Hydrogen Atom & Strong-Field Dynamics

Coulomb Baseline, Time-Dependent External Field & Strong-Field Observables

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Foundations Root · Single-Electron Systems & Ultrafast Laser Excitation

Public Status Boundary. The unperturbed Hydrogen spectrum is exact and anchored to CODATA/NIST ASD values. Strong-field laser interactions are modeled via the Time-Dependent Schrödinger Equation (TDSE). Numerical output spectra are computational models and must be distinguished from detector-acquired experimental data.

$$\boxed{{\displaystyle \mathbf{\text{Analytical baseline}}\ne \mathbf{\text{strong-field TDSE solver}}}}$$$$\boxed{{\displaystyle \mathbf{\text{Numerical solver residual}}\ne \mathbf{\text{laboratory detector signal}}}}$$

Figure 5.2 — Strong-Field Measurement Chain: Causal progression from laser pulse parameters and time-dependent Hamiltonian H(t) to atomic state evolution, detector acquisition, calibration, and experimental uncertainty decomposition.

Details the causal physical chain from ultrafast laser pulse envelope and time-dependent Hamiltonian H(t) to TDSE evolution, atomic observable, instrumental detector, and calibration uncertainty.

Credit: Ivan Pasev / GILC Research·CC BY-NC-SA 4.0·SCHEMATIC

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1. Established External Baseline: Analytical Coulomb System

The unperturbed single-electron Hydrogen atom serves as the primary theoretical anchor:

1.1 Nonrelativistic Coulomb Model (Infinite Nuclear Mass)

In atomic units ($\hslash =e=m_e=4\pi {ϵ}_0=1$):

$$H_0=-\frac{1}{2}{\mathrm{\nabla }}^2-\frac{1}{r}$$

Under the NONRELATIVISTIC_COULOMB_MODEL with INFINITE_NUCLEAR_MASS_APPROXIMATION, the analytical bound-state eigenenergies are:

$$E_n=-\frac{1}{2n^2}\text{ a.u.}=-\frac{hc{R}_{\mathrm{\infty }}}{n^2}=-\frac{13.605693009(84)\text{ eV}}{n^2}$$

1.2 Versioned External Reference Anchor: Experimental Ground-State Ionization

In physical reality, finite proton mass ($m_p/m_e\approx 1836.15$), Dirac relativistic spin-orbit coupling, and QED radiative corrections (Lamb shift) shift the physical binding energy. The consensus experimental value recorded in the NIST Atomic Spectra Database (NIST ASD v5.11, DOI: 10.18434/T4W30F / CODATA 2018) is:

$$I_1(\text{H})=13.598434599702(12)\text{ eV}$$$$\text{Uncertainty }{\sigma }_{\text{ref}}=0.000000000012\text{ eV}(\text{Relative uncertainty: }8.8\times {10}^{-13})$$

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2. Time-Dependent External Laser Field Model

Under intense coherent laser irradiation, the atom is described by the time-dependent Hamiltonian:

$$H(t)=H_0+V_{\text{laser}}(t)$$

2.1 Gauge Formulations

In the dipole approximation (${\lambda }_{\text{laser}}\gg a_0$):

• Length Gauge: $V_{\text{laser}}(t)=\mathbf{E}(t)\cdot \mathbf{r}=E(t)z$ (for linearly polarized light along $\hat{z}$).
• Velocity Gauge: $V_{\text{laser}}(t)=\mathbf{A}(t)\cdot \mathbf{p}+\frac{1}{2}\mathbf{A}(t{)}^2$, where $\mathbf{E}(t)=-{\partial }_t\mathbf{A}(t)$.

2.2 Laser Pulse Parameterization

The laser electric field is parameterized with a well-defined carrier frequency ${\omega }_0$, peak intensity $I_0$, duration $\tau$, and carrier-envelope phase ${\varphi }_{\text{CEP}}$:

$$E(t)=E_{0}f(t)\cos ({\omega }_{0}t+{\varphi }_{\text{CEP}})$$

where $E_0=\sqrt{I_0/(I_a)}$ and $f(t)={\sin }^2(\pi t/\tau )$ for $t\in [0,\tau ]$.

