{"id":"c23cf65b-0293-44cd-a35a-d92c44117ef1","arxiv_id":"2507.14329","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Numerically exact TEMPO simulations of the spin-boson model with negative spectral exponents reveal a coherent-to-pseudo-coherent crossover with no incoherent regime, and yield an empirical formula for the zero-temperature dephasing rate.","lead":"This paper studies a standard model of a qubit coupled to an environment whose noise grows at low frequencies, called 1/f noise. Using an exact numerical method, it maps out when qubit oscillations survive and gives a simple formula for how quickly the qubit loses coherence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The dephasing formula Eq. (6) is extracted by fitting the ansatz Eq. (4), which is not justified for s<0; at T=0 the influence Re Q(t) grows like t^{1-s}, so the fitted exponential rate may be a window-dependent artifact.","rationale":"Agree with the reader: the unsupported use of Eq. (4) for s<0 is the weakest load-bearing step. The additional scaling observation makes the concern concrete rather than stylistic: the T=0 influence phase is superlinear in time, so an exponential dephasing rate is not the generic weak-coupling behavior for negative spectral exponents. This does not undermine the numerically exact TEMPO results or the qualitative regime classification; it specifically threatens the quantitative empirical formula unless the fitting stage is checked. Since the original verdict was already CONDITIONAL, my read leaves that verdict unchanged.","tokens_in":9693,"tokens_out":12755,"duration_ms":145354,"concrete_test":"Use the T=0, s=-0.5 TEMPO data at the weakest coupling shown in the inset of Fig. 3 and fit Eq. (4) over the windows Omega t in [0,2], [0,5], and [0,10], recording gamma, its confidence interval, and the reduced chi-squared. If gamma shifts by more than its statistical uncertainty, or if a stretched-exponential envelope P(t)=1-A cos(Omega_0 t) exp[-(gamma t)^beta] yields beta significantly different from 1, the ansatz is invalid for s<0 and Eq. (6) cannot be claimed. Repeating the same check at s=-0.25 and s=-0.75 would map the range of validity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is Eq. (6), obtained by fitting TEMPO polarization data to Eq. (4), a damped-cosine ansatz whose parameter gamma is interpreted as a dephasing rate. For s=1, Eq. (4) has a controlled NIBA derivation; for s<0 none is supplied, and the paper only states that it 'fits well.' This is not a harmless gap. At T=0 with omega_ir=0, the real part of the bath influence in Eq. (3) is Re Q(t) = (2 alpha/(pi omega_c^{s-1})) integral d omega omega^{s-2} e^{-omega/omega_c} (1-cos omega t). For -1<s<0, this integral is finite and scales as alpha t^{1-s} times a constant when omega_c t >> 1. Since 1-s>1, the natural weak-coupling envelope is exp[-const alpha t^{1-s}], not exp[-gamma t]. Fitting such a stretched/Gaussian-like envelope with Eq. (4) can produce a 'rate' that depends on the fit window, on omega_c, and on omega_ir; the alpha-linearity of the fitted value does not distinguish a true rate from an artifact because both are linear in alpha. The paper reports no residuals, no confidence intervals, and no window-dependence check before promoting the fitted slope to the empirical law Eq. (6). The regime diagram and the qualitative pseudo-coherent statement are far less affected, since they rely on the shape of the time traces rather than on this ansatz.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the spin-boson model to spectral exponents s<0 (quantum 1/f^η noise with η=-s) and studies polarization and coherence dynamics using the TEMPO method. It reports a dynamical regime diagram in the (s,α) plane, finding that for s<0 the system crosses directly from coherent to pseudocoherent dynamics without an incoherent regime. At weak coupling the authors extract a dephasing rate γ from fits of Eq. (4) and propose the empirical formula γ ≈ 14.5 α e^{-2.4s} Ω (Eq. (6)). They also derive the infrared-cutoff dependence of the reorganization energy and argue that it explains the measurement-time dependence of low-frequency dephasing.","tokens_in":10101,"tokens_out":13204,"duration_ms":144327,"significance":"If confirmed, the regime diagram and the empirical dephasing formula would be useful for modeling superconducting qubits and other solid-state devices limited by 1/f^η noise. The use of numerically exact TEMPO is appropriate and the paper makes explicit, falsifiable predictions. The reorganization-energy analysis and the connection to an infrared cutoff related to the measurement time are interesting and physically motivated. However, the