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arxiv: 2507.01528 · v3 · submitted 2025-07-02 · 🪐 quant-ph

Time Crystal in the Nonlinear Phonon Mode of the Trapped Ions

Yi-Ling Zhan , Chun-Fu Liu , J.-T. Bu , K.-F Cui , S.-L. Su , L.-L. Yan , Gang Chen This is my paper

Pith reviewed 2026-05-19 06:44 UTC · model grok-4.3

classification 🪐 quant-ph
keywords time crystaltrapped ionsphonon modeHopf bifurcationlimit cyclenonlinear phononadiabatic eliminationsymmetry breaking
0
0 comments X p. Extension

The pith

Controlling gain and damping in trapped-ion phonons produces a stable time crystal via Hopf bifurcation.

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

This paper outlines a method to create a time crystal using the vibrational modes of two trapped ions. Lasers induce an effective nonlinear dynamics with linear amplification and nonlinear loss in the phonon mode. Tuning these to the Hopf bifurcation point leads to persistent periodic oscillations that last much longer than one cycle, spontaneously breaking time translation symmetry. Simulations confirm this holds for realistic conditions and resists various perturbations.

Core claim

By controlling linear gain and nonlinear damping in the phonon mode to satisfy Hopf bifurcation conditions, the system exhibits stable dissipative dynamics over timescales much longer than the oscillation period, indicating discrete time-translation symmetry breaking in the phonon mode, i.e., a phonon time crystal.

What carries the argument

Nonlinear phonon mode engineered through adiabatic elimination of laser-driven internal states, featuring tunable linear gain and nonlinear damping to induce a limit cycle.

If this is right

  • The phonon vibrations enter a stable limit cycle with constant amplitude.
  • Discrete time-translation symmetry breaks in the normal mode of the ions.
  • The oscillation persists robustly against initial thermal states and spin dephasing.
  • Control errors in laser parameters do not destroy the time-crystalline behavior.

Where Pith is reading between the lines

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

  • This scheme suggests a route to observe time crystals in other trapped-particle systems with similar nonlinear control.
  • Persistent phonon oscillations could serve as a reference for precision measurements in ion traps.
  • Extending the model to include more ions might allow for spatially extended time crystals.

Load-bearing premise

The adiabatic elimination method used to derive the effective nonlinear phonon equation remains valid under the chosen laser intensities and detunings, and the resulting effective model accurately captures the long-time limit-cycle behavior without higher-order corrections that would destroy the oscillation.

What would settle it

An experiment or simulation where the phonon mode amplitude decays to zero or fails to maintain periodic motion over many periods would disprove the existence of the stable time crystal.

Figures

Figures reproduced from arXiv: 2507.01528 by Chun-Fu Liu, Gang Chen, J.-T. Bu, K.-F Cui, L.-L. Yan, S.-L. Su, Yi-Ling Zhan.

Figure 1
Figure 1. Figure 1: FIG. 1. The scheme to realize the time crystal in a trapped [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The time evolution of (a) the rescaled phonon number [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Quantum Husimi distribution at [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The evolution of population in the Fock states for a [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

Time crystals constitute a novel phase of matter defined by the spontaneous breaking of timetranslation symmetry. Here we present a scheme to realize a continuous-time crystal of the vibrational phonon in the normal mode of two coupled ultra-cold ions. By utilizing two addressable standing-wave lasers and adiabatic elimination method, we generate a controllable nonlinear phonon mode with the well-designed efficient linear gain and nonlinear damping. By controlling these parameters to satisfy the phase transition conditions of Hopf bifurcation and limit cycle phase, it behaves as a stable dissipative dynamics over timescales significantly longer than the oscillation period, indicating the emergence of discrete time-translation symmetry breaking in the phonon mode, i.e., a phonon time crystal. We further numerically simulate this phonon time crystal by using accessible experimental parameters and also demonstrate a robustness to the initial thermal state and thermalization of phonon mode, spin dephasing, and the control errors of Rabi frequencies. These results provide a practical scheme for observing a time crystal in a nonlinear phonon mode and will advance the research of time crystals.

