Pith. sign in

REVIEW 3 major objections 4 minor 62 references

Statically coupled lossy qubits cool a periodically driven system into a low-energy steady state, preserving order that would otherwise heat away.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 21:04 UTC pith:JT5JNIOU

load-bearing objection Solid cooling result, but the time-crystal claim is overreached: the period-doubled response is driven by explicit 2T-symmetric noise, not spontaneous symmetry breaking. the 3 major comments →

arxiv 2607.16391 v2 pith:JT5JNIOU submitted 2026-07-17 quant-ph cond-mat.quant-gascond-mat.str-el

Dissipative Stabilization of Floquet-Engineered Many-Body Order

Shreyas Raman , Robin Sch\"afer , Alicia J. Kollar , Anushya Chandran This is my paper
classification quant-ph cond-mat.quant-gascond-mat.str-el
keywords Floquet engineeringdissipative stabilizationsteady stateFermi's Golden Rulediscrete time crystaltransverse-field Ising modelreservoir engineeringLindblad dynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to show that the unavoidable heating of periodically driven (Floquet) many-body systems can be counteracted by coupling the system to simple dissipative elements—spin-1/2 auxiliaries with a preferred decay direction. In the rotating frame defined by the drive, each auxiliary resonantly absorbs excitations of the effective Floquet Hamiltonian near its own energy splitting, acting as a narrow-band cold bath. The steady state is set by a rate balance between this cooling and the various heating channels, and the excess energy density can be made controllably small when the auxiliary is weak and narrow-band. Numerical simulations of driven Ising chains confirm the cooling in both phases and show that a period-doubled time-crystalline response can persist in the steady state. If correct, the scheme gives a practical, autonomous route to stabilizing Floquet-engineered phases without relying on prethermal windows.

Core claim

The central claim: a finite density of statically coupled lossy auxiliary spins cools a Floquet-driven system toward the ground state of its effective Hamiltonian H_eff, yielding a steady state that can exhibit order impossible in equilibrium, including a period-doubled response. Each spin-1/2 auxiliary (splitting Δ, decay κ) preferentially decays downward and resonantly absorbs excitations of H_eff near Δ, while off-resonant and sideband heating stay small at high frequency and weak coupling. Eigenstate populations then obey a parameter-free Fermi Golden Rule rate equation, and the steady-state energy offset is small when the coupling hierarchy holds. Lindblad simulations of driven Ising ch

What carries the argument

The central object is the dissipative auxiliary: a static spin-1/2 with energy splitting Δ, relaxation rate κ, and weak coupling (g/4) τ^y O to a local system operator O. In the rotating frame its m=0 term resonantly transfers excitations of H_eff near energy Δ into the auxiliary, which then decays; m≠0 sidebands are exponentially suppressed at high frequency. The Fermi Golden Rule rate equation for eigenstate populations of H_eff—combining intrinsic Floquet heating, period-jitter noise, and auxiliary transitions—determines the steady state. The paper's design rules are a hierarchy of scales: κ ~ ν ≲ Δ ~ Λ, g/4 ≲ κ, and Δ, κ, g ≪ ω, with Λ and ν the center and width of the low-energy excitat

Load-bearing premise

The discrete-time-crystal claim assumes period-two jitter breaks the symmetry between the two ferromagnetic sectors; at zero jitter the cooling balances both sectors equally and no period-doubled response appears, so the stabilized order is not spontaneous but noise-selected.

What would settle it

Run the long-range Ising simulation without period-two jitter (ε=0) starting from a generic (non-polarized) initial state and check whether the stroboscopic magnetization still period-doubles; the paper's own mechanism predicts it does not, which would confirm the response is an artifact of the explicitly 2T-periodic noise rather than spontaneous order.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A single auxiliary cools a finite chain to a low-energy steady state in a time that grows roughly linearly with system size, and the steady-state energy offset grows linearly with L, consistent with bulk heating balanced by one local cooler.
  • The FGR rate-equation prediction quantitatively matches the full Lindblad simulation without any fitted parameters, giving a rare analytical handle on driven-dissipative steady states.
  • With a finite density of auxiliaries, the excess energy density is controllably small in the high-frequency, weak-coupling regime, so the scheme can stabilize order in the thermodynamic limit.
  • In the long-range Ising model, the steady state exhibits a period-doubled magnetization response that persists at infinite time, whereas without the auxiliary the response decays.
  • The scheme works on both sides of the effective Ising transition and does not require fine tuning of the auxiliary parameters, with a broad minimum near Δ≈Λ and κ≈ν.

