# OEP 8 — The bound two-photon resonances of the noble gases for propagation | | | |---|---| | **OEP** | 8 | | **Title** | The bound two-photon resonances of the noble gases for propagation | | **Author** | John Travers | | **Drafting agent** | Opus 5.5 (2026-10-09); GPT-6 (review revisions, 2026-10-12) | | **Status** | Draft | | **Review status** | Reviewed | | **Reviewers** | John Travers (human); GPT-6 (AI) | | **Type** | Feature | | **Created** | 2026-10-09 | | **Resolution** | — | | **Requires** | [OEP 1](archive/oep-0001-propagation-interface.md): model identity, digests and the declared public surface; [OEP 5](archive/oep-0005-hyperpolarizability-support.md): the fitted $\gamma$ and its resonance limits; [OEP 6](archive/oep-0006-two-photon-ionisation.md): the continuum model, the bound-level records and the pair-spectral identity | | **Supersedes** | — | | **Superseded-by** | — | | **Implementation** | — | | **Manual pages** | — | ## Part I — Proposal ### 1. Abstract Below half its ionisation energy a noble gas absorbs two ultraviolet photons only at its even-parity bound levels: He 1s2s, Xe 6p and the like. The excited atom may then absorb a third photon and ionise. OEP 6 curated those levels and their two-photon strengths (Phase 2a), but deferred the response a propagation code needs (its question Q9). The reason was that no derivation yet tied a time-dependent response to the curated amplitudes, its units, its residues and its energy balance. This OEP supplies that derivation and proposes what optmat should publish from it. - **The model.** An effective-level Bloch model, obtained by eliminating the odd intermediate states. Its two-photon coupling is $-\tfrac12M_f(E^+)^2$, and its conservative polarisation is $-\partial\langle H_R\rangle/\partial E^-$, with photoionisation loss accounted for separately. The local model obeys the energy identity in section 4.4. General frequency-dependent dynamics are deferred (Q8). - **What optmat publishes, per qualified level:** - the residue $\delta^{TP}$, with its uncertainty and polarisation factors; - an empirical estimate of the error of the pair-frequency form; - the excited level's photoionisation cross section; - where curated, the two-colour amplitude. The propagation code owns the resulting polarisation, electron accounting, propagation energy balance and integration. The derivation here checks the material coefficients and their conventions; it is not an optmat propagation interface. - **Coefficient qualification proposed:** - He 1s2s strengths and photoionisation; strong-field dynamics additionally need the light shifts to be curated in Phase 1; - Xe 6p and 6p′, for weak-field absorption and populations. Their excited-level light shifts are not curated, and the dispersive part of their response fails Lehmberg et al.'s measured $n_2$ by a factor of 2.2 to 2.8; - Ar 4p, the $J=0$ and $J=2$ levels, for weak-field absorption and populations within the limits of section 6, on calculated strengths only (Q7). Their radiative widths are measured, and their photoionisation at the drive is calculated. Their light shifts and pressure broadening are not curated. - **Refused:** Ne and Kr. **Success** is judged by three reproductions and one honesty test: - the weak-field limit reproduces OEP 6's pair-spectral yield; - a published broadband He 1s2s calculation (28 % transfer) is reproduced; - the specified model closes its energy budget to a converged numerical tolerance; - every unqualified case is refused. ### 2. Motivation and scope **The need.** Resonant dispersive-wave emission and soliton self-compression in gas-filled hollow-core fibres reach the vacuum ultraviolet. Their spectra cross the two-photon resonances of the filling gas: - He 1s2s at 120.3 nm; - Xe 6p[1/2]₀ at 249.6 nm; - the 6p and 6p′ levels of Xe between 222 and 256 nm. **What a resonance does to the pulse.** - It depletes the pulse. - It leaves excited atoms that a third photon ionises: the (2+1) channel, which a plasma model must count. - Near the resonance, it changes the Kerr response. **Why linear absorption is not enough.** A femtosecond pulse at these intensities drives the level coherently. In Kumar et al.'s case [1] the peak two-photon Rabi frequency exceeds both the level's decay rate and the pulse's bandwidth, so rate equations fail (section 4.5). **The paired solprop proposal** (solprop's two-photon ionisation proposal) asked optmat for: - the continuum cross section, which OEP 6 supplies; - the bound resonances, deferred until optmat can say what they are. **In scope:** - the theory needed to establish the conventions and qualification of the curated two-photon amplitudes and material coefficients; - the coefficients optmat publishes for it, their uncertainties and qualification; - the reproducible checks; - the refusals. The first dynamical scope is the local one-beam model (one field, one polarisation state across its supported spectrum), with its spectral and field-strength conditions checked. A pulse may be broad compared with the final-state linewidth while narrow enough relative to intermediate-state detunings. The two-colour helium kernel is restricted to perturbative on-shell yields, not a general pump-plus-ultraviolet evolution model (Q8, TC-1055). **Out of scope:** - **The propagation response.** solprop owns the equations and implementation for the resulting polarisation, electron accounting, plasma current and propagation energy balance, as well as integration. optmat supplies material properties and basic data, as it supplies ionisation rates without a plasma polarisation and Raman responses without the resulting field polarisation (TC-1057). The single-atom derivations and internal fixtures here check material conventions and coefficients; they do not transfer ownership of the propagation response to optmat. - **The continuum's time-resolved loss.** No published small-split continuum amplitude is curated. Tier-3 TDSE runs now support the pair form for helium and argon (section 4.2), but its propagation response belongs in a solprop SEP. A coupled optmat OEP is needed only if that SEP requires new material quantities, models or interfaces beyond OEP 6. This does not extend OEP 8's phases (Q6, TC-1057). - **The plasma kinetics after the pulse.** - **Higher members of the even series,** without curated strengths. - **Molecules.** - **The ionic fine structure of the (2+1) electron,** a matter for the plasma model. #### 2.1 The initial experiment and the simplest useful test **The maintainer's use case (2026-10-12; TC-1056).