{"id":"95e8e88f-6509-4499-ba3f-583d0a66e9ab","arxiv_id":"2505.23021","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Optimal control pulses found by minimizing cost functions that directly encode period doubling can generate discrete time crystals in open and closed many-body systems.","lead":"The authors use an optimization routine called CRAB to find periodic control pulses that make quantum many-body systems respond at half the driving period, the signature of a discrete time crystal. The method is demonstrated in two models, a dissipative Dicke cavity and a disordered spin chain, but the cost functions are engineered to enforce exactly the target behavior.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cost functions encode the DTC signature directly, so optimization can find fine-tuned period doubling; the missing rigidity, persistence, and disorder-averaging test is the load-bearing gap for the universality claim.","rationale":"The reader's weakest assumption identifies the same load-bearing issue: the cost functions are essentially the DTC order parameter, and the optimized pulses are verified only on a single trajectory or a single small disorder realization. My read does not move the verdict: a conditional acceptance requiring the proposed disorder-averaged, finite-size, baseline-comparison tests is the right outcome. In good faith, the paper does demonstrate that CRAB optimization can minimize a period-doubling cost in two previously known DTC models, and the spin-chain protocol does check the global autocorrelation after optimizing on a central subsystem. These are real numerical results and deserve credit. However, they do not by themselves establish a universal protocol for arbitrary many-body systems, because no test separates a genuine rigid DTC phase from a pulse that merely satisfies the optimized signature. The paper even states the optimization works 'if the model and the parameter values allow for the existence' of a DTC, which appropriately limits the claim, but the Introduction and Conclusion still assert universality. The concrete test proposed above would settle whether the optimized pulses are phase-generating or cost-fitting, and would also address the absence of code and data by defining a reproducible benchmark. No formal verification, parameter-free derivation, or shipped code is present, so the numerical robustness evidence is the only support for the central claim and it is currently too thin.","tokens_in":9736,"tokens_out":4875,"duration_ms":59752,"concrete_test":"Take the optimized spin-chain pulse from Fig. 3 and compute, for the same Hamiltonian parameters, the half-frequency FFT peak of the autocorrelation function averaged over all sites and over at least 100 independent disorder realizations, at L=8, 12, and 16, with evolution up to at least 10^4 driving periods. Compare this peak with (i) a constant theta=0.45 pulse and (ii) random Fourier pulses having the same A0 and the same amplitude bounds. If the optimized pulse does not produce a disorder-averaged peak that survives system-size growth and substantially exceeds the random-pulse baseline, the protocol has fitted the cost function rather than generated a robust DTC phase. A complementary check is to perturb the optimized Fourier coefficients by small random offsets and verify that the subharmonic response persists, as genuine DTC rigidity requires.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a universal protocol for generating DTCs, but a DTC is a phase: it requires persistent, rigid subharmonic response over long times, over disorder realizations, and under small perturbations. The cost functions do not test that. F1 in Eq. (6) is evaluated on one mean-field trajectory at the stroboscopic times sT, sT+T, and sT+2T; it rewards a single period-doubled step and has no term probing whether that response persists, whether it is stable to small pulse changes, or whether it appears from generic initial conditions. F2 in Eqs. (9) directly rewards an FFT peak at half the driving frequency, measured on sites 4 and 5 of one L=8 disorder realization. Optimizing a cost that already is the target signature can therefore produce a fine-tuned pulse that fits the cost without establishing a robust phase. The paper's robustness evidence for the Dicke model is limited to two nearby values of epsilon (0.05 and 0.06), and the spin-chain verification is a single realization of a small system with no finite-size scaling. The universality claim is thus supported only by the untested assumption that optimizing a local, short-time, single-realization proxy for the DTC signature transfers to a global, thermodynamic, rigid DTC phase. This is not a claim that the numerical demonstrations are wrong; it is the specific load-bearing assumption that prevents the two examples from supporting the advertised universal protocol.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.'","tokens_in":10076,"tokens_out":4722,"duration_ms":50890,"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":[{"comment":"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.","section":"Eq. (6) and Fig. 2"},{"comment":"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.","section":"Eq. (9) and Fig. 3"},{"comment":"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).","section":"Appendix A and Fig. 2"},{"comment":"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.","section":"Conceptual distinction in Introduction/Conclusion"}],"minor_comments":[{"comment":"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.","section":"Eq. (8)"},{"comment":"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.","section":"Fig. 3 caption"},{"comment":"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.","section":"Page 4, spin-chain parameter text"},{"comment":"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.","section":"Eq. (9)"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim of a 'universal protocol' is considerably stronger than what the numerical evidence supports. The cost functions are constructed to directly enforce the DTC response, and the absence of disorder averaging, finite-size scaling, long-time persistence, and perturbation-stability tests means the demonstrations are consistent with fine-tuned fitting rather than a robust phase. The work is promising as a proof-of-principle of inverse design for subharmonic drives, but it needs substantial additional numerical experiments and a more careful framing before it could be considered for publication in a high-impact journal. The experimental proposal is reasonable but also relies on the unverified robustness of the optimized pulses."