Where Pith is reading between the lines

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

  • The paper's own numerics show that at ε=0 (no period-two jitter) the steady state has no net S^z_tot; the period-doubled response is therefore selected by an explicitly 2T-periodic noise term, not by spontaneous breaking of the one-period time-translation symmetry. A reader should not take the 'discrete time crystal' claim as evidence of spontaneous time-crystalline order in the usual sense.
  • The same cooling mechanism could be combined with nonlocal or spatially engineered couplings to prepare topological Floquet states or error-correctable logical states, an extension the authors mention but do not implement.
  • A testable prediction is that the steady-state cooling rate should depend on the local spectral function of the coupled operator O; measuring this in a superconducting circuit or trapped-ion setup would verify the FGR picture beyond the specific Ising realization.
  • If the rate-equation description holds, the scheme should apply to any Floquet system with a gapped effective spectrum and local quasiparticles, not just Ising chains; the challenge is to maintain the hierarchy of scales when the gap is small, as near a critical point.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes and analyzes an autonomous scheme to cool a Floquet-driven many-body system into a low-energy state of its effective (Floquet-engineered) Hamiltonian by coupling it to statically coupled dissipative spin-1/2 auxiliaries. The authors derive a Floquet Fermi golden rule rate theory for the combined effects of intrinsic Floquet heating, period-jitter noise, and auxiliary-mediated cooling (Eqs. 8–10), and use it to obtain a closed-form estimate of the steady-state excess energy density (Eq. 4). They test the rate theory against trajectory simulations of a driven transverse-field Ising chain with a single auxiliary at the edge (L up to 18, 250–1000 trajectories), reporting parameter-free agreement. They then consider a long-range Ising Floquet drive and show that, with period-two jitter, the dissipative steady state exhibits a period-doubled magnetization response that persists at infinite time, which they label discrete time-crystalline order. The paper emphasizes that without the auxiliary the response decays and that period-two jitter is needed to break the symmetry between the two ferromagnetic sectors. The authors present the scheme as a broadly applicable solution to Floquet heating.

Significance. The cooling mechanism is a potentially important advance: it is experimentally simple (lossy qubits/cavities), autonomous, and comes with an analytic rate description that captures the numerics without fitted parameters. The parameter-free comparison between FGR rate theory and trajectory simulations is a real strength, and the authors are transparent about the single-auxiliary limitation. However, the headline result—'discrete time-crystalline order in the steady state'—does not live up to the paper's own definition of spontaneous time-translation symmetry breaking. The observed period-doubled response is produced by period-two jitter, an explicitly 2T-periodic perturbation, and the reported uniqueness of the steady state is inconsistent with the expected symmetry-broken degeneracy. The dissipative stabilization of a period-doubled response remains a meaningful and publishable result, but the claim of discrete time-crystalline order needs major revision, as does the support for finite-density cooling.