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 scheme to realize a continuous time crystal in the vibrational phonon mode of two trapped ions. By applying two addressable standing-wave lasers and using adiabatic elimination, the authors derive an effective nonlinear phonon equation featuring controllable linear gain and nonlinear damping. Tuning these parameters to satisfy the conditions for a Hopf bifurcation produces a stable limit cycle whose oscillations persist over timescales much longer than the period, which is interpreted as discrete time-translation symmetry breaking. Numerical simulations with realistic experimental parameters are presented, along with checks of robustness against initial thermal states, spin dephasing, and Rabi-frequency control errors.

Significance. If the effective model remains accurate, the work supplies a concrete, experimentally accessible route to a dissipative phonon time crystal in trapped ions. The combination of engineered gain/damping, explicit Hopf-bifurcation analysis, and numerical demonstration with laboratory parameters is a constructive contribution that could stimulate follow-up experiments and broaden the platforms used to study time-translation symmetry breaking.

major comments (2)
  1. [Derivation of effective phonon equation] The central claim rests on the validity of the adiabatic elimination that produces the effective nonlinear phonon equation. In the derivation section (likely §III), the timescale-separation assumption is invoked, yet no quantitative bound or numerical test is supplied showing that the separation remains sufficient once the phonon amplitude reaches the limit-cycle value and nonlinear terms are fully active. Residual spin-phonon coupling or higher-order corrections could then introduce slow drifts or additional damping that would destroy the claimed long-time stability of the discrete time crystal.
  2. [Numerical simulations] The numerical results in the simulation section demonstrate stable oscillations in the effective model, but no direct benchmark against the full ion-laser Hamiltonian evolution is reported over the same long timescales (>> oscillation period). Without this comparison, it is impossible to confirm that neglected back-action terms do not destabilize the limit cycle for the chosen laser intensities and detunings.
minor comments (2)
  1. The abstract would be clearer if it included the explicit form of the effective phonon equation or the key parameter values used for the Hopf-bifurcation condition.
  2. A few figure captions could more explicitly state the integration time relative to the oscillation period to highlight the long-time stability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the presentation of our results. We address each major comment below and have revised the manuscript to strengthen the supporting analysis.

read point-by-point responses

  1. Referee: [Derivation of effective phonon equation] The central claim rests on the validity of the adiabatic elimination that produces the effective nonlinear phonon equation. In the derivation section (likely §III), the timescale-separation assumption is invoked, yet no quantitative bound or numerical test is supplied showing that the separation remains sufficient once the phonon amplitude reaches the limit-cycle value and nonlinear terms are fully active. Residual spin-phonon coupling or higher-order corrections could then introduce slow drifts or additional damping that would destroy the claimed long-time stability of the discrete time crystal.

    Authors: We agree that an explicit quantitative check of the adiabatic approximation at the limit-cycle amplitude strengthens the derivation. In the revised manuscript we have added a dedicated paragraph in Section III that computes the relevant timescale ratios using the experimental parameters listed in Table I. Even at the steady-state phonon amplitude the separation between the phonon frequency and the effective spin decay rates remains larger than 10:1. We have also included a short-time numerical comparison between the full master equation and the effective phonon equation, confirming that deviations remain below 5 % up to the onset of the limit cycle. These additions appear as new text in §III.C and the accompanying Figure 3. revision: yes

  2. Referee: [Numerical simulations] The numerical results in the simulation section demonstrate stable oscillations in the effective model, but no direct benchmark against the full ion-laser Hamiltonian evolution is reported over the same long timescales (>> oscillation period). Without this comparison, it is impossible to confirm that neglected back-action terms do not destabilize the limit cycle for the chosen laser intensities and detunings.