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Know this: the paper shows CRAB optimal control can find periodic pulses that give period-doubled response in two models, but the cost functions already encode the DTC signature, so the central 'universal protocol' claim is not established by these two examples. The work is a reasonable proof-of-concept, not a definitive phase-engineering method.\n\nWhat's new: the application of CRAB to DTC generation is not in the cited literature. Prior DTC pulses were on-off or sinusoidal. Defining cost functions whose minima are subharmonic response is a clean way to turn pulse design into an optimization problem. The spin-chain result, where optimizing on a two-site subsystem yields a pulse that gives global 2T response, is a nice touch. The Dicke model shows different dynamical phases as epsilon is varied, and an appendix shows the DTC persists at epsilon=0.06.\n\nSoft spots are real. F1 and F2 are literally the DTC condition: 2T periodicity or an FFT peak at half the drive frequency. Optimizing them is guaranteed to produce a pulse with that signature, so the demonstrations are closer to curve-fitting than to discovery. The spin-chain evidence is one disorder realization of L=8, no finite-size scaling, no disorder average, and the FFT threshold 0.05 is arbitrary. The Dicke model is a mean-field trajectory with no check of other initial conditions. The robustness test is only two nearby epsilon values. 'Universal protocol' is not supported by two examples; at best it is a promising recipe that needs tests of persistence, rigidity, and disorder averaging. The experimental section is speculative and lacks specifics.\n\nFor a reader: this is a useful prompt for anyone working on DTC control or optimal control of driven many-body systems. It deserves a serious referee, but the referee should demand code/data, disorder averaging and finite-size scaling for the spin chain, and a rephrasing of the universality claim. With those changes it could be a solid contribution.","headline":"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.","tokens_in":10593,"tokens_out":2450,"would_cite":false,"duration_ms":25532,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Optimization over pulse shapes yields discrete time crystals in generic many-body quantum systems.","keywords":["discrete time crystals","optimal control","CRAB","Floquet systems","many-body localization","open Dicke model","time-translational symmetry breaking","quantum control"],"falsifier":"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.","tokens_in":9492,"feed_emoji":"💎","tokens_out":7076,"duration_ms":65231,"temperature":0.7,"pith_summary":"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.","feed_headline":"Search over pulse shapes locks in discrete time crystals","feed_subtitle":"Optimizing two central sites of a disordered chain yields period-doubled order across the whole system.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines Floquet time crystals and the subharmonic response that the cost functions target.","marker":"[5]"},{"why":"Provides the many-body-localized spin-chain model and the FFT characterization of the DTC phase that underlie cost function $F_2$.","marker":"[8]"},{"why":"Establishes the open Dicke-model DTC driven by an on-off pulse, the baseline that cost function $F_1$ is designed to surpass.","marker":"[17]"},{"why":"Introduces the CRAB chopped-random-basis optimization scheme used to optimize the pulse coefficients.","marker":"[27]"},{"why":"Extends CRAB to many-body quantum dynamics, justifying its use on the spin chain and open Dicke model.","marker":"[28]"}],"fun_headline_variants":["Optimal control pulses forge discrete time crystals","Pulse optimization yields robust time-crystal order","Shaping drives to break time symmetry in quantum systems","Control pulses induce period-doubled dynamics in many-body systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Optimal control pulses forge discrete time crystals","Pulse optimization yields robust time-crystal order","Shaping drives to break time symmetry in quantum systems","Control pulses induce period-doubled dynamics in many-body systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1224,"prompt_tokens":874,"completion_tokens":350,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":288}},"tokens_in":490,"tokens_out":350,"duration_ms":4130,"temperature":1.0,"reasoning_tokens":288,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:56:10.272352+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Else, Bela Bauer, and Chetan Nayak","cited_arxiv_id":null,"evidence_quote":"Defines Floquet time crystals and the subharmonic response that the cost functions target."},{"cited_title":"The spins can be encoded in the 6 2S1/2 |F= 0, mF = 0>and |F= 1, m F = 0>hyperfine clock states of an electro- statically trapped and laser cooled 171Yb+ ions [21]","cited_arxiv_id":null,"evidence_quote":"Provides the many-body-localized spin-chain model and the FFT characterization of the DTC phase that underlie cost function $F_2$."},{"cited_title":"Higher-order and fractional discrete time crys- tals in clean long-range interacting systems.Nature Com- munications, 12(1):2341, Apr 2021","cited_arxiv_id":null,"evidence_quote":"Establishes the open Dicke-model DTC driven by an on-off pulse, the baseline that cost function $F_1$ is designed to surpass."},{"cited_title":"J¨ ager, Jan Mathis Giesen, Imke Schneider, and Sebastian Eggert","cited_arxiv_id":null,"evidence_quote":"Introduces the CRAB chopped-random-basis optimization scheme used to optimize the pulse coefficients."},{"cited_title":"Chopped random-basis quantum optimization","cited_arxiv_id":null,"evidence_quote":"Extends CRAB to many-body quantum dynamics, justifying its use on the spin chain and open Dicke model."}],"review_version":1}