsdispersion gapGoldstone modedynamical instabilitysupersolid modulationexciton-polariton
0
0 comments X

The pith

An external coherent drive opens a gap in the collective excitation spectrum of a driven-dissipative condensate by fixing its phase.

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

The paper builds a minimal model showing that an added coherent phase-locking drive can open a gap in the dispersion of collective excitations in a driven-dissipative condensate. This happens because the drive fixes the otherwise free phase, replacing the usual gapless Goldstone mode with either a purely imaginary gap or one that has a finite real part. The authors map the full phase diagram in drive amplitude and frequency, marking regions where the spectrum stays gapped, where the gap closes, and where finite-wavevector instabilities appear that favor supersolid-like spatial modulation. A sympathetic reader cares because the same framework applies directly to exciton-polariton condensates and to a range of optical parametric oscillators or lasers that use external injection.

Core claim

We build a minimal theoretical model to describe the opening of a gap in the dispersion of the collective excitations of a driven-dissipative condensate when the condensate phase is fixed by an additional coherent phase-locking drive. We map out the phase diagram as a function of the amplitude and frequency of the coherent drive, identifying distinct regions corresponding to different steady-state regimes. For each region, we analyze the dispersion of the collective excitations and determine whether the spectrum is gapped, with either a purely imaginary gap or a finite real part. When the coherent drive is unable to lock the condensate phase, a gapless Goldstone mode is recovered. Within the

What carries the argument

The additional coherent phase-locking drive term that fixes the condensate phase, combined with linearization of the minimal mean-field equations around the resulting steady state.

If this is right

  • When the coherent drive successfully locks the phase, the excitation spectrum acquires a gap that is either purely imaginary or carries a finite real part.
  • When the drive is too weak to lock the phase, the gapless Goldstone mode is recovered.
  • Distinct regions of the drive-amplitude versus frequency plane show finite-wavevector dynamical instability, where the condensate tends to develop a supersolid-like spatial modulation.
  • The same phase diagram and gap analysis apply to spatially extended optical parametric oscillators and laser devices under external injection.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • External phase locking could be used as a practical knob to suppress long-wavelength fluctuations in extended driven-dissipative systems.
  • The instability regions suggest a route to engineer supersolid order by tuning only the frequency and strength of an injected beam.
  • Similar gap-opening effects may appear in other driven open systems whenever an external field breaks the continuous phase symmetry.

Load-bearing premise

The minimal mean-field model together with linearization around the steady state captures the essential physics without higher-order fluctuations or spatial inhomogeneities changing the gap or instability thresholds.

What would settle it

Measure the dispersion relation of collective excitations in an exciton-polariton condensate while sweeping the amplitude and frequency of an added coherent drive; the gap should open and close exactly where the calculated phase diagram predicts, and finite-wavevector instabilities should appear at the calculated thresholds.

Figures

Figures reproduced from arXiv: 2512.20459 by E. Stazzu, G. A. P. Sacchetto, I. Carusotto.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of the steady-state intensity [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Left: flow lines of the dynamic evolution of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Flow lines of the dynamic evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Flow lines of the dynamic evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Left: absence of real gap in the Bogoliubov spectrum for the stable solution in the small ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Floquet-Bogoliubov spectrum around a limit cycle in a stable (left) configurations, which turns unstable (right) as the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fig.5. As a main feature, the gap at [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Phase diagram for the stationary solutions in the pa [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fig.3. As the radius of the limit cycle grows from [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Phase diagram in the parameter space ( [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Top: flow lines describing the dynamic evolution of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

We build a minimal theoretical model to describe the opening of a gap in the dispersion of the collective excitations of a driven-dissipative condensate when the condensate phase is fixed by an additional coherent phase-locking drive. We map out the phase diagram as a function of the amplitude and frequency of the coherent drive, identifying distinct regions corresponding to different steady-state regimes. For each region, we analyze the dispersion of the collective excitations and determine whether the spectrum is gapped, with either a purely imaginary gap or a finite real part. When the coherent drive is unable to lock the condensate phase, a gapless Goldstone mode is recovered. Within the same phase diagram, we further identify regions of finite-wavevector dynamical instability, where the condensate tends to develop a supersolid-like spatial modulation. While our theoretical framework is directly related to recent experiments with exciton-polariton condensates, it can be applied to describe the effect of external injection also in a variety of spatially extended optical parametric oscillators or laser devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs a minimal mean-field model for a driven-dissipative condensate subject to an additional coherent phase-locking drive. It maps the steady-state phase diagram versus drive amplitude and frequency, linearizes to obtain the collective-mode dispersion in each regime, and shows that phase locking produces a gapped spectrum (purely imaginary gap or finite real part) while failure to lock recovers the gapless Goldstone mode. Regions of finite-wavevector dynamical instability that favor supersolid-like spatial modulations are also identified. The framework is related to exciton-polariton experiments and stated to apply to optical parametric oscillators and lasers.