** A 6 fs, 800 nm pump self-compresses in argon and emits a weaker VUV resonant dispersive wave (RDW). When pressure and phase matching tune the RDW towards the two-photon ionisation threshold, the RDW disappears. Pump-driven ionisation disrupting phase matching can explain the observations. The scientific question is whether VUV two-photon absorption or ionisation can also explain them, and which contribution dominates. Helium is a second target. These are the maintainer's reported context and competing hypotheses, not a curated experimental dataset or a conclusion of this OEP. Distinguish departure from a phase-matching curve as the pump blue-shifts from the shortest observable RDW across optimised operating conditions. Evidence for the first mechanism does not by itself establish the cause of the latter limit. Frequency-selective loss can also shift the measured RDW centroid; distinguish that centroid, the blue edge and the detection limit when testing the hypotheses. Reconcile the input duration for each experimental or simulated dataset before defining a fixture; the 6 fs context above is not a universal input specification (TC-1057). **Continuum first.** The manual's NIST-derived thresholds ({doc}`../two_photon`; TC-943) give 157.34 nm for two equal photons in Ar and 100.85 nm in He. These are OEP 6's continuum thresholds. OEP 8's first bound levels are elsewhere: Ar 4p at about 185–192 nm and He 1s2s at 120.28 nm. Include them when the actual pair spectrum overlaps them. The higher bound series approaching threshold still lack curated strengths; a null result for the retained low levels does not exclude all bound absorption. The continuum diagnostic need not wait for this OEP's implementation. At the Ar threshold, an 800 nm photon plus one 157.34 nm photon carries about 9.43 eV, below the 15.76 eV ground-state ionisation energy. Thus that particular mixed two-photon channel is closed for field-free atoms; the near-threshold direct channel uses two VUV photons. This does not exclude pump dressing, higher-order pump-assisted processes, excited-state ionisation, or other spectral components. The intense pump can shift levels even when this mixed channel is closed. **Recommended solprop investigation, not a new optmat propagator:** 1. Generate matched pressure scans with pump-driven ionisation disabled and enabled, holding the other input parameters fixed. Save the complex VUV field versus propagation distance, its transverse distribution or mode normalisation, the local neutral density and the pump-induced electron density. The input's 6 fs is not a proxy for the RDW duration, its bandwidth or its intensity at generation. 2. On these prescribed fields, evaluate OEP 6's perturbative pair-spectral probability and energy loss. With $E_V^+$ the declared VUV band and $S_{2,V}$ the transform of $(E_V^+)^2$, use the existing kernel $K$ of OEP 6 section 4: $$ P_2(z)=\frac{1}{2\pi}\int_0^\infty K(\Omega)|S_{2,V}(z,\Omega)|^2d\Omega,\qquad \Delta\mathcal E_2(z)=\frac{1}{2\pi}\int_0^\infty\hbar\Omega K(\Omega) |S_{2,V}(z,\Omega)|^2d\Omega . $$ These are per-atom diagnostics within that approximation. Integrating $N_g\Delta\mathcal E_2$ over transverse area and distance estimates the lost pulse energy on the prescribed field. Likewise $N_gP_2$ estimates the added electron density only when depletion is negligible and the neutral population can be treated as fixed during the VUV pulse. A time-varying pump-depleted population needs a specified dynamical model; it cannot simply be inserted into an integrated yield. Track unsupported spectral weight and vary the band split and available source envelope. In particular, the Ar model has a gap immediately above threshold and unresolved fine-structure detail between 157.34 and 155.59 nm. Neither a gap nor an unqualified contribution is zero. 3. Where the whole relevant band has a slowly varying supported cross section, the cheaper estimate $\beta=2N_g\sigma^{(2)}/(\hbar\omega_V)$ and $dI_V/dz=-\beta I_V^2$ checks the scale of loss. Its dimensionless scale is $\beta I_VL$, with the actual intensity and interaction length. A centroid coefficient is not reliable when the pair spectrum straddles the threshold. 4. If the diagnostic indicates a potentially relevant effect, compare four propagation cases: neither channel, pump ionisation only, VUV two-photon ionisation only, and both. The two-photon channel includes its field-energy loss and its electron contribution; an additional loss-only run can isolate direct absorption from plasma feedback. Avoid counting the VUV band again in a pump strong-field rate. Record any assumed continuum timing and spectral-loss rule: OEP 6's integrated constraints do not supply a unique time-dependent current (Q6). A phenomenological closure in solprop is an exploratory model, not a newly qualified optmat response. The post-processing step tests the scale and spectral onset; it cannot by itself predict disappearance or establish the dominant mechanism. Small direct loss also need not imply a small phase-matching change from the added electrons. If alternative admissible continuum timing models change the conclusion, or pump dressing is large, that is the evidence needed to prioritise the deferred dynamics. No particular outcome is asserted without the simulated fields and experimental geometry. **The continuum follow-up belongs in a solprop SEP (TC-1057).** Its deliverable is the propagation response: continuum polarisation, electron accounting, plasma coupling and energy-balance tests. optmat's contribution is the cross sections and other material properties, their provenance, uncertainty and validity limits. First identify which additional material inputs the SEP actually needs; a coupled optmat OEP would cover only those additions, with material validation fixtures for parity. An effective continuum driven by $(E_V^+)^2$, with coupling strengths matched to OEP 6's cross sections, is a candidate for that SEP to derive and validate. The absence of published nondegenerate amplitudes does not forbid such an approximate model. However, matching integrated yields does not uniquely fix the transient response, validate electron production during the pulse, or resolve missing near-threshold data. The candidate is not an approved or qualified model. Its detailed design and tests belong in the SEP; new material-data requirements may motivate the coupled OEP. ### 3. Context **What optmat supplies today** ({doc}`../two_photon`): - `optmat.gas.two_photon`: the degenerate continuum cross section $\sigma^{(2)}(\omega)$ of He, Ne, Ar and Xe, with both thresholds, labelled regions, an envelope and an identity (OEP 6 Phase 1; TC-943, TC-944). - `optmat.gas.two_photon_bound`: the even $J = 0, 1, 2$ levels below threshold from NIST, with every curated strength in one convention and the excited levels' photoionisation cross sections (OEP 6 Phase 2a; TC-952, TC-953). - **The convention** (TC-952): $\delta^{TP}$ per level, summed over $M_J$, parallel linear polarisation, equal photons, in atomic units, with $\sigma_0^{(2)}=2\pi^3\alpha^2a_0^4\omega_\mathrm{au}^2\delta^{TP}$. - **The assessments:** He and Xe checked; Ne, Ar and Kr unresolved. - `optmat.gas.hyperpolarizability`: the fitted Taylor-series $\gamma$ with its support, below the band edge $E_2/2$ (OEP 5). - **The near-infrared NIST lines of Ne, Ar, Kr and Xe** (TC-983): the transition probabilities among excited levels. They feed the sums over intermediate states and the radiative widths (section 5). **What OEP 6 deferred:** - its Phase 2b, the complex evaluator, until Q9 is answered (TC-945); - its optional Phase 3, consistency of a real part with $\gamma$ (Q8); - the widths and their conventions (Q7). Its own fallback is to report the evaluator as deferred to a later proposal. This is that proposal. It answers Q9 for the bound levels. It takes up Q7 and Q8 where they concern the bound levels. OEP 6 is Final with Phases 1 and 2a implemented (TC-1053). **The roadmap item answered:** OEP 6's Phase 2 evaluator, and the "bound resonances" deferred by the paired solprop proposal. ### 4. Theory SI throughout, with the manual's conventions ({doc}`../conventions`): - $E=E^++E^-$, with $E^+(t)=\int_0^\infty\frac{d\omega}{2\pi}\tilde E(\omega)e^{-i\omega t}$; - a monochromatic $E_0\cos\omega t$ has $E^+=(E_0/2)e^{-i\omega t}$; - $\gamma$ is the Taylor-series hyperpolarizability, so $\mu_\omega=\gamma E_0^3/8$; - $\Phi=I/\hbar\omega$, and $W=\sigma_0^{(2)}g(2\omega)\Phi^2$ for a bound line of unit-area profile $g$ in the two-photon angular frequency. $N$ is the initial atomic number density. These equations use the dilute-gas approximation in which the field acting on an atom equals the propagated field and $I=2\varepsilon_0c|E^+|^2$. Finite-density local-field corrections are not established by this derivation. Coherence rates describe homogeneous decay: Doppler broadening requires an average over velocity-dependent detunings, not insertion of a Gaussian width as $\gamma^\phi$. The numerical results below were reported during drafting from source records and NIST data. They are targets for reproducible Phase 1 fixtures, not tests already in the public tree. Section 8 distinguishes the independent review checks (TC-1054). #### 4.1 The exact second-order amplitude **The amplitude.** With $H=H_0-\mathbf d\cdot\mathbf E$, take second order and keep the $E^+E^+$ terms (the two-photon rotating-wave approximation). Symmetrising over the two photons gives the final-state amplitude: $$ c_f(\infty)=\frac{i}{2\hbar}\int_0^{\Omega_f}\frac{d\omega}{2\pi}\, \tilde E_\alpha(\Omega_f-\omega)\tilde E_\beta(\omega)\,M^f_{\alpha\beta}(\Omega_f-\omega,\omega), \tag{1} $$ **The kernel:** $$ M^f_{\alpha\beta}(\omega_a,\omega_b)=\sum_n\Big[\frac{d^\alpha_{fn}d^\beta_{ng}}{E_{ng}-\hbar\omega_b} +\frac{d^\beta_{fn}d^\alpha_{ng}}{E_{ng}-\hbar\omega_a}\Big]. \tag{2} $$ **The denominators** carry $-i0$, or $-i\hbar\Gamma_n/2$ for an intermediate level of width $\Gamma_n$. That matters only when a pole lies inside the spectrum. There the pair form fails, and phase or amplitude shaping can raise the transfer above the transform-limited value, as Dudovich et al. measured [20]. This OEP excludes that regime (section 4.3). **The omitted $E^-E^+$ terms** reach $f$ only if the field holds two components separated by $\Omega_f$. **Check 1, the monochromatic limit.** It gives $W=\frac{\pi}{32\hbar^2}E_0^4|M_{zz}|^2g(\Omega)$. Hence: $$ \sigma_0^{(2)}=\frac{\pi\omega^2|M_{zz}|^2}{8\varepsilon_0^2c^2} =2\pi^3\alpha^2a_0^4\omega_\mathrm{au}^2\delta, $$ with $\delta=|M_{zz}(m_f=0)|^2$ in atomic units. This is TC-952's per-level convention, reached independently. Here $M$ has SI units C m²/V, $\sigma_0^{(2)}$ has units m⁴, and $g$ has units s; the au strength is converted before an SI calculation. **Check 2, the continuum.** If $M$ does not depend on the split, (1) summed over an energy-normalised continuum gives OEP 6 §4's identity: - $P=\frac1{2\pi}\int K|S_2|^2d\Omega$; - with $S_2(\Omega)=\int e^{i\Omega t}(E^+)^2dt$. **Split variables.** Write $\omega_{a,b}=\Omega/2\pm\nu$ and $D_n=E_{ng}/\hbar-\Omega/2$. The kernel separates into a part even in $\nu$, from the symmetric product $S^n$ of the two dipoles, and a part odd in $\nu$, from their antisymmetric product $A^n$: $$ \hbar M^f_{\alpha\beta}=\sum_n\Big[S^n_{\alpha\beta}\frac{2D_n}{D_n^2-\nu^2}+A^n_{\alpha\beta}\frac{2\nu}{D_n^2-\nu^2}\Big]. \tag{3} $$ **Tensor structure from the $J_g=0$ ground state:** - **Rank 0** (the trace of $S$) reaches $J_f=0$. - **Rank 2** (the traceless part of $S$) reaches $J_f=2$. - **Rank 1** ($A$) reaches $J_f=1$. **For one field of fixed polarisation,** $\epsilon_a=\epsilon_b$, so the $A$ term vanishes at every split: no $J=1$ level is excited. - **Linear parallel:** $J=0$, and $J=2$ with $M=0$, at strength $\delta$ per level. - **Co-rotating circular:** $J=0$ is 0; $J=2$ is 3/2 of the linear value (TC-928). - **Unequal polarisations** (two beams, or a polarisation that varies across the spectrum) open $J=1$, at order $(\nu/D)^2$ in probability. - **Crossed linear beams** give no $J=0$. #### 4.2 When the pair-frequency form holds **When it is exact.** Within the two-photon rotating-wave approximation, Eq. (1) reduces to OEP 6's pair form, $c_f=\frac{i}{2\hbar}M^f(\Omega_f)S_2(\Omega_f)$, for arbitrary spectral amplitudes when: - one polarisation state holds across the spectrum; - and $M^f$ does not depend on $\nu$ over the pair spectrum's support. For a physical kernel with finite intermediate distances $D_n$, this generally requires an approximation; agreement for a particular pulse is not an equivalence for all fields. **The error.** Expand $M(\nu)=M(0)[1+\kappa\nu^2+O(\nu^4)]$: $$ \kappa=\frac{\sum_n w_n/D_n^2}{\sum_n w_n},\qquad w_n=S^n_{zz}/D_n . \tag{4} $$ For a Gaussian spectrum of rms width $\sigma$ (of $|\tilde E|^2$, in photon frequency) and group-delay dispersion $\phi''$: $$ \langle\nu^2\rangle=\frac{\sigma^2}{1-2i\phi''\sigma^2},\qquad \frac{P-P_0}{P_0}=2\,\mathrm{Re}(\kappa\langle\nu^2\rangle)+O(\sigma^4),\qquad \left|\frac{P-P_0}{P_0}\right|\lesssim2|\kappa|\sigma^2 . \tag{5} $$ - **If every $w_n$ has one sign,** $0<\kappa\le D_\mathrm{min}^{-2}$. - **If they cancel,** $\kappa$ is unbounded, and it can be negative: OEP 6's "cancellations" warning, made quantitative. A negative $\kappa$ means bandwidth lowers the transfer. Hydrogen's exact amplitudes show both signs: $\kappa=36.8$ hartree⁻² for 1s–2s and $-16.3$ hartree⁻² for 1s–6s (TC-1034). The rule below estimates $|\kappa|$. - **For real $\kappa$ and quadratic spectral phase,** chirp reduces the magnitude of this leading Gaussian correction. This is not a bound for arbitrary shaping, complex $\kappa$, cancellations of the pair amplitude, or higher-order terms. - **For a mixed pair** (pump plus ultraviolet), $\nu$ is half the colours' separation, (5) does not apply, and the two-variable $M$ is required. **Tested against curated data:** | level | evidence | $D_\mathrm{eff}=\lvert\kappa\rvert^{-1/2}$ against $D_1$ | |---|---|---| | He 1s2s | Drake et al.'s variational two-photon decay spectrum [2], against Hättig et al.'s small-split