{"id":"4d38fc4b-8f33-4a81-897e-00f1bf5813fd","arxiv_id":"2607.19154","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"By solving the tracking equation E + α|E|^{1/ν} sgn(E) = ⟨F_z⟩ at every time step, the authors make hydrogen's dipole acceleration follow sublinear power laws for several exponents.","lead":"Using tracking control, this paper shows numerically that a hydrogen atom can be driven so that its emitted response scales as a fractional power of the applied field — e.g., R ∝ E^{1/3} — instead of the usual integer powers of nonlinear optics. The self-consistent driving fields are found with a compact continuum basis, pointing toward programmable, non-polynomial optical nonlinearities.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The tracking demonstration is enforced by construction via Eq. (7); the physical sublinear-response claim therefore rests entirely on the unverified convergence of the WPCD basis in the self-consistent tracking regime.","rationale":"The paper's central claim is that tracking control can engineer a sublinear optical response in hydrogen. The demonstration uses Eq. (7) as a feedback law: at every time step the field is chosen so that the Ehrenfest expression for the dipole acceleration equals the target power law. Within the model, this equality is therefore automatic; the curves in Figs. 3 and 4 collapsing onto the target curve confirm that the root-finding and time-stepping are self-consistent, not that the physical atom obeys the sublinear law independently. The only way the central claim can be true for a real hydrogen atom is if the finite WPCD basis accurately represents the true quantum dynamics under the self-consistent tracking field. This is exactly the reader's weakest assumption. The HHG benchmark at E0=0.02 a.u. provides some evidence that the basis works for moderate fields, but tracking fields are not benchmarked; they may be more intense and have broader spectral content, and the paper gives no convergence study for the tracking trajectories. The Ehrenfest check in Fig. 2 is also performed for a different (benchmark) pulse, so it does not bound the truncation error in the control regime. Other weaknesses—overstated 'proof', missing code URLs, unsupported 'speedup' claim, and the partly tautological validation—are real but secondary. The tautology is inherent to tracking control and does not invalidate the method; the missing code affects reproducibility but not the physics; the speedup claim is separable from the central claim. The decisive test is a systematic convergence study of the WPCD basis in the tracking regime. If the tracking field and response are stable under basis enlargement, the conditional accept is justified. If they change substantially, the demonstrated sublinear response is likely a basis artifact. Therefore the reader's conditional verdict is appropriate and should remain unchanged.","tokens_in":8201,"tokens_out":11024,"duration_ms":118911,"concrete_test":"Run the ν=3, α=0.03 tracking case with at least two systematically larger WPCD bases (e.g., nmax=25/Lmax=8/80 bins and nmax=30/Lmax=10/100 bins) and, if feasible, with a converged grid TDSE (e.g., Ref. [44]). Compare E_track(t), R(t) computed independently as a numerical derivative of ⟨p_z⟩(t) (not via Eq. (6)), and the final-state populations. If the tracking field or R(t) changes by more than a few percent, or if significant population accumulates in the highest-energy continuum bins, the sublinear-response claim is not converged. Also report max|E_track| and the ionization fraction to demonstrate that the dynamics remain within the basis capacity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result is obtained by solving Eq. (7), E(t)+α sgn(E)|E|^{1/ν}=⟨Ψ(t)|F_z|Ψ(t)⟩, at every time step and then propagating with that E(t). Therefore R(t)=d⟨p_z⟩/dt = ⟨F_z⟩−E(t) equals the target α sgn(E)|E|^{1/ν} by construction in the model. Figs. 3(a) and 4 mostly verify that the algebraic equation was solved; they do not independently confirm that hydrogen physically realizes a sublinear response. The non-tautological content is that the E_track(t) generated by this feedback loop, when applied to a real atom, would produce the target response. This requires the 512-state WPCD basis (nmax=17, Lmax=6, 60 continuum bins per ℓ) to faithfully represent the ionization and continuum dynamics induced by E_track, which can be more intense and spectrally broader than the 0.02 a.u. HHG benchmark. The paper reports no convergence study for the tracking trajectories (varying nmax, Lmax, or N bins), and the Ehrenfest verification in Fig. 2 is performed for a benchmark pulse, not for a tracking pulse, so it does not bound the truncation error in the control regime. If high-lying continuum states are populated, ⟨F_z⟩ and hence the required E(t) are distorted, so the demonstrated sublinear relation could be a basis artifact. This is the load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to use quantum tracking control to