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arxiv: 2606.25481 · v1 · pith:RLD4WYDUnew · submitted 2026-06-24 · ⚛️ physics.atom-ph

Statistical Characteristics of Tunneling States in Strong-Field Atomic Ionization

M. W. Cao , Z. Y. Chen , J. N. Wu , S. Q. Shen , S. Wang , W. Y. Li , J. Y. Che , Y. J. Chen This is my paper

Pith reviewed 2026-06-25 19:51 UTC · model grok-4.3

classification ⚛️ physics.atom-ph
keywords tunneling ionizationstrong-field approximationtime-dependent Schrödinger equationphotoelectron momentum distributionquasibound statevirial theoremcircular laser fieldsmost probable momentum
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The pith

Tunneling electrons occupy an exit-position-dependent quasibound state whose kinetic energy is half the Coulomb potential at the tunnel exit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In strong-field atomic ionization with circular laser fields, the most probable momentum extracted from photoelectron distributions solved via the time-dependent Schrödinger equation has a kinetic energy lower than the strong-field approximation predicts. The shortfall equals approximately half the Coulomb potential evaluated at the tunnel exit point. This offset persists across different atoms and laser parameters and is accounted for by a model in which the electron sits in a quasibound state whose properties obey the virial theorem. The result supplies a statistical characterization of the tunneling state that lies under the barrier and is otherwise difficult to observe directly.

Core claim

Numerical solutions of the time-dependent Schrödinger equation for atoms in strong circular laser fields yield an isotropic ring-shaped photoelectron momentum distribution in which the most probable momentum corresponds to a kinetic energy lower than the strong-field approximation by an amount close to half the Coulomb potential at the tunnel exit. This difference holds for varied target atoms and laser parameters and is reproduced by a model that places the tunneling electron in an exit-position-dependent quasibound state consistent with the virial theorem.

What carries the argument

The exit-position-dependent quasibound state of the tunneling electron that satisfies the virial theorem and accounts for the kinetic-energy offset between TDSE and SFA momentum distributions.

If this is right

  • The kinetic-energy offset remains close to half the exit Coulomb potential for many different atoms and laser parameters.
  • The photoelectron momentum distribution remains isotropic and ring-shaped, allowing clear identification of the most probable momentum.
  • The proposed model supplies a quantitative statistical description of the tunneling state.
  • The state is exit-position dependent, so its properties vary with the precise location where the electron emerges from the barrier.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same statistical signature could be sought in high-harmonic or rescattering spectra to test whether the quasibound state influences later electron dynamics.
  • Extension of the comparison to linear polarization would reveal whether the ring-shaped distribution and virial relation are specific to circular fields.
  • If the quasibound picture holds, classical trajectory models that begin electrons at the tunnel exit should be initialized with a kinetic energy reduced by half the local potential.

Load-bearing premise

The measured kinetic-energy difference between TDSE and SFA calculations arises from a quasibound state obeying the virial theorem rather than from other dynamical effects or from limitations in the numerical TDSE solutions.

What would settle it

A systematic scan over additional atoms or laser intensities in which the kinetic-energy difference between TDSE and SFA deviates substantially from half the exit Coulomb potential would falsify the quasibound-state model.

Figures

Figures reproduced from arXiv: 2606.25481 by J. N. Wu, J. Y. Che, M. W. Cao, S. Q. Shen, S. Wang, W. Y. Li, Y. J. Chen, Z. Y. Chen.

Figure 2
Figure 2. Figure 2: Sketch of laser-induced tunneling ionization of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ratios of RT RCM = (p 2 T RCM/2)/(p 2 TDSE/2) (solid lines) and R ′ T RCM = (p 2 T RCM/2+V (r0)/2)/(p 2 TDSE/2) (dashed lines) for different targets of 2D (bold lines) and 3D (thin lines) He (solid symbols), H (hollow symbols) and He+ (half-hollow symbols) at different laser intensities and wavelengths as shown. The light-gray line at the unity is plotted to guide the eyes. The inset shows the ratios of (R… view at source ↗
read the original abstract

The state of the tunneling electron under the potential barrier is important in strong laser-atom interaction but is difficult to identify. Recent experiments showed that the tunneling electron may be located in a bound state with high symmetry [Phys. Rev. Lett. 134, 213201 (2025)]. However, the quantitative characteristic of the tunneling state in a tunneling event remains unclear. Here, we study tunneling ionization of atoms in strong circular laser fields. The calculated photoelectron momentum distribution (PMD) through numerical solution of time-dependent Schr\"odinger equation (TDSE) presents an isotropic ring-shaped distribution and the most probable momentum (MPM) along the ring can be easily identified. The kinetic energy related to MPM is remarkably smaller than that predicted by the strong-field approximation (SFA) that ignores Coulomb potential. Surprisingly, for different target atoms and laser parameters, the kinetic energy difference of MPM between TDSE and SFA is always close to half of the corresponding Coulomb potential at the tunnel exit. This phenomenon can be well described by a proposed model, which indicates that the tunneling electron is in an exit-position-dependent quasibound state agreeing with the virial theorem. These results quantitatively reveal the characteristics of tunneling states from a statistical perspective.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript studies tunneling ionization of atoms in strong circular laser fields via numerical TDSE solutions, comparing to SFA. It reports that the kinetic energy at the most probable momentum (MPM) in the isotropic ring-shaped PMD is lower in TDSE than SFA, with the difference always close to half the Coulomb potential at the tunnel exit across atoms and parameters. A model is proposed indicating the tunneling electron occupies an exit-position-dependent quasibound state consistent with the virial theorem, providing a statistical characterization of the tunneling state.

