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arxiv: 2507.10862 · v3 · submitted 2025-07-14 · 🪐 quant-ph · cond-mat.quant-gas· gr-qc· nlin.PS· physics.optics

Time Crystal from Self-Amplification of Spontaneous Analog Hawking Radiation

Juan Ram\'on Mu\~noz de Nova , Fernando Sols This is my paper

Pith reviewed 2026-05-19 03:57 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gasgr-qcnlin.PSphysics.optics
keywords time crystalHawking radiationblack hole laserparametric amplifierFloquet statespontaneous symmetry breakinganalog gravitycorrelation function
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The pith

Self-amplification of spontaneous Hawking radiation in a black-hole laser forms a time crystal.

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

The paper establishes that a time crystal can arise in a quantum black-hole laser through the self-amplification of spontaneous analog Hawking radiation. This process leads to spontaneous symmetry breaking with a random phase, resulting in a Hawking time crystal where out-of-time density-density correlations are periodic but equal-time observables appear time-independent due to averaging over realizations. The authors show that any parametric amplifier possesses an associated time operator, allowing the time-crystal formation to be characterized by two time operators tied to the initial laser setup and the final Floquet state. This offers a concrete physical mechanism for realizing time crystals in analog gravity systems.

Core claim

We propose a time crystal based on a quantum black-hole laser, where the genuinely spontaneous character of the symmetry breaking stems from the self-amplification of spontaneous Hawking radiation. The resulting Hawking time crystal is characterized by the periodic dependence of the out-of-time density-density correlation function, while equal-time observables are time independent because they embody averages over different realizations with a random oscillation phase. We prove that any parametric amplifier has associated a time operator, which leads to a unique characterization of the time-crystal formation in terms of two time operators: one associated with the initial black-hole laser and

What carries the argument

The time operator associated to any parametric amplifier, which provides a unique characterization of the time-crystal formation via two time operators linked to the initial black-hole laser and the final spontaneous Floquet state.

If this is right

  • The Hawking time crystal exhibits anticorrelation bands resulting from the spontaneous quantum emission of pairs of dispersive waves and solitons into the upstream and downstream regions.
  • It serves as a nonlinear periodic analog of the Andreev-Hawking effect.
  • Equal-time observables average to time-independent values while out-of-time correlations remain periodic.
  • Time-crystal formation is uniquely determined by the two time operators.

Where Pith is reading between the lines

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

  • This mechanism could be realized in experiments with flowing Bose-Einstein condensates to observe the predicted periodic correlations.
  • The connection between parametric amplifiers and time operators may apply to other driven quantum systems beyond analog gravity.
  • Ensemble averaging over random phases highlights the importance of statistical measurements in detecting such symmetry-broken states.

Load-bearing premise

The self-amplification produces a genuinely spontaneous symmetry breaking whose phase is random across realizations, causing equal-time observables to average to time-independent values.

What would settle it

A measurement showing that out-of-time density-density correlations are periodic in time while equal-time correlations remain constant, or the absence of random phase variation between experimental runs.

Figures

Figures reproduced from arXiv: 2507.10862 by Fernando Sols, Juan Ram\'on Mu\~noz de Nova.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Spatial profile of sound (solid blue) and flow [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Expectation values for a FPBHL with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a)-(c) Histogram of the oscillation phase [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We propose a time crystal based on a quantum black-hole laser, where the genuinely spontaneous character of the symmetry breaking stems from the self-amplification of spontaneous Hawking radiation. The resulting Hawking time crystal (HTC) is characterized by the periodic dependence of the out-of-time density-density correlation function, while equal-time observables are time independent because they embody averages over different realizations with a random oscillation phase. The HTC provides a nonlinear periodic analog of the Andreev-Hawking effect, exhibiting anticorrelation bands resulting from the spontaneous, quantum emission of pairs of dispersive waves and solitons into the upstream and downstream regions. Remarkably, we prove that any parametric amplifier has associated a time operator, which leads to a unique characterization of the time-crystal formation in terms of two time operators: one associated with the initial black-hole laser and another associated with the final spontaneous Floquet state.

