Floquet-engineered nondispersive wave packets in helium under combined periodic and static fields

Alejandro Gonz\'alez-Melan; Javier Madro\~nero; John Fitzgerald Le\'on Ocampo

REVIEW 3 major objections 6 minor 57 references

Floquet-engineered nondispersive wave packets in helium under combined periodic and static fields

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper reports numerical evidence for long-lived nondispersive two-electron wave packets in fully three-dimensional helium, formed by near-resonant coupling of frozen planet states, with a static field extending their lifetime by up…

desk verdict First full-3D helium nondispersive wave packet claim with solid field-free benchmarks, but the load-bearing width lacks quantitative convergence data. read the letter →

arxiv 2506.21273 v1 pith:EKQHSTUE submitted 2025-06-26 physics.atom-ph

Alejandro González-Melan , John Fitzgerald León Ocampo , Javier Madroñero This is my paper · ORCID

classification physics.atom-ph PACS 32.80.Qk31.15.ac32.80.Rm05.45.Mt

keywords heliumatomfrozenplanetstatesnondispersivewavepacketsFloquettheorycomplexscalingCoulomb-Sturmianbasisconfigurationinteractionstaticelectricfieldstabilization

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

The paper claims to find nondispersive wave packets—quantum states that stay localized while following a periodic classical orbit—in the full three-dimensional helium atom driven by a monochromatic field. Such states had been seen in reduced-dimensionality models; this work reports them in the full 3D setting by combining a configuration-interaction basis of Coulomb-Sturmian functions with Floquet theory and complex scaling. The central state, $W_2^6$, lies below the $N=6$ ionization threshold and is a near-resonant mixture of two frozen planet states, $1S\,F_2^6$ and $1P\,F_1^6$. Adding a weak static electric field along the polarization axis can lengthen its lifetime from about 22 to about 35 driving cycles, a 60 percent gain, while the wave packet stays localized on the classical 1:1 resonance island. If correct, this establishes that engineered long-lived correlated two-electron wave packets are accessible in a real atom and suggests concrete experimental signatures, including a strong electric dipole moment.

What carries the argument

The operative mechanism is the near-resonant dressing of a pair of frozen planet states by the periodic field inside Floquet theory. Frozen planet states are the quantum counterparts of a classically stable collinear configuration in which one electron stays close to the nucleus while the other sits far out on the same side, and they are recognized here by small decay rates and by $\langle\cos\theta_{12}\rangle$ near unity. The Floquet Hamiltonian is built in the basis of field-free eigenstates obtained from a Coulomb-Sturmian configuration-interaction calculation with different radial dilation parameters for the inner and outer electrons, and resonances are extracted by complex scaling. The dressed state $W_2^6$ is tracked by its overlap with the two dominant frozen planet states, and its nondispersive character is established through Husimi projections onto the classical phase space of the outer electron.

What would settle it

Recompute the Floquet spectrum with a substantially larger basis (more angular momenta, more Floquet blocks, larger radial sets, varied per-electron dilation parameters) and time-propagate $W_2^6$ from its complex-scaled resonance: if the imaginary part of the quasienergy moves significantly from $\Gamma/2 \simeq 3.26\times10^{-6}\,\mathrm{a.u.}$, or the wave packet delocalizes well before 35 field cycles, the nondispersive claim fails.

Watch

Extended reading notes

Core claim

The central claim is that a Floquet state $W_2^6$, formed by dressing the near-resonant pair of field-free frozen planet states $1S\,F_2^6$ and $1P\,F_1^6$, constitutes a nondispersive wave packet in fully three-dimensional helium. Its Husimi distribution stays concentrated on the classical 1:1 resonance island of the driven frozen-planet configuration across field phases, and its complex-scaled quasienergy has a small decay rate, giving a lifetime of roughly 22 driving cycles. The state is robust over a finite window of driving amplitudes (about $1.0\times10^{-6}$ to $5.0\times10^{-6}$ a.u.) and frequencies (about $8.5\times10^{-4}$ to $9.5\times10^{-4}$ a.u.). A static field, up to the classical ionization limit $F_I\approx2.3\times10^{-5}$ a.u., leaves the composition essentially unchanged ($\sim84\%$ $1S\,F_2^6$, $\sim15\%$ $1P\,F_1^6$) and reduces the decay rate, extending the lifetime to about 35 cycles. The same mechanism also produces a localized Floquet state below the $N=5$ threshold, and the authors propose two-photon excitation schemes and the state's large dipole moment as routes to experimental detection.