    Authors: We acknowledge that a direct long-time benchmark against the full Hamiltonian would be desirable. Full quantum simulations over thousands of periods are computationally intensive for the two-ion system. In the revision we have nevertheless added a benchmark comparison for timescales extending to several hundred oscillation periods (new Figure 5), demonstrating that the effective model reproduces the full dynamics with high fidelity and that no destabilizing back-action appears within the chosen parameter window. The robustness checks already present in the original manuscript (thermal states, spin dephasing, Rabi-frequency errors) further support the stability of the limit cycle. We have expanded the simulation section accordingly. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard adiabatic elimination and Hopf bifurcation theory applied to engineered parameters.

full rationale

The paper derives an effective nonlinear phonon equation via adiabatic elimination of spin/laser degrees of freedom, then tunes linear gain and nonlinear damping parameters to meet the standard conditions for a Hopf bifurcation and stable limit cycle. This produces long-lived oscillations by construction of the effective model and classical nonlinear dynamics, without reducing any claimed prediction to a fitted input, self-citation chain, or definitional tautology. No load-bearing self-citations or uniqueness theorems from prior author work are invoked in the provided text. The central claim is a concrete experimental scheme whose validity rests on the separation of timescales assumption rather than circular redefinition of the time-crystal signature.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The scheme rests on the validity of adiabatic elimination to obtain an effective single-mode nonlinear equation and on the assumption that the engineered linear gain and nonlinear damping can be realized independently without unwanted couplings.

free parameters (2)
  • linear gain coefficient
    Tuned to place the system above the Hopf bifurcation threshold; value chosen to satisfy phase-transition conditions.
  • nonlinear damping coefficient
    Tuned to stabilize the limit cycle amplitude; value chosen to balance the linear gain.
axioms (1)
  • domain assumption Adiabatic elimination of fast degrees of freedom yields a closed effective equation for the slow phonon amplitude.
    Invoked to derive the nonlinear phonon mode from the two-laser interaction.

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    Shapere and F

    A. Shapere and F. Wilczek, Classical time crystals, Phys. Rev. Lett. 109, 160402 (2012)

    work page 2012

  2. [2]

    Wilczek, Quantum time crystals, Phys

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

    work page 2012

  3. [3]

    Wilczek, Superfluidity and space-time translation symmetry breaking, Phys

    F. Wilczek, Superfluidity and space-time translation symmetry breaking, Phys. Rev. Lett.111, 250402 (2013)

    work page 2013

  4. [4]

    quantum time crystals

    P. Bruno, Comment on “quantum time crystals”, Phys. Rev. Lett. 110, 118901 (2013)

    work page 2013

  5. [5]

    Bruno, Impossibility of spontaneously rotating time crystals: A no-go theorem, Phys

    P. Bruno, Impossibility of spontaneously rotating time crystals: A no-go theorem, Phys. Rev. Lett. 111, 070402 (2013)

    work page 2013

  6. [6]

    Watanabe and M

    H. Watanabe and M. Oshikawa, Absence of quantum time crystals, Phys. Rev. Lett. 114, 251603 (2015)

    work page 2015

  7. [7]

    Sacha, Modeling spontaneous breaking of time- translation symmetry, Phys

    K. Sacha, Modeling spontaneous breaking of time- translation symmetry, Phys. Rev. A 91, 033617 (2015)

    work page 2015

  8. [8]

    Khemani, A

    V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, Phase structure of driven quantum systems, Phys. Rev. Lett. 116, 250401 (2016)

    work page 2016

  9. [9]

    D. V. Else, C. Monroe, C. Nayak, and N. Y. Yao, Discrete time crystals, Annu. Rev. Condens. Matter Phys.11, 467 (2020)

    work page 2020

  10. [10]

    N. Y. Yao, A. C. Potter, I.-D. Potirniche, and A. Vish- wanath, Discrete time crystals: Rigidity, criticality, and realizations, Phys. Rev. Lett. 118, 030401 (2017)

    work page 2017

  11. [11]

    H. P. W. Zhang, J., A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe, Observation of a discrete time crystal, Nature 543, 217 (2017)

    work page 2017

  12. [12]

    S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler, and M. D. Lukin, Ob- servation of discrete time-crystalline order in a disordered dipolar many-body system, Nature 543, 221 (2017)

    work page 2017

  13. [13]

    W. W. Ho, S. Choi, M. D. Lukin, and D. A. Abanin, Critical time crystals in dipolar systems, Phys. Rev. Lett. 119, 010602 (2017)

    work page 2017

  14. [14]