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

2 major / 2 minor

Summary. The manuscript proposes a Hawking time crystal (HTC) in a quantum black-hole laser setup, where self-amplification of spontaneous Hawking radiation produces spontaneous symmetry breaking. Equal-time observables are claimed to be time-independent due to averaging over random oscillation phases across realizations, while out-of-time density-density correlation functions exhibit periodicity. The paper claims to prove that any parametric amplifier has an associated time operator, enabling a characterization of the time-crystal formation via two such operators: one linked to the initial black-hole laser and one to the final spontaneous Floquet state. It also predicts anticorrelation bands as a nonlinear periodic analog of the Andreev-Hawking effect.

Significance. If the central claims hold, this work could provide a novel mechanism for realizing time crystals in analog gravity systems, bridging Hawking radiation, parametric amplification, and Floquet physics in platforms such as Bose-Einstein condensates. The claimed proof associating time operators with any parametric amplifier represents a potential conceptual advance with possible broader utility in analyzing time-dependent quantum systems. The predicted anticorrelation bands offer concrete, falsifiable signatures. These elements, if rigorously established, would strengthen the case for spontaneous time-crystal formation without external driving.

major comments (2)
  1. [Discussion of spontaneous symmetry breaking and HTC signature (near the description of equal-time vs. out-of-time correl] The central distinction between time-independent equal-time observables and periodic out-of-time correlations rests on the assumption that self-amplification produces a genuinely spontaneous symmetry breaking with a phase that is uniformly random across independent realizations. No explicit demonstration is provided that the nonlinear evolution (or the associated time operators) selects phases ergodically rather than locking them to the initial black-hole laser or Floquet state. If phases are correlated or biased, equal-time observables acquire explicit time dependence and the claimed time-crystal signature collapses. This assumption is load-bearing for the HTC characterization.
  2. [Section presenting the proof of time operators for parametric amplifiers] The claimed proof that any parametric amplifier has an associated time operator, leading to a unique characterization with two time operators, requires more detail on the derivation steps and how it guarantees the spontaneous, random-phase nature of the final Floquet state. Without this, it is unclear whether the construction is general or reduces to the specific model assumptions.
minor comments (2)
  1. [Notation and definitions] Clarify the notation for the two time operators with explicit definitions or equations in the main text to improve readability.
  2. [Introduction] Add a brief comparison to existing proposals for time crystals in driven or undriven systems to better contextualize the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have helped us strengthen the presentation of the spontaneous symmetry breaking and the generality of the time-operator construction. We address each major comment below and have revised the manuscript to incorporate additional derivations and clarifications.

read point-by-point responses

  1. Referee: The central distinction between time-independent equal-time observables and periodic out-of-time correlations rests on the assumption that self-amplification produces a genuinely spontaneous symmetry breaking with a phase that is uniformly random across independent realizations. No explicit demonstration is provided that the nonlinear evolution (or the associated time operators) selects phases ergodically rather than locking them to the initial black-hole laser or Floquet state. If phases are correlated or biased, equal-time observables acquire explicit time dependence and the claimed time-crystal signature collapses. This assumption is load-bearing for the HTC characterization.

    Authors: We agree that an explicit demonstration of the ergodic phase selection is necessary for rigor. The spontaneous character originates from the vacuum fluctuations of the initial Hawking radiation, whose phase is uniformly random. In the nonlinear regime the self-amplification is governed by a phase-insensitive parametric process; the quantum noise seeds a random initial phase that is subsequently preserved by the Floquet evolution rather than locked to any classical reference. To make this explicit we have added a new subsection that derives the phase distribution from the Bogoliubov coefficients of the linear stage and shows that the nonlinear terms induce phase diffusion, ensuring that the ensemble average over realizations remains time-independent while out-of-time correlators retain periodicity. revision: yes

  2. Referee: The claimed proof that any parametric amplifier has an associated time operator, leading to a unique characterization with two time operators, requires more detail on the derivation steps and how it guarantees the spontaneous, random-phase nature of the final Floquet state. Without this, it is unclear whether the construction is general or reduces to the specific model assumptions.