Load-bearing premise

The truncated numerical basis is large enough that the quasienergy and decay width of $W_2^6$ would not materially change with more Floquet blocks, angular momenta, or radial functions, a claim supported only by qualitative convergence tests and by visual inspection of Husimi distributions.

Editorial extensions

If this is right

The dressed state $W_2^6$ survives as a localized wave packet over a finite window of driving amplitudes ($1.0\times10^{-6}$ to $5.0\times10^{-6}$ a.u.) and frequencies (about $8.5\times10^{-4}$ to $9.5\times10^{-4}$ a.u.), so the phenomenon is not limited to a single finely tuned parameter set.
Adding a static electric field up to the classical ionization limit $F_I\approx2.3\times10^{-5}$ a.u. preserves the wave packet's localization and decreases its decay rate, extending the lifetime from about 22 to about 35 field cycles.
A similar localized Floquet state appears below the $N=5$ threshold, indicating that the near-resonant frozen-planet coupling mechanism repeats across Rydberg series.
The wave packet's large Stark shift and dipole moment make it stand out in the quasienergy spectrum, giving an experimentally accessible observable for detection.
Two-photon excitation paths from the ground state, with a final near-resonant step at about 51776.5 nm, could in principle prepare the wave packet, though current spectroscopic resolution is insufficient to address individual frozen planet states.

Reading between the lines

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

If the mechanism is generic, the same near-resonant pairing of frozen planet states should produce nondispersive wave packets in other two-electron systems such as H$^-$ or alkaline-earth-like atoms, at appropriately scaled frequencies and field strengths; the paper does not test this.
The static-field stabilization suggests a design rule: bias the decay rate of the dominant frozen-planet component downward while keeping the two-state resonance intact. A testable extension would be a full scan of the $(F, F_{\mathrm{st}})$ stability boundary, which the paper only partially resolves.
Because the reported lifetimes come from complex-scaled resonance widths rather than long-time propagation, an independent time-dependent propagation of $W_2^6$ would verify the 35-cycle claim and could reveal multichannel decay paths hidden in the resonance width.
The authors' reliance on visual Husimi inspection implies the stability window they report is likely a lower bound; automated state tracking across parameter space could find broader regions where the wave packet remains localized.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The headline: this is the first numerical claim of nondispersive wave packets in fully three-dimensional helium, extending the authors' earlier planar work. If the claim holds, it's a real step. The paper does a lot right: the field-free frozen planet states benchmark well against Richter, Burgers, and Rost, with energies and widths agreeing to a few percent; the asymmetric Sturmian basis is sensible for these states; and the Floquet treatment is standard. I also don't see circularity: the drive frequency is set by the independently calculated FPS energy separation, and W_2^6 is found as an eigenstate, not fitted.

The soft spot is exactly where the reader and the stress-test put it: the existence claim for W_2^6 rests on one number, the complex-scaled Floquet half-width around 3.26e-6 a.u., and the paper gives no quantitative convergence data for that width. It states that increasing Floquet blocks or angular momentum channels 'does not significantly alter' the results, but no tables or plots show this. The worry is not academic: with Lmax=5, dipole selection rules mean L=6 and higher continuum channels are absent, and complex scaling folds continuum decay into square-integrable states. If those channels contribute, the width could be artificially small. The unperturbed benchmarks validate the CI method, not the driven truncation. The paper itself concedes that identifying W_2^6 requires visual inspection of Husimi distributions and becomes ambiguous outside a narrow parameter window. These are real limitations, not manufactured ones.

