Isomer production by multi-photon excitation

Weitao Liu; Yuanbin Wu

REVIEW 3 major objections 5 minor 85 references

Isomer production by multi-photon excitation

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Two-photon lasers can excite the thorium-229m nuclear isomer

desk verdict Useful parameter scan for multiphoton 229mTh excitation, but the long-pulse saturation claims rest on an unproven and apparently incorrect numerical identity, and the initial magnetic-substate averaging is unspecified. read the letter →

arxiv 2504.18828 v1 pith:N5AZRQEI submitted 2025-04-26 nucl-th

Weitao Liu , Yuanbin Wu This is my paper · ORCID

classification nucl-th

keywords multiphotonexcitationthorium-229isomerlaser-nucleusinteractiontime-dependentSchrödingerequationM1nucleartransitionclockhigh-intensitylaserpulses

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Multi-photon excitation is usually discussed for atoms and molecules; this paper asks whether a bare nucleus can absorb several laser photons at once and reach an excited isomer. The authors study thorium-229, whose first excited state, the 229m isomer, lies only 8.36 eV above the ground state, and solve the time-dependent Schrödinger equation with a scheme that sorts the wavefunction by how many photons have been absorbed. They find that for short, intense pulses the n-photon excitation probability follows a clean power law: proportional to the square of the pulse duration and to the n-th power of the peak intensity. For long pulses, laser-induced photon emission interrupts the climb, and the two-photon excitation probability saturates near 0.3 no matter how high the intensity is. If these numbers hold, current and near-future high-intensity lasers might create measurable amounts of 229mTh directly in a bare-nucleus target, a new path toward studying laser-nucleus interactions and nuclear clocks.

What carries the argument

The computational engine is a time-dependent Schrödinger equation solver in which the wavefunction is split into components labelled by the net number of absorbed photons, so that the n-photon excitation amplitude can be followed separately. The transfer matrix for one time step is tridiagonal in this photon-number ladder, and a two-step approximation factors a long Gaussian pulse into intervals, reducing the number of matrix multiplications from linear to logarithmic; this makes nanosecond-scale pulses tractable. The laser-nucleus coupling is treated in the multipole expansion, and for 229Th the M1 transition dominates over E2, giving a 10-by-10 coupling matrix between the six ground and four isomeric magnetic substates. The identity that carries the argument is the simple scaling $P_f^{(n)} \propto \Gamma_I^2 I^n$ in the leading-order regime, which the numerics establishes and which the authors present as the key rule for estimating isomer yield.

What would settle it

Measure the 229mTh yield from a bare-nucleus target irradiated by a short, intense pulse at half the isomer energy; if the two-photon probability does not scale as $\Gamma_I^2 I^2$ in the low-probability regime, or if the saturation limit is found to exceed about 0.3, the central scaling claim would be contradicted.

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Extended reading notes

Core claim

The central claim is that degenerate multi-photon absorption can drive the 229mTh isomer in the direct laser-nucleus interaction, with a calculable probability. In the leading-order regime where the process is dominated by the net absorption of n photons and emission is negligible, the final excitation probability satisfies $P_f^{(n)} \propto \Gamma_I^2 I^n$, with $\Gamma_I$ the FWHM pulse duration and $I$ the peak intensity. This scaling is verified numerically for the 2-, 3-, and 4-photon cases under the assumption of an isolated two-level nucleus coupled by the M1 transition between the $J=5/2$ ground and $J=3/2$ isomer manifolds. At long pulse durations, higher-order effects become significant and the excitation probability saturates; for the 2-photon case the saturation value is approximately 0.3 across intensities from $10^{20}$ to $10^{24}\,\text{W/cm}^2$, while for 3- and 4-photon cases the maximum grows with intensity. The paper further argues that these results imply that high-intensity short-pulse lasers, rather than low-intensity long-pulse ones, are the promising regime for experimental multiphoton isomer production.

Load-bearing premise

The calculation starts from a single, fixed magnetic substate of the 229Th ground state, and the reported probabilities assume that this represents the real excitation probability; an unpolarized sample would need an incoherent average over ground substates, and no such average is given.