quantitative content currently rests on an unvalidated fitting ansatz and on numerical runs without documented convergence or uncertainty analysis, so the main quantitative claim needs substantial additional support.","major_comments":[{"comment":"The extraction of γ is based on an ansatz whose validity for s<0 is not established. In the Ohmic limit Eq. (4) follows from NIBA, but for s<0 the paper only states that it 'fits well.' The concern is concrete: at T=0 the real part of Q(t) in Eq. (3) grows as α t^{1-s} for -1<s<0 and ω_c t >> 1, so the natural weak-coupling envelope is exp[-const α t^{1-s}], which is superlinear, not exponential. Fitting an exponential over a fixed window then yields a 'rate' that is linear in α but depends on the fit window and on ω_c and ω_ir. The paper reports no residuals, no window-dependence test, no comparison with a stretched-exponential fit, and no confidence intervals for the slopes or for the parameters 14.5 and -2.4 in Eq. (6). Since Eq. (6) is fitted to rates obtained from the same ansatz, it is a compact representation of the simulated data rather than an independent empirical law. I recommend testing whether the exponential form survives changes of t_max and ω_c, and either deriving a proper functional form for the envelope or restricting the claim to a clearly delimited time window.","section":"Section VI, Eq. (4)-(6)"},{"comment":"The dynamical regime diagram and the claim that no incoherent regime exists for s<0 are based on visual classification of time traces and on the disappearance of the first maximum, but the paper provides no convergence analysis for TEMPO. The text states that s=-0.75 is the smallest exponent for which numerical convergence could be achieved, yet no time-step, bond-dimension, or truncation-error data are shown. The boundary line in Fig. 2 is drawn without error bars and without displaying the transition points. To support the 'numerically exact' claim, the paper should include convergence tests and an uncertainty estimate for the boundary α(s).","section":"Section V, Fig. 2"},{"comment":"The linear dependence γ ∝ α is established on a very small range of α (in Fig. 3 the data extend only to α ≈ 0.003) and without error bars on the individual extracted values of γ. The slopes γ'(0) displayed in Fig. 4 are therefore fits to fits. The paper should report the goodness of fit (residuals or χ²), the number of data points, whether the linear fits were forced through the origin, and stability of the slopes under changes of the fit window. Without this information, the empirical formula Eq. (6) is not robustly supported.","section":"Section VI, Fig. 3"}],"minor_comments":[{"comment":"In the discussion of Fig. 1(c) the text refers to coherent dynamics at α=0.1, while the caption lists α=0.01, 0.02, and 0.04; this is presumably a typo and should be corrected.","section":"Section IV"},{"comment":"Equation (4) is typeset ambiguously; the fraction is unclear. Please rewrite it so that the reader can see which terms are divided by (1-α̃).","section":"Equation (4)"},{"comment":"The text says the model is considered for -1≤s≤1, but the numerical results and Eq. (6) cover only s≥-0.75 and s≥-0.5, respectively; clarify whether s=-1 is included and under which conditions (e.g., ω_ir>0).","section":"Section II"},{"comment":"Please clarify why a rate extracted from the polarization dynamics P(t) is called a dephasing rate; in the unbiased spin-boson model the damping of P(t) is not identical to the pure dephasing of the off-diagonal coherence and the distinction should be stated explicitly.","section":"Section VI"},{"comment":"The connection between ω_ir=2π/t_meas and the experimental flux-noise data of Ref. [22] is qualitative; a brief statement of the parameter mapping would make the claim more concrete.","section":"Section VII"}],"recommendation":"major_revision","confidential_remarks":"I see the paper as potentially publishable after a major revision. The qualitative regime diagram may be the most durable contribution, while the quantitative dephasing formula needs substantially more evidence or a reanalysis with a properly justified envelope. Given the authors' track record, the requested tests seem feasible; I would not reject the paper on the current evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the regime diagram for s<0 is probably right and worth having; the quantitative dephasing law Eq. (6) is on shakier ground, because the exponential fit it comes from is not justified for s<0. I'd take the paper with that caveat.