CCSD value [3] (1.00073 against 1.00075) | 0.75–0.86 (a second, opposite-signed effective pole) | | H 1s–2s, 3s, 6s | the exact hydrogen amplitude against the split (Quattropani et al., Drachman et al.; TC-1034) | 1s–2s: 0.88; 1s–3s: 2.4 (a cancelling path); 1s–6s: $\kappa<0$ | | He 1s2s, two colours | Kumar et al.'s two-colour coupling [1] at 19.06 + 1.54 eV: $\lvert M/M_\mathrm{deg}\rvert^2=13.83$, against Drake's 14.04–14.11 | a check of the full two-variable $M$, 1.5–2 % | | Ne 3p ¹S, ¹D | Hättig et al. at $\nu=0$ and at $\omega_2=10.49$ eV [3] | 0.89–0.99 | | Ar 4p ¹S, ¹D | Hättig et al. [3]; Sundholm et al. [4] | 0.93–1.01 | | Xe 6p, Kr 5p | single-path estimates from measured oscillator strengths, checked against Shekhtman et al.'s [5] and Gornik et al.'s [6] | at least 0.77, assuming the second path's sign | Here $D_1=E_1/\hbar-\Omega_f/2$ is an angular frequency, with $E_1$ the gas's lowest odd $J=1$ level; the table below gives $\hbar D_1$ in eV. For negative $\kappa$, define $D_\mathrm{eff}=|\kappa|^{-1/2}$. **The adopted empirical rule** is $D_\mathrm{eff}=0.75\,D_1$ (Q3, TC-984). The cases above support this estimate under their stated assumptions; they do not prove a universal bound on cancellations or the $\nu^4$ remainder. The estimated rms width for a leading fractional error $\varepsilon$ is $\sigma_\mathrm{max}=0.75\,D_1\sqrt{\varepsilon/2}$. | final levels | $\hbar D_1$ (eV) | estimated shortest transform-limited intensity FWHM for 1 % (fs) | |---|---|---| | He 1s2s | 10.9 | 1.3 | | Ne 3p | 7.2–7.4 | 2.0 | | Ar 4p | 4.9–5.1 | 2.9–3.0 | | Kr 5p | 3.9–4.3 | 3.4–3.7 | | Xe 6p; 7p and 6p′ | 3.5–3.6; 2.9–3.0 | 4.1–4.2; 4.9–5.1 | These estimates suggest a small split correction for smooth Gaussian pulses longer than about 5 fs. Qualification still requires the actual spectrum, intermediate-pole distances, cancellations and field-strength limit to be checked. A shaped pulse with $S_2(\Omega_f)$ near zero needs an absolute amplitude error, not a relative bound. **The continuum is different.** Its one curated nondegenerate value is McCann et al.'s Xe 118 + 355 nm [7]. It lies past the 5d[3/2]₁ pole, where the pair form fails by $2.7\times10^4$. It shows the size of mixed-pair errors and tests nothing at small splits. **Tier-3 evidence at small splits** comes from the maintainer's tRecX runs (TC-1048, TC-1049). With a one-colour reference at $\Omega/2$, $P_\mathrm{cross}/P_\mathrm{ref} = 4|M(\omega_1,\omega_2)/M(\Omega/2,\Omega/2)|^2$ directly. - **Helium**, two-electron haCC at $\Omega/2$ = 14, 17 and 20 eV: $\kappa>0$ everywhere, and $D_\mathrm{eff}$ = 1.1–2.2 $D_1$. The one-electron model gives 1.03–1.8 $D_1$. - **Argon**, one electron at 8.5 eV: $D_\mathrm{eff}\approx1.9\,D_1$. These runs support the 0.75 $D_1$ estimate at the sampled continuum energies. They do not qualify the continuum's transient response. A solprop SEP must derive and validate that response; missing published amplitudes do not preclude an explicit approximate model, but its additional material inputs need separate qualification (Q6, TC-1057). #### 4.3 The effective-level dynamics **Adiabatic elimination.** Remove the odd intermediate states. This needs two conditions: - **spectral:** no spectral component lies within a few $\sigma$ of an odd resonance, and $|\nu|\ll D$ for the local pair approximation (section 4.2); - **field strength:** $|d_{ng}E_0|/(2\hbar D_n)\ll1$ for every intermediate $n$. Its error is the admixture of the intermediate states, $2(d E_0/2\hbar D)^2$ in probability in the cited three-level example, not a general multilevel bound. The excited-to-intermediate couplings must also be small compared with their detunings; a test of $d_{ng}$ alone is insufficient. In Kaufman et al.'s three-level case [21] that error is 3.1 % of the transfer, against 0.2 % from the spectral estimate (5). It grows with peak intensity at fixed bandwidth (TC-990). The basis is then the ground level $g$ and the even levels $f$ inside the pair spectrum: $$ H_{gg}=-\alpha_g^R|E^+|^2,\quad H_{ff}=\hbar\Omega_f-\alpha_f^R|E^+|^2,\quad H_{fg}=-\tfrac12\mathcal Q_f(t),\quad \mathcal Q_f=M_fE^+(t)^2 . \tag{6} $$ - **The Hamiltonian in (6) is Hermitian**, denoted $H_R$ below, with $H_{gf}=H_{fg}^*$ and $\alpha_j^R=\mathrm{Re}\,\alpha_j(\omega_c)$ at a specified carrier. $M_f$ and these real shifts are constant coefficients in this local model. Loss enters the density equation separately, once. A frequency-dependent kernel or polarizability is nonlocal in time: substituting a convolution into (6) does not establish (8) or the energy identity without a corresponding functional derivation (deferred by Q8). - **Ionisation of the excited level:** $\mathrm{Im}\,\alpha_f=\varepsilon_0c\,\sigma^{PI}_f/\omega$, so its rate is $\Gamma^{PI}_f=2\alpha_f^I|E^+|^2/\hbar=\sigma^{PI}_f\Phi$, where $\alpha_f^I=\mathrm{Im}\,\alpha_f$. The ground-state one-photon loss is excluded here. Putting this imaginary part in $H_R$ as well would count loss twice. - **Kumar et al.'s Eq. 7 [1] is (6),** with $\Omega_{ge}=\sqrt{2\sigma_0/\pi}\,\Phi$. Kaufman et al. [21] and Tóth et al. have the same form (TC-990, TC-991). - **A naming trap:** Kaufman et al. and Tóth et al. call their "two-photon Rabi frequency" is the off-diagonal element itself, $|H_{fg}|/\hbar$. That is half of this OEP's $\Omega_R$ and of Kumar et al.'s $\Omega_{ge}$. **The population and coherence equations** for independent loss channels, with $\rho_{ab}=\langle c_a c_b^*\rangle$ and no direct effective coupling between excited levels: $$ \begin{aligned} \dot\rho_{ff}&=\hbar^{-1}\mathrm{Im}[\mathcal Q_f^*\rho_{fg}]-(\Gamma^{PI}_f+\Gamma^{rad}_f)\rho_{ff},\qquad \dot\rho_{gg}=-\textstyle\sum_f\hbar^{-1}\mathrm{Im}[\mathcal Q_f^*\rho_{fg}],\\ \dot\rho_{fg}&=-i\big[\Omega_f-(\alpha_f^R-\alpha_g^R)|E^+|^2/\hbar\big]\rho_{fg} +\tfrac{i}{2\hbar}\mathcal Q_f(\rho_{gg}-\rho_{ff}) -\Gamma^{coh}_f\rho_{fg}-\tfrac{i}{2\hbar}\textstyle\sum_{f'\ne f}\mathcal Q_{f'}\rho_{ff'} . \end{aligned} \tag{7} $$ - **Coherence decay:** $\Gamma^{coh}_f=\tfrac12(\Gamma^{PI}_f+\Gamma^{rad}_f)+\gamma^\phi_f$. Half of the population decay goes to the coherence; pure (collisional) dephasing $\gamma^\phi_f$ is separate (OEP 6, Q7). For $f\ne f'$, the missing equation is fixed by the same commutator: $$ \dot\rho_{ff'}=-\frac{i}{\hbar}(H_{ff}-H_{f'f'})\rho_{ff'} +\frac{i}{2\hbar}\left(\mathcal Q_f\rho_{gf'}-\rho_{fg}\mathcal Q_{f'}^*\right) -\left[\frac{\Gamma_f+\Gamma_{f'}}2+\gamma_f^\phi+\gamma_{f'}^\phi\right]\rho_{ff'}, \qquad \Gamma_f=\Gamma_f^{PI}+\Gamma_f^{rad}. $$ This dephasing convention is the Lindblad choice $L_f=\sqrt{2\gamma_f^\phi}|f\rangle\langle f|$, with independent level noise. Correlated collision shifts or transfer between retained levels require a specified relaxation operator; single-line widths alone do not determine it. The sign of the last term of (7) follows from $-i[H_R,\rho]/\hbar$ (TC-1054). - **The coherences between excited levels** $\rho_{ff'}$ are needed when two levels lie inside one pair spectrum. The Xe 6p levels are 0.03–0.25 eV apart. - **The radiative decay of np goes to ns,** not to $g$, and is tracked separately. He 1s2s instead decays to $g$ by two-photon emission: for times comparable with its lifetime, add $+\Gamma^{rad}_f\rho_{ff}$ to $\dot\rho_{gg}$. The short-pulse fixture may neglect this decay only with its timescale stated. - **The star-shaped Hamiltonian is an approximation.