Editorial extensions

If this is right

At intensities around $10^{23}\,\text{W/cm}^2$ and pulse durations of a few hundred femtoseconds, the two-photon channel could produce a non-negligible $^{229\text{m}}$Th population in a bare-nucleus target, making direct laser-nucleus multiphoton excitation experimentally testable.
The scaling law $P_f^{(n)} \propto \Gamma_I^2 I^n$ gives a simple calibration rule: once one point is measured, the n-photon yield at other intensities and pulse lengths is predicted.
The 3- and 4-photon channels are strongly suppressed ($I^3$ and $I^4$) and will likely remain below detection except at the very highest planned intensities.
The saturation at about 0.3 for two-photon excitation means that simply raising intensity or extending pulse length cannot push the isomer population above that in this model; different mechanisms, such as involving atomic electrons or multiple pulses, would be needed.
For nuclear clock applications, the authors advise low-intensity, narrow-linewidth long pulses, but note that practical issues such as linewidth and control remain unsolved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Outside the paper: an incoherent density-matrix treatment over the ground magnetic substates is the natural next step; if the average differs from the single-state result, the quoted 0.3 saturation should be reinterpreted as an upper bound for polarized targets only.
Outside the paper: the same photon-number-resolved TDSE method could be applied to other low-lying nuclear isomers or to transitions in highly charged ions where the nucleus and shell interact, possibly revealing enhanced multiphoton rates via electronic bridge mechanisms.
Outside the paper: if the scaling law is confirmed experimentally, the $^{229\text{m}}$Th yield could serve as a diagnostic for the peak intensity and temporal shape of extreme laser pulses, since the probability depends sensitively on both $\Gamma_I^2$ and $I^n$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This paper is worth a read if you are tracking 229mTh excitation, but treat the quantitative long-pulse curves as provisional. The core scaling law—P_f^(n) ∝ I^n Γ_I^2 for short pulses—is exactly what lowest-order perturbation theory gives, and the authors correctly say so. The new content is the photon-number-resolved TDSE implementation and the first scan of 2-, 3-, and 4-photon excitation probabilities with realistic laser parameters. The parameter inputs (B(M1), moments, energy) come from prior experiments; there is no fitting to the target result. That part is clean.

The soft spots are two. First, Eq. (13) is not a minor off-by-one: the claimed identity T_{I,n}...T_{I,1} = S^{n+1}[S^{-1}T_{I,1}]^n does not follow from T_{I,i+1}=S T_{I,i} S^{-1}. Even for n=2 the two sides differ by a factor of S, and the statement that the approximation vanishes for n=1 is plainly false for the printed formula. The long-pulse saturation curves (Figs. 4, 6, 8) rely on this fast-propagation trick, and no code or convergence data are provided. As printed, the numerical method is not checkable.

Second, the initial state is underspecified. Eq. (7) picks one basis vector |φ_i> among the six ground substates, and P_f^(n) is reported for that single state. The appendix says the nucleus is “not polarized”, but that does not define the statistical operator. A real unpolarized sample requires averaging over initial M_G and summing over final M_I; the 3j factors do make the total two-photon strength from M_G=5/2 differ from M_G=1/2 because the J=3/2 manifold truncates the final-state sum. So the ~0.3 saturation limit in the conclusion may be specific to the chosen substate. If the authors intended a single-substate calculation, they should say so; if they meant an unpolarized target, they need the incoherent average.

The bare two-level nucleus in vacuum is a stated idealization, so I would not count that against them.

This deserves a serious referee—the topic is timely and the parameter scan is genuinely new—but the referee should demand a derivation or correction of Eq. (13), a specification of the initial substate with a proper unpolarized average (or an explicit single-substate disclaimer), and convergence tests. I would condition acceptance on those.

Best.

Referee Report

3 major / 5 minor

Summary. The manuscript presents a theoretical study of degenerate multi-photon excitation of the 8-eV isomeric state 229mTh from the 229Th ground state in the direct laser-nucleus interaction. The authors solve the time-dependent Schrödinger equation with a numerical method that sorts contributions by the net photon absorption number, and they use a 10x10 M1 interaction matrix built from measured B(M1), magnetic moments, and excitation energy. They report that, for short pulses, the n-photon excitation probability scales as the square of the FWHM pulse duration and the n-th power of the peak laser intensity, and that for long pulses high-order effects impose an intensity-independent saturation plateau near 0.3 for the 2-photon case, while the 3- and 4-photon limits depend on intensity. The paper argues that current and near-future high-intensity lasers could produce non-negligible isomer populations.