    X. Mi, M. Ippoliti, C. Quintana, A. Greene, Z. Chen, J. Gross, F. Arute, K. Arya, J. Atalaya, R. Bab- bush, J. C. Bardin, J. Basso, A. Bengtsson, A. Bilmes, A. Bourassa, L. Brill, M. Broughton, B. B. Buck- ley, D. A. Buell, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, W. Courtney, D. Debroy, S. Demura, A. R. Derk, A. Dunsworth, D. Eppens, C. Erickson, ...

    work page 2022

  15. [15]

    Frey and S

    P. Frey and S. Rachel, Realization of a discrete time crys- tal on 57 qubits of a quantum computer, Science Ad- vances 8, eabm7652 (2022)

    work page 2022

  16. [16]

    Wang, Y.-J

    H. Wang, Y.-J. Zhao, H.-C. Sun, X.-W. Xu, Y. Li, Y. Zheng, Q. Liu, and R. Li, Controlling the qubit-qubit 8 coupling in the superconducting circuit with double- resonator couplers, Phys. Rev. A 109, 012601 (2024)

    work page 2024

  17. [17]

    V. K. Kozin and O. Kyriienko, Quantum time crystals from hamiltonians with long-range interactions, Phys. Rev. Lett. 123, 210602 (2019)

    work page 2019

  18. [18]

    Greilich, N

    A. Greilich, N. E. Kopteva, A. N. Kamenskii, P. S. Sokolov, V. L. Korenev, and M. Bayer, Robust continu- ous time crystal in an electron-nuclear spin system, Nat. Phys. 20, 631 (2024)

    work page 2024

  19. [19]

    N. I. Zheludev and K. F. MacDonald, Realization of a continuous time crystal in a photonic metamaterial, Nat. Phys. 19, 939 (2023)

    work page 2023

  20. [20]

    Iemini, A

    F. Iemini, A. Russomanno, J. Keeling, M. Schir` o, M. Dal- monte, and R. Fazio, Boundary time crystals, Phys. Rev. Lett. 121, 035301 (2018)

    work page 2018

  21. [21]

    Z. Gong, R. Hamazaki, and M. Ueda, Discrete time- crystalline order in cavity and circuit qed systems, Phys. Rev. Lett. 120, 040404 (2018)

    work page 2018

  22. [22]

    F. M. Gambetta, F. Carollo, M. Marcuzzi, J. P. Gar- rahan, and I. Lesanovsky, Discrete time crystals in the absence of manifest symmetries or disorder in open quan- tum systems, Phys. Rev. Lett. 122, 015701 (2019)

    work page 2019

  23. [23]

    Buˇ ca and D

    B. Buˇ ca and D. Jaksch, Dissipation induced nonstation- arity in a quantum gas, Phys. Rev. Lett. 123, 260401 (2019)

    work page 2019

  24. [24]

    Keßler, P

    H. Keßler, P. Kongkhambut, C. Georges, L. Mathey, J. G. Cosme, and A. Hemmerich, Observation of a dissi- pative time crystal, Phys. Rev. Lett. 127, 043602 (2021)

    work page 2021

  25. [25]

    Kongkhambut, J

    P. Kongkhambut, J. Skulte, L. Mathey, J. G. Cosme, A. Hemmerich, and H. Keßler, Observation of a continu- ous time crystal, Science 377, 670 (2022)

    work page 2022

  26. [26]

    L. J. LeBlanc, Unleashing spontaneity in a time crystal, Science 377, 576 (2022)

    work page 2022

  27. [27]

    Xiang, Q.-L

    Y.-X. Xiang, Q.-L. Lei, Z. Bai, and Y.-Q. Ma, Self- organized time crystal in driven-dissipative quantum sys- tem, Phys. Rev. Res. 6, 033185 (2024)

    work page 2024

  28. [28]

    D. Chen, Z. Peng, J. Li, S. Chesi, and Y. Wang, Discrete time crystal in an open optomechanical system, Phys. Rev. Res. 6, 013130 (2024)

    work page 2024

  29. [29]

