pith. sign in

arxiv: 2511.07140 · v2 · submitted 2025-11-10 · ❄️ cond-mat.mes-hall · cond-mat.str-el· quant-ph

Synchronizing microwave cQED limit-cycle oscillators

Cecilie Hermansen , Jens Paaske This is my paper

Pith reviewed 2026-05-17 23:46 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-elquant-ph
keywords quantum synchronizationlimit-cycle oscillatorscircuit quantum electrodynamicsdouble quantum dotHopf bifurcationKeldysh action
0
0 comments X

The pith

Two microwave resonators coupled through a voltage-biased double quantum dot synchronize their limit-cycle oscillations for small frequency detunings.

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

The paper investigates a hybrid system of superconducting microwave resonators coupled resonantly to a voltage-biased double quantum dot. It shows that a Hopf bifurcation occurs at a critical electron-photon coupling strength, after which effective negative friction sustains steady limit-cycle oscillations in each resonator. When two resonators share the same DQD, their oscillations synchronize provided the frequency detuning stays small enough. The analysis derives a nonlinear photon Keldysh action through perturbation theory in the circuit fine-structure constant and solves the resulting saddle-point and Fokker-Planck equations; these agree with the Lindblad master equation in the infinite-bias Markovian limit.

Core claim

A Hopf bifurcation occurs at a critical value of the electron-photon coupling, beyond which an effective negative friction sustains steady limit-cycle oscillations of individual resonators. Two such limit-cycle resonators coupled via the same voltage-biased DQD synchronize for small enough frequency detuning. A nonlinear photon Keldysh action is derived by perturbation theory in the effective circuit fine-structure constant, and the limit-cycle dynamics is analyzed in terms of resulting saddle-point and Fokker-Planck equations that agree with the Lindblad master equation in the Markovian limit of infinite bias voltage.

What carries the argument

Nonlinear photon Keldysh action derived by perturbation theory in the effective circuit fine-structure constant, which produces saddle-point and Fokker-Planck equations for the limit-cycle dynamics.

If this is right

  • Individual resonators develop self-sustained limit-cycle oscillations once the electron-photon coupling exceeds the critical value for the Hopf bifurcation.
  • Two resonators coupled through the shared DQD lock their phases when frequency detuning is sufficiently small.
  • The Keldysh saddle-point and Fokker-Planck descriptions reproduce the synchronization and match the Lindblad master-equation solution in the infinite-bias limit.
  • Effective negative friction from the driven DQD is what sustains the steady oscillations.

Where Pith is reading between the lines

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

  • The same shared-DQD coupling could be used to synchronize larger arrays of resonators in circuit-QED devices.
  • Varying the DQD bias voltage in experiment would provide a direct test of the Markovian approximation used here.
  • The synchronization mechanism offers a concrete quantum-electronic route to study analogs of classical coupled-oscillator phenomena.

Load-bearing premise

The perturbative expansion in the effective circuit fine-structure constant remains valid near the Hopf bifurcation and the Markovian limit of infinite bias voltage captures the essential physics.

What would settle it

An experiment that measures resonator amplitude versus coupling strength and finds no onset of sustained oscillations at the predicted critical value, or that finds loss of phase locking at a detuning smaller than expected from the Fokker-Planck analysis.

Figures

Figures reproduced from arXiv: 2511.07140 by Cecilie Hermansen, Jens Paaske.

Figure 1
Figure 1. Figure 1: FIG. 1. Voltage-biased double quantum dot with each dot [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Real and imaginary parts of the retarded, and imagi [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Real and imaginary parts of the photon interaction [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Limit cycle radius, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Wigner function (blue) overlaid by the limit-cycle [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Real (a) and imaginary (b) part of eigenfrequencies [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. From left to right column: Wigner function for the left and right mode, joint position and phase difference probability [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Power spectral density for the left/right mode (or [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Occupation probabilities of the eigenstates of the [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
read the original abstract

