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REVIEW 4 major objections 5 minor 47 references

Generating discrete time crystals through optimal control

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Optimization over pulse shapes yields discrete time crystals in generic many-body quantum systems.

desk verdict A plausible proof-of-concept that optimal control can find DTC-generating pulses, but the cost functions encode the target response and the universality claim outruns the evidence. read the letter →

arxiv 2505.23021 v1 pith:EKS7L4CL submitted 2025-05-29 quant-ph cond-mat.stat-mech

Mrutyunjaya Sahoo , Rahul Ghosh , Bandita Das , Shishira Mahunta , Bodhaditya Santra , Victor Mukherjee This is my paper
classification quant-phcond-mat.stat-mech
keywords discretetimecrystalsoptimalcontrolCRABFloquetsystemsmany-bodylocalizationopenDickemodeltime-translationalsymmetrybreakingquantum
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that discrete time crystals — phases whose response to a periodic drive repeats with double the drive period — can be produced on demand by treating their creation as an optimization problem. The authors use the CRAB optimal-control method to search over the Fourier coefficients of a periodic pulse, minimizing cost functions that reward period-doubled ($2T$) response and punish period-one ($T$) response. They demonstrate the approach in two very different settings: a dissipative open Dicke model and a many-body-localized spin chain with disorder. In both, the optimized pulse drives the system into a stable DTC phase, and in the spin chain optimizing over just two central sites produces phase locking across the whole chain. If right, this replaces model-by-model hand-designed pulses with a general recipe for generating discrete time crystals and for mapping their parameter regimes.

What carries the argument

The machinery is the CRAB (chopped random basis) optimal-control scheme: the time-dependent modulation $\lambda(t)$ is expanded as a truncated Fourier series with coefficients $\{A_n, B_n\}$, and these coefficients are numerically minimized against a cost function built to detect DTC order. For the Dicke model the cost function $F_1$ rewards $2T$-periodicity of the collective spin $j_x(t)$ while penalizing $T$-periodicity; for the spin chain $F_2$ rewards a Fourier peak at half the driving frequency $\Omega_0/2$, suppresses all other peaks, and enforces a minimum peak height. The optimization is constrained by coefficient bounds $|A_n|, |B_n| \leq \chi$, which encode experimental limitations on pulse shapes. The method works because a DTC's defining observable signature—period-doubling under a periodic drive—can be written directly as a target for an optimizer.

What would settle it

Measure, numerically or experimentally, the subharmonic peak height at $\Omega_0/2$ in the optimized spin-chain pulse averaged over at least 100 disorder realizations for system sizes $L = 8, 12, 16$: if the peak height decreases with $L$, the optimized pulse does not produce a genuine many-body DTC.

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Extended reading notes

Core claim

The central discovery is that spontaneous breaking of discrete time-translational symmetry can be engineered by numerically optimizing the Fourier coefficients of a periodic drive against a cost function that encodes the DTC signature: a $2T$-periodic response to a $T$-periodic drive, equivalently a single Fourier peak at half the driving frequency. In the open Dicke model, minimizing $F_1 = |j_x(sT) - j_x(sT+2T)| + 1/|j_x(sT) - j_x(sT+T)|$ takes an initial non-DTC guess pulse to an optimized pulse that produces period-doubled oscillation, robust to detuning variations (persisting for a range of $\epsilon$). In a many-body-localized spin chain, minimizing $F_2$, which pins the autocorrelation FFT peak at $\Omega_0/2$ with a threshold height and suppresses other peaks, yields a smaller optimized $\theta$ that produces a global DTC; optimizing over just two central sites (4 and 5) of an 8-site chain suffices for the whole chain. The paper presents this as a universal protocol for generating DTCs in arbitrary many-body quantum systems.

Load-bearing premise

The entire protocol rests on the assumption that the cost functions $F_1$ and $F_2$ correctly encode genuine DTC order, so that minimizing them on a single trajectory (Dicke model) or on two sites of one disorder realization (spin chain) yields a pulse whose period doubling is stable and system-wide; if the cost functions are merely fitting the target response, the central claim fails.