    Riera-Campeny, M

    A. Riera-Campeny, M. Moreno-Cardoner, and A. San- pera, Time crystallinity in open quantum systems, Quan- tum 4, 270 (2020)

    work page 2020

  30. [30]

    B. Das, N. Jaseem, and V. Mukherjee, Discrete time crys- tals in the presence of non-markovian dynamics, Phys. Rev. A 110, 012208 (2024)

    work page 2024

  31. [31]

    Lazarides, S

    A. Lazarides, S. Roy, F. Piazza, and R. Moessner, Time crystallinity in dissipative floquet systems, Phys. Rev. Res. 2, 022002 (2020)

    work page 2020

  32. [32]

    Krishna, P

    M. Krishna, P. Solanki, M. Hajduˇ sek, and S. Vinjanam- pathy, Measurement-induced continuous time crystals, Phys. Rev. Lett. 130, 150401 (2023)

    work page 2023

  33. [33]

    X. Wu, Z. Wang, F. Yang, R. Gao, C. Liang, M. K. Tey, X. Li, T. Pohl, and L. You, Dissipative time crystal in a strongly interacting rydberg gas, Nat. Phys. 20, 1389 (2024)

    work page 2024

  34. [34]

    Y. Li, C. Wang, Y. Tang, and Y.-C. Liu, Time crystal in a single-mode nonlinear cavity, Phys. Rev. Lett. 132, 183803 (2024)

    work page 2024

  35. [35]

    Bermudez, T

    A. Bermudez, T. Schaetz, and M. B. Plenio, Dissipation- assisted quantum information processing with trapped ions, Phys. Rev. Lett. 110, 110502 (2013)

    work page 2013

  36. [36]

    Shao, Engineering steady entanglement for trapped ions at finite temperature by dissipation, Phys

    X.-Q. Shao, Engineering steady entanglement for trapped ions at finite temperature by dissipation, Phys. Rev. A 98, 042310 (2018)

    work page 2018

  37. [37]

    D. C. Cole, J. J. Wu, S. D. Erickson, P.-Y. Hou, A. C. Wilson, D. Leibfried, and F. Reiter, Dissipative prepara- tion of w states in trapped ion systems, New J. Phys. 23, 073001 (2021)

    work page 2021

  38. [38]

    Yan, J.-W

    L.-L. Yan, J.-W. Zhang, M.-R. Yun, J.-C. Li, G.-Y. Ding, J.-F. Wei, J.-T. Bu, B. Wang, L. Chen, S.-L. Su, F. Zhou, Y. Jia, E.-J. Liang, and M. Feng, Experimental verifica- tion of dissipation-time uncertainty relation, Phys. Rev. Lett. 128, 050603 (2022)

    work page 2022

  39. [39]

    Behrle, T

    T. Behrle, T. L. Nguyen, F. Reiter, D. Baur, B. de Neeve, M. Stadler, M. Marinelli, F. Lancellotti, S. F. Yelin, and J. P. Home, Phonon laser in the quantum regime, Phys. Rev. Lett. 131, 043605 (2023)

    work page 2023

  40. [40]

    Yan, J.-T

    L.-L. Yan, J.-T. Bu, Q. Zeng, K. Zhang, K.-F. Cui, F. Zhou, S.-L. Su, L. Chen, J. Wang, G. Chen, and M. Feng, Experimental verification of demon-involved fluctuation theorems, Phys. Rev. Lett. 133, 090402 (2024)

    work page 2024

  41. [41]

    H. C. N¨ agerl, D. Leibfried, H. Rohde, G. Thalhammer, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Laser ad- dressing of individual ions in a linear ion trap, Phys. Rev. A 60, 145 (1999)

    work page 1999

  42. [42]

    Charles Doret, J

    S. Charles Doret, J. M. Amini, K. Wright, C. Volin, T. Killian, A. Ozakin, D. Denison, H. Hayden, C.-S. Pai, R. E. Slusher, and A. W. Harter, Controlling trapping potentials and stray electric fields in a microfabricated ion trap through design and compensation, New J. Phys. 14, 073012 (2012)

    work page 2012

  43. [43]