Self-sustained oscillators play a central role in the stabilization and synchronization of complex dynamical systems. A number of different physical systems are currently being investigated to clarify the importance of such active components in the quantum realm. Here we explore the properties of a driven dissipative electron-photon hybrid system based on superconducting microwave resonators coupled resonantly to a voltage-biased double quantum dot (DQD). First, we establish a Hopf bifurcation at a critical value of the electron-photon coupling, beyond which an effective negative friction sustains steady limit-cycle oscillations of individual resonators. Second, we show that two such limit-cycle resonators coupled via the same voltage-biased DQD synchronize for small enough frequency detuning. A nonlinear photon Keldysh action is derived by perturbation theory in the effective circuit fine-structure constant, and the limit-cycle dynamics is analyzed in terms of resulting saddle-point, and Fokker-Planck equations. In the Markovian limit of infinite bias voltage, these results are shown to agree well with the solution of a corresponding Lindblad master equation for the DQD resonator system.

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 paper explores self-sustained oscillators in a driven dissipative electron-photon hybrid system with superconducting microwave resonators coupled to a voltage-biased double quantum dot (DQD). It establishes a Hopf bifurcation at a critical electron-photon coupling, resulting in limit-cycle oscillations for individual resonators. Furthermore, it demonstrates that two such resonators coupled via the same DQD synchronize for small frequency detuning. The analysis uses a nonlinear photon Keldysh action derived from perturbation theory in the effective circuit fine-structure constant, examined through saddle-point and Fokker-Planck equations. In the Markovian limit of infinite bias voltage, these findings align with solutions from a corresponding Lindblad master equation.

Significance. This research advances the field of quantum synchronization in circuit quantum electrodynamics by providing a detailed theoretical model for limit-cycle behavior and synchronization in hybrid systems. The perturbative Keldysh approach combined with master equation validation offers a robust method for analyzing such non-equilibrium quantum phenomena. Should the results be confirmed, they may inform experimental designs for synchronized quantum oscillators.

major comments (2)
  1. [Derivation of the nonlinear photon Keldysh action] In the derivation of the nonlinear photon Keldysh action by perturbation theory in the effective circuit fine-structure constant (as described in the abstract), the expansion is applied to obtain the saddle-point and Fokker-Planck equations used for the synchronization analysis. Near the Hopf bifurcation, however, the limit-cycle amplitude is small and fluctuations become parametrically large; this regime can invalidate the truncation even for weak bare coupling. This is load-bearing for the central synchronization claim for small detuning.
  2. [Comparison with Lindblad master equation] The agreement with the Lindblad master equation is demonstrated only in the Markovian limit of infinite bias voltage. It remains open whether finite-bias effects or non-Markovian corrections alter the synchronization threshold, which is central to the claim that the two resonators synchronize for small enough frequency detuning.
minor comments (2)
  1. [Abstract] The abstract states that the results 'agree well' with the Lindblad equation but does not specify quantitative measures of agreement (e.g., via a figure or table comparing thresholds or amplitudes).
  2. Clarify the typical experimental range of the effective circuit fine-structure constant to better contextualize the validity of the perturbative regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: In the derivation of the nonlinear photon Keldysh action by perturbation theory in the effective circuit fine-structure constant (as described in the abstract), the expansion is applied to obtain the saddle-point and Fokker-Planck equations used for the synchronization analysis. Near the Hopf bifurcation, however, the limit-cycle amplitude is small and fluctuations become parametrically large; this regime can invalidate the truncation even for weak bare coupling. This is load-bearing for the central synchronization claim for small detuning.