Significance. If the results hold, the paper provides a falsifiable prediction: degenerate two-photon excitation of 229mTh could reach probabilities on the order of 0.1 at 10^24 W/cm^2 with picosecond pulses, and the n-photon scaling law is a clean, parameter-free consequence of the interaction Hamiltonian. A notable strength is that no fitting to the target result is performed: the inputs, B(M1; I→G), the magnetic moments, and the excitation energy, are taken from earlier experiments. The photon-number-resolved TDSE method is a useful conceptual tool, provided the fast-propagation identity is correct. The central short-pulse scaling is analytically sound, but the quantitative saturation curves currently rest on an unverified numerical identity and an unspecified magnetic-substate averaging procedure, so the paper's quantitative claims are not yet fully substantiated.

major comments (3)

[Sec. III.A, Eq. (7), Figs. 2–8] The initial condition in Eq. (7) fixes a single basis state |φ_i⟩ and P_f^(n) is computed as |c^(n)|^2 for that state, but the index i is never specified and no average over the sixfold ground-state manifold is performed. The statement in Sec. III.A that "the nucleus is not polarized" only sets the beam axis as z; it does not define a statistical operator. Because the 10×10 M1 matrix in Appendix A contains within-manifold couplings and the 3j symbols make the two-photon amplitude depend on M_G, an unpolarized sample requires the incoherent average over initial M_G and a sum over final M_I. The quantitative curves and the ≈0.3 saturation plateau in Fig. 4 and the Conclusions may therefore be specific to a single magnetic substate; please state which state was used and provide the ensemble-averaged results or a quantitative bound on the difference.
[Sec. II.B, Eq. (13)] The fast-propagator identity is algebraically incorrect as printed. From the stated recurrence T_{I,i+1}=S T_{I,i} S^{-1} one obtains T_{I,i}=S^{i-1}T_{I,1}S^{-(i-1)}, so ∏_{i=1}^n T_{I,i}=S^{n-1}(T_{I,1}S^{-1})^{n-1}T_{I,1}, which is not equal to S^{n+1}(S^{-1}T_{I,1})^n in general. The printed formula also fails to reduce to T_{I,1} when n=1, contradicting the statement that the approximation vanishes for n=1. Since this identity is the basis of the long-pulse saturation calculations (Figs. 3, 4, 6, 8), the derivation must be supplied and the numerical results must be either recomputed with the corrected expression or shown to be unaffected.
[Secs. II.B and III.C] The two-step approximation and the photon-number truncation (λ_high=12–14, λ_low=−10) are central to the reported numbers, but no quantitative convergence test is provided; the sentence "By varying the parameters, we ensure the convergency" is not a numerical error statement. Please report, for at least one representative case, the dependence of P_f^(n) on δt, N, n, and the photon-number cutoffs.

minor comments (5)

[Sec. III.A] The phrase "isomeric state isomeric state" is duplicated; also clarify that the "two-level system" includes the magnetic substates of the ground and isomeric levels, since the numerical basis is 10-dimensional.
[Figs. 2, 5, 7 captions] The figure captions contain LaTeX artifacts such as "I /greaterorequalslant 10^20 W/cm^2"; these should be typeset correctly.
[Eq. (12)] The displayed definition of S is incomplete: the block structure should be written explicitly, e.g., each diagonal block is \tilde{S} e^{-iλωδt} with λ ranging from λ_high to λ_low.
[References] References [55] and [64] contain corrupted author names ("P/suppress l´ ociennik" and "G/suppress lowacki"); these should be corrected.
[Sec. IV] The paper should add an explicit limitations paragraph listing the assumptions of a bare nucleus in vacuum, no atomic electrons or hyperfine coupling, and no decoherence, since these are relevant to the experimental interpretation of the predicted yields.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model's inputs are independent measured nuclear constants, and the predicted scaling laws are direct consequences of the TDSE Hamiltonian and Gaussian pulse envelope, not fitted outputs.