\n\nWhat's actually new: This is the first numerically exact TEMPO treatment of the spin-boson model with negative spectral exponents. The claim that pseudo-coherent dynamics persists for -0.75≤s≤0.45, with no incoherent regime for s<0, is new and supported by the time traces. The reorganization energy discussion (Λ diverges as ωir→0 for s<0) is correct and relevant to experiments. The IR-cutoff dependence is an honest treatment of a tricky limit.\n\nSoft spots: The dephasing rate is the problem. Eq. (4) has a controlled NIBA derivation only for s=1. For s<0 it's an ansatz, and the paper tells us only that it 'fits well.' At T=0, the real part of the influence function Re Q(t) grows like t^{1-s} for -1<s<0. That means the natural weak-coupling envelope is exp[-c α t^{1-s}], not exp[-γ t]. Fitting the former with an exponential can produce a 'rate' that depends on the fitting window, the UV cutoff, and the low-frequency cutoff, even though the fitted value remains linear in α. The paper shows no residuals, no confidence intervals, no check against an alternative envelope, and no TEMPO convergence tests anywhere. So Eq. (6) is best read as a compact fit to the authors' data under a particular functional form, not as a demonstrated physical law.\n\nThe regime diagram is less affected by this, because it is defined by shape features of P(t) (first maximum vanishing) and not by the exponential fit. The inflection point in α(s) at s≈-0.5 is interesting but has no error bar.\n\nWho this is for: quantum-optics and condensed-matter people working on 1/f noise in superconducting qubits, and open-quantum-systems people who want TEMPO applied to an awkward regime. They will get a plausible qualitative picture and a formula to treat with caution.\n\nRecommendation: I'd send it to review, but with a clear request: add convergence analysis, show residuals, compare the t^{1-s} envelope against Eq. (4), and clearly label Eq. (6) as an empirical fit whose functional form needs justification. The paper is not wasted; it just overreaches a bit on the rate.","headline":"New regime diagram for s<0 is credible, but the exponential dephasing law rests on an unjustified fit and needs a major revision before I'd trust Eq. (6).","tokens_in":10556,"tokens_out":4518,"would_cite":true,"duration_ms":535005,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper extends the spin-boson model to quantum $1/f^\\eta$ noise and finds a direct coherent-to-pseudo-coherent transition with a zero-temperature dephasing rate $\\gamma\\approx14.5\\,\\alpha\\,e^{-2.4s}\\,\\Omega$.","keywords":["spin-boson model","quantum 1/f noise","negative spectral exponent","dephasing rate","pseudo-coherent dynamics","TEMPO","infrared cutoff","qubit decoherence"],"falsifier":"Recompute the zero-temperature polarization for $s=-0.5$ at $\\alpha=0.03$ with a converged TEMPO bond dimension and monitor the first local maximum of $P(t)$: the coherent-to-pseudo-coherent transition is defined by the loss of that maximum, so an intermediate window of exponential, non-oscillatory decay between the two regimes would refute the regime diagram. Alternatively, measure the weak-coupling slope $\\gamma/\\alpha$ in a qubit with engineered $1/f^\\eta$ noise at fixed $s$ and compare it with $14.5\\,e^{-2.4s}\\,\\Omega$; a nonlinear dependence on $\\alpha$ or a different exponential factor would falsify Eq. (6).","tokens_in":9543,"feed_emoji":"⚛️","tokens_out":13098,"duration_ms":137568,"temperature":0.7,"pith_summary":"The paper extends the spin-boson model, the standard two-state model of quantum dissipation, to baths with negative spectral exponents, i.e. quantum $1/f^\\eta$ noise whose spectral density grows toward low frequencies. Using the numerically exact TEMPO path-integral method, it maps the dynamical regimes for exponents $-0.75 \\le s \\le 1$ and finds that for negative $s$ the polarization passes directly from damped coherent oscillations to \"pseudo-coherent\" motion locked to the bath cutoff, with no intermediate incoherent regime. At weak coupling, the decay is still exponential, and the dephasing rate is linear in the coupling $\\alpha$, collapsing onto the empirical zero-temperature law $\\gamma \\approx 14.5\\, \\alpha\\, e^{-2.4s}\\, \\Omega$. Because the bath reorganization energy diverges as the infrared cutoff vanishes for $s\\le 0$, the model predicts that dephasing grows with the duration of the measurement window. The result matters for qubit platforms whose dominant noise is low-frequency $1/f^\\eta$ noise.","feed_headline":"Quantum 1/f noise skips the incoherent regime","feed_subtitle":"Exact simulations map spin-boson dephasing for negative spectral exponents and tie it to measurement time.","key_machinery":"The load-bearing object is the spectral density $J(\\omega)=2\\alpha\\,\\omega^s\\,\\omega_c^{1-s}e^{-\\omega/\\omega_c}\\Theta(\\omega-\\omega_{\\rm