    Authors: We thank the referee for highlighting this subtlety in the perturbative regime. The expansion is performed in the small effective circuit fine-structure constant α, which parametrizes the electron-photon interaction strength and is independent of the oscillation amplitude. This expansion yields the effective nonlinear Keldysh action before any dynamical analysis. The saddle-point equations then determine the mean-field limit-cycle amplitude (which vanishes continuously at the Hopf bifurcation), while the Fokker-Planck equation provides a systematic treatment of fluctuations around this solution within the same perturbative framework. The synchronization for small detuning follows from the phase-locking dynamics encoded in the coupled Fokker-Planck equations and does not rely on a finite amplitude per se. We maintain that the truncation remains valid because parametrically large fluctuations are already incorporated via the stochastic terms rather than requiring higher orders in α. We will add a clarifying paragraph on the range of validity near the bifurcation in the revised manuscript. revision: partial

  2. Referee: The agreement with the Lindblad master equation is demonstrated only in the Markovian limit of infinite bias voltage. It remains open whether finite-bias effects or non-Markovian corrections alter the synchronization threshold, which is central to the claim that the two resonators synchronize for small enough frequency detuning.

    Authors: We agree that the explicit numerical comparison with the Lindblad master equation is restricted to the Markovian (infinite-bias) limit. The Keldysh action formalism employed in the manuscript is formulated for arbitrary bias voltage and retains non-Markovian contributions through the frequency-dependent self-energies. While a full finite-bias comparison lies beyond the scope of the present work, the structure of the effective equations indicates that the synchronization threshold is set by the dissipative coupling strength mediated by the DQD, which remains operative at finite bias. We will add a short discussion of the expected finite-bias corrections and the robustness of the synchronization claim in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation self-contained via perturbation theory and independent cross-check

full rationale

The paper derives the nonlinear photon Keldysh action from perturbation theory in the effective circuit fine-structure constant, obtains the limit-cycle dynamics from the resulting saddle-point and Fokker-Planck equations, and explicitly cross-validates the synchronization threshold against an independent Lindblad master equation in the Markovian limit. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the synchronization for small detuning follows from the dynamical equations rather than being imposed by the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard perturbative expansion of a Keldysh action and the validity of the Markovian limit; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Perturbation theory in the effective circuit fine-structure constant is valid near the Hopf bifurcation
    Invoked to derive the nonlinear photon action
  • domain assumption Infinite bias voltage renders the system Markovian
    Used to obtain agreement with the Lindblad master equation

pith-pipeline@v0.9.0 · 5484 in / 1225 out tokens · 23906 ms · 2026-05-17T23:46:43.689754+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    The electronic Keldysh matrix Green function reads ˆGij(t−t ′) = GR ij(t−t ′)G K ij(t−t ′) 0G A ij(t−t ′) ,(8) with inverse component (L/R-matrix) Green functions given in the wide-band limit, respectively, by the Fourier transforms GR/A(ω) −1 ij = ω−ε L ±iΓ L −t −t ω−ε R ±iΓ R ij G−1(ω) K ij = 2iΓLFL(ω) 0 0 2iΓ RFR(ω) ij .(9) The tunneling rates Γ i =πρ ...

    work page

  2. [2]

    The rightmost column shows the power spectral density (PSD), calculated from the auto- correlation function ofX(t) (cf

    The corresponding phase-space trajectories are shown in Fig.6 together with the Lissajous curves traced out by (X(t), F(t)), revealing frequency entrainment when the amplitude of the force exceeds a critical strength, found here to beA c = 0.044. The rightmost column shows the power spectral density (PSD), calculated from the auto- correlation function of...

    work page

  3. [3]

    S. H. Strogatz,Nonlinear Dynamics and Chaos(Rout- ledge & CRC Press, Boca Raton, FL, USA, 2024)

    work page 2024

  4. [4]

    Pikovsky, M

    A. Pikovsky, M. Rosenblum, and J. Kurths,Synchroniza- tion: A Universal Concept in Nonlinear Sciences(Cam- bridge University Press, Cambridge, England, UK, 2001)

    work page 2001

  5. [5]

    Minorsky,Introduction to Nonlinear Mechanics: Topological Methods, Analytical Methods, Nonlinear Res- onance, Relaxation Oscillations(J

    N. Minorsky,Introduction to Nonlinear Mechanics: Topological Methods, Analytical Methods, Nonlinear Res- onance, Relaxation Oscillations(J. W. Edwards, Ann Arbor, MI, USA, 1947)

    work page 1947

  6. [6]