major comments (3)
  1. [Stabilizing a period-doubled response (paragraph beginning 'The period-doubled response should surprise...')] The paper's own definition of a Z2 discrete time crystal is that it 'spontaneously breaks the discrete time-translation symmetry of the drive' (previous paragraph). The mechanism proposed here instead uses period-two jitter, 'in which a new ζ_n is chosen every two periods,' which is an explicitly 2T-periodic perturbation of the T-periodic drive. The paper concedes that at ε=0 the two ferromagnetic sectors are cooled symmetrically and the steady state has no net S^z_tot; the period-doubled response appears only because this 2T-periodic noise makes the sectors heat at different rates. This is a driven response to an explicitly symmetry-breaking perturbation, not spontaneous discrete time-crystalline order. The statement that 'the steady state appears to be unique' further reinforces the absence of the symmetry-broken degeneracy expected for a genuine time crystal. The abstract's claims of
  2. [End Matter (Eqs. 11–13) and Fig. 2] The numerical verification of cooling is carried out only with N_aux=1 for L≤18. The paper itself states that a single auxiliary 'cannot cool as L→∞; that requires a finite density of auxiliaries.' The central scaling estimate, Eq. (4)/Eq. (13), assumes a finite auxiliary density n_aux, but no trajectory simulation with n_aux>0 is presented; the single-auxiliary data show a steady-state energy offset that increases with L (insets to Fig. 2(a,c)), consistent with boundary cooling. The abstract's 'excess energy density is controllably small' is therefore an analytic prediction, not a numerically confirmed many-body result. Please add finite-density simulations or at least finite-density rate-theory curves, and state this limitation explicitly.
  3. [Stabilizing a period-doubled response (final paragraph)] The assertion that generic initial states exhibit the same even-period magnetization 'as the steady state appears to be unique' is supported only by numerical evidence deferred to the Supplemental Material. This claim is load-bearing for the 'generic initial states' part of the abstract and for the uniqueness interpretation of the steady state. Without seeing the data, I cannot assess whether the Lindblad dynamics truly has a unique steady state or whether this apparent uniqueness is a finite-size artifact. Please include the relevant numerical evidence—e.g., even-period magnetization for several distinct initial states with trajectory-count information—either in the main text or in a self-contained supplementary appendix.
minor comments (4)
  1. [Fig. 2 and Fig. 3] No error bars or statistical uncertainties are shown for the trajectory averages. The text states 250–1000 trajectories are used; adding representative error bars (or shaded regions) would make the comparison with the FGR curves more convincing.
  2. [Fig. 2 insets] The label 'SS diff.' is undefined; the caption should state explicitly that this is (E_ss - E_0)/|E_0| and which quantities are plotted.
  3. [References / related work] Ref. [31] (Ritter et al., Phys. Rev. X 15, 031028) appears to be closely related prior work by two of the same authors on autonomous stabilization of Floquet states with static dissipation. The text should state more concretely what is new beyond Ref. [31], e.g., the many-body cooling-to-ordered-state aspect and the period-doubled steady state.
  4. [End Matter, Eq. (8)] The Lorentzian prefactor κ/((Eα−Eβ+mω−Δ)^2+κ^2/4) treats κ as a rate; please clarify the units/conventions so that the expression is manifestly a rate when ℏ=1.

Circularity Check

0 steps flagged

Cooling derivation is self-contained; DTC label overstates but is not a circular step.

full rationale

The FGR cooling derivation is self-contained: rates in Eqs. 8–10 are computed from H_eff and compared to trajectory numerics with 'No parameters are fitted' (End Matter), so the central cooling claim is not extracted from the data it explains. The main concern is labeling, not circularity. In 'Stabilizing a period-doubled response,' the paper defines a Z2 DTC as spontaneously breaking one-period time-translation symmetry, yet the demonstrated period-doubled steady state is obtained with 'period-two jitter, in which a new ζ_n is chosen every two periods,' and the paper concedes that at ε=0 there is 'no net S^z_tot.' A 2T-periodic noise input makes the 2T-periodic response a driven response rather than a spontaneous time crystal; the abstract's 'discrete time-crystalline order' therefore overstates what is shown. This is a correctness/interpretation limitation, not a circular derivation: the existence of a steady-state period-doubled response is a real consequence of cooling plus the chosen noise, and the paper is transparent about the role of jitter. Ref. [31] shares two authors and already contains the static-dissipation stabilization idea, but the present derivation and numerics stand independently, so this self-citation is not load-bearing. Score 2 reflects the minor self-citation; no circular step rises to 4 or higher.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The central claim rests on the dilute-quasiparticle FGR rate balance and on the specific period-jitter noise model. The most important ad hoc input is the period-two jitter for the DTC section, which explicitly breaks one-period time-translation symmetry and selects the ferromagnetic sector. No new particles or forces are introduced; the auxiliary is a physical spin-1/2/cavity with standard T1 decay.