Significance. If the minimal model holds, the work supplies a transparent derivation of how an external coherent drive gaps the collective excitations by fixing the condensate phase, recovering the expected Goldstone mode in the unlocked regime and flagging instability thresholds. This provides a compact, falsifiable phase diagram with direct experimental relevance to polariton condensates and related driven-dissipative systems. The strength is the direct construction from standard ingredients that yields the gapped/gapless distinction without extra fitting parameters.

minor comments (2)
  1. [Collective excitations analysis] The linearization procedure around the steady state is central to the dispersion results; a short appendix or inline derivation of the Bogoliubov-de Gennes matrix would improve reproducibility.
  2. [Phase diagram] The boundaries of the phase diagram are described qualitatively; explicit analytic expressions or scaling relations for the locking threshold versus drive frequency would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending acceptance. The referee's summary accurately captures the construction of the minimal mean-field model, the phase diagram, the gapped versus gapless collective-mode spectra, and the identification of finite-wavevector instabilities.

Circularity Check

0 steps flagged

Minimal model constructed from standard ingredients with no circular reduction

full rationale

The paper constructs a minimal mean-field model for a driven-dissipative condensate plus coherent phase-locking drive directly from standard ingredients. Steady-state regimes are obtained by solving the equations of motion versus drive amplitude and frequency; collective-mode dispersion follows by linearization around each steady state. The gapped spectrum appears precisely when the drive locks the phase, while a gapless Goldstone mode is recovered otherwise; these outcomes are direct algebraic consequences of the model equations rather than fits or self-referential definitions. No load-bearing self-citation, imported uniqueness theorem, or ansatz smuggling is required for the central claim. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard mean-field and linearization assumptions common to the driven-dissipative condensate literature without introducing new free parameters or entities beyond the drive amplitude and frequency treated as external controls.

free parameters (2)
  • amplitude of coherent drive
    Treated as a tunable parameter that defines the phase diagram regions
  • frequency of coherent drive
    Treated as a tunable parameter that defines the phase diagram regions
axioms (2)
  • domain assumption The condensate can be described by a mean-field wavefunction whose steady state is found by balancing drive, dissipation, and interactions
    Standard starting point for driven-dissipative condensates
  • domain assumption Collective excitations are obtained by linearizing the equations of motion around the steady state
    Standard technique for dispersion relations in the field

pith-pipeline@v0.9.0 · 5492 in / 1396 out tokens · 37127 ms · 2026-05-16T20:12:16.276221+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    as a consequence of the explicit breaking of theU(1) symmetry by theE inc term. 4 C. Limit cycles and Floquet-Bogoliubov spectrum of collective excitations The stationary solutions discussed so far correspond to configurations in which the condensate is locked in frequency and phase to the incident field. But other forms of steady-state solutions are poss...

    work page

  2. [2]

    Some analyt- ical considerations on the existence of multiple solutions at a givenI inc in a general ∆̸= 0 case are given in Ap- pendix A

    Steady state: stationary solutions and limit cycles For general values of ∆, the equation (5) for the sta- tionary state has the form ∆− i 2 P 1 +|E ss|2/ns −γ Ess =iE inc : (17) the phase difference betweenE ss andE inc can have ar- bitrary values ∆ϕEss,Einc = π 2 + arctan 1 2∆ P 1 +|E ss|2/ns −γ (18) and, by taking the squared modulus of (17), the relat...

    work page

  3. [3]

    In this Section we will focus on collective excitations around stationary solutions, while in the next Section we will consider collective excitations around a limit cycle