ir})$ continued to negative exponents $s$, together with the time-evolving matrix product operator (TEMPO) path integral, based on the quasiadiabatic propagator path integral (QUAPI), that integrates out the bath exactly through the pair-correlation function $Q(t)$; a carefully added $-i\\omega t$ integration constant widens convergence into the $s<0$ regime. The argument then runs through a weak-coupling fit, $P(t)\\approx(1-\\tilde\\alpha\\cos(\\omega_0t)e^{-\\gamma t}-\\tilde\\alpha)/(1-\\tilde\\alpha)$, whose rigorous anchor is the Ohmic NIBA result, but which is applied as an ansatz across the negative-exponent range. From fits of this form the paper extracts $\\gamma(\\alpha)$, establishes the empirical law $\\gamma\\approx14.5\\,\\alpha\\,e^{-2.4s}\\,\\Omega$, and classifies coherent, incoherent, and pseudo-coherent regimes by the presence and frequency of oscillations.","core_discovery":"The paper's central finding is that negative spectral exponents do not produce a new incoherent phase: for $-0.75 \\le s \\le 0.45$ the spin-boson polarization goes directly from the coherent to the pseudo-coherent dynamical regime as coupling increases, and only for $s \\gtrsim 0.45$ does an incoherent regime exist. In the pseudo-coherent regime the oscillation frequency is set by the bath high-frequency cutoff $1/\\omega_c$, so the central spin is effectively enslaved to short-lived coherent bath oscillations. At weak coupling the polarization is well fit by a damped cosine envelope, yielding a dephasing rate that depends linearly on $\\alpha$ with slope $\\gamma'(0) \\approx 14.5\\, e^{-2.4s}\\, \\Omega$ at zero temperature and $\\omega_{\\rm ir}=0$. The reorganization energy $\\Lambda = 2\\alpha\\omega_c\\,\\Gamma(s,\\omega_{\\rm ir}/\\omega_c)$ diverges for $s\\le0$ as $\\omega_{\\rm ir}\\to0$, which the paper connects to the measurement-time dependence observed for $1/f$ flux noise in superconducting qubits.","pith_inferences":["Editorial inference: if Eq. (6) holds beyond the fitted range, the factor $e^{-2.4s}$ means the dephasing slope grows by roughly an order of magnitude from $s=0$ to $s=-1$ at fixed $\\alpha$, so a small low-frequency noise component can dominate over an Ohmic contribution.","Editorial inference: the infrared-cutoff sensitivity implies that quoting an intrinsic dephasing rate for $1/f^\\eta$ environments is incomplete; reported rates should state the measurement window or low-frequency cutoff used.","Editorial inference: a direct experimental test would tune a qubit's engineered noise environment to a known negative exponent and measure $\\gamma(\\alpha)$ across two decades of coupling, checking both the linearity in $\\alpha$ and the exponential $s$ dependence.","Editorial inference: testing whether the damped-cosine ansatz of Eq. (4) remains accurate for $s<0$ under other initial preparations or stronger coupling would clarify how far the extracted $\\gamma$ can be trusted beyond the weak-coupling window studied."],"forward_implications":["For qubits dominated by low-frequency quantum $1/f^\\eta$ noise, stronger coupling to the bath should produce bath-controlled pseudo-coherent oscillations rather than pass through the incoherent decay phase expected from classical-noise treatments.","At weak coupling the dephasing time is $T_\\phi\\approx e^{2.4s}/(14.5\\,\\alpha\\,\\Omega)$, so for fixed $\\alpha$ a more negative exponent $s$ shortens coherence, emphasizing the need to suppress the lowest-frequency bath modes.","Dephasing becomes measurement-time dependent: identifying $\\omega_{\\rm ir}\\sim 2\\pi/t_{\\rm meas}$ predicts that longer averaging in an experiment degrades coherence, consistent with flux-noise observations.","The numerically exact TEMPO method stays convergent for $s$ down to $-0.75$ at zero temperature, making the same tensor-network treatment applicable to strongly non-Markovian low-frequency baths in larger superconducting circuits."],"supporting_citations":[{"why":"Supplies the spin-boson Hamiltonian, the spectral-density conventions, the pair-correlation function Q(t), and the Ohmic NIBA result behind the weak-coupling fitting form Eq. (4).","marker":"[1]"},{"why":"Establishes the standard coherent, incoherent, and localized classifications of spin-boson dynamics that the paper extends to negative spectral exponents.","marker":"[2]"},{"why":"Defines pseudo-coherent dynamics and the hidden phase of the spin-boson model for the nonnegative spectral-exponent range, the regime diagram the paper continues to s<0.","marker":"[3]"},{"why":"Documents 1/f noise as a dominant decoherence source in solid-state qubits, motivating the extension of the