Editorial extensions

If this is right

  • DTC generation no longer requires analytically designed on-off or sinusoidal pulse shapes; a generic numerical search over pulse coefficients suffices.
  • The same optimization protocol produces DTCs in both an open dissipative system and a closed disordered system, supporting the claim of universality across many-body platforms.
  • In the spin chain, optimizing the autocorrelation over only two central lattice sites yields a pulse that produces a DTC over the entire 8-site chain, suggesting the protocol is practical for larger systems where global measurements are infeasible.
  • The optimized value of $\theta$ sets a characteristic scale such that the DTC phase persists for $|\theta| \lesssim |\theta_{\rm opt}|$, giving an efficient route to reconstruct dynamical phase diagrams.
  • Constraint bounds on the Fourier coefficients allow the search to respect experimental limitations, so the pulses found are realistic to implement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same cost-function logic could be applied to target other Floquet phases, such as period-$3T$ responses or fractional subharmonics, by changing the target frequency and threshold in the spectral cost function.
  • Because $F_1$ requires only stroboscopic knowledge of a collective observable, the scheme could be run as a closed-loop experiment that finds a DTC drive without a Hamiltonian model, by feeding measured subharmonic amplitudes back into the optimizer.
  • A single-trajectory cost in the open Dicke model may not detect multi-partite entanglement signatures of genuine time-crystalline order; a stricter test would optimize against a cost built from the quantum Fisher information or entanglement entropy.
  • If DTC phases prove to be typical attractors of this optimization landscape rather than fine-tuned solutions, that would strengthen the universality claim and suggest many models host hidden DTC regimes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes using Chopped RAndom Basis (CRAB) optimal control to generate discrete time crystals (DTCs) in generic many-body quantum systems. The method defines cost functions whose minimization yields periodic driving pulses that produce period-doubled stroboscopic dynamics. The authors demonstrate the protocol in two settings: an open Dicke model in the thermodynamic mean-field limit, with a cost function F1 (Eq. 6) that penalizes T-periodic j_x and rewards 2T-periodic j_x at a fixed late time sT; and a disordered spin chain of length L=8, with a cost function F2 (Eq. 9) that directly rewards an FFT peak at half the driving frequency and suppresses other spectral weight, evaluated using only sites 4 and 5 of one disorder realization. They report optimized pulses that yield period-doubled response, claim robustness to parameter changes (ε for the Dicke model, and a global DTC for the spin chain), and discuss experimental implementations in BEC-cavity and trapped-ion platforms. The central claim is that this constitutes a 'universal protocol for generating DTCs in arbitrary many-body quantum systems.'

Significance. If the universality and robustness claims were properly substantiated, the work would introduce a valuable inverse-design tool: instead of manually constructing DTC-generating pulses, one could optimize over a control landscape to discover non-trivial drives. The numerical demonstrations are plausible and the experimental discussion is reasonable. However, the current evidence does not establish that the optimized pulses produce a genuine DTC phase rather than a fine-tuned response. The cost functions explicitly encode the DTC signature, so the demonstrations are to some extent expected outcomes of fitting. The paper would be significantly strengthened by tests of persistence, rigidity, disorder averaging, and finite-size scaling. As it stands, the central 'universal protocol' claim is not supported by the presented numerics.