    Bogoliubov spectrum around a stationary state These rich features reflect into different forms of the Bogoliubov dispersion of the collective excitations in the different cases. In this Section we will focus on collective excitations around stationary solutions, while in the next Section we will consider collective excitations around a limit cycle. The Bo...

    work page

  4. [4]

    This requires taking the logarithm (12) of the eigenvalues of the linearized propagatorU(T) for small perturbations around the limit cycle

    Floquet-Bogoliubov spectrum around a limit cycle As discussed in Sec.II C, the collective excitations around a limit cycle can be stroboscopically studied by monitoring the field at discrete times separated by the limit cycle periodT. This requires taking the logarithm (12) of the eigenvalues of the linearized propagatorU(T) for small perturbations around...

    work page

  5. [5]

    Nozieres and D

    P. Nozieres and D. Pines,Theory Of Quantum Liquids, Advanced Books Classics (Avalon Publishing, 1999)

    work page 1999

  6. [6]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari,Bose-Einstein condensa- tion and superfluidity, Vol. 164 (Oxford University Press, 2016)

    work page 2016

  7. [7]

    Carusotto and C

    I. Carusotto and C. Ciuti, Quantum fluids of light, Re- views of Modern Physics85, 299 (2013)

    work page 2013

  8. [8]

    Bloch, I

    J. Bloch, I. Carusotto, and M. Wouters, Non-equilibrium bose–einstein condensation in photonic systems, Nature Reviews Physics4, 470 (2022)

    work page 2022

  9. [9]

    Bogoliubov, On the theory of superfluidity, J

    N. Bogoliubov, On the theory of superfluidity, J. Phys 11, 23 (1947)

    work page 1947

  10. [10]

    J. D. Gunton and M. J. Buckingham, Condensation of the ideal bose gas as a cooperative transition, Phys. Rev. 166, 152 (1968)

    work page 1968

  11. [11]

    Claude, M

    F. Claude, M. J. Jacquet, R. Usciati, I. Carusotto, E. Gi- acobino, A. Bramati, and Q. Glorieux, High-resolution coherent probe spectroscopy of a polariton quantum fluid, Physical Review Letters129, 103601 (2022)

    work page 2022

  12. [12]

    Carusotto and C

    I. Carusotto and C. Ciuti, Probing microcavity polariton superfluidity through resonant rayleigh scattering, Phys. Rev. Lett.93, 166401 (2004)

    work page 2004

  13. [13]

    Wouters and I

    M. Wouters and I. Carusotto, Absence of long-range co- herence in the parametric emission of photonic wires, Phys. Rev. B74, 245316 (2006)

    work page 2006

  14. [14]

    M. H. Szyma´ nska, J. Keeling, and P. B. Littlewood, Nonequilibrium quantum condensation in an incoher- ently pumped dissipative system, Phys. Rev. Lett.96, 230602 (2006)

    work page 2006

  15. [15]

    Wouters and I

    M. Wouters and I. Carusotto, Goldstone mode of opti- cal parametric oscillators in planar semiconductor micro- cavities in the strong-coupling regime, Phys. Rev. A76, 043807 (2007)

    work page 2007

  16. [16]

    Claude, M

    F. Claude, M. J. Jacquet, Q. Glorieux, M. Wouters, E. Giacobino, I. Carusotto, and A. Bramati, Observa- tion of the diffusive nambu–goldstone mode of a non- equilibrium phase transition, Nature Physics , 1 (2025)

    work page 2025

  17. [17]

    Siegman,Lasers, G - Reference,Information and In- terdisciplinary Subjects Series (University Science Books, 1986)

    A. Siegman,Lasers, G - Reference,Information and In- terdisciplinary Subjects Series (University Science Books, 1986)

    work page 1986

  18. [18]

    Adler, A study of locking phenomena in oscillators, Proceedings of the IRE34, 351 (1946)

    R. Adler, A study of locking phenomena in oscillators, Proceedings of the IRE34, 351 (1946)

    work page 1946

  19. [19]

    Paciorek, Injection locking of oscillators, Proceedings of the IEEE53, 1723 (1965)

    L. Paciorek, Injection locking of oscillators, Proceedings of the IEEE53, 1723 (1965)

    work page 1965

  20. [20]

    Stover and W

    H. Stover and W. Steier, Locking of laser oscillators by light injection, applied physics letters8, 91 (1966)

    work page 1966

  21. [21]