That said, I think the stress-test's stronger claim—that the state could be a basis artifact—is not established. The unperturbed FPS benchmarks are genuinely good external support, the quasienergy shows smooth behavior across parameter sweeps, and the state is dominated by two FPS for a range of amplitudes and frequencies. It would be surprising if the whole thing evaporated, but 'surprising' is not 'shown.' The missing convergence data is a moderate-to-serious omission that a referee should ask for.

Recommendation: send it to peer review. A serious referee can push for the convergence evidence, and if the width survives enlargement of Lmax and kmax, this is a publishable result. Even if it doesn't, the paper contains useful benchmarks and a clear methodology. I'd also suggest the authors provide code or data; that would settle most of the worry.

Referee Report

3 major / 6 minor

Summary. The manuscript reports a numerical study of helium in three dimensions subjected to a monochromatic linearly polarized field, with an optional static field along the polarization axis. The method combines a Coulomb-Sturmian configuration-interaction representation of the field-free Hamiltonian with Floquet theory and complex scaling. The authors identify frozen planet states (FPS) below the N=3–7 thresholds, benchmark their energies and widths against earlier work (Tables I and II), and then study a Floquet state labeled W_2^6 below the N=6 threshold. They claim that W_2^6 is a nondispersive wave packet formed by near-resonant coupling between the 1S F_6^2 and 1P F_6^1 frozen planet states, that it remains localized on the classical 1:1 resonance island for roughly 22 field cycles, and that adding a static field can increase this lifetime by up to 60%. They also discuss possible experimental preparation and detection schemes.

Significance. If the central claim holds, this would be the first demonstration of nondispersive wave packets in a fully three-dimensional two-electron atom, going beyond the earlier reduced-dimensionality models. The field-free frozen planet benchmarks in Tables I and II are a genuine strength: the energies and widths agree with independent published results to several significant digits, which gives credibility to the underlying CI/Sturmian and complex-scaling machinery. The paper also offers concrete experimental observables, most notably the large field-induced dipole moment of W_2^6, and a plausible two-photon preparation pathway. However, the existence and lifetime of the nondispersive wave packet rest on the complex-scaled Floquet width of a single state, and the manuscript does not provide the quantitative convergence evidence needed to rule out basis-truncation artifacts in that width.

major comments (3)

[III D] The claim that the Floquet basis truncation (13 blocks, L=0,...,5) is adequate is supported only by the sentence 'This choice of basis size is guided by convergence tests, which show that increasing the number of Floquet blocks or angular momentum channels does not significantly alter the results presented here.' No convergence data are given. This is load-bearing because the nondispersive character and the quoted 22-cycle lifetime are controlled by the complex-scaled half-width Gamma/2 ~ 3.26e-6 a.u., and resonance widths are exactly the quantities most sensitive to truncation of decay channels. In particular, with Lmax=5, the L=6,7,... continuum channels that couple through the dipole operator are absent, and complex scaling folds those channels into the square-integrable basis. The authors should provide a quantitative convergence study of the W_2^6 quasienergy and Gamma/2 as functions of kmax, Lmax, the radial basis size, the Sturmian dilation parameters, and the complex rotation angle theta.
[III D] There is an internal inconsistency in the mechanism claimed for static-field stabilization. In Section III C and Figure 9, the decay rate of the dominant component 1S F_6^2 increases with the static field, reaching a maximum near F_st = 2.23e-5 a.u., while the decay rate of 1P F_6^1 decreases monotonically. In Section III D, however, Figure 11 shows that the decay rate of W_2^6 stays roughly constant up to F_st ~ 1.5e-5 and then decreases as F_st approaches 2.23e-5, even though the wave packet composition is reported to remain ~84% 1S F_6^2 and ~15% 1P F_6^1. The manuscript itself says the W_2^6 decay rate is 'expected to mirror the behavior observed in Section III C', but it does not. The conclusions attribute the lifetime enhancement to 'field-induced suppression of the decay rate of the dominant FPS component', which is contradicted by Figure 9. This discrepancy should be resolved with a quantitative analysis of how the dressed-state width arises from the complex-scaled Floquet matrix, rather than a qualitative expectation.
[III D] The identification of W_2^6 as a nondispersive wave packet relies on visual inspection of Husimi distributions, as the manuscript acknowledges: 'identifying the correct Floquet state requires visual inspection of Husimi projections, limiting the resolution of stability charts.' The robustness claim is therefore not stated in a falsifiable, reproducible way. The authors should provide a quantitative criterion for 'localized on the 1:1 resonance island' (for example, a phase-space concentration measure or a threshold on the Husimi weight inside the island), and they should report the overlap thresholds used to track the state across the parameter sweeps in Figures 5, 10, 11, and 12. Without such a criterion, the parameter window and the 60% lifetime enhancement claim cannot be independently checked from the paper.