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3. Numerical Realization & TDSE Solvers

To calculate non-perturbative ionization and harmonic emission, the Time-Dependent Schrödinger Equation is integrated numerically:

$$i\frac{\partial }{\partial t}\psi (r,\theta ,t)=[-\frac{1}{2}{\mathrm{\nabla }}^2-\frac{1}{r}+E(t)r\cos \theta ]\psi (r,\theta ,t)$$

3.1 Discretization & Grid Parameters

• Coordinate Representation: Spherical harmonics expansion $\psi (r,\theta ,t)=\sum _{\ell =0}^{{\ell }_{\text{max}}}\frac{u_{\ell }(r,t)}{r}{Y}_{\ell 0}(\theta )$.
• Radial Grid: Uniform or logarithmic grid $r\in [0,R_{\text{max}}]$ with $R_{\text{max}}=300\text{ a.u.}$, $\mathrm{\Delta }r=0.05\text{ a.u.}$, and maximum angular momentum ${\ell }_{\text{max}}=64$.
• Time Propagator: Crank-Nicolson or Split-Operator FFT method with time step $\mathrm{\Delta }t=0.02\text{ a.u.}$ ($\approx 0.48\text{ as}$).
• Boundary Conditions: Complex Absorbing Potential (CAP) $V_{\text{CAP}}(r)=-i\eta (r-R_{\text{abs}}{)}^n\mathrm{\Theta }(r-R_{\text{abs}})$ to prevent artificial reflection of the continuum wavepacket.

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4. Authorial Extension: Relational Boundary Constraints

In the Science of Fabric Reality, the interaction of the electron cloud with the laser field is modeled as a dynamic deformation of relational boundary constraints ($\partial \mathcal{S}$):

• Invariant Tracking: Monitoring the preservation of gauge-invariant phase curvature across multiscale transitions.
• Non-Perturbative Recombination: Modeling the recollision step in the three-step model ($x(t)\to 0$) as an admissible boundary restoration.

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5. Experimental Observables & Metrology Interfaces

Numerical TDSE simulations compute physical observables that interface with laboratory instruments:

5.1 High-Harmonic Generation (HHG) Power Spectrum

Computed from the expectation value of electron acceleration $a(t)=⟨\psi (t)|-\mathrm{\nabla }V|\psi (t)⟩$:

$$S(\omega )={|\frac{1}{\sqrt{2\pi }}{\int }_0^{\tau }a(t)e^{-i\omega t}dt|}^2$$

• Classical Cutoff Law: $\hslash {\omega }_c=I_p+3.17U_p$, where $U_p=\frac{E_0^2}{4{\omega }_0^2}$ is the ponderomotive energy.

5.2 Photoelectron Spectra (ATI)

Computed via projection onto continuum Coulomb wavefunctions $|{\varphi }_{\mathbf{k}}^{(-)}⟩$:

$$P(\mathbf{k})=\underset{t\to \mathrm{\infty }}{lim}|⟨{\varphi }_{\mathbf{k}}^{(-)}|\psi (t)⟩{|}^2$$

5.3 Laboratory Instrumentation & Uncertainty Budget

• EUV Grating Spectrometer: Resolution $\mathrm{\Delta }\lambda /\lambda <{10}^{-3}$, calibrated against noble gas emission lines.
• Velocity Map Imaging (VMI) / Time-of-Flight (ToF): Energy resolution $\mathrm{\Delta }E/E\approx 2\mathrm{\%}$.
• Systematic Error Sources: Laser peak intensity calibration ($\pm 10\mathrm{\%}$), pulse duration jitter ($\pm 5\mathrm{\%}$), focal volume averaging.

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6. Canonical Descent & Navigation

Explore adjacent atomic and experimental sectors:

Related foundations: Atomic Physics Root $\parallel$ LPFR Interface $\parallel$ Simulation Atlas.

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SOURCE AUTHORITY & BOUNDARY LOCK

This route enforces strict cryptographic and epistemic boundaries between consensus reference data, comparator theoretical literature, and authorial candidate predictions.

ESTABLISHED BASELINE

• NIST ASD v5.12 (2024-11-07)
  Hydrogen 1s ground-state ionization energy = 13.598434599702(12) eV.
• CODATA 2022 (2025)
  Rydberg constant R_inf = 10973731.568157(12) m^-1.

COMPARATOR LITERATURE

• Corkum (1993) (1993)
  Three-step model: field ionization, continuum acceleration, and recollision cutoff Ip + 3.17 Up.
• Lewenstein et al. (1994) (1994)
  Quantum Strong-Field Approximation (SFA) dipole integral.

METROLOGY / DATA STANDARDS

• JCGM 100:2008 (GUM)
  Decomposition of numerical grid convergence and detector systematic uncertainty.

AUTHORIAL EXTENSION BOUNDARY

VERSIONED_EXTERNAL_REFERENCE_PLUS_MODEL_BENCHMARK

Hydrogen ground state is an established external reference. Non-perturbative relational boundary extensions are theoretical models requiring prospective empirical testing.

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