    Markovi´ c, J

    D. Markovi´ c, J. Pillet, E. Flurin, N. Roch, and B. Huard, Injection locking and parametric locking in a supercon- ducting circuit, Phys. Rev. Appl.12, 024034 (2019)

    work page 2019

  22. [22]

    Wouters and I

    M. Wouters and I. Carusotto, Excitations in a nonequi- librium bose-einstein condensate of exciton polaritons, Phys. Rev. Lett.99, 140402 (2007)

    work page 2007

  23. [24]

    L. A. Lugiato, F. Prati, E. Brambilla, and A. Gatti, The cavity kerr medium model and the surprising his- tory around it, inQuantum Fluids of Light and Matter (IOS Press, 2025) pp. 5–21

    work page 2025

  24. [25]

    I. S. Aranson and L. Kramer, The world of the complex ginzburg-landau equation, Reviews of modern physics 74, 99 (2002)

    work page 2002

  25. [26]

    Since the analytical study of limit cycles is difficult, we used a specialized numerical software of MATLAB, namedMatcont[30]: given an initial cycle found ‘by hand’ at fixed parameters, keeping its period fixed, it evaluates the limit cycle continuation in the space of pa- rameters

    work page

  26. [27]

    Viebahn, Introduction to floquet theory, Institute for Quantum Electronics, ETH Zurich8093(2020)

    K. Viebahn, Introduction to floquet theory, Institute for Quantum Electronics, ETH Zurich8093(2020)

    work page 2020

  27. [28]

    Recati and S

    A. Recati and S. Stringari, Supersolidity in ultracold dipolar gases, Nature Reviews Physics5, 735 (2023)

    work page 2023

  28. [29]

    Nigro, D

    D. Nigro, D. Trypogeorgos, A. Gianfrate, D. Sanvitto, I. Carusotto, and D. Gerace, Supersolidity of polariton condensates in photonic crystal waveguides, Physical Re- view Letters134, 056002 (2025)

    work page 2025

  29. [30]

    Trypogeorgos, A

    D. Trypogeorgos, A. Gianfrate, M. Landini, D. Nigro, D. Gerace, I. Carusotto, F. Riminucci, K. W. Baldwin, L. N. Pfeiffer, G. I. Martone,et al., Emerging supersolid- ity in photonic-crystal polariton condensates, Nature , 1 (2025)

    work page 2025

  30. [31]

    Columbo, M

    L. Columbo, M. Piccardo, F. Prati, L. Lugiato, M. Bram- billa, A. Gatti, C. Silvestri, M. Gioannini, N. Opaˇ cak, B. Schwarz,et al., Unifying frequency combs in active and passive cavities: Temporal solitons in externally driven ring lasers, Physical Review Letters126, 173903 (2021)

    work page 2021

  31. [32]

    Contractor, W

    R. Contractor, W. Noh, W. Redjem, W. Qarony, E. Mar- tin, S. Dhuey, A. Schwartzberg, and B. Kant´ e, Scalable single-mode surface-emitting laser via open-dirac singu- larities, Nature608, 692 (2022)

    work page 2022

  32. [33]

    Carusotto, How to exploit driving and dissipation to stabilize and manipulate quantum many-body states, Comptes Rendus

    I. Carusotto, How to exploit driving and dissipation to stabilize and manipulate quantum many-body states, Comptes Rendus. Physique26, 533 (2025)

    work page 2025

  33. [34]

    Dhooge, W

    A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Mei- jer, and B. Sautois, New features of the software matcont for bifurcation analysis of dynamical systems, Mathemat- ical and Computer Modelling of Dynamical Systems14, 147 (2008)

    work page 2008

  34. [35]

    C. J. Pethick and H. Smith,Bose–Einstein condensation in dilute gases(Cambridge university press, 2008). 14

    work page 2008

  35. [36]

    R. K. Pathria and P. D. Beale,Statistical Mechanics, 3rd ed. (Elsevier/Academic Press, Amsterdam ; Boston, 2011). Appendix A: Analytical considerations on multiple solutions forg= 0 In theg= 0 case, analytical conditions for the exis- tence of multiple solutions at a givenIinc can be obtained studying the sign of the derivative dIinc dIss = ∆2 + 1 4 P 1 +...

    work page 2011