    Balanov, N

    A. Balanov, N. Janson, D. Postnov, and O. Sosnovtseva, Synchronization(Springer, Berlin, Germany, 2009)

    work page 2009

  7. [7]

    Marquardt and S

    F. Marquardt and S. M. Girvin, Optomechanics, Physics 2(2009)

    work page 2009

  8. [8]

    L¨ orch, J

    N. L¨ orch, J. Qian, A. Clerk, F. Marquardt, and K. Ham- merer, Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime, Phys. Rev. X4, 011015 (2014)

    work page 2014

  9. [9]

    A. Chia, L. C. Kwek, and C. Noh, Relaxation oscillations and frequency entrainment in quantum mechanics, Phys. Rev. E102, 042213 (2020)

    work page 2020

  10. [10]

    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)

    work page 2021

  11. [11]

    A. J. Sudler, J. Talukdar, and D. Blume, Driven general- ized quantum Rayleigh–van der Pol oscillators: Phase localization and spectral response, Phys. Rev. E109, 054207 (2024)

    work page 2024

  12. [12]

    Dutta, S

    S. Dutta, S. Zhang, and M. Haque, Quantum Origin of Limit Cycles, Fixed Points, and Critical Slowing Down, Phys. Rev. Lett.134, 050407 (2025)

    work page 2025

  13. [13]

    A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, Measures of Quantum Synchronization in Con- tinuous Variable Systems, Phys. Rev. Lett.111, 103605 (2013)

    work page 2013

  14. [14]

    T. E. Lee and H. R. Sadeghpour, Quantum Synchroniza- tion of Quantum van der Pol Oscillators with Trapped Ions, Phys. Rev. Lett.111, 234101 (2013)

    work page 2013

  15. [15]

    Walter, A

    S. Walter, A. Nunnenkamp, and C. Bruder, Quantum Synchronization of a Driven Self-Sustained Oscillator, Phys. Rev. Lett.112, 094102 (2014)

    work page 2014

  16. [16]

    Walter, A

    S. Walter, A. Nunnenkamp, and C. Bruder, Quantum synchronization of two Van der Pol oscillators, Ann. Phys.527, 131 (2015)

    work page 2015

  17. [17]

    L¨ orch, E

    N. L¨ orch, E. Amitai, A. Nunnenkamp, and C. Bruder, Genuine Quantum Signatures in Synchronization of An- harmonic Self-Oscillators, Phys. Rev. Lett.117, 073601 (2016)

    work page 2016

  18. [18]

    Tindall, C

    J. Tindall, C. S. Mu˜ noz, B. Buˇ ca, and D. Jaksch, Quan- tum synchronisation enabled by dynamical symmetries and dissipation, New J. Phys.22, 013026 (2020)

    work page 2020

  19. [19]

    Buˇ ca, C

    B. Buˇ ca, C. Booker, and D. Jaksch, Algebraic theory of quantum synchronization and limit cycles under dissipa- tion, SciPost Phys.12, 097 (2022)

    work page 2022

  20. [20]

    Kehrer, T

    T. Kehrer, T. Nadolny, and C. Bruder, Quantum syn- chronization through the interference blockade, Phys. Rev. A110, 042203 (2024)

    work page 2024

  21. [21]

    relaxation-oscillations

    B. van der Pol, On “relaxation-oscillations”, London, Ed- inburgh, and Dublin Philosophical Magazine and Journal of Science2, 978 (1926)

    work page 1926

  22. [22]

    Ginoux and C

    J.-M. Ginoux and C. Letellier, Van der Pol and the his- tory of relaxation oscillations: Toward the emergence of a concept, Chaos22, 023120 (2012)

    work page 2012

  23. [23]