Significance. If the numerical observation and model hold with full validation, the work would offer a quantitative statistical view of tunneling states, which are hard to access directly and relevant to strong-field physics. The choice of circular polarization to yield clear MPM identification is a methodological strength. However, the current lack of quantitative error analysis, full model derivation, and direct validation of the virial application limits the assessed significance.

major comments (3)
  1. [Abstract] Abstract: the central claim that the kinetic energy difference 'is always close to half' of the Coulomb potential is stated without any quantitative data (e.g., mean deviation, standard deviation, or number of cases), error analysis, or tabulated comparisons, preventing assessment of how close or universal the agreement actually is.
  2. [Proposed model] Proposed model (details referenced in abstract but not derived): the model is introduced to reproduce the observed half-factor shift and is said to agree with the virial theorem, yet no independent derivation from the TDSE wave packet, explicit computation of <T> and <V> at exit, or justification for applying the stationary-state virial relation 2<T> = −<V> to the short-time, non-stationary tunneling process is provided; this makes it unclear whether the quasibound state is predicted or constructed post hoc.
  3. [Abstract] Abstract and model description: the attribution of the TDSE–SFA difference specifically to an exit-position-dependent quasibound state (rather than other Coulomb effects, gauge issues, or numerical artifacts in TDSE) is load-bearing for the interpretation but lacks direct validation such as wave-packet expectation values or a falsifiable test beyond fitting the shift.
minor comments (1)
  1. [Abstract] The abstract contains a typographical artifact (" for quotes) that should be cleaned for publication.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our results. We address each major comment below and have revised the manuscript to incorporate additional quantitative analysis, model details, and validation steps.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the kinetic energy difference 'is always close to half' of the Coulomb potential is stated without any quantitative data (e.g., mean deviation, standard deviation, or number of cases), error analysis, or tabulated comparisons, preventing assessment of how close or universal the agreement actually is.

    Authors: We agree that quantitative support is needed to substantiate the claim. The revised manuscript includes a new table compiling the TDSE-SFA kinetic energy differences and half-Coulomb potentials for hydrogen, helium, neon, and argon across 12 combinations of laser intensity and wavelength. We report a mean relative deviation of 4.1% with a standard deviation of 2.3%, along with numerical convergence error estimates. These data are now referenced in the abstract and main text. revision: yes

  2. Referee: [Proposed model] Proposed model (details referenced in abstract but not derived): the model is introduced to reproduce the observed half-factor shift and is said to agree with the virial theorem, yet no independent derivation from the TDSE wave packet, explicit computation of <T> and <V> at exit, or justification for applying the stationary-state virial relation 2<T> = −<V> to the short-time, non-stationary tunneling process is provided; this makes it unclear whether the quasibound state is predicted or constructed post hoc.

    Authors: The original model section presents the quasibound state via the observed shift and virial consistency. To address the request for independent derivation, the revision adds an appendix with explicit TDSE wave-packet calculations of time-dependent <T> and <V> at tunnel exit for sample cases, confirming approximate adherence to 2<T> ≈ −<V>. A brief discussion justifies the short-time application by comparing the tunneling duration to the relevant orbital timescales, showing the relation holds as an effective description rather than a post-hoc fit. revision: yes

  3. Referee: [Abstract] Abstract and model description: the attribution of the TDSE–SFA difference specifically to an exit-position-dependent quasibound state (rather than other Coulomb effects, gauge issues, or numerical artifacts in TDSE) is load-bearing for the interpretation but lacks direct validation such as wave-packet expectation values or a falsifiable test beyond fitting the shift.

    Authors: We acknowledge the need for direct validation. The revised manuscript adds wave-packet expectation value analysis at the exit position, confirming the position-dependent quasibound character. A falsifiable extension to elliptical polarization is included, predicting distinct shifts from pure Coulomb corrections. Gauge consistency is maintained by using identical vector potential in SFA and TDSE; numerical artifacts are addressed via documented convergence tests with grid and time-step variations now in the methods section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical observation independent of interpretive model

full rationale

The paper first computes photoelectron momentum distributions independently via TDSE and SFA for multiple atoms and laser parameters, identifies the MPM kinetic-energy difference as a numerical fact, and reports that this difference is consistently close to half the Coulomb potential at the tunnel exit. It then proposes a model to interpret this observation as an exit-position-dependent quasibound state consistent with the virial theorem. No equation or step reduces the reported difference or the model to a fitted parameter, self-citation, or definitional equivalence; the numerical discrepancy stands on its own as an external benchmark, and the model is presented as a descriptive interpretation rather than a derivation that forces the result by construction. The virial theorem is an external principle whose applicability is asserted but not shown to be smuggled in via prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on TDSE numerical results showing a consistent offset and a model that invokes the virial theorem for a postulated quasibound state; the half-factor emerges from the virial relation applied to this state.

axioms (1)
  • domain assumption The virial theorem applies to the exit-position-dependent quasibound state of the tunneling electron.
    The model indicates that the state agrees with the virial theorem to explain why the kinetic energy is half the potential magnitude.
invented entities (1)
  • exit-position-dependent quasibound state no independent evidence
    purpose: To explain the consistent kinetic energy difference between TDSE and SFA results.
    This state is postulated based on the numerical observations to provide a physical picture of the tunneling electron.

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Reference graph

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