full rationale

The paper contains no circular derivation. The time-dependent Schrödinger equation (Eq. 1) with the laser-nucleus interaction (Eqs. A1-A10) is solved numerically, and the only physical inputs are external experimental values: E_I = 8.35574 eV, tau_I = 1740 s, B(M1, I -> G) = 0.022 W.u. from Ref. [33], and mu_G = 0.360 mu_N, mu_I = -0.37 mu_N from Refs. [61-64]. None of these inputs is adjusted to reproduce the reported excitation probabilities. The central conclusion that P_f^(n) is proportional to Gamma_I^2 and I^n is not a fitted relation: it is the standard perturbative consequence of an n-photon transition amplitude built from the Gaussian pulse envelope, and the paper explicitly verifies the method by recovering Göppert-Mayer's two-photon result (Eqs. 8-9). The high-order saturation plateau near 0.3 for the 2-photon case comes from solving the full TDSE including emission terms, not from an ansatz fitted to that plateau. There are no load-bearing self-citations or imported uniqueness theorems; the cited prior work supplies parameters and context only. The main numerical caveat, namely the lack of magnetic-substate averaging over an unpolarized initial ensemble, is a physical-domain concern about the applicability of the model to a specific target, not a circularity in the derivation. Hence the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central prediction rests on measured nuclear inputs (B(M1) and the two magnetic moments), on a two-level isolated-nucleus model, and on an unverified numerical fast-propagation identity. No new particles or forces are introduced.

free parameters (3)

B(M1; I → G) = 0.022 W.u. (taken from Ref [33])
This reduced M1 transition probability sets the off-diagonal coupling strength; the 2-photon probability scales as its square, so all quantitative results depend on this measured input.
µ_G (ground-state magnetic moment) = 0.360 µ_N (Refs [61-63])
Diagonal M1 couplings in the ground 6-fold subspace affect the rotations within the degenerate manifold and the saturation dynamics.
µ_I (isomeric-state magnetic moment) = -0.37 µ_N derived from µ_I/µ_G = -1.04 (Ref [64])
Diagonal M1 couplings in the isomeric 4-fold subspace affect internal rotations and the high-order response.

assumptions (5)

domain assumption The 229Th nucleus can be treated as an isolated two-level system with only M1 coupling between ground and isomeric states; E2 and all other nuclear levels are neglected.
Stated in Sec III.A and Appendix A; E2 is said to be three orders of magnitude smaller, but no quantitative comparison or uncertainty is shown.
domain assumption The interaction is the direct coupling of a classical Gaussian laser field to the nuclear current in vacuum, V = -∫ j·A; atomic electrons, plasma screening, and environmental shifts are absent.
Appendix A, Eq (A1); at intensities 10^20 to 10^24 W/cm^2 a real target is ionized, so this premise may not hold in experiment.
domain assumption The initial nuclear state is represented by a single magnetic substate via Eq (7), with no stated averaging over the degenerate ground manifold.
Sec II.A Eq (7); an unpolarized sample requires an incoherent average over six ground substates, which is not described.
ad hoc to paper The two-step interval propagator, Eq (13), with T_I = S^{n+1}(S^{-1}T_{I,1})^n, correctly evolves through n slices.
The identity is asserted without proof; for n=1 it does not reduce to T_{I,1}, indicating a likely off-by-one error in the printed formula. Long-pulse saturation results depend on it.
ad hoc to paper The photon-number basis is truncated at λ_high=12, 13, and 14 and λ_low=-10, and the chosen time steps give converged results.
Sec III.A states convergence was ensured by varying parameters, but no convergence data are shown.

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Pith. "Pith review of Isomer production by multi-photon excitation." pith.science (2026). https://pith.science/paper/N5AZRQEI

@misc{pith2026250418828,
  author       = {Pith},
  title        = {Pith review of: Isomer production by multi-photon excitation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N5AZRQEI}},
  note         = {Machine review of arXiv:2504.18828}
}

abstract

The multi-photon excitation to the $8$-eV nuclear isomeric state $^{229\text{m}}$Th in the direct laser-nucleus interaction is investigated theoretically. We solve the time-dependent Schr\"odinger equation with the method which allows us to study the $n$-photon absorption in the nuclear excitation in the direct laser-nucleus interaction. Based on the laser facilities available currently or in the near future, we analyze the impact of the laser parameters on the excitation probability of the multi-photon excitation. The possibilities of the $2$-, $3$- and $4$-photon excitations to the isomeric state $^{229\text{m}}$Th from the ground state are discussed in details. Our results show the strong impact of the laser intensity and pulse duration on the multi-photon excitation probability. The onset of high-order effects in the multi-photon excitation in the direct laser-nucleus interaction is also revealed. Our findings open new possibilities to study the multi-photon laser-nucleus interaction in high-power laser facilities.