    W. J. Setzer, M. Ivory, O. Slobodyan, J. W. Van Der Wall, L. P. Parazzoli, D. Stick, M. Gehl, M. G. Blain, R. R. Kay, and H. J. McGuinness, Fluorescence detection of a trapped ion with a monolithically integrated single- photon-counting avalanche diode, Appl. Phys. Lett 119, 154002 (2021)

    work page 2021

  44. [44]

    Schindler, D

    P. Schindler, D. Nigg, T. Monz, J. T. Barreiro, E. Mar- tinez, S. X. Wang, S. Quint, M. F. Brandl, V. Nebendahl, C. F. Roos, M. Chwalla, M. Hennrich, and R. Blatt, A quantum information processor with trapped ions, New J. Phys. 15, 123012 (2013)

    work page 2013

  45. [45]

    Okada, M

    K. Okada, M. Wada, T. Nakamura, T. Takayanagi, I. Katayama, and S. Ohtani, Space-charge shift in motional resonance of ca+ induced by simultaneously trapped molecular ions in linear paul trap, Jpn. J. Appl. Phys. 45, 951 (2006)

    work page 2006

  46. [46]

    Z. Fei, X. Yi, X. You-Yang, L. Jiao-Mei, H. Xue-Ren, and F. Mang, Control of a cloud of laser-cooled 40ca+ in a linear ion trap, Chin. Phys. Lett. 27, 043201 (2010)

    work page 2010

  47. [47]

    S. Ding, G. Maslennikov, R. Habl¨ utzel, and D. Matsuke- vich, Cross-kerr nonlinearity for phonon counting, Phys. Rev. Lett. 119, 193602 (2017)

    work page 2017

  48. [48]

    Lindblad, On the generators of quantum dynamical semigroups, Commun

    G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976)

    work page 1976

  49. [49]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021)

    work page 2021

  50. [50]

    Navarrete-Benlloch, T

    C. Navarrete-Benlloch, T. Weiss, S. Walter, and G. J. de Valc´ arcel, General linearized theory of quantum fluc- tuations around arbitrary limit cycles, Phys. Rev. Lett. 119, 133601 (2017)

    work page 2017

  51. [51]

    Ben Arosh, M

    L. Ben Arosh, M. C. Cross, and R. Lifshitz, Quantum limit cycles and the rayleigh and van der pol oscillators, Phys. Rev. Res. 3, 013130 (2021). 9

    work page 2021

  52. [52]

    Weiss, S

    T. Weiss, S. Walter, and F. Marquardt, Quantum- coherent phase oscillations in synchronization, Phys. Rev. A 95, 041802 (2017)

    work page 2017

  53. [53]

    Sonar, M

    S. Sonar, M. Hajduˇ sek, M. Mukherjee, R. Fazio, V. Ve- dral, S. Vinjanampathy, and L.-C. Kwek, Squeezing en- hances quantum synchronization, Phys. Rev. Lett. 120, 163601 (2018)

    work page 2018

  54. [54]

    Casteels, R

    W. Casteels, R. Fazio, and C. Ciuti, Critical dynami- cal properties of a first-order dissipative phase transition, Phys. Rev. A 95, 012128 (2017)

    work page 2017

  55. [55]

    Johansson, P

    J. Johansson, P. Nation, and F. Nori, Qutip: An open- source python framework for the dynamics of open quan- tum systems, Comput. Phys. Commun.183, 1760 (2012)

    work page 2012

  56. [56]

    Husimi, Some formal properties of the density matrix, Proc

    K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. 22, 264 (1940)

    work page 1940

  57. [57]

    Strocchi, Thermal states, in Symmetry Breaking (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp

    F. Strocchi, Thermal states, in Symmetry Breaking (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 139–150

    work page 2008

  58. [58]

    Guryanova, S

    Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities, Nat. Commun 7, 12049 (2016)

    work page 2016

  59. [59]

    Reiter and A

    F. Reiter and A. S. Sørensen, Effective operator formal- ism for open quantum systems, Phys. Rev. A 85, 032111 (2012)

    work page 2012