    T. Frey, P. J. Leek, M. Beck, A. Blais, T. Ihn, K. Ensslin, and A. Wallraff, Dipole Coupling of a Double Quantum Dot to a Microwave Resonator, Phys. Rev. Lett.108, 046807 (2012)

    work page 2012

  24. [24]

    K. D. Petersson, C. G. Smith, D. Anderson, P. Atkin- son, G. A. C. Jones, and D. A. Ritchie, Charge and Spin State Readout of a Double Quantum Dot Coupled to a Resonator, Nano Lett.10, 2789 (2010)

    work page 2010

  25. [25]

    Y.-Y. Liu, K. D. Petersson, J. Stehlik, J. M. Taylor, and J. R. Petta, Photon Emission from a Cavity-Coupled Double Quantum Dot, Phys. Rev. Lett.113, 036801 (2014)

    work page 2014

  26. [26]

    Y.-Y. Liu, J. Stehlik, C. Eichler, M. J. Gullans, J. M. Taylor, and J. R. Petta, Semiconductor double quantum dot micromaser, Science347, 285 (2015)

    work page 2015

  27. [27]

    Y.-Y. Liu, J. Stehlik, M. J. Gullans, J. M. Taylor, and J. R. Petta, Injection locking of a semiconduc- tor double-quantum-dot micromaser, Phys. Rev. A92, 053802 (2015)

    work page 2015

  28. [28]

    Y.-Y. Liu, J. Stehlik, C. Eichler, X. Mi, T. R. Hartke, M. J. Gullans, J. M. Taylor, and J. R. Petta, Threshold Dynamics of a Semiconductor Single Atom Maser, Phys. Rev. Lett.119, 097702 (2017)

    work page 2017

  29. [29]

    Stockklauser, V

    A. Stockklauser, V. F. Maisi, J. Basset, K. Cujia, C. Re- ichl, W. Wegscheider, T. Ihn, A. Wallraff, and K. Ensslin, Microwave Emission from Hybridized States in a Semi- conductor Charge Qubit, Phys. Rev. Lett.115, 046802 (2015)

    work page 2015

  30. [30]

    X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, and J. R. Petta, Strong coupling of a single electron in silicon to a microwave photon, Science355, 156 (2016)

    work page 2016

  31. [31]

    Stockklauser, P

    A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Strong Coupling Cavity QED with Gate-Defined Double Quantum Dots Enabled by a High Impedance Resonator, Phys. Rev. X7, 011030 (2017)

    work page 2017

  32. [32]

    L. E. Bruhat, T. Cubaynes, J. J. Viennot, M. C. Darti- ailh, M. M. Desjardins, A. Cottet, and T. Kontos, Cir- cuit QED with a quantum-dot charge qubit dressed by Cooper pairs, Phys. Rev. B98, 155313 (2018)

    work page 2018

  33. [33]

    Scarlino, D

    P. Scarlino, D. J. van Woerkom, A. Stockklauser, J. V. Koski, M. C. Collodo, S. Gasparinetti, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, All-Microwave Control and Dispersive Readout of Gate- Defined Quantum Dot Qubits in Circuit Quantum Elec- trodynamics, Phys. Rev. Lett.122, 206802 (2019)

    work page 2019

  34. [34]

    Scarlino, J

    P. Scarlino, J. H. Ungerer, D. J. van Woerkom, M. Mancini, P. Stano, C. M¨ uller, A. J. Landig, J. V. 13 Koski, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, In situ Tuning of the Electric-Dipole Strength of a Double-Dot Charge Qubit: Charge-Noise Protection and Ultrastrong Coupling, Phys. Rev. X12, 031004 (2022)

    work page 2022

  35. [35]

    J. H. Ungerer, A. Pally, A. Kononov, S. Lehmann, J. Ridderbos, P. P. Potts, C. Thelander, K. A. Dick, V. F. Maisi, P. Scarlino, A. Baumgartner, and C. Sch¨ onenberger, Strong coupling between a microwave photon and a singlet-triplet qubit, Nat. Commun.15, 1 (2024)

    work page 2024

  36. [36]