    Authors: We thank the referee for this observation. The original manuscript presented the time-operator construction in a condensed form. We have now expanded the relevant section with a step-by-step derivation that begins from the general squeezing Hamiltonian of a parametric amplifier, constructs the associated time operator as the generator conjugate to the squeezing phase, and applies it first to the initial black-hole laser and subsequently to the emergent spontaneous Floquet state. The construction relies only on the structure of parametric amplification and the vacuum input state; it therefore remains general and automatically incorporates the random-phase property inherited from the initial quantum fluctuations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated assumptions

full rationale

The paper proposes a Hawking time crystal arising from self-amplification of spontaneous Hawking radiation in a quantum black-hole laser. The key step is a claimed proof that any parametric amplifier possesses an associated time operator, which then furnishes a characterization via two such operators (initial black-hole laser and final Floquet state). This is presented as an independent mathematical result rather than a redefinition or fit. The time-crystal signature—periodic out-of-time density-density correlations with time-independent equal-time observables—is explicitly tied to an averaging assumption over random oscillation phases across realizations; the abstract and description treat this as an input modeling choice, not something derived by construction from the dynamics or reduced to a fitted parameter. No equations or self-citations are shown to force the central claims back onto the inputs. The construction therefore remains self-contained against external benchmarks and the listed assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The proposal rests on standard quantum-field-theory-in-curved-space assumptions for analog Hawking radiation and on the existence of a parametric-amplifier regime in the black-hole laser; no explicit free parameters or new invented particles are named in the abstract, but the time operator for any parametric amplifier is introduced as a new mathematical object.

axioms (2)
  • domain assumption Analog Hawking radiation arises from quantum vacuum fluctuations near an effective event horizon in a flowing medium or optical system.
    Invoked to justify the spontaneous emission of wave pairs that then self-amplify.
  • domain assumption The black-hole laser supports a parametric-amplifier regime in which small fluctuations grow exponentially.
    Required for the self-amplification step that turns spontaneous radiation into a time crystal.
invented entities (2)
  • Hawking time crystal (HTC) no independent evidence
    purpose: A time crystal whose periodicity is visible only in out-of-time density correlations and originates from spontaneous Hawking radiation.
    New physical state proposed in the paper; no independent experimental signature is given in the abstract.
  • Time operator associated with a parametric amplifier no independent evidence
    purpose: Mathematical object that characterizes the time-crystal formation together with a second operator for the final Floquet state.
    Introduced as part of the proof; the abstract states that any parametric amplifier possesses such an operator.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Collective excitations in quantum gravity condensates

    gr-qc 2026-05 unverdicted novelty 6.0

    Collective excitations analogous to phonons are derived in quantum gravity condensates within a group field theory model, yielding leading beyond-mean-field corrections to emergent Friedmann dynamics.

Reference graph

Works this paper leans on

101 extracted references · 101 canonical work pages · cited by 1 Pith paper

  1. [1]

    W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981)

    work page 1981

  2. [2]

    L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000)

    work page 2000

  3. [3]

    Lahav, A

    O. Lahav, A. Itah, A. Blumkin, C. Gordon, S. Rinott, A. Zayats, and J. Steinhauer, Phys. Rev. Lett. 105, 240401 (2010)

    work page 2010

  4. [4]

    Belgiorno, S

    F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala, and D. Fac- cio, Phys. Rev. Lett. 105, 203901 (2010)

    work page 2010

  5. [5]

    Drori, Y

    J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, and U. Leonhardt, Phys. Rev. Lett. 122, 010404 (2019)

    work page 2019

  6. [6]

    Horstmann, B

    B. Horstmann, B. Reznik, S. Fagnocchi, and J. I. Cirac, Phys. Rev. Lett. 104, 250403 (2010)

    work page 2010

  7. [7]

    Wittemer, F

    M. Wittemer, F. Hakelberg, P. Kiefer, J.-P. Schr¨ oder, C. Fey, R. Sch¨ utzhold, U. Warring, and T. Schaetz, Phys. Rev. Lett. 123, 180502 (2019)

    work page 2019

  8. [8]

    Weinfurtner, E

    S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Phys. Rev. Lett. 106, 021302 (2011)

    work page 2011

  9. [9]

    Euv´ e, F

    L.-P. Euv´ e, F. Michel, R. Parentani, T. G. Philbin, and G. Rousseaux, Phys. Rev. Lett. 117, 121301 (2016)

    work page 2016

  10. [10]

    Carusotto and C

    I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85, 299 (2013)

    work page 2013

  11. [11]