major comments (4)
  1. [Eq. (6) and Fig. 2] The cost function F1 in Eq. (6) is evaluated at three stroboscopic times only (sT, sT+T, sT+2T) and directly rewards 2T periodicity while penalizing T periodicity. This means the optimization is fitting the DTC signature by construction, not discovering it. To support the claim that the optimized pulse generates a robust DTC phase, the authors should demonstrate persistence of the period-doubled response over many periods (e.g., 10^4 or more), not just the short window shown in Fig. 2 (t from 4970 to 5000). They should also test stability under small perturbations of the optimized pulse coefficients and the Hamiltonian parameters, and show that the response is not fine-tuned to the specific initial steady state used in the optimization.
  2. [Eq. (9) and Fig. 3] The cost function F2 directly rewards an FFT peak at half the drive frequency, suppresses all other spectral peaks, and includes an ad hoc amplitude threshold (x = FFT(Ω0/2) − 0.05 in Eq. (9)). Optimizing such a cost can produce a pulse that yields the desired Fourier peak without necessarily producing a rigid, many-body DTC. The spin-chain demonstration uses a single disorder realization of L=8 and evaluates the cost only on sites 4 and 5. The claim that the optimized pulse yields a 'global DTC' is based on one full-chain FFT for that single realization. To establish a DTC phase, the authors need to average over many disorder realizations, perform finite-size scaling (L=8, 10, 12, ...), and verify that the optimized pulse parameters (e.g., θ_opt) are stable across realizations. Without this, the result could be a fine-tuned response for one particular sample.
  3. [Appendix A and Fig. 2] The robustness evidence for the Dicke model is too limited to justify the statement that the DTC phase 'persists for a finite range of ε' (page 3). The optimization is performed at ε=0.05, and the only additional DTC check is ε=0.06 in Appendix A. Fig. 2 shows a limit cycle at ε=0.04 and a thermal phase at ε=0.1, but this does not map out a finite DTC region. The authors should systematically scan ε (and ideally κ and the pulse amplitude bound χ) to identify the extent of the DTC phase, and also test sensitivity to the initial condition, since the cost function starts from a specific symmetry-broken steady state of Eq. (4).
  4. [Conceptual distinction in Introduction/Conclusion] A discrete time crystal is a phase of matter characterized by spontaneous breaking of time-translational symmetry, which implies rigidity of the subharmonic response against generic perturbations. Here, the drive is explicitly optimized to produce subharmonic response, so the protocol is closer to entrainment or direct subharmonic driving unless additional rigidity tests are provided. To support the claim of spontaneous symmetry breaking and a 'universal protocol', the authors should show that the optimized pulse produces period doubling from a variety of initial states (not only the symmetry-broken steady state or the specific product-state ensemble) and that the response survives weak local perturbations to the Hamiltonian or the drive. These tests are standard in the DTC literature (e.g., Ref. [8]) and would distinguish a true DTC from a fine-tuned response.
minor comments (5)
  1. [Eq. (8)] The definition of f(t) is incomplete in the manuscript; the sine term is cut off after 'A_n cos(ν_n t)+'. Please complete the expression.
  2. [Fig. 3 caption] The caption states the cost function is 'averaging over the lattice sites i=4,5 and all possible initial product states.' For L=8 this is 2^8=256 product states; please specify whether the FFT is computed for each state and then averaged, and over what time window the FFT is taken.
  3. [Page 4, spin-chain parameter text] The statement 'we start from an initial guess pulse with a large θ=0.65' is ambiguous because θ is time-dependent in Eq. (8). It would be clearer to specify the initial Fourier coefficients (e.g., A_0=0.65, other coefficients zero) and the corresponding optimized coefficients that yield θ_opt.
  4. [Eq. (9)] The penalty function Θ(x) is discontinuous at x=0, which may complicate the numerical optimization. Please comment on how CRAB handles this non-smoothness, or whether a smooth approximation was used.
  5. [Throughout] The phrase 'spontaneous breaking of time-translational symmetry' is used repeatedly, but the paper does not discuss the distinction between a symmetry-broken phase and a response that is explicitly imposed by the drive. A short discussion of this distinction, and how the optimized pulse relates to the Floquet eigenstate order, would help place the results in context.

Circularity Check

2 steps flagged · score 7.0 of 10

Cost functions F1 and F2 encode the DTC signature itself, so the optimized pulses reproduce the objective by construction; the universality claim rests on an unverified proxy.

  1. self definitional [DTC in the modulated open Dicke model, Eq. (6) and surrounding text]
    "F1({An, Bn}) = |j_x(sT) - j_x(sT + 2T)| + 1/|j_x(sT) - j_x(sT + T)| ... In the above equation (6), the first term on the r.h.s. ensures that as a result of the optimization, j_x(t) becomes 2T periodic, whereas the second term on the r.h.s. denotes a penalty for pulses which result in T periodic j_x's."

    The minimized function is exactly the stroboscopic DTC criterion for j_x: making the first term small is 2T periodicity, and the reciprocal penalty forbids T-periodic stroboscopic response. Thus the optimized pulse necessarily produces the reported period-doubled j_x (Figs. 1-2), because that is what the optimizer was asked to do. The only 'robustness' test is one additional detuning value (epsilon=0.06, Fig. 4); no term in F1 requires persistence over long times, stability under perturbation, or generic initial conditions. Calling this a robust DTC generated by optimal control is therefore restating the fitted objective rather than an independent prediction.

  2. fitted input called prediction [DTC in a many-body-localized spin chain system, Eq. (9)]
    "F2({An, Bn}) = F2^(1) + F2^(2) + F2^(3), F2^(1) = |F_CMA - 0.5Ω0|, F2^(2) = Σ_{ω≠0.5Ω0} |FFT(ω)|, F2^(3) = Θ(x)/|x|; x = FFT(Ω0/2) - 0.05 ... The first ... and second ... ensure that the peak occurs at half of the driving frequency and no other peaks exist at any other frequencies, respectively, whereas, the third part ... defines a lower threshold value (=0.05) for the FFT peak at Ω=Ω0/2, for the phase to be considered as a DTC."