free parameters (3)
  • period-jitter strength ε = 0.075–0.212 (Fig 1c); 0.114–0.149 (Fig 2); 0.1 (Fig 3); period-two variant in DTC section
    Tunable noise/heating strength; chosen by hand to set the bulk heating rate and, in the period-two form, to break Z2 symmetry. Not fitted to the FGR curves but directly controls the steady-state energy offset and the DTC sector selection.
  • auxiliary parameters (Δ, κ, g) = Δ=Λ, κ=ν/4, g=ν (Fig 2); Δ=0.5, κ=0.125, g=0.5 (Fig 3)
    Chosen by hand to satisfy the design hierarchy Eq. 3; the FGR rates are not fitted, but the cooling performance is contingent on this weak-coupling/narrow-band regime.
  • auxiliary density n_aux = N_aux/L = 1/L in the numerics (N_aux=1)
    The analytic estimate Eq. 4 uses n_aux as a control parameter; the numerics only test n_aux=1/L, leaving the finite-density (n_aux=O(1)) scaling unverified.
axioms (7)
  • domain assumption High-frequency Schrieffer-Wolff expansion (Eq. 1) is controlled up to optimal order n* ~ ω/Ω with exponentially small residual heating.
    Standard Floquet prethermalization result [21,22,61]; used to define H_eff and residual heating terms.
  • domain assumption Auxiliary decay is Markovian and described by a Lindblad jump operator L_aux=√κ τ^- (Eq. 14).
    Standard open-quantum-systems modeling for a lossy qubit/cavity; used throughout the analytic and numerical work.
  • domain assumption Fermi's Golden Rule rates (Eqs. 8–10) can be added independently and eigenstate populations obey a Pauli rate equation (Eq. 2).
    Weak-coupling/high-frequency regime; neglects coherent multi-process interference and sideband processes.
  • domain assumption Near the ground state the system is a dilute quasiparticle gas with density n_qp ~ (E_eff-E_0)/(ΛL).
    Used to derive Eq. 11 and the steady-state energy estimate Eq. 13.
  • ad hoc to paper Period jitter enters as a stochastic Hamiltonian linear in ζ_n, with the period-two variant updating every two periods and breaking Z2 symmetry.
    This is a specific noise model introduced to give finite-size heating and, in the DTC section, to select a ferromagnetic sector; the DTC result depends on the period-two variant.
  • ad hoc to paper The steady state is unique and the even-period magnetization is identical for generic initial states (DTC section).
    Stated in main text with support deferred to Supplemental Material.
  • domain assumption Sideband (m≠0) auxiliary transitions are negligible because local operators have exponentially small spectral weight at frequencies ~mω.
    Used to keep only m=0 resonant cooling; standard Floquet prethermal suppression.

pith-pipeline@v1.3.0-alltime-deepseek · 13363 in / 18924 out tokens · 165969 ms · 2026-08-01T21:04:56.992280+00:00 · methodology

0 comments
read the original abstract

Floquet driving underlies Hamiltonian and gate engineering, and produces dynamical orders with no equilibrium counterpart. These phenomena are, however, transient in well-isolated systems. We show that statically coupled dissipative auxiliaries cool the system toward low-energy states of the Floquet-engineered Hamiltonian, stabilizing orders in the steady state. The excess energy density is controllably small at high drive frequency and weak coupling and is captured by rate theory based on Fermi's Golden Rule. We numerically confirm robust cooling in a Floquet-engineered transverse-field Ising chain in both phases, and demonstrate discrete time-crystalline order, with a period-doubled magnetization response, in the steady state of a long-range Ising chain. Our results provide a rare analytical handle on the steady states of driven dissipative systems.

Figures

Figures reproduced from arXiv: 2607.16391 by Alicia J. Kollar, Anushya Chandran, Robin Sch\"afer, Shreyas Raman.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) A driven system coupled to a dissipative auxil [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Cooling into low-energy states of a Floquet-engineered transverse-field Ising chain using a single auxiliary coupled [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Steady-state stabilization of a discrete time crys [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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

62 extracted references · 3 linked inside Pith

  1. [1]

    Bukov, L

    M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to floquet engineering, Ad- vances in Physics64, 139 (2015)

    2015

  2. [2]

    Goldman and J

    N. Goldman and J. Dalibard, Periodically driven quan- tum systems: Effective hamiltonians and engineered gauge fields, Physical Review X4, 031027 (2014)

    2014

  3. [3]

    Oka and S

    T. Oka and S. Kitamura, Floquet engineering of quantum materials, Annual Review of Condensed Matter Physics 10, 387 (2019)