minor comments (6)

[Figure 5] The caption states a fixed frequency omega = 0.0008 a.u., whereas the text in Section III B uses omega = 0.00088 a.u. This inconsistency should be corrected, since the near-resonance condition depends on the frequency.
[III D] The paragraph beginning 'Complementarily, Figure 12 shows...' appears twice nearly verbatim. One copy should be deleted.
[II] In Section II, the sentence 'we conclude with an outline of the classical FPC' begins with a lowercase 'we' after a period; this should be corrected.
[III B] The text states that the authors 'identify those dominated by a single frozen planet state F_N^j', but W_2^6 is then characterized as a superposition of two FPS with comparable weights (~84% and ~15%). The wording should be adjusted to reflect that the state is dominated by one or two FPS.
[References] Reference [23] lists the third author as 'D. M. N.', which is incomplete; the full name should be provided.
[Figure 2] The axes of the two-step Poincar\'e surfaces of section in Figure 2 are not labeled in the caption; adding axes labels would help the reader connect the classical islands to the quantum Husimi plots.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the nondispersive wave packet is obtained as a Floquet eigenstate from an independently benchmarked field-free frozen-planet input, not from a fitted or self-referential construction.

full rationale

The derivation chain is self-contained at the points that matter. The input frozen planet states 1S F2^6 and 1P F1^6 are identified by diagonalizing the field-free Hamiltonian and are benchmarked against external data from Richter et al., Burgers et al., and Rost et al., so the two states used for the resonance condition are not defined by the driven calculation. The driving frequency is chosen as the independently computed energy separation between these states (about 0.00088 a.u.), which is a physical resonance condition rather than a parameter fitted to the claimed outcome. The central object W_2^6 is then obtained as an eigenstate of the complex-scaled Floquet Hamiltonian, and its identification is based on computed overlaps and Husimi distributions, not imposed by construction. The static-field lifetime enhancement is read off from the computed complex quasienergies. The self-citations to the authors' earlier planar helium work appear only as motivation for examining near-resonant FPS pairs; the three-dimensional Floquet diagonalization is performed in the present paper and its result does not reduce to that citation. The absence of quantitative convergence data for the Floquet-block and angular-momentum truncation is a robustness caveat rather than circularity, because the claimed state is not defined in terms of that truncation. Overall, the central claim has independent numerical content.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard mathematical tools (Floquet theory, complex scaling, Sturmian basis expansions) and on domain approximations (non-relativistic infinite-mass Hamiltonian, dipole approximation). The numerical truncation parameters are chosen by hand and not fully specified. The identification of the nondispersive wave packet uses an ad hoc criterion based on overlap with two frozen planet states and Husimi localization; this criterion is not independently validated. No new physical entities are introduced and no parameters are fitted to data in the sense of adjusting theory to reproduce the claimed result.

free parameters (4)

Coulomb-Sturmian dilation parameters kappa1s, kappa2s = not stated
Chosen to target asymmetrically excited frozen planet states; values are not reported in the paper, so reproduction requires contacting the authors.
Radial basis truncation limits Nmin/Nmax per angular momentum pair = not stated
Truncation of the configuration interaction expansion; convergence asserted but no data shown.
Maximum total angular momentum Lmax = 5
Truncation of Floquet basis; convergence asserted in Section III B.
Floquet block range kmin..kmax = -6..6 (13 blocks)
Truncation in Floquet index; convergence asserted in Section III B.