** Elimination can also produce Raman couplings among the excited levels and common-continuum loss terms. Neglecting them needs a separate check when several levels are populated (Q9). - **The direct continuum** has an integrated pair-spectral yield in OEP 6, not a qualified time-local depletion rate. If a propagation model independently supplies a justified ground-state loss rate $\Gamma_g$, it must add $-\Gamma_g\rho_{gg}$ and $-\Gamma_g\rho_{fg}/2$ consistently. OEP 8 does not supply that missing continuum closure (Q6). **The polarisation** in the same local, narrowband rotating-wave model: $$ P_c^+=N\Big[\sum_fM_f^*E^-\rho_{fg}+\alpha_g^R\rho_{gg}E^++\sum_f\alpha_f^R\rho_{ff}E^+\Big] =-N\,\partial\langle H_R\rangle/\partial E^- ,\qquad P_{PI}^+=iN\sum_f\alpha_f^I\rho_{ff}E^+,\quad P=2\,\mathrm{Re}(P_c^++P_{PI}^+) . \tag{8} $$ **Why (8) follows from the amplitudes,** and is not chosen: 1. The coupling in (6) is fixed by (1), and reproduces its constant-kernel limit in the weak field. 2. The two-photon term of (8) equals the direct third-order dipole: both $\langle\psi^{(1)}|d|\psi^{(2)}\rangle$ and $\langle\psi^{(0)}|d|\psi^{(3)}\rangle$ contribute $\tfrac12M^*E^-\rho_{fg}$ at positive frequency. 3. The conservative part, as $-\partial H_R/\partial E^-$ at fixed $\rho$, obeys the energy identity below. The absorptive term has its own field-work term. Products labelled $P^+$ must remain in the positive-frequency band under these approximations. An arbitrary broadband product need not do so; a positive-frequency projection and a derivation of its work balance belong to Q8's deferred extension. A spectral factor with a chosen phase, the candidate OEP 6 §4 rejected, has none of these properties. #### 4.4 The energy theorem and the absence of double counting **The local-model identity.** Let $\dot\rho=-i[H_R,\rho]/\hbar+\mathcal D\rho$, where $\mathcal D$ contains every retained loss, return and dephasing term, and let $U=N\operatorname{Tr}(\rho H_R)$. For a pulse whose field and interaction energy vanish at the endpoints, integration by parts and (8) give $$ W_c=\int E\dot P_c\,dt =\Delta U-N\int\operatorname{Tr}(H_R\mathcal D\rho)\,dt . $$ The unitary commutator contributes zero to $\dot U$; the remaining explicit field derivative is $-P_c\dot E$. This proves the identity for the stated local model, including multilevel coherences and dephasing. It does not prove the accuracy of the elimination or of a broadband extension. For one excited level with an outgoing population sink, no return and no pure dephasing, initially in $g$, write $\Gamma=\Gamma^{PI}+\Gamma^{rad}$. The total work is $$ W=N\left\{\hbar\Omega_f\rho_{ff}(t_{\rm end}) +\int\Gamma\left[\rho_{ff}(\hbar\Omega_f-\alpha_f^R|E^+|^2) +\operatorname{Re}(\rho_{fg}H_{gf})\right]dt\right\}+W_{PI}, \qquad W_{PI}=-2\operatorname{Re}\int P_{PI}^+\dot E^-dt . $$ For constant $\alpha_f^I$ this last term is $W_{PI}=N\int\Gamma^{PI}\rho_{ff}\hbar\omega_{\rm inst}(t)dt$, where $\omega_{\rm inst}=-\operatorname{Im}(\dot E^+/E^+)$ wherever $E^+\ne0$; use the field-work form at a zero. For a narrowband pulse this is the absorbed third-photon energy. It is not a general broadband photon-energy prescription. With pure dephasing, $W_c$ also contains $2N\int\gamma_f^\phi\operatorname{Re}(\rho_{fg}H_{gf})dt$. A multilevel or returning-decay model must use the trace identity, not the one-level sink formula (TC-1054). The outgoing sink term includes the dressed energy and coupling energy lost from the retained states. For radiative decay to an excited lower state, it includes both the emitted photon and the energy retained by that lower state; it is not all fluorescence. A plasma model must separately track the ionisation threshold, electron kinetic energy and these sink populations. **Reported during drafting** with Kumar et al.'s helium coefficients (10 TW cm⁻², 203 fs), using the one-level, zero-dephasing local model: - relative residual $3.8\times10^{-11}$; - counting ionisation at the bare level instead gives $2.2\times10^{-4}$. Phase 1 must regenerate this result and demonstrate convergence in integration and quadrature tolerances. A small residual checks consistency of the chosen model, not its physical accuracy. **No double counting:** - **The linear response:** (8) includes the ground-state linear polarisation. If propagation already includes $N\alpha_g^R E^+$ in its linear index, add only the correction $P_c^+-N\alpha_g^R E^+$ and the absorptive part. Excited, ionic and radiative-sink populations need their own response accounting; they cannot also remain neutral ground-state atoms in the linear model. - **The continuum and bound terms** have disjoint final states: pairs above $I_p$ against pairs near $\Omega_f0$. Do not infer signs across a zero, phases across a pole or off-shell values. The endpoints require a demonstrated limit, not division by $y^3(1-y)^3$. This kernel qualifies perturbative yields; a time-dependent two-colour Hamiltonian and its polarisation are deferred by Q8. In particular, Kumar's strong two-colour scheme also drives 1s2s–1s4s and is not a two-level test. Drake's spectrum and the helium decay rates currently occur only in source-record notes. Phase 1 must add extraction scripts, datasets and register entries before they become shipped coefficients. Q10 now specifies their quantities and normalisation. Use Drake Table II for the normalised shape and Bondy Table VI's finite-mass, relativistically corrected helium rate, 50.93 ± 0.07 s⁻¹, for decay. Preserve the estimated theoretical uncertainty without assigning it a statistical confidence level. This combines sources with different approximations, explicitly; the relative kernel shape is not promoted to Bondy's rate accuracy. Drake's half-spectrum quadrature gives 51.1457 s⁻¹ against its printed 51.3 s⁻¹; keep this check, without silently rescaling raw data. Bondy's extracted Table VI is scrambled and must be repaired and checked against its table image before ingestion (TC-1055). **Cost:** milliseconds per record. No new dependencies. ### 8. Verification and validation These are the drafting calculations reported as Phase 1 targets. Their original residuals are not acceptance tolerances: integration, quadrature, interpolation and finite-field errors must be separated by convergence studies. A comparison using a paper's own input coefficients reproduces that model; it is