    2019

  4. [4]

    Eckardt, Atomic quantum gases in periodically driven optical lattices, Reviews of Modern Physics89, 011004 (2017)

    A. Eckardt, Atomic quantum gases in periodically driven optical lattices, Reviews of Modern Physics89, 011004 (2017)

    2017

  5. [5]

    M. S. Rudner and N. H. Lindner, Band structure engi- neering and non-equilibrium dynamics in floquet topo- logical insulators, Nature Reviews Physics2, 229 (2020)

    2020

  6. [6]

    Rahav, I

    S. Rahav, I. Gilary, and S. Fishman, Effective hamilto- nians for periodically driven systems, Physical Review A 68, 013820 (2003)

    2003

  7. [7]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zil- berberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys.91, 015006 (2019)

    2019

  8. [8]

    Kuwahara, T

    T. Kuwahara, T. Mori, and K. Saito, Floquetmagnus the- ory and generic transient dynamics in periodically driven many-body quantum systems, Annals of Physics367, 96124 (2016)

    2016

  9. [9]

    D. A. Abanin, W. De Roeck, W. W. Ho, and F. m. c. Hu- veneers, Effective hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems, Phys. Rev. B95, 014112 (2017)

    2017

  10. [10]

    Huang, P

    Z. Huang, P. S. Mundada, A. Gyenis, D. I. Schuster, A. A. Houck, and J. Koch, Engineering dynamical sweet spots to protect qubits from 1/fnoise, Phys. Rev. Appl. 15, 034065 (2021)

    2021

  11. [11]

    Wang, B.-J

    Y.-S. Wang, B.-J. Liu, S.-L. Su, and M.-H. Yung, Error- resilient floquet geometric quantum computation, Phys. Rev. Res.3, 033010 (2021)

    2021

  12. [12]

    Weitenberg and J

    C. Weitenberg and J. Simonet, Tailoring quantum gases by floquet engineering, Nature Physics17, 1342 (2021)

    2021

  13. [13]

    M. P. Zaletel, M. Lukin, C. Monroe, C. Nayak, F. Wilczek, and N. Y. Yao, Colloquium: Quantum and classical discrete time crystals, Rev. Mod. Phys.95, 031001 (2023)

    2023

  14. [14]

    Khemani, R

    V. Khemani, R. Moessner, and S. L. Sondhi, A brief his- tory of time crystals (2019), arXiv:1910.10745

    Pith/arXiv arXiv 2019

  15. [15]

    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, Observation of discrete time-crystalline order in a dis- ordered dipolar many-body system, Nature543, 221225 (2017)

    2017

  16. [16]

    Zhang, P

    J. Zhang, P. W. Hess, 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, Nature543, 217 (2017)

    2017

  17. [17]

    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, ...

    2022

  18. [18]

    D. V. Else, B. Bauer, and C. Nayak, Prethermal phases of matter protected by time-translation symmetry, Phys. Rev. X7, 011026 (2017)

    2017

  19. [19]

    Wintersperger, C

    K. Wintersperger, C. Braun, F. N. ¨Unal, A. Eckardt, M. D. Liberto, N. Goldman, I. Bloch, and M. Aidels- burger, Realization of an anomalous floquet topological system with ultracold atoms, Nature Physics16, 1058 (2020)

    2020

  20. [20]

    Roy and F

    R. Roy and F. Harper, Periodic table for floquet topo- logical insulators, Phys. Rev. B96, 155118 (2017)

    2017

  21. [21]

    D. A. Abanin, W. De Roeck, and F. m. c. Huveneers, Exponentially slow heating in periodically driven many- body systems, Phys. Rev. Lett.115, 256803 (2015)

    2015

  22. [22]

    T. Mori, T. Kuwahara, and K. Saito, Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems, Physical Review Letters116, 120401 (2016)

    2016

  23. [23]

    W. W. Ho, T. Mori, D. A. Abanin, and E. G. Dalla Torre, Quantum and classical floquet prethermalization, Annals of Physics454, 169297 (2023)

    2023

  24. [24]

    D’Alessio and M

    L. D’Alessio and M. Rigol, Long-time behavior of isolated periodically driven interacting lattice systems, Phys. Rev. X4, 041048 (2014)