make the dipole acceleration of a hydrogen atom follow a prescribed sublinear power-law response, R_target(t) = α sgn(E)|E|^{1/ν}, with ν>1. The field is not prescribed; instead, at each time step the algebraic equation α sgn(E)|E|^{1/ν} + E = ⟨Ψ|F_z|Ψ⟩ is solved and the resulting E(t) is used to propagate the Schrödinger equation. The propagation is performed in a wave-packet continuum discretization (WPCD) basis containing 512 states. The authors benchmark the WPCD basis against a converged TDSE for HHG at E0=0.02 a.u., verify the Ehrenfest relations for a benchmark pulse, and then show that for several α and ν the computed response R(t)=d⟨p_z⟩/dt collapses onto the target curve. They conclude that tracking control is a general route to engineering optical responses beyond conventional polynomial nonlinearities.","tokens_in":8602,"tokens_out":3776,"duration_ms":40077,"significance":"If the numerical demonstration is supported by a convergence study in the tracking regime, the paper provides a valuable proof-of-principle: a feedback-determined waveform can make a single quantum system exhibit a sublinear optical response that lies outside the standard integer-order susceptibility expansion. The derivation from Ehrenfest's theorem is clean, and the WPCD basis is a computationally efficient tool that appears well suited to this problem. The paper also includes useful validation steps, namely the HHG benchmark and the Ehrenfest consistency check. The main limitation is that the headline result, the collapse of the response onto the target power law, is enforced by construction through Eq. (7); the physically nontrivial content is whether the self-consistent tracking field, when applied to the real atom, would produce the same response. That question hinges on the fidelity of the 512-state WPCD basis in the tracking regime, which is not yet established.","major_comments":[{"comment":"The agreement between R(t) and R_target(t) in Figs. 3(a) and 4 is imposed by construction. Since the propagation uses the field E(t) obtained by solving Eq. (7), and since Eq. (6) is the Ehrenfest relation in the same finite basis, R(t)=⟨F_z⟩−E(t) equals α sgn(E)|E|^{1/ν} automatically (up to numerical solution of the algebraic equation and the ODE integrator). Thus Figs. 3–4 verify that the solver works, not that hydrogen physically realizes a sublinear response. This is not fatal, but the physical claim must be reframed: the nontrivial assertion is that the E_track(t) generated by this procedure, when applied to the true atom, would indeed produce the target response. That requires the WPCD basis to be faithful for the tracking trajectories, which is only indirectly supported by the current benchmarks.","section":"Sec. III, Eq. (7) and Figs. 3–4"},{"comment":"No convergence study is reported for the tracking trajectories. The WPCD basis is fixed at nmax=17, Lmax=6, and 60 continuum bins per ℓ, and the only external benchmark is an HHG spectrum at E0=0.02 a.u. (Fig. 1). The tracking fields shown in Fig. 3(b) are self-consistent and can be more intense and spectrally broader than the benchmark pulse, so the HHG validation does not bound the truncation error in the tracking regime. The authors should report how the R-versus-E_track curves in Fig. 4 change as nmax, Lmax, and the number of continuum bins are increased, and ideally compare one tracking trajectory against a converged TDSE calculation. Without this, the sublinear response could be a basis artifact.","section":"Sec. II A and Sec. III"},{"comment":"The Ehrenfest verification in Fig. 2 is performed for a benchmark pulse, not for a tracking pulse, and the text does not specify the driving parameters, the basis, or the magnitude of the discrepancy. Since the tracking algorithm relies sensitively on the identity R=⟨F_z⟩−E, and since finite bases violate Ehrenfest theorems (as the text itself notes), the relevant check is the Ehrenfest violation for the actual E_track(t) used in Figs. 3–4. A quantification of the discrepancy for a tracking run is needed to support the claim that the tracking equation is physically meaningful.","section":"Sec. II B and Fig. 2"}],"minor_comments":[{"comment":"Please specify the pulse parameters and basis used for the Ehrenfest verification. The statement that 'the small discrepancy visible in Fig. 2 can be further reduced by using a lower intensity' is vague; the discrepancy should be quantified.","section":"Sec. III, Fig. 2"},{"comment":"The claim of 'orders-of-magnitude speedup over converged three-dimensional TDSE calculations' is not supported by any timing or memory comparison in the paper. Please provide quantitative performance data or soften the claim.","section":"Sec. IV"},{"comment":"References [45] and [50] have empty parentheses; the Jupyter notebook and GitHub repository URLs or DOIs are missing. These are needed to verify the reproducibility claims.","section":"References [45] and [50]"},{"comment":"The statement 'a solution of Eq. (7) always exists' is true, but