    Childress, A

    L. Childress, A. S. Sørensen, and M. D. Lukin, Meso- scopic cavity quantum electrodynamics with quantum dots, Phys. Rev. A69, 042302 (2004)

    work page 2004

  37. [37]

    D. J. van Woerkom, P. Scarlino, J. H. Ungerer, C. M¨ uller, J. V. Koski, A. J. Landig, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Microwave Photon- Mediated Interactions between Semiconductor Qubits, Phys. Rev. X8, 041018 (2018)

    work page 2018

  38. [38]

    Cottet, M

    A. Cottet, M. C. Dartiailh, M. M. Desjardins, T. Cubaynes, L. C. Contamin, M. Delbecq, J. J. Viennot, L. E. Bruhat, B. Dou¸ cot, and T. Kontos, Cavity QED with hybrid nanocircuits: from atomic-like physics to condensed matter phenomena, J. Phys.: Condens. Mat- ter29, 433002 (2017)

    work page 2017

  39. [39]

    Jan´ ık, K

    M. Jan´ ık, K. Roux, C. Borja-Espinosa, O. Sagi, A. Bagh- dadi, T. Adletzberger, S. Calcaterra, M. Botifoll, A. Garz´ on Manj´ on, J. Arbiol, D. Chrastina, G. Isella, I. M. Pop, and G. Katsaros, Strong charge-photon cou- pling in planar germanium enabled by granular alu- minium superinductors, Nat. Commun.16, 1 (2025)

    work page 2025

  40. [40]

    Y.-Y. Liu, T. R. Hartke, J. Stehlik, and J. R. Petta, Phase locking of a semiconductor double-quantum-dot single-atom maser, Phys. Rev. A96, 053816 (2017)

    work page 2017

  41. [41]

    Johansson, P

    J. Johansson, P. Nation, and F. Nori, Qutip: An open- source python framework for the dynamics of open quan- tum systems, Computer Physics Communications183, 1760 (2012)

    work page 2012

  42. [42]

    Johansson, P

    J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Computer Physics Communications184, 1234 (2013)

    work page 2013

  43. [43]

    Lambert, E

    N. Lambert, E. Gigu` ere, P. Menczel, B. Li, P. Hopf, G. Su´ arez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, QuTiP 5: The Quantum Toolbox in Python, Phys. Rep.1153, 1 (2026)

    work page 2026

  44. [44]

    Kirˇ sanskas, M

    G. Kirˇ sanskas, M. Francki´ e, and A. Wacker, Phenomeno- logical position and energy resolving Lindblad approach to quantum kinetics, Phys. Rev. B97, 035432 (2018)

    work page 2018

  45. [45]

    Hermansen, M

    C. Hermansen, M. Caltapanides, V. Meden, and J. Paaske, Simulating electron-vibron energy transfer with quantum dots and resonators, Phys. Rev. B110, 205424 (2024)

    work page 2024

  46. [46]

    Kamenev,Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, UK, 2023)

    A. Kamenev,Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, UK, 2023)

    work page 2023

  47. [47]

    Mozyrsky, M

    D. Mozyrsky, M. B. Hastings, and I. Martin, Intermit- tent polaron dynamics: Born-Oppenheimer approxima- tion out of equilibrium, Phys. Rev. B73, 035104 (2006)

    work page 2006

  48. [48]

    Schir´ o and K

    M. Schir´ o and K. Le Hur, Tunable hybrid quantum elec- trodynamics from nonlinear electron transport, Phys. Rev. B89, 195127 (2014)

    work page 2014

  49. [49]

    Cottet, Z

    A. Cottet, Z. Leghtas, and T. Kontos, Theory of inter- actions between cavity photons induced by a mesoscopic circuit, Phys. Rev. B102, 155105 (2020)

    work page 2020

  50. [50]

    Thompson and A

    F. Thompson and A. Kamenev, Qubit decoherence and symmetry restoration through real-time instantons, Phys. Rev. Res.4, 023020 (2022)

    work page 2022

  51. [51]