    2014

  25. [25]

    Lazarides, A

    A. Lazarides, A. Das, and R. Moessner, Equilibrium states of generic quantum systems subject to periodic driving, Phys. Rev. E90, 012110 (2014)

    2014

  26. [26]

    Ponte, A

    P. Ponte, A. Chandran, Z. Papi, and D. A. Abanin, Pe- riodically driven ergodic and many-body localized quan- tum systems, Annals of Physics353, 196 (2015)

    2015

  27. [27]

    Rubio-Abadal, M

    A. Rubio-Abadal, M. Ippoliti, S. Hollerith, D. Wei, J. Rui, S. L. Sondhi, V. Khemani, C. Gross, and I. Bloch, Floquet prethermalization in a bose-hubbard system, Phys. Rev. X10, 021044 (2020)

    2020

  28. [28]

    Wanckel and A

    L. Wanckel and A. Eckardt, Dissipative floquet engineer- ing of gapped many-body phases using thermal baths (2026), arXiv:2604.01291

    arXiv 2026

  29. [29]

    Petiziol and A

    F. Petiziol and A. Eckardt, Cavity-based reservoir engi- neering for floquet-engineered superconducting circuits, Phys. Rev. Lett.129, 233601 (2022)

    2022

  30. [30]

    L. C. Steinfadt, A. Eckardt, and F. Petiziol, Dissipation- assisted preparation of floquet-laughlin states in super- conducting circuits (2026), arXiv:2605.18377

    Pith/arXiv arXiv 2026

  31. [31]

    Ritter, D

    M. Ritter, D. M. Long, Q. Yue, A. Chandran, and A. J. Koll´ ar, Autonomous stabilization of floquet states using static dissipation, Phys. Rev. X15, 031028 (2025)

    2025

  32. [32]

    Shankar, M

    S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L. Frunzio, M. Mir- rahimi, and M. H. Devoret, Autonomously stabilized en- tanglement between two superconducting quantum bits, Nature504, 419422 (2013)

    2013

  33. [33]

    Hacohen-Gourgy, V

    S. Hacohen-Gourgy, V. V. Ramasesh, C. De Grandi, I. Siddiqi, and S. M. Girvin, Cooling and autonomous feedback in a bose-hubbard chain with attractive inter- actions, Physical Review Letters115, 240501 (2015)

    2015

  34. [34]

    K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. Girvin, and I. Siddiqi, Cavity-assisted quantum bath en- gineering, Physical Review Letters109, 183602 (2012)

    2012

  35. [35]

    J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum Reser- voir Engineering with Laser Cooled Trapped Ions, Phys- ical Review Letters77, 4728 (1996)

    1996

  36. [36]

    Diehl, A

    S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. B¨ uchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nature Physics4, 878 (2008)

    2008

  37. [37]

    Verstraete, M

    F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quan- tum computation and quantum-state engineering driven by dissipation, Nature Physics5, 633 (2009)

    2009

  38. [38]

    K. I. Seetharam, C.-E. Bardyn, N. H. Lindner, M. S. Rudner, and G. Refael, Controlled Population of Floquet- Bloch States via Coupling to Bose and Fermi Baths, Physical Review X5, 041050 (2015)

    2015

  39. [39]

    Iadecola, T

    T. Iadecola, T. Neupert, and C. Chamon, Occupation of topological floquet bands in open systems, Phys. Rev. B 91, 235133 (2015)

    2015

  40. [40]

    Shirai, J

    T. Shirai, J. Thingna, T. Mori, S. Denisov, P. Hnggi, and S. Miyashita, Effective floquetgibbs states for dissipative quantum systems, New Journal of Physics18, 053008 (2016)

    2016

  41. [41]

    Mori, Floquet states in open quantum systems, An- nual Review of Condensed Matter Physics14, 35 (2023)

    T. Mori, Floquet states in open quantum systems, An- nual Review of Condensed Matter Physics14, 35 (2023)

    2023

  42. [42]

    X. Mi, A. A. Michailidis, S. Shabani, K. C. Miao, P. V. Klimov, J. Lloyd, E. Rosenberg, R. Acharya, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. As- faw, J. Atalaya, J. C. Bardin, A. Bengtsson, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bush- nell, Z. Chen, B. Chiaro,...

