{
  "schema_version": 1,
  "artifact_id": "qft.artifact.curved-spacetime.quantum-fields-cosmology.adiabatic-pulse-spectrum",
  "title": "Adiabatic pulse spectrum: numerical, exact, and Born comparison",
  "generated_by": "figures-src/curved-spacetime/adiabatic-pulse-spectrum.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-23",
  "reader_question": "How accurately does first-Born particle production describe a smooth localized frequency pulse, and which numerical checks distinguish physical mixing from integration error?",
  "takeaway": "The exact and numerical occupations fall exponentially with q; the weak-pulse Born curve has the correct scale and shape but a measurable finite-g truncation error.",
  "conventions": {
    "dimensionless_time": "x=eta/tau",
    "dimensionless_frequency": "q=omega*tau",
    "dimensionless_pulse_strength": "g=deltaV*tau^2",
    "mode_equation": "u_xx+[q^2+g*sech^2(x)]u=0",
    "positive_frequency_mode": "w=exp(-i*q*x)/sqrt(2*q)",
    "decomposition": "u=alpha*w+beta*conjugate(w)",
    "extraction": {
      "alpha": "conjugate(w)*(q*u+i*u_x)",
      "beta": "w*(q*u-i*u_x)"
    },
    "wronskian": "u*conjugate(u_x)-u_x*conjugate(u)=i"
  },
  "analytic_results": {
    "lambda": 0.019615242270663247,
    "lambda_definition": "lambda*(lambda+1)=g",
    "exact_occupation": "sin(pi*lambda)^2/sinh(pi*q)^2",
    "first_born_beta": "+i*pi*g/sinh(pi*q) for the stated centered-pulse phase convention",
    "first_born_occupation": "pi^2*g^2/sinh(pi*q)^2",
    "born_over_exact_ratio": 1.0409321895642316
  },
  "scientific_references": [
    {
      "authors": "G. Pöschl and E. Teller",
      "title": "Bemerkungen zur Quantenmechanik des anharmonischen Oszillators",
      "journal": "Zeitschrift für Physik",
      "volume": "83",
      "pages": "143–151",
      "year": 1933,
      "doi": "10.1007/BF01331132",
      "url": "https://doi.org/10.1007/BF01331132",
      "use": "Historical source for the exactly solvable potential family; the scattering normalization and occupation formula used here are independently checked by the numerical fixture."
    }
  ],
  "fixture": {
    "g": 0.02,
    "qMin": 0.5,
    "qMax": 3,
    "qCount": 101,
    "halfDomain": 12,
    "comparisonHalfDomain": 10,
    "fineStep": 0.0025,
    "coarseStep": 0.005
  },
  "numerical_method": {
    "integrator": "dependency-free fixed-step classical fourth-order Runge-Kutta",
    "state_vector": "[Re(u), Im(u), Re(u_x), Im(u_x)]",
    "initial_data": "u(-L)=exp(+i*q*L)/sqrt(2*q), u_x(-L)=-i*q*u(-L)",
    "fine_run": {
      "half_domain": 12,
      "requested_step_size": 0.0025
    },
    "step_comparison": {
      "half_domain": 12,
      "requested_step_size": 0.005
    },
    "domain_comparison": {
      "half_domain": 10,
      "requested_step_size": 0.0025
    }
  },
  "thresholds": {
    "exactRelativeError": 0.000001,
    "wronskianResidual": 1e-9,
    "unitarityResidual": 1e-9,
    "stepConvergenceRelative": 0.000001,
    "domainConvergenceRelative": 0.000002,
    "zeroPulseBetaMagnitude": 1e-8,
    "reflectionlessBetaMagnitude": 1e-7,
    "centeredPositivePulseRealBetaMagnitude": 1e-8
  },
  "diagnostics": {
    "maximum_numerical_exact_relative_error": 2.7852174713746304e-8,
    "maximum_wronskian_residual": 2.3756219214021712e-11,
    "maximum_unitarity_residual": 2.3755553080206937e-11,
    "maximum_step_convergence_relative": 2.075196361129106e-9,
    "maximum_domain_convergence_relative": 0.000001428613067949379,
    "centered_positive_pulse_maximum_beta_real_magnitude": 1.1136057354033113e-14,
    "centered_positive_pulse_minimum_beta_imaginary": 0.000009939615614823779
  },
  "checks": {
    "exact_agreement": true,
    "wronskian": true,
    "bogoliubov_unitarity": true,
    "step_convergence": true,
    "domain_convergence": true,
    "centered_positive_pulse_phase": true,
    "zero_pulse": true,
    "reflectionless_pulse": true
  },
  "negative_controls": {
    "zero_pulse": {
      "purpose": "A vanishing pulse must produce beta=0.",
      "g": 0,
      "rows": [
        {
          "q": 0.5,
          "beta_re": 0,
          "beta_im": 0,
          "beta_magnitude": 0,
          "wronskian_residual": 5.995204332975845e-15,
          "unitarity_residual": 6.217248937900877e-15
        },
        {
          "q": 1,
          "beta_re": 0,
          "beta_im": 0,
          "beta_magnitude": 0,
          "wronskian_residual": 2.042810365310288e-14,
          "unitarity_residual": 2.042810365310288e-14
        },
        {
          "q": 2,
          "beta_re": 0,
          "beta_im": 0,
          "beta_magnitude": 0,
          "wronskian_residual": 2.085442929455894e-12,
          "unitarity_residual": 2.085442929455894e-12
        },
        {
          "q": 3,
          "beta_re": -2.2577629932027217e-15,
          "beta_im": -1.6280425156592622e-15,
          "beta_magnitude": 2.7835258515540777e-15,
          "wronskian_residual": 2.3726243192356833e-11,
          "unitarity_residual": 2.372568808084452e-11
        }
      ],
      "maximum_beta_magnitude": 2.7835258515540777e-15,
      "pass": true
    },
    "reflectionless_pulse": {
      "purpose": "The nonperturbative Pöschl-Teller fixture g=2 (lambda=1) must be reflectionless.",
      "g": 2,
      "lambda": 1,
      "rows": [
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          "wronskian_residual": 2.1094237467877974e-14,
          "unitarity_residual": 2.0650148258027912e-14
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        {
          "q": 1,
          "beta_re": -4.0798061005567846e-17,
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          "beta_magnitude": 3.6061653280435076e-11,
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          "unitarity_residual": 9.059419880941277e-14
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          "wronskian_residual": 2.4457103009467573e-12,
          "unitarity_residual": 2.445599278644295e-12
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          "beta_im": -1.0249186777166656e-12,
          "beta_magnitude": 1.0249276078750874e-12,
          "exact_occupation": 3.9068215748103224e-40,
          "wronskian_residual": 2.5267010705931625e-11,
          "unitarity_residual": 2.5266677639024238e-11
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      ],
      "maximum_beta_magnitude": 1.39990132983381e-10,
      "pass": true
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  },
  "columns": [
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    "born_n",
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    "domain_convergence_relative"
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