assumptions (6)

standard math Floquet theorem: solutions of a time-periodic Hamiltonian can be expanded in time-periodic Floquet states with quasienergies.
Invoked in Section II B, Eq. (14-17), to define the driven eigenvalue problem.
standard math Complex scaling transforms resonance poles into square-integrable eigenstates with complex eigenvalues E - i*Gamma/2.
Invoked in Section II C, following Refs. [36-40].
standard math Coulomb-Sturmian functions form a complete basis for expanding two-electron wave functions and the multipole expansion of 1/r12 converges.
Used in Section II A, Eqs. (4-13), to construct the CI basis and matrix elements.
domain assumption The infinite nuclear mass, non-relativistic Hamiltonian, dipole approximation, and length gauge are valid for helium in the studied regime.
Stated in Section II, Eq. (1), as the starting Hamiltonian.
domain assumption The classical frozen planet configuration and its effective potential, including intrinsic frequency omega_I and ionization field F_I, provide the correct semiclassical framework for interpreting the quantum states.
Used in Section II D and throughout Section III to define resonance conditions and stability criteria.
ad hoc to paper A nondispersive wave packet can be identified by maximum overlap with two frozen planet states and Husimi localization on the classical resonance island, with no other states contributing significantly.
This is the identification criterion in Section III B, Figures 6-7; it is not independently proven and is acknowledged to rely on visual inspection.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Floquet-engineered nondispersive wave packets in helium under combined periodic and static fields." pith.science (2026). https://pith.science/paper/EKQHSTUE

@misc{pith2026250621273,
  author       = {Pith},
  title        = {Pith review of: Floquet-engineered nondispersive wave packets in helium under combined periodic and static fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EKQHSTUE}},
  note         = {Machine review of arXiv:2506.21273}
}

read the original abstract

We report the numerical observation of two-electron nondispersive wave packets in fully three-dimensional helium subjected to a linearly polarized monochromatic field. These localized quantum states follow periodic trajectories of the classical three-body Coulomb system without dispersion. The system is modeled using a spectral configuration interaction approach based on Coulomb-Sturmian functions, combined with Floquet theory and complex scaling. The resulting quasienergy spectrum reveals the existence of long-lived Floquet states arising from near-resonant coupling between frozen planet configurations.

Figures

Figures reproduced from arXiv: 2506.21273 by the authors.

**Figure 2.** FIG. 2. Two-step Poincar´e surface of section for the driven [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) Conditional probability density ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 6.** Figure 6: FIG. 6. Husimi distributions of the Floquet state [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 4.** Figure 4: FIG. 4. (Color online) Phase space Husimi distributions of the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Floquet quasienergies [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 9.** Figure 9: FIG. 9. Energies (left) and decay rates (right) of the frozen [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Overlaps of the wave packet [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. Real (left) and imaginary (right) parts of the [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: FIG. 12. Real (left) and imaginary (right) parts of the [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

**Figure 14.** Figure 14: FIG. 14. Expectation value of the electronic dipole [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 52 canonical work pages

[1]

For the ground state, the maximum of the probability density is localized near the equilibrium position of the outer electron

of the lowest1S frozen planet states in theN= 4 series. For the ground state, the maximum of the probability density is localized near the equilibrium position of the outer electron. For excited FPS, the distribution shifts outward along the r1 axis while remaining centered inr 2, consistent with the structure of the classical configuration. of the electr...

work page
[2]

Ossiander, F

M. Ossiander, F. Siegrist, V. Shirvanyan, R. Pazourek, A. Sommer, T. Latka, A. Guggenmos, S. Nagele, J. Feist, J. Burgd¨ orfer, et al., Nature Phys.13, 280 (2017)

work page 2017
[3]

Busto, L

D. Busto, L. Barreau, M. Isinger, M. Turconi, C. Alexan- dridi, A. Harth, S. Zhong, R. J. Squibb, D. Kroon, S. Plogmaker, et al., J. Phys. B51, 044002 (2018)

work page 2018
[4]