    2024

  43. [43]

    R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Si- mon, and D. I. Schuster, A dissipatively stabilized mott insulator of photons, Nature566, 51 (2019)

    2019

  44. [44]

    Schnell, C

    A. Schnell, C. Weitenberg, and A. Eckardt, Dissipative preparation of a floquet topological insulator in an op- tical lattice via bath engineering, SciPost Phys.17, 052 (2024)

    2024

  45. [45]

    Shavitt and L

    I. Shavitt and L. T. Redmon, Quasidegenerate pertur- bation theories. a canonical van vleck formalism and its relationship to other approaches, The Journal of Chemi- cal Physics73, 5711 (1980)

    1980

  46. [46]

    Bukov, M

    M. Bukov, M. Heyl, D. A. Huse, and A. Polkovnikov, Heating and many-body resonances in a periodically driven two-band system, Phys. Rev. B93, 155132 (2016)

    2016

  47. [47]

    Fleckenstein and M

    C. Fleckenstein and M. Bukov, Prethermalization and thermalization in periodically-driven many-body systems away from the high-frequency limit, Physical Review B 103, L140302 (2021)

    2021

  48. [48]

    Khemani, A

    V. Khemani, A. Lazarides, R. Moessner, and S. Sondhi, Phase Structure of Driven Quantum Systems, Physical Review Letters116, 250401 (2016)

    2016

  49. [49]

    D. V. Else, B. Bauer, and C. Nayak, Floquet time crys- tals, Phys. Rev. Lett.117, 090402 (2016)

    2016

  50. [50]

    C. W. von Keyserlingk, V. Khemani, and S. L. Sondhi, Absolute stability and spatiotemporal long-range order in Floquet systems, Physical Review B94, 085112 (2016)

    2016

  51. [51]

    N. Yao, A. Potter, I.-D. Potirniche, and A. Vishwanath, Discrete Time Crystals: Rigidity, Criticality, and Real- izations, Physical Review Letters118, 030401 (2017)

    2017

  52. [52]

    Khemani, R

    V. Khemani, R. Moessner, and S. L. Sondhi, A comment on ”discrete time crystals: rigidity, criticality, and real- izations” (2021), arXiv:2109.00551

    Pith/arXiv arXiv 2021

  53. [53]

    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, Observation of a prethermal discrete time crystal, Science372, 1192 (2021)

    2021

  54. [54]

    F. J. Dyson, Existence of a phase-transition in a one-dimensional ising ferromagnet, Communications in Mathematical Physics12, 91 (1969)

    1969

  55. [55]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002)

    2002

  56. [56]

    M. B. Hastings and J. Haah, Dynamically generated log- ical qubits, Quantum5, 564 (2021)

    2021

  57. [57]

    Polla, Y

    S. Polla, Y. Herasymenko, and T. E. O’Brien, Quantum digital cooling, Physical Review A104, 012414 (2021)

    2021

  58. [58]

    Sch¨ afer and D

    R. Sch¨ afer and D. J. Luitz, DanceQ: High-performance library for number conserving bases, SciPost Phys. Code- bases , 48 (2025)

    2025

  59. [59]

    Sch¨ afer and D

    R. Sch¨ afer and D. J. Luitz, Codebase release 1.0 for DanceQ, SciPost Phys. Codebases , 48 (2025)

    2025

  60. [60]

    T. N. Ikeda and A. Polkovnikov, Fermi’s golden rule for heating in strongly driven floquet systems, Physical Re- view B104, 134308 (2021)

    2021

  61. [61]

    O’Dea, F

    N. O’Dea, F. Burnell, A. Chandran, and V. Khemani, Prethermal stability of eigenstates under high frequency floquet driving, Physical Review Letters132, 100401 (2024)

    2024

  62. [62]

    H. Carmichael,An Open Systems Approach to Quan- tum Optics: Lectures Presented at the Universit Libre de Bruxelles October 28 to November 4, 1991, Lecture Notes in Physics Monographs, Vol. 18 (Springer, 1993). 8 End Matter FGR Transition Rates and Steady-State Energy.Let Heff |α⟩=E α |α⟩, and defineδE βα =E β −E α. In the weak-coupling and high-frequency ...

    1991