    The optimization variable θ(t) is fitted by minimizing F2, whose components directly reward an FFT maximum at Ω0/2, penalize every other frequency component, and impose the 0.05 threshold. Consequently, the post-optimization FFT shown in Fig. 3 is the objective function value, not an independent measurement of DTC order. Moreover, F2 is evaluated only on sites 4 and 5 of one L=8 disorder realization (no disorder averaging or finite-size scaling is reported); the subsequent 'global DTC' check is the FFT of the same realization. Thus the protocol's universality claim is supported by a fit to the target signature, not by a prediction that could have failed independently.

full rationale

The central reduction is by construction: the cost functions are not independent probes of a DTC phase but direct encodings of the reported signatures. F1 minimizes the stroboscopic 2T-periodicity condition plus a penalty for T-periodicity, so the optimized Dicke pulse is guaranteed to show period-doubled j_x at the sampled times. F2 rewards an FFT peak at half the drive frequency, suppresses all other peaks, and enforces a threshold, so the optimized spin-chain pulse is likewise guaranteed to produce that FFT peak at the sites used for optimization. The additional checks—one neighboring detuning value in the Dicke model and a global-site FFT on the same single disorder realization in the spin chain—do not supply the missing rigidity, persistence, or disorder-averaged evidence that would turn a fitted pulse into a robust DTC phase. There is no load-bearing self-citation chain here; the paper's own equations exhibit the circularity. However, the optimizer genuinely searches a constrained pulse space and could have failed to satisfy the constraints, so the numerical demonstrations retain some nontrivial content. The score is therefore 7 rather than 8: the advertised universal protocol reduces substantially to the chosen cost functions, though not every step is purely tautological.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four domain or ad hoc assumptions plus several hand-chosen cost parameters. No invented entities, particles, or forces are introduced. The Fourier coefficients are optimization variables fitted to the target DTC conditions, which is reflected in the high circularity burden.

free parameters (4)
  • Control pulse Fourier coefficients {A_0, A_n, B_n} for open Dicke model = not reported
    Optimized via CRAB to minimize F1 in Eq. (6); these coefficients are the control parameters fitted to produce period-doubled j_x.
  • Control pulse Fourier coefficients for spin chain = theta_opt = 0.45 (reported)
    Optimized via CRAB to minimize F2 in Eq. (9), with constraints |A0|<1, |An|,|Bn|<0.0005, Nc=6; the resulting near-constant theta=0.45 defines the reported DTC regime.
  • FFT peak threshold in cost function F2^(3) = 0.05
    Hand-chosen threshold in Eq. (9); a half-frequency peak below this value is heavily penalized, so the definition of DTC is tied to this ad hoc number.
  • Penalty magnitude Theta(x) = 10^4 for x <= 0, 1 for x > 0
    Hand-chosen weight in the threshold term of Eq. (9); no justification is given for the specific value or for the divergence at x=0.
assumptions (4)
  • domain assumption Mean-field approximation in the thermodynamic limit N -> infinity accurately describes the open Dicke model dynamics used to evaluate the cost function in Eq. (6).
    The paper uses mean-field steady states from Eq. (4) and evaluates j_x(t) without including finite-N or photon-number fluctuations; if these fluctuations destroy the subharmonic response, the DTC claim fails.
  • domain assumption Lindblad master equation with Markovian photon loss (Eq. (3)) captures the dissipative dynamics.
    Standard open quantum system modeling, but the paper does not assess non-Markovian corrections, which are known to matter in related DTC studies (Ref. [19]).
  • ad hoc to paper The truncated Fourier basis in Eq. (1) with finite N_c is sufficient to represent an experimentally realizable, DTC-generating control pulse.
    The choices of N_c and the bound chi are not justified by convergence or sensitivity analysis; the optimized pulse is only optimal within this restricted ansatz.
  • ad hoc to paper For the spin chain, a single disorder realization and a cost function evaluated only on sites 4 and 5 are representative enough to yield a global DTC phase for L=8.
    No disorder averaging or finite-size scaling is reported, so the generality of the optimized pulse across disorder realizations and system sizes is assumed.

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Cite this review

Pith. "Pith review of Generating discrete time crystals through optimal control." pith.science (2026). https://pith.science/paper/EKS7L4CL

@misc{pith2026250523021,
  author       = {Pith},
  title        = {Pith review of: Generating discrete time crystals through optimal control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EKS7L4CL}},
  note         = {Machine review of arXiv:2505.23021}
}
read the original abstract

In this work we use optimal control to generate Discrete Time Crystals (DTC) in generic many-body quantum systems. We define appropriate cost functions, which, when optimized, result in the formation of DTCs. This hitherto unexplored method represents DTCs as an optimization problem, and allows us to find non-trivial realistic periodic control pulses and parameter regimes which result in spontaneous breaking of time-translational symmetry in quantum systems. We exemplify our approach using many-body quantum systems in the presence, as well as absence of dissipation. We also discuss possible experimental realization of the control protocol for generating DTCs.

Figures

Figures reproduced from arXiv: 2505.23021 by the authors.

Figure 1
Figure 1. FIG. 1. Plot (a) shows the initial guess pulse and the final op [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. We show the behaviour of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We show the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Plot (a) shows the DTC signature for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.