    Falque, A

    K. Falque, A. Delhom, Q. Glorieux, E. Giacobino, A. Bramati, and M. J. Jacquet, Phys. Rev. Lett. 135, 023401 (2025)

    work page 2025

  12. [12]

    Shi, R.-Q

    Y.-H. Shi, R.-Q. Yang, Z. Xiang, Z.-Y. Ge, H. Li, Y.-Y. Wang, K. Huang, Y. Tian, X. Song, D. Zheng, et al. , Nature Communications 14, 3263 (2023)

    work page 2023

  13. [13]

    ˇSvanˇ cara, P

    P. ˇSvanˇ cara, P. Smaniotto, L. Solidoro, J. F. MacDon- ald, S. Patrick, R. Gregory, C. F. Barenghi, and S. We- infurtner, Nature 628, 66 (2024)

    work page 2024

  14. [14]

    Jaskula, G

    J.-C. Jaskula, G. B. Partridge, M. Bonneau, R. Lopes, J. Ruaudel, D. Boiron, and C. I. Westbrook, Phys. Rev. Lett. 109, 220401 (2012)

    work page 2012

  15. [15]

    C.-L. Hung, V. Gurarie, and C. Chin, Science 341, 1213 (2013)

    work page 2013

  16. [16]

    Torres, S

    T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford, and S. Weinfurtner, Nature Physics 13, 833 (2017)

    work page 2017

  17. [17]

    Eckel, A

    S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell, Phys. Rev. X 8, 021021 (2018)

    work page 2018

  18. [18]

    J. Hu, L. Feng, Z. Zhang, and C. Chin, Nature Physics 15, 785 (2019)

    work page 2019

  19. [19]

    Torres, S

    T. Torres, S. Patrick, M. Richartz, and S. Weinfurtner, Phys. Rev. Lett. 125, 011301 (2020)

    work page 2020

  20. [20]

    Patrick, H

    S. Patrick, H. Goodhew, C. Gooding, and S. Weinfurt- ner, Phys. Rev. Lett. 126, 041105 (2021)

    work page 2021

  21. [21]

    Steinhauer, M

    J. Steinhauer, M. Abuzarli, T. Aladjidi, T. Bienaim´ e, C. Piekarski, W. Liu, E. Giacobino, A. Bramati, and Q. Glorieux, Nat. Commun. 13, 2890 (2022)

    work page 2022

  22. [22]

    Viermann, M

    C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, ´A. Parra-L´ opez, M. Tolosa-Sime´ on, N. S´ anchez-Kuntz, T. Haas, H. Strobel, et al., Nature 611, 260 (2022)

    work page 2022

  23. [23]

    Leonhardt, T

    U. Leonhardt, T. Kiss, and P. ¨Ohberg, J. Opt. B: Quan- tum Semiclass. Opt. 5, S42 (2003)

    work page 2003

  24. [24]

    Balbinot, A

    R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, and I. Carusotto, Phys. Rev. A 78, 21603 (2008)

    work page 2008

  25. [25]

    Carusotto, S

    I. Carusotto, S. Fagnocchi, A. Recati, R. Balbinot, and A. Fabbri, New J. Phys. 10, 103001 (2008)

    work page 2008

  26. [26]

    Macher and R

    J. Macher and R. Parentani, Phys. Rev. A 80, 43601 (2009)

    work page 2009

  27. [27]

    Recati, N

    A. Recati, N. Pavloff, and I. Carusotto, Phys. Rev. A 80, 43603 (2009)

    work page 2009

  28. [28]

    Zapata, M

    I. Zapata, M. Albert, R. Parentani, and F. Sols, New J. Phys. 13, 63048 (2011)

    work page 2011

  29. [29]

    P. E. Larr´ e, A. Recati, I. Carusotto, and N. Pavloff, Phys. Rev. A 85, 13621 (2012)

    work page 2012

  30. [30]

    J. R. M. de Nova, F. Sols, and I. Zapata, Phys. Rev. A 89, 043808 (2014)

    work page 2014

  31. [31]

    Finazzi and I

    S. Finazzi and I. Carusotto, Phys. Rev. A 90, 033607 (2014)

    work page 2014

  32. [32]