not an independent test of those coefficients. | check | quantity | result for this draft | tolerance set by | |---|---|---|---| | per-level convention | $\sigma_0$ from (1) against TC-952's constant | $10^{-11}$ | rounding | | weak-field limit | (7)–(8) yield against OEP 6's pair-spectral $P$ (He, 0.01 TW cm⁻², 50 fs) | 0.99973 | convergence in step and intensity separately; Stark shifts and ionisation also vanish only in the weak-field limit | | Kumar et al.'s one-colour case | $P_e$ after the pulse (10 TW cm⁻², 9 meV, $\Delta=+14.3$ meV) | 0.276; text "about 28 %" | the paper's printed precision | | local energy identity | relative residual, one level without pure dephasing | $3.8\times10^{-11}$ | time integration and work quadrature convergence | | coupling | Kumar et al. against Kaprálová-Žďánská [18] | 1.9 % | their spread | | two-colour kernel | Drake et al. against Kumar et al.'s Table II | 1.5–2 % | their spread | | split error | quadrature against (5), He, 10 fs and 1 fs | $1.353\times10^{-4}$ both; 1.38 % against 1.35 % | the $\nu^4$ term | | field-strength error | Kaufman et al.'s three-level model against its eliminated form [21] | $P_e$ 0.843 against 0.869 (3.1 %; 2.5 % printed) | the admixture $2(dE_0/2\hbar D)^2$ | | Xe strengths | Henriksson et al. against Kröll and Bischel's and Drag et al.'s measurements | 0.83–1.30 | the measurements' errors | | Xe 6p single path | against Gornik et al.'s estimates | $1.20\times10^4$ against $1.2\times10^4$ s⁻¹ | their two digits | | real part | (9) against Lehmberg et al.'s $n_2$ | fails by 2.2–2.8 | recorded as a failure (Q1) | **Review checks (2026-10-12; TC-1054):** the multilevel terms were compared directly with the matrix commutator; Kumar's Table I and Lehmberg's Table 1 were checked against the primary table images. Independent integration of Kumar's printed Table I coefficients gives $P_e=0.27612491$ for a 202.77234 fs Gaussian with 9 meV spectral FWHM and $\hbar(2\omega-\Omega_f)=+14.3$ meV; TC-1054 records the solver and convergence settings. Phase 1 must additionally test Hermiticity, positivity, trace plus sink populations, two excited levels, nonzero pure dephasing, radiative return, invalid inputs and spectral masks. Preserve the paper's coefficients for its reproduction fixture, and separately report results with the preferred curated inputs. **Independent validators:** - tier-3 TDSE runs of He 1s2s dynamics at 1–10 fs and with chirp (TC-1048, TC-1051). - **At 10¹² W cm⁻², at haCC's propagated resonance:** the 1s2s populations equal the weak-field form to ≤ 1 % for every duration and chirp (3 % at 1 fs). - **The fitted coupling:** |M|² = 12.31 au², 0.04 % from Hättig et al.'s δ [3]. - **Beyond lowest order (10¹³, 10¹⁴ W cm⁻²):** the departures, down to 0.08 of lowest order at 10 fs, 10¹⁴, are Phase 1's Bloch-model targets; - solprop parity on the stored fixtures. ### 9. Documentation **The manual page** {doc}`../two_photon` gains a section *Bound two-photon resonances in propagation*. It covers: - the material conventions established by equations (6)–(9), with single-atom checks distinguished from the propagation equations owned by solprop; - the material data optmat supplies and the response solprop constructs from them; - material energy conventions and checks, with propagation energy bookkeeping in solprop; - the validity limits; - the failed real-part test; - the refusals; - the RDW diagnostic of section 2.1, separating continuum loss from bound excitation, and approved scope decisions from deferred scientific qualification. **Also:** - {doc}`../propagation` gains a pointer; - docstrings carry units; - a gallery example: He 1s2s transfer against duration for a 120 nm pulse, with the weak-field identity overlaid. ### 10. Compatibility and impact - **Additive only.** No recommended value or default changes. - **OEP 5's $\gamma$ is unchanged.** The subtraction is the code's, and is not applied in optmat. - **OEP 6:** its Phase 2b and optional Phase 3 have passed to this proposal. OEP 6 is recorded as Final with Phases 1 and 2a implemented (TC-1053). - **solprop:** new parity fixtures. ### 11. Alternatives considered - **A spectral factor $H(\Omega)$ multiplying the pair spectrum, with a chosen phase.** Rejected by OEP 6's review, and here because it has no energy theorem and is not the amplitude's consequence (section 4.3). - **Linear complex $\chi_2$ (option C).** It is correct only below saturation and with field-independent widths. Kumar et al.'s case violates both. - **Rate equations with a Lorentzian line.** They fail for the coherent femtosecond example (section 4.5). They remain valid in the fast-dephasing or suitably chaotic limits stated there. - **Using all five gases with labelled best-effort strengths.** Rejected for Kr, whose measurements and calculations disagree by up to 94 times. Accepted for Ar (Q7), whose convention is settled and whose calculations agree to about 25 %. Ne could follow argon's route, from calculations alone, but is not proposed here. ### 12. Phases 1. **Coefficients, helium dynamics and the weak-field handoff.** Deliverables: - `two_photon_bound_response` and its records for He, Xe and Ar, with separate coefficient, perturbative and nonlinear-dynamical qualifications (Q8–Q9); - the Q10 quantities and energy-fraction abscissa, documented in the vocabulary and data manual before ingestion; extraction of Drake's spectrum and Bondy's corrected helium decay rate, with the Table VI extraction repaired; - the radiative widths: measured lifetimes where curated (Bruce et al. for Xe), otherwise NIST's sum of A (TC-983), and Bondy's He 1s2s rate; - the He 1s2s on-shell kernel, restricted to perturbative yields; - He 1s2s's light shifts from Kumar et al.'s Table I, including the separate bound and continuum contributions and TC-989's limitation, moved here from Phase 2; - argon's 4p strengths: hattig_1998b's ¹De rows taken as data (`one_component`, ×4/3), and argon's assessment changed to "calculated", by the author's decision (TC-1053); - `gamma_resonant` for the supported perturbative response; no qualified coupled strong-field Xe or Ar model and no arbitrary broadband kernel; - the internal single-atom reference integrator in `tools/`, the local energy-identity tests of material conventions and stored parity fixtures; both the exact Kumar input reproduction and the preferred curated-input comparison, with their distinct purposes; - the tests of section 8, including decay normalisation, kernel symmetry, domain masks and convergence; the manual section and gallery example; - a documented handoff for section 2.1's continuum-first RDW investigation, using OEP 6's existing coefficients and integrated constraints. This is guidance for solprop; its continuum diagnostic does not wait for completion of this phase. A continuum-response SEP owns the polarisation, plasma coupling and propagation energy accounting; a coupled optmat OEP would cover any additional material inputs it requires, rather than adding a continuum phase here (TC-1057); - register entries and a changelog line. Extraction and public-surface changes require the library and release lanes in addition to the full gate. 2. **Further light shifts and collision data.