spin-boson model to negative exponents.","marker":"[21]"},{"why":"Provides the superconducting flux-noise data whose measurement-time dependence the infrared-cutoff sensitivity is invoked to explain.","marker":"[22]"},{"why":"Motivates the s=-1 regime with an approximate hybrid treatment of extremely sub-Ohmic spin-boson dynamics.","marker":"[34]"},{"why":"Introduces the time-evolving matrix product operator (TEMPO) method used to obtain the numerically exact dynamics.","marker":"[35]"},{"why":"Supplies the tensor-network formulation of TEMPO on which the simulations are built.","marker":"[36]"},{"why":"Gives the quasiadiabatic propagator path integral (QUAPI) that TEMPO implements.","marker":"[37]"},{"why":"Provides the numerical methodology of the QUAPI propagator underlying the simulations.","marker":"[38]"}],"fun_headline_variants":["Negative 1/f noise bypasses incoherent spin-boson regime","Pseudo-coherent dynamics replace incoherent phase for 1/f noise","Spin-boson dephasing maps directly to pseudo-coherent regime","Negative 1/f noise yields no incoherent phase in spin-boson"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the polarization's decay envelope is a damped cosine for negative spectral exponents, even though that shape is rigorously justified only for Ohmic baths in the NIBA scaling limit; if the envelope has a different functional form for s<0, the extracted dephasing rate, its linear dependence on α, and Eq. (6) collapse.","fun_headline_variants_meta":{"raw":{"variants":["Negative 1/f noise bypasses incoherent spin-boson regime","Pseudo-coherent dynamics replace incoherent phase for 1/f noise","Spin-boson dephasing maps directly to pseudo-coherent regime","Negative 1/f noise yields no incoherent phase in spin-boson"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000913,"raw_usage":{"total_tokens":3904,"prompt_tokens":908,"completion_tokens":2996,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":2925}},"tokens_in":524,"tokens_out":2996,"duration_ms":21717,"temperature":1.0,"reasoning_tokens":2925,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:58:18.178858+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the zero-temperature polarization for $s=-0.5$ at $\\alpha=0.03$ with a converged TEMPO bond dimension and monitor the first local maximum of $P(t)$: the coherent-to-pseudo-coherent transition is defined by the loss of that maximum, so an intermediate window of exponential, non-oscillatory decay between the two regimes would refute the regime diagram. Alternatively, measure the weak-coupling slope $\\gamma/\\alpha$ in a qubit with engineered $1/f^\\eta$ noise at fixed $s$ and compare it with $14.5\\,e^{-2.4s}\\,\\Omega$; a nonlinear dependence on $\\alpha$ or a different exponential factor would falsify Eq. (6).","supporting_citations":[{"cited_title":"Weiss, Quantum Dissipative Systems, 5th ed","cited_arxiv_id":null,"evidence_quote":"Supplies the spin-boson Hamiltonian, the spectral-density conventions, the pair-correlation function Q(t), and the Ohmic NIBA result behind the weak-coupling fitting form Eq. (4)."},{"cited_title":"Otterpohl, P","cited_arxiv_id":null,"evidence_quote":"Defines pseudo-coherent dynamics and the hidden phase of the spin-boson model for the nonnegative spectral-exponent range, the regime diagram the paper continues to s<0."},{"cited_title":"Paladino, Y","cited_arxiv_id":null,"evidence_quote":"Documents 1/f noise as a dominant decoherence source in solid-state qubits, motivating the extension of the spin-boson model to negative exponents."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the superconducting flux-noise data whose measurement-time dependence the infrared-cutoff sensitivity is invoked to explain."},{"cited_title":"Abdi and M","cited_arxiv_id":null,"evidence_quote":"Motivates the s=-1 regime with an approximate hybrid treatment of extremely sub-Ohmic spin-boson dynamics."},{"cited_title":"Strathearn, P","cited_arxiv_id":null,"evidence_quote":"Introduces the time-evolving matrix product operator (TEMPO) method used to obtain the numerically exact dynamics."},{"cited_title":"Strathearn, Modelling non-Markovian quantum sys- tems using tensor networks (Springer International Pub- lishing, Cham, 2020)","cited_arxiv_id":null,"evidence_quote":"Supplies the tensor-network formulation of TEMPO on which the simulations are built."},{"cited_title":"Makri and D","cited_arxiv_id":null,"evidence_quote":"Gives the quasiadiabatic propagator path integral (QUAPI) that TEMPO implements."},{"cited_title":"Makri and D","cited_arxiv_id":null,"evidence_quote":"Provides the numerical methodology of the QUAPI propagator underlying the simulations."}],"review_version":1}