    Busch and R

    X. Busch and R. Parentani, Phys. Rev. D 89, 105024 (2014)

    work page 2014

  33. [33]

    J. R. M. de Nova, F. Sols, and I. Zapata, New J. Phys. 17, 105003 (2015)

    work page 2015

  34. [34]

    Michel, R

    F. Michel, R. Parentani, and R. Zegers, Phys. Rev. D 93, 065039 (2016)

    work page 2016

  35. [35]

    Isoard, N

    M. Isoard, N. Milazzo, N. Pavloff, and O. Giraud, Phys. 6 Rev. A 104, 063302 (2021)

    work page 2021

  36. [36]

    C. C. H. Ribeiro, S.-S. Baak, and U. R. Fischer, Phys. Rev. D 105, 124066 (2022)

    work page 2022

  37. [37]

    C. C. Holanda Ribeiro and U. R. Fischer, Phys. Rev. D 107, L121502 (2023)

    work page 2023

  38. [38]

    Ciliberto, S

    G. Ciliberto, S. Emig, N. Pavloff, and M. Isoard, Phys. Rev. A 109, 063325 (2024)

    work page 2024

  39. [39]

    Steinhauer, Nature Physics 12, 959 (2016)

    J. Steinhauer, Nature Physics 12, 959 (2016)

    work page 2016

  40. [40]

    J. R. M. de Nova, K. Golubkov, V. I. Kolobov, and J. Steinhauer, Nature 569, 688 (2019)

    work page 2019

  41. [41]

    V. I. Kolobov, K. Golubkov, J. R. M. de Nova, and J. Steinhauer, Nature Physics 17, 362 (2021)

    work page 2021

  42. [42]

    Zapata and F

    I. Zapata and F. Sols, Phys. Rev. Lett. 102, 180405 (2009)

    work page 2009

  43. [43]

    Walls and G

    D. Walls and G. Milburn, Quantum Optics , Springer- Link: Springer e-Books (Springer, 2008)

    work page 2008

  44. [44]

    J. R. M. de Nova, P. F. Palacios, P. A. Guerrero, I. Za- pata, and F. Sols, Comptes Rendus. Physique 25, 1 (2024)

    work page 2024

  45. [45]

    Corley and T

    S. Corley and T. Jacobson, Phys. Rev. D 59, 124011 (1999)

    work page 1999

  46. [46]

    Leonhardt, T

    U. Leonhardt, T. Kiss, and P. ¨Ohberg, Phys. Rev. A 67, 33602 (2003)

    work page 2003

  47. [47]

    Barcel´ o, A

    C. Barcel´ o, A. Cano, L. J. Garay, and G. Jannes, Phys. Rev. D 74, 024008 (2006)

    work page 2006

  48. [48]

    P. Jain, A. S. Bradley, and C. Gardiner, Phys. Rev. A 76, 23617 (2007)

    work page 2007

  49. [49]

    Coutant and R

    A. Coutant and R. Parentani, Phys. Rev. D 81, 84042 (2010)

    work page 2010

  50. [50]

    Finazzi and R

    S. Finazzi and R. Parentani, New J. Phys. 12, 095015 (2010)

    work page 2010

  51. [51]

    Faccio, T

    D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, Classical and Quantum Gravity 29, 224009 (2012)

    work page 2012

  52. [52]

    J. R. M. de Nova, S. Finazzi, and I. Carusotto, Phys. Rev. A 94, 043616 (2016)

    work page 2016

  53. [53]

    Peloquin, L.-P

    C. Peloquin, L.-P. Euv´ e, T. Philbin, and G. Rousseaux, Phys. Rev. D 93, 084032 (2016)

    work page 2016

  54. [54]

    Berm´ udez and U

    D. Berm´ udez and U. Leonhardt, Classical and Quantum Gravity 36, 024001 (2018)

    work page 2018

  55. [55]

    B¨ urkle, A

    R. B¨ urkle, A. Gaidoukov, and J. R. Anglin, New Jour- nal of Physics 20, 083020 (2018)

    work page 2018

  56. [56]

    J. R. M. de Nova, P. F. Palacios, I. Carusotto, and F. Sols, New Journal of Physics 23, 023040 (2021)

    work page 2021

  57. [57]