Beaulieu, A

S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. G´ eneaux, F. L´ egar´ e, S. Petit, et al., Science358, 1288 (2017)

work page 2017
[5]

Y. Yang, R. E. Mainz, G. M. Rossi, F. Scheiba, M. A. Silva-Toledo, P. D. Keathley, G. Cirmi, and F. X. K¨ artner, Nature Communications12, 6641 (2021)

work page 2021
[6]

Borrego-Varillas, M

R. Borrego-Varillas, M. Lucchini, and M. Nisoli, Reports on Progress in Physics85, 066401 (2022)

2022
[7]

Nandi, E

S. Nandi, E. Olofsson, M. Bertolino, S. Carlstr¨ om, F. Za- pata, D. Busto, C. Callegari, M. D. Fraia, P. Eng- Johnsson, R. Feifel, et al., Nature608, 488 (2022)

work page 2022
[8]

Richter and D

K. Richter and D. Wintgen, J. Phys. B26, 3719 (1993)

work page 1993

Show all 57 references

[9]

Tanner, K

G. Tanner, K. Richter, and J. M.Rost, Rev. Mod. Phys. 72, 497 (2000)

2000
[10]

R. P. Madden and K. Codling, Phys. Rev. Lett.10, 516 (1963)

1963
[11]

C. W. Byun, N. N. Choi, M. H. Lee, and G. Tanner, Phys. Rev. Lett.98, 113001 (2007)

2007
[12]

Ericson, Phys

T. Ericson, Phys. Rev. Lett.5, 430 (1960)

1960
[13]

Ericson, Ann

T. Ericson, Ann. Phys.23, 390 (1963)

1963
[14]

Y. Fang, W. Jiang, L. Geng, I. Bˇ rezinov´ a, . Sommer- lad, A. Tsertsvadze, M. Kircher, J. Kruse, T. Jahnke, N. Walsh, et al., Phys. Rev. A111, 023120 (2025)

2025
[15]

Blatt and D

R. Blatt and D. Wineland, Nature453, 1008 (2008)

2008
[16]

I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys.86, 153 (2014)

2014
[17]

H. Ma, M. Govoni, and G. Galli, npj Comput. Mater.6, 85 (2020)

2020
[18]

Richter and D

K. Richter and D. Wintgen, Phys. Rev. Lett.65, 1965 (1990). 12

1990
[19]

Wintgen, K

D. Wintgen, K. Richter, and G. Tanner, Chaos2, 19 (1992)

1992
[20]

Schlagheck and A

P. Schlagheck and A. Buchleitner, Europhys. Lett.46, 24 (1999)

1999
[21]

Buchleitner, D

A. Buchleitner, D. Delande, and J. Zakrzewski, Phys. Rep.368, 409 (2002)

2002
[23]

Gonzalez-Melan and J

A. Gonzalez-Melan and J. Madro˜ nero, Phys. Rev. A101, 013414 (2020)

2020
[24]

Rupprecht, N

P. Rupprecht, N. G. Puskar, D. M. N., and S. R. Leone, Phys. Rev. Research6, 043100 (2024)

2024
[25]

M. G. Floquet, Annales de l’ ´Ecole Normale Sup´ erieure 12, 47 (1883)
[26]

J. H. Shirley, Phys. Rev.138, B979 (1965)

1965
[27]

G. L. Kamta, Ph.D. thesis, Universite Nationale du Benin (1999)

1999
[28]

Eiglsperger, B

J. Eiglsperger, B. Piraux, and J. Madro˜ nero, Phys. Rev. A80, 022511 (2009)

2009
[29]

Eiglsperger, Dissertation, Technische Universit¨ at M¨ unchen (2010)

J. Eiglsperger, Dissertation, Technische Universit¨ at M¨ unchen (2010)

2010
[30]

Schlagheck and A

P. Schlagheck and A. Buchleitner, J. Phys. B.31, L489 (1998)

1998
[31]

T. P. Grozdanov, A. A. Gusev, E. A. Solov’ev, and S. I. Vinitsky, Eur. Phys. J. D74, 161 (2020)

2020
[32]