Figures

Figures reproduced from arXiv: 2504.18828 by the authors.

**Figure 3.** FIG. 3. The 2-photon excitation with selected peak laser [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. Leading-order and complete results of the 2-photon [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 7.** Figure 7: FIG. 7. The 4-photon excitation with selected peak laser [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 6.** Figure 6: FIG. 6. Leading-order and complete results of the 3-photon [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: FIG. 8. Leading-order and complete results of the 4-photon [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

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Reference graph

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The 2-photon excitation In Fig. 2, we demonstrate the relation between the 2-photon excitation probability and the laser pulse duration (FWHM) with the intensity ranging from 1020 W/ cm2 to 1024 W/ cm2, assuming the photon energy ω =EI/ 2. The nuclear excitation probability is propor- tional to the square of the peak laser intensity. When the laser pulse ...
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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

The 2-photon excitation In Fig. 2, we demonstrate the relation between the 2-photon excitation probability and the laser pulse duration (FWHM) with the intensity ranging from 1020 W/ cm2 to 1024 W/ cm2, assuming the photon energy ω =EI/ 2. The nuclear excitation probability is propor- tional to the square of the peak laser intensity. When the laser pulse ...

[2] [2]

In the case of the isomeric excitation of 229mTh, the exci- tation energy corresponds to a time period about 0

requires ∆ Eδt ≪ 1. In the case of the isomeric excitation of 229mTh, the exci- tation energy corresponds to a time period about 0 . 5 fs. The time slice δt is required to be much smaller than this value. For long pulses around several tens of nanosec- onds, with δt = 1 as, this requirement leads to 10 10 times matrix multiplication, which makes the numer...

2019

[3] [3]

shows that the variation of ck(t) comes from two paths: 1-photon absorption via the interac- tion U + and 1-photon emission via the interaction U − . This observation allows us to make the decomposition ck(t) = ∑ λ high n=λ low c(n) k (t), where n is the photon absorp- tion number, and λ high(low) the high (low) cutoﬀ of the photon absorption number. To s...

[4] [4]

These diﬀerent intensities lead to a similar shape of the line curves, except that the less intense laser beam re- quires longer pulse to achieve the same probability

We analyze a few peak laser intensities from I = 10 24 W/ cm2 to I = 10 20 W/ cm2. These diﬀerent intensities lead to a similar shape of the line curves, except that the less intense laser beam re- quires longer pulse to achieve the same probability. We can observe from Fig. 4 that the results of the 2-photon excitation with the laser intensities consider...

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[5] [5]

In the case of pulsed lasers, similar problems exist if the laser pulse du- ration is very long

This is because the leading-order derivations do not include the laser-induced photon emissions. In the case of pulsed lasers, similar problems exist if the laser pulse du- ration is very long. By setting U − = 0 instead of the Her- mitian conjugate of U +, we can extract the leading-order contribution. The diﬀerences between the leading-order results and...

[6] [6]

This is a result from the balance be- tween the long time evaluation of the photon absorptions and the laser-induced photon emissions

Unlike the 2-photon excitation case, in the 3-photon excitation, the maximum value of the nu- clear excitation probability has a strong dependence on the laser intensity. This is a result from the balance be- tween the long time evaluation of the photon absorptions and the laser-induced photon emissions. 6 0 2 4 6 0 1 2 3 4 5 0 2 4 6 0 1 2 3 4 5 0 2 4 6 0...

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[7] [7]

The 3-photon excitation The 3-photon excitation process is evaluated with the photon energy ω = EI/ 3. In Fig. 5, we present the re- lation between the 3-photon excitation probability and the laser pulse duration FWHM with selected peak laser intensities, for short laser pulse durations. Since the 3-photon excitation process involves the 3rd or higher ord...

work page

[8] [8]

The corresponding scale of the laser pulse duration is proportional to the in- verse of the peak laser intensity. Similar to the 3-photon excitation case, the maximum probability depends on the peak laser intensity, but it is proportional to the inverse of the square of the peak laser intensity. 1 10 100 1 ,000 10− 35 10− 31 10− 27 10− 23 10− 19 10− 15 10...

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