** Deliverables: - Xe 6p's pressure coefficients from Gornik et al.; - additional excited-level shifts, quenching and dephasing where primary sources qualify them; the light-shift sensitivity stated per level. These additions do not by themselves qualify strong-field multilevel dynamics. 3. **More levels and improved coefficients.** Ne 3p and Kr 5p as their strengths are resolved, each with its own assessment. Argon's measured strengths, photoionisation, light shifts and pressure broadening as curated. Strong-field multilevel qualification requires the deferred Q9 evidence and an explicit scope revision. 4. **(Optional) The background Kerr term.** A windowed subtraction from OEP 5's $\gamma$, if Q1 and Q2 are settled. General frequency-dependent/two-colour dynamics (Q8), strong-field multilevel qualification (Q9), and the continuum's time dependence (Q6) are not silently added to these phases. They remain on the roadmap. Approval of Q8–Q10 resolves the initial scope and data choices, not those scientific extensions or acceptance of the whole OEP. ### 13. Open questions Q1 to Q6 resolved by the maintainer on 2026-10-09, as recommended (TC-984); Q7 on 2026-10-12 (TC-1053). - **Q1 (the real part; resolved).** `gamma_resonant` publishes the dispersive part, labelled as failing Lehmberg et al.'s xenon $n_2$ by 2.2–2.8. The qualification of the absorptive and dispersive parts is stated separately. The earlier xenon $n_2$ literature is to be requested. - **Q2 (the background; resolved).** Deferred to Phase 4. Until then optmat offers no subtracted $\gamma$; with fixed residues the tail is 63 % of $\gamma_\mathrm{fit}$ at 1064 nm, with falling residues 15 %. - **Q3 (the split rule; resolved).** $D_\mathrm{eff}=0.75\,D_1$ is adopted, labelled as a bound from the curated cases. The review clarifies that this is an empirical leading-order estimate, not a mathematical guarantee for arbitrary spectra or cancelling amplitudes (TC-1054). The Kr and Xe single-path cases rest on an assumed sign of the second path, and the label says so. - **Q4 (the integrator; resolved).** In `tools/` only, with stored fixtures for parity; no public integrator. - **Q5 (photon statistics; resolved).** The residues are for coherent light. Chaotic or partially coherent light is the code's responsibility; optmat states this and supplies nothing more. - **Q6 (the continuum's time dependence; resolved).** Not a phase of this OEP. The response belongs in a solprop SEP; a coupled optmat OEP would cover any new material quantities, models or interfaces it requires. Published small-split amplitudes would provide additional material information, but are not a prerequisite to proposing an explicitly approximate response. Such a response must be derived and validated; OEP 6's integrated constraints alone do not qualify it (TC-1057). - **Q7 (argon; resolved 2026-10-12).** Should argon be qualified without a decisive measurement? **Decided by the author, as recommended:** - argon's 4p J = 0 and J = 2 levels are qualified on henriksson_2005's J-resolved TDDHF strengths; - the uncertainty is −5 % and +30 %, from the correlated calculations, labelled "calculated only"; - for absorption and populations only. See section 5 and TC-1053. Q8–Q10 were resolved by the maintainer on 2026-10-12, approving the AI review's recommendations (TC-1055). The experimental priority is recorded in TC-1056 and section 2.1. The wider scientific work remains deferred, not declared validated. - **Q8 (broadband dynamics; resolved).** The first dynamical response is the local effective model, within its spectral and field-strength limits. Publish helium's two-colour kernel for perturbative on-shell yields only. A general frequency-dependent Hamiltonian, polarisation and work balance are deferred to a later extension. This scope still permits short coherent pulses; width relative to an intermediate-state detuning is the relevant elimination condition, not width relative to the final state's natural linewidth. An intense overlapping pump needs a separate dressing check; a closed pump–VUV two-photon channel does not establish negligible dressing. - **Q9 (several excited levels; resolved).** Qualify He 1s2s as the first nonlinear dynamical example, with its light shifts in Phase 1. Retain the approved Xe and Ar coefficients and their weak-field multilevel applications. Defer qualified coupled strong-field populations and ion yields until effective excited-state couplings, shared continuum loss and correlated dephasing are supplied or shown negligible. A maintainer decision about scope cannot establish that scientific condition. - **Q10 (helium source data; resolved).** Add the two decay quantities and the dimensionless energy-fraction abscissa specified in section 7. Use Drake's Table II for the spectrum and Bondy's Table VI for the preferred helium decay rate. Preserve printed values, counting conventions and uncertainty meanings. Repair the scrambled Bondy table extraction before ingestion; derive lifetimes and kernel ratios rather than treating them as printed data. This is Phase 1 curation, not new atomic physics. ### 14. References Curated as source records unless marked (the record's id in brackets). 1. S. Kumar, J. D. Piel, C. H. Greene and N. Shivaram, "Broadband femtosecond lasers enable efficient two-photon excitation of the ultranarrow linewidth singlet 1s2s state in helium," *Phys. Rev. Research* **8**, 013009 (2026). doi:10.1103/nfbf-h6hc [kumar_2026] 2. G. W. F. Drake, G. A. Victor and A. Dalgarno, "Two-photon decay of the singlet and triplet metastable states of helium-like ions," *Phys. Rev.* **180**, 25 (1969). doi:10.1103/PhysRev.180.25 [drake_1969] 3. C. Hättig, O. Christiansen and P. Jørgensen, "Coupled cluster response calculations of two-photon transition probability rate constants for helium, neon and argon," *J. Chem. Phys.* **108**, 8355 (1998). doi:10.1063/1.476262 [hattig_1998b] 4. D. Sundholm, A. Rizzo and P. Jørgensen, "Multiconfiguration self-consistent field quadratic response calculations of the two-photon transition probability rate constants for argon," *J. Chem. Phys.* **101**, 4931 (1994). doi:10.1063/1.467415 [sundholm_1994] 5. D. Shekhtman, M. A. Mustafa and N. J. Parziale, "Two-photon cross-section calculations for krypton in the 190–220 nm range," *Appl. Opt.* **59**, 10826 (2020). doi:10.1364/AO.410806 [shekhtman_2020] 6. W. Gornik, S. Kindt, E. Matthias and D. Schmidt, "Two-photon excitation of xenon atoms and dimers in the energy region of the 5p⁵6p configuration," *J. Chem. Phys.