it would be helpful to give a one-line argument: the function E ↦ E+α sgn(E)|E|^{1/ν} is continuous, odd-like, and surjective onto R. The uniqueness of the solution is also worth stating explicitly.","section":"Sec. II B"},{"comment":"The word 'prove' in Section IV and 'prove that a quantum system can be driven' in the Introduction overstate the numerical demonstration. The existence of a tracking field is established by construction, but the physical realization is a computational demonstration. I suggest using 'show' or 'demonstrate' throughout.","section":"Abstract and Sec. IV"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the gap between the constructive enforcement of the tracking equation and the physical claim. The paper is not circular in the sense that the tracking equation has nontrivial dynamical content, but the current evidence for basis fidelity in the tracking regime is indirect. A convergence study for the tracking trajectories, plus a quantitative Ehrenfest check for a tracking field, would address the main concern and likely make the paper publishable. The literature context and the WPCD application are appropriate for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper’s real news is narrow but real: it’s the first time tracking control has been aimed at a fractional-power response, and the first time WPCD has been used in strong-field optics. The Ehrenfest-based derivation of the tracking equation is clean, and the HHG benchmark against a converged 3D TDSE is a solid piece of validation for the basis at moderate intensity. The handling of the singular zero-crossing via the u variable is a neat touch.\n\nThe soft spot is the one the stress-test flags: by construction, Eq. (7) forces R(t) to be α sgn(E)|E|^{1/ν} if the solver works. So Figs. 3 and 4 mostly show that the algebraic equation was solved each step, not that a physical hydrogen atom displays a sublinear response. The physical content lies in whether the 512-state basis faithfully represents the continuum during tracking, where fields are more complex and possibly more intense than the benchmark pulse. On that, the paper is silent: no convergence study for the tracking runs, and the Ehrenfest verification is done for the benchmark pulse, not for a tracking trajectory. So the headline 'proof' and 'order-of-magnitude speedup' claims outrun what is shown. The speedup is plausible given the compact basis, but no timing comparison is given.\n\nThat said, the paper is not circular. The WPCD-to-TDSE benchmark and the Ehrenfest check give some independent evidence that the basis is trustworthy in a related regime, and the parametric plots show the method can enforce several exponents and strengths. The authors are aware of the finite-basis limitations and cite the right literature on Ehrenfest violations.\n\nWho wants this? Groups working on quantum control of observables and strong-field simulation. A serious referee could push for a convergence study and a direct check of the tracking regime; if that comes back clean, the paper would be a useful method contribution. I’d take a chance on it in review, but I’d want the authors to tone down the 'proof' and speedup language.\n\nBest.","headline":"Novel target family and WPCD benchmark, but the sublinear-response claim rests on an unverified basis in the tracking regime — worth a serious referee.","tokens_in":9057,"tokens_out":2221,"would_cite":true,"duration_ms":23923,"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":"The paper demonstrates that tracking control can force a hydrogen atom to emit light whose intensity scales as a fractional power of the driving field, a response lying outside the conventional integer-order nonlinear optics hierarchy.","keywords":["sublinear optical response","tracking control","hydrogen atom","wave-packet continuum discretization","nonlinear optics","strong-field physics","dipole acceleration","fractional power-law"],"falsifier":"An independent converged three-dimensional time-dependent Schrödinger equation simulation of the same tracking field, using a substantially larger or different basis, that fails to reproduce the target power law would cast doubt on the claim. Alternatively, a laboratory measurement of the emitted field scaling showing a deviation from the prescribed fractional power for a driven hydrogen-like system would falsify the result.","tokens_in":8111,"feed_emoji":"⚛️","tokens_out":2399,"duration_ms":29957,"temperature":0.7,"pith_summary":"The paper shows that a hydrogen atom can be driven by a self-consistently determined laser field to produce an optical response that scales as a sublinear power of the applied field, such as a cube root. This is achieved by solving a nonlinear algebraic equation at each time step that enforces a prescribed relation between the dipole acceleration and the driving field. The authors argue this constitutes a general route to engineering optical nonlinearities beyond the perturbative expansion. The demonstration relies