    J. D. Rinc´ on-Estrada and D. Berm´ udez, Annalen der Physik 533, 2000239 (2021)

    work page 2021

  58. [58]

    Katayama, Scientific Reports 11, 19137 (2021)

    H. Katayama, Scientific Reports 11, 19137 (2021)

    work page 2021

  59. [59]

    Steinhauer, Phys

    J. Steinhauer, Phys. Rev. D 106, 102007 (2022)

    work page 2022

  60. [60]

    J. R. M. de Nova and F. Sols, Phys. Rev. Res. 5, 043282 (2023)

    work page 2023

  61. [61]

    J. R. M. de Nova and F. Sols, Phys. Rev. A 105, 043302 (2022)

    work page 2022

  62. [62]

    J. R. M. de Nova and F. Sols, arXiv preprint arXiv:2402.10784 (2024)

    work page arXiv 2024

  63. [63]

    Wilczek, Phys

    F. Wilczek, Phys. Rev. Lett. 109, 160401 (2012)

    work page 2012

  64. [64]

    Sacha and J

    K. Sacha and J. Zakrzewski, Reports on Progress in Physics 81, 016401 (2017)

    work page 2017

  65. [65]

    Sacha, Phys

    K. Sacha, Phys. Rev. A 91, 033617 (2015)

    work page 2015

  66. [66]

    D. V. Else, B. Bauer, and C. Nayak, Phys. Rev. Lett. 117, 090402 (2016)

    work page 2016

  67. [67]

    Bhowmick, H

    D. Bhowmick, H. Sun, B. Yang, and P. Sengupta, Phys. Rev. B 108, 014434 (2023)

    work page 2023

  68. [68]

    Syrwid, J

    A. Syrwid, J. Zakrzewski, and K. Sacha, Phys. Rev. Lett. 119, 250602 (2017)

    work page 2017

  69. [69]

    Iemini, A

    F. Iemini, A. Russomanno, J. Keeling, M. Schir` o, M. Dalmonte, and R. Fazio, Phys. Rev. Lett. 121, 035301 (2018)

    work page 2018

  70. [70]

    Buˇ ca, J

    B. Buˇ ca, J. Tindall, and D. Jaksch, Nature Communi- cations 10, 1 (2019)

    work page 2019

  71. [71]

    Booker, B

    C. Booker, B. Buˇ ca, and D. Jaksch, New Journal of Physics 22, 085007 (2020)

    work page 2020

  72. [72]

    Daviet, C

    R. Daviet, C. P. Zelle, A. Rosch, and S. Diehl, Phys. Rev. Lett. 132, 167102 (2024)

    work page 2024

  73. [73]

    S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, et al., Nature 543, 221 (2017)

    work page 2017

  74. [74]

    Zhang, P

    J. Zhang, P. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, et al., Nature 543, 217 (2017)

    work page 2017

  75. [75]

    Autti, V

    S. Autti, V. B. Eltsov, and G. E. Volovik, Phys. Rev. Lett. 120, 215301 (2018)

    work page 2018

  76. [76]

    Smits, L

    J. Smits, L. Liao, H. T. C. Stoof, and P. van der Straten, Phys. Rev. Lett. 121, 185301 (2018)

    work page 2018

  77. [77]

    Rovny, R

    J. Rovny, R. L. Blum, and S. E. Barrett, Phys. Rev. Lett. 120, 180603 (2018)

    work page 2018

  78. [78]

    Kyprianidis, F

    A. Kyprianidis, F. Machado, W. Morong, P. Becker, K. S. Collins, D. V. Else, L. Feng, P. W. Hess, C. Nayak, G. Pagano, N. Y. Yao, and C. Monroe, Science 372, 1192 (2021)

    work page 2021

  79. [79]

    Randall, C

    J. Randall, C. E. Bradley, F. V. van der Gron- den, A. Galicia, M. H. Abobeih, M. Markham, D. J. Twitchen, F. Machado, N. Y. Yao, and T. H. Taminiau, Science 374, 1474 (2021)

    work page 2021

  80. [80]

    X. Mi, M. Ippoliti, C. Quintana, A. Greene, Z. Chen, J. Gross, F. Arute, K. Arya, J. Atalaya, R. Babbush, et al., Nature 601, 531 (2022)

    work page 2022

Showing first 80 references.