* **75**, 68 (1981). doi:10.1063/1.441856 [gornik_1981] 7. M. P. McCann, C. H. Chen and M. G. Payne, "Two-photon (vacuum ultraviolet + visible) spectroscopy of argon, krypton, xenon, and molecular hydrogen," *J. Chem. Phys.* **89**, 5429 (1988). doi:10.1063/1.455595 [mccann_1988] 8. J. Henriksson, P. Norman and H. J. Aa. Jensen, "Two-photon absorption in the relativistic four-component Hartree-Fock approximation," *J. Chem. Phys.* **122**, 114106 (2005). doi:10.1063/1.1869469 [henriksson_2005] 9. R. H. Lehmberg, C. J. Pawley, A. V. Deniz, M. Klapisch and Y. Leng, "Two-photon resonantly-enhanced negative nonlinear refractive index in xenon at 248 nm," *Opt. Commun.* **121**, 78 (1995). doi:10.1016/0030-4018(95)00433-9 [lehmberg_1995] 10. S. Kröll and W. K. Bischel, "Two-photon absorption and photoionization cross-section measurements in the 5p⁵6p configuration of xenon," *Phys. Rev. A* **41**, 1340 (1990). doi:10.1103/PhysRevA.41.1340 [kroll_1990] 11. C. Drag, F. Marmuse and C. Blondel, "Measurement of the two-photon excitation cross-section of the 6p′[3/2]₂ and 6p′[1/2]₀ levels of Xe I at the wavelengths 224.3 and 222.6 nm," *Plasma Sources Sci. Technol.* **30**, 075026 (2021). doi:10.1088/1361-6595/abfbeb [drag_2021] 12. M. S. Pindzola, M. G. Payne and W. R. Garrett, "Two-photon excitation of xenon," *Phys. Rev. A* **24**, 3115 (1981). doi:10.1103/PhysRevA.24.3115 [pindzola_1981b] 13. A. Kramida, Yu. Ralchenko, J. Reader and NIST ASD Team, NIST Atomic Spectra Database (version 5.12), https://physics.nist.gov/asd. doi:10.18434/T4W30F [nist_asd_2026] 14. A. T. Bondy, D. C. Morton and G. W. F. Drake, "Two-photon decay rates in heliumlike ions: Finite-nuclear-mass effects," *Phys. Rev. A* **102**, 052807 (2020). doi:10.1103/PhysRevA.102.052807 [bondy_2020] 15. M. S. Pindzola, "Photoionization of excited-state xenon," *Phys. Rev. A* **23**, 201 (1981). doi:10.1103/PhysRevA.23.201 [pindzola_1981] 16. M. G. Payne, C. H. Chen, G. S. Hurst, S. D. Kramer, W. R. Garrett and M. Pindzola, "Resonance ionization spectroscopy schemes for Ar, Kr and Xe," *Chem. Phys. Lett.* **79**, 142 (1981). doi:10.1016/0009-2614(81)85306-7 [payne_1981] 17. D. G. Hummer and P. J. Storey, "Recombination of helium-like ions — I. Photoionization cross-sections and total recombination and cooling coefficients for atomic helium," *Mon. Not. R. Astron. Soc.* **297**, 1073 (1998). doi:10.1046/j.1365-8711.1998.2970041073.x [hummer_1998] 18. P. R. Kaprálová-Žďánská, "Coherent superposition mediating energy flow between fundamental and third-harmonic modes," *Phys. Rev. A* **111**, 043105 (2025). doi:10.1103/PhysRevA.111.043105 [kapralova_zdanska_2025] 19. M. A. Dakka et al., "Laser-based metastable krypton generation," *Phys. Rev. Lett.* **121**, 093201 (2018). doi:10.1103/PhysRevLett.121.093201 [dakka_2018] 20. N. Dudovich, B. Dayan, S. M. Gallagher Faeder and Y. Silberberg, "Transform-limited pulses are not optimal for resonant multiphoton transitions," *Phys. Rev. Lett.* **86**, 47 (2001). doi:10.1103/PhysRevLett.86.47 [dudovich_2001] 21. B. Kaufman, T. Rozgonyi, P. Marquetand and T. Weinacht, "Adiabatic elimination in strong-field light-matter coupling," *Phys. Rev. A* **102**, 063117 (2020). doi:10.1103/PhysRevA.102.063117 [kaufman_2020] 22. P. L. Knight, M. A. Lauder and B. J. Dalton, "Laser-induced continuum structure," *Phys. Rep.* **190**, 1 (1990). doi:10.1016/0370-1573(90)90089-K [knight_1990] 23. M. R. Bruce, W. B. Layne, C. A. Whitehead and J. W. Keto, "Radiative lifetimes and collisional deactivation of two-photon excited xenon in argon and xenon," *J. Chem. Phys.* **92**, 2917 (1990). doi:10.1063/1.457939 [bruce_1990] 24. W. L. Wiese, J. W. Brault, K. Danzmann, V. Helbig and M. Kock, "Unified set of atomic transition probabilities for neutral argon," *Phys. Rev. A* **39**, 2461 (1989). doi:10.1103/PhysRevA.39.2461 [wiese_1989] 25. K. Dzierżęga, U. Volz, G. Nave and U. Griesmann, "Accurate transition rates for the 5p–5s transitions in Kr I," *Phys. Rev. A* **62**, 022505 (2000). doi:10.1103/PhysRevA.62.022505 [dzierzega_2000] 26. F. Brandi, W. Hogervorst and W. Ubachs, "High-resolution vacuum-ultraviolet and ultraviolet photoionization spectroscopy of krypton," *J. Phys. B* **35**, 1071 (2002). doi:10.1088/0953-4075/35/4/329 [brandi_2002] ## Part II — Implementation record None yet. ## Part III — History - 2026-10-09: Drafted, from the derivation that answers OEP 6's Q9 for the bound levels. - 2026-10-09: Q1 to Q6 resolved by the maintainer, as recommended (TC-984). - 2026-10-09: Revised in Draft with the literature of TC-985 to TC-1010: the field-strength condition, the pole prescription, the xenon strengths divided in two, the measured lifetimes and the light-shift sources. - 2026-10-09: The radiative, light-shift and quenching quantities approved (TC-1011); sections 5, 7 and 12 updated to use them. - 2026-10-09: Revised with the literature of TC-1021 to TC-1041: hydrogen's exact kernel and the sign of $\kappa$, no third xenon strength, the xenon lifetimes against NIST, the pressure-line and amplitude quantities. - 2026-10-09: The xenon 6p′ lifetimes settled by Inoue et al. (TC-1042). - 2026-10-11: The two-photon tRecX runs added as tier-3 evidence (TC-1048, TC-1049). - 2026-10-11: Run 5's helium 1s2s discrepancy resolved; the weak-field form and the 1s2s strength confirmed (TC-1051). - 2026-10-12: Argon's 4p levels qualified on calculated strengths, by the author (Q7, TC-1053). - 2026-10-12: John Travers confirmed his human review. GPT-6 completed the AI review and revised the coherence sign, local energy identity, bandwidth-error qualification, coefficient availability and validation contract (TC-1054). Review status is Reviewed; lifecycle status remains Draft. Q8–Q10 record the remaining design gaps. The review did not rerun the TDSE calculations or reproduce every drafting numerical claim. - 2026-10-12: The maintainer approved the recommended resolutions of Q8–Q10 (TC-1055). The local dynamical scope and perturbative kernel restriction are now explicit; helium shifts and the approved decay-data schema are in Phase 1. Broader dynamics remain deferred. The initial Ar/He RDW experiment and a continuum-first diagnostic were added to section 2.1 (TC-1056). Status remains Draft; Reviewed. - 2026-10-12: The maintainer clarified the package boundary (TC-1057): optmat supplies material properties and basic data; solprop owns the propagation polarisation, electron accounting and energy balance. A continuum-response SEP may require a coupled material-data OEP. Section 2.1 distinguishes pump detuning from the ultimate observed cut-off and records the effective-continuum suggestion as an unqualified candidate. No new phase, OEP or SEP is created; status remains Draft; Reviewed. - 2026-10-12: Argon's radiative widths, quenching and photoionisation at the drive curated (TC-1068). - 2026-10-12: Argon's quenching reconciled and its photoionisation scale questioned (TC-1074).