on a compact wave-packet representation of the continuum that treats bound and ionized states on equal footing.","feed_headline":"Hydrogen emits light with a fractional-power response","feed_subtitle":"A self-consistent tracking field makes the optical response scale as a cube root of the drive, bypassing integer-order nonlinear optics.","key_machinery":"The tracking equation (Eq. 7) is the central object: a nonlinear algebraic relation that determines the instantaneous driving field from the expectation value of the Coulomb force, ⟨F_z⟩ = ⟨Ψ|−∂V_0/∂z|Ψ⟩. Solving this equation self-consistently at every time step converts a desired response law into a concrete field waveform. The computation of ⟨F_z⟩, including continuum contributions, is made feasible by the wave-packet continuum discretization (WPCD) basis, which represents continuum states as square-integrable energy-bin superpositions, allowing bound and continuum states to be treated in a single finite Hilbert space.","core_discovery":"The central claim is that sublinear optical response, described by R_target(t) = α sgn(E)|E|^{1/ν} with ν > 1, can be realized by tracking control. By requiring the dipole acceleration R(t) = d⟨p_z⟩/dt to equal the target, the driving field is obtained at each instant from the tracking equation α sgn(E)|E|^{1/ν} + E = ⟨Ψ|F_z|Ψ⟩, whose right-hand side depends on the evolving quantum state. The paper reports that the achieved response collapses onto the target power law for multiple exponents (ν = 3, 5, 7) and strengths (α = 0.03, 0.04, 0.05), demonstrating that a single quantum system can exhibit a family of distinct sublinear responses.","pith_inferences":["The same self-consistent tracking mechanism could be applied to engineer other non-analytic response functions, such as logarithmic or saturating scalings, by replacing the power-law target in Eq. (5).","The requirement of full knowledge of ⟨F_z⟩ from the instantaneous wavefunction suggests the scheme may be sensitive to decoherence or many-body effects; in real experiments, feedback or robust control might be needed.","A natural extension is to translate the engineered sublinear response from the atomic dipole to macroscopic media, where collective effects and propagation could alter the effective field–response relation.","The divergence of the derivative of |E|^{1/ν} at zero crossing is circumvented here by a variable transformation; similar regularizations could be important for other singular targets."],"forward_implications":["If the claim holds, optical responses can be designed to follow arbitrary power laws, including fractional and sign-dependent scalings, in a single atomic system rather than by fabricating new materials.","Sublinear responses enhance the relative strength of weak fields while compressing strong ones, which could enable dynamic-range compression and improved weak-signal detection in optical sensing.","The tracking-control framework offers a systematic way to analyze non-integer optical responses that may emerge naturally in engineered materials, such as strained graphene.","The WPCD basis provides an efficient numerical tool for strong-field optics and quantum control, potentially making converged continuum-inclusive simulations routine."],"fun_headline_variants":["Hydrogen atoms emit light with fractional power law","Sublinear optics: hydrogen's fractional response","Tracking control gives hydrogen a cube-root light response","Hydrogen's optical response: fractional, not integer","Hydrogen atoms break the integer-order optics rule"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The finite 512-state wave-packet basis is assumed to faithfully represent the hydrogen continuum throughout the tracking dynamics, a regime that may involve more intense and complex fields than the modest benchmark pulse.","fun_headline_variants_meta":{"raw":{"variants":["Hydrogen atoms emit light with fractional power law","Sublinear optics: hydrogen's fractional response","Tracking control gives hydrogen a cube-root light response","Hydrogen's optical response: fractional, not integer","Hydrogen atoms break the integer-order optics rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000138,"raw_usage":{"total_tokens":971,"prompt_tokens":704,"completion_tokens":267,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":196}},"tokens_in":448,"tokens_out":267,"duration_ms":3654,"temperature":1.0,"reasoning_tokens":196,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:17:16.615926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent converged three-dimensional time-dependent Schrödinger equation simulation of the same tracking field, using a substantially larger or different basis, that fails to reproduce the target power law would cast doubt on the claim. Alternatively, a laboratory measurement of the emitted field scaling showing a deviation from the prescribed fractional power for a driven hydrogen-like system would falsify the result.","supporting_citations":[],"review_version":1}