    J. T. Stuart, On the non-linear mechanics of wave distur- bances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow, J. Fluid Mech. 9, 353 (1960)

    work page 1960

  52. [52]

    L. D. Landau, On the theory of turbulence, Proc. Russian Acad. Sci.44, 311 (1944)

    work page 1944

  53. [53]

    T. Ota, T. Hayashi, K. Muraki, and T. Fujisawa, Wide- band capacitance measurement on a semiconductor dou- ble quantum dot for studying tunneling dynamics, Appl. Phys. Lett.96, 032104 (2010)

    work page 2010

  54. [54]

    The calculations were performed using the Mathematica function NDeigensystem, where the area of the cells in the discrete spatial grid can be specified

    work page

  55. [55]

    H. J. Carmichael,Statistical Methods in Quantum Optics 1(Springer, Berlin, Germany)

    work page

  56. [56]

    A. P. Kuznetsov, N. V. Stankevich, and L. V. Turukina, Coupled van der Pol–Duffing oscillators: Phase dynamics and structure of synchronization tongues, Physica D238, 1203 (2009)

    work page 2009

  57. [57]

    N. R. Bernier, L. D. T´ oth, A. K. Feofanov, and T. J. Kip- penberg, Level attraction in a microwave optomechanical circuit, Phys. Rev. A98, 023841 (2018)

    work page 2018

  58. [58]

    C. Lu, M. Kim, Y. Yang, Y. S. Gui, and C.-M. Hu, Syn- chronization of dissipatively coupled oscillators, J. Appl. Phys.134, 221101 (2023)

    work page 2023

  59. [59]

    Bourcin, A

    G. Bourcin, A. Gardin, J. Bourhill, V. Vlaminck, and V. Castel, Level attraction in a quasiclosed cavity: An- tiresonance in magnonic devices, Phys. Rev. Appl.22, 064036 (2024)

    work page 2024

  60. [60]

    S. M. Barnett and D. T. Pegg, Quantum theory of optical phase correlations, Phys. Rev. A42, 6713 (1990)

    work page 1990

  61. [61]

    M. R. Jessop, W. Li, and A. D. Armour, Phase synchro- nization in coupled bistable oscillators, Phys. Rev. Res. 2, 013233 (2020)

    work page 2020

  62. [62]

    T. E. Lee, C.-K. Chan, and S. Wang, Entanglement tongue and quantum synchronization of disordered os- cillators, Phys. Rev. E89, 022913 (2014)

    work page 2014

  63. [63]

    Christiansen, V

    H. Christiansen, V. V. Baran, and J. Paaske, Reduced ba- sis method for driven-dissipative quantum systems, Phys. Rev. Lett. 10.1103/7429-w2mx (2025)

    work page doi:10.1103/7429-w2mx 2025

  64. [64]

    Dorda, M

    A. Dorda, M. Nuss, W. von der Linden, and E. Arrigoni, Auxiliary master equation approach to nonequilibrium correlated impurities, Phys. Rev. B89, 165105 (2014)

    work page 2014

  65. [65]

    Haldar, D

    S. Haldar, D. Zenelaj, P. P. Potts, H. Havir, S. Lehmann, K. A. Dick, P. Samuelsson, and V. F. Maisi, Microwave power harvesting using resonator-coupled double quan- tum dot photodiode, Phys. Rev. B109, L081403 (2024)

    work page 2024

  66. [66]

    Haldar, H

    S. Haldar, H. Havir, W. Khan, D. Zenelaj, P. P. Potts, S. Lehmann, K. A. Dick, P. Samuelsson, and V. F. Maisi, High-efficiency microwave photodetection by cavity-coupled double quantum dots with single-cavity- photon sensitivity, Phys. Rev. Appl.24, 044074 (2025)

    work page 2025

  67. [67]

    14.2, cham- paign, IL (2024)

    Wolfram research inc., mathematica, vs. 14.2, cham- paign, IL (2024)

    work page 2024