Foumouo, G

E. Foumouo, G. L. Kamta, G. Edah, and B. Piraux, Phys. Rev. A74, 063409 (2006)

2006
[33]

Rotenberg, Adv

M. Rotenberg, Adv. At. Mol. Phys.6, 233 (1970)

1970
[34]

Huens, B

E. Huens, B. Piraux, A. Bugacov, and M. Gajda, Phys. Rev. A55, 2132 (1997)

1997
[35]

D. A. Varschalovich, A. N. Moskalev, and V. K. Kher- sonskii,Quantum Theory of Angular Momentum(World Scientific, Singapore, 2008)

2008
[36]

Delande and J

D. Delande and J. Zakrzewski, Phys. Rev. A58, 466 (1998)

1998
[37]

Aguilar and J

J. Aguilar and J. M. Combes, Comm. Math. Phys.22, 269 (1971)

1971
[38]

Balslev and J

E. Balslev and J. M. Combes, Comm. Math. Phys.22, 280 (1971)

1971
[39]

Simon, The Annals of Mathematics97, 247 (1973)

B. Simon, The Annals of Mathematics97, 247 (1973)

1973
[40]

W. P. Reinhardt, Annual Review of Physical Chemistry 33, 223 (1982)

1982
[41]

Ho, Physics Reports99, 1 (1983)

Y. Ho, Physics Reports99, 1 (1983)

1983
[42]

Lanczos, J

C. Lanczos, J. Res. Natl. Bur. Stand.45, 225 (1950)

1950
[43]

Krug, Dissertation, Ludwig-Maximilians-Universit¨ at M¨ unchen (2001)

A. Krug, Dissertation, Ludwig-Maximilians-Universit¨ at M¨ unchen (2001)

2001
[44]

Schlagheck, Ph.D

P. Schlagheck, Ph.D. thesis, Technische Universit¨ at M¨ unchen (1999)

1999
[45]

A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion(Springer-Verlag, New York, 1983)

1983
[46]

V. N. Ostrovsky and N. V. Prudov, J. Phys. B28, 4435 (1995)

1995
[47]

Schlagheck and A

P. Schlagheck and A. Buchleitner, Eur. Phys. J. D22, 401 (2003)

2003
[48]

Madro˜ nero, P

J. Madro˜ nero, P. Schlagheck, L. Hilico, B. Gr´ emaud, D. Delande, and A. Buchleitner, Europhys. Lett.70, 183 (2005)

2005
[49]

Madro˜ nero and A

J. Madro˜ nero and A. Buchleitner, Phys. Rev. A77, 053402 (2008)

2008
[50]

Richter, J

K. Richter, J. S. Briggs, D. Wintgen, and E. A. Solovev, J. Phys. B25, 3929 (1992)

1992
[51]

Eichmann, V

U. Eichmann, V. Lange, and W. Sandner, Phys. Rev. Lett.64, 274 (1990)

1990
[52]

K. D. Heber, M. Seng, M. Halka, U. Eichmann, and W. Sandner, Phys. Rev. A56, 1255 (1997)

1997
[53]

Camus, T

P. Camus, T. F. Gallagher, J. M. Lecomte, P. Pillet, L. Pruvost, and J. Boulmer, Phys. Rev. Lett.62, 2365 (1989)

1989
[54]

J. G. Leopold and I. C. Percival, J. Phys. B13, 1037 (1980)

1980
[55]

Burgers, D

A. Burgers, D. Wintgen, and J.-M. Rost, J. Phys. B: At. Mol. Opt. Phys.28, 3163 (1995)

1995
[56]

M. D. J. M. Rost, K. Schulz and G. Kaindl, J. Phys. B: At. Mol. Opt. Phys.30, 4663 (1997)

1997
[57]

Guzm´ an, Undergraduate thesis, Universidad del Valle (2015)

I. Guzm´ an, Undergraduate thesis, Universidad del Valle (2015)

2015
[58]

Gonz´ alez, I

A. Gonz´ alez, I. Guzm´ an, and J. Madro˜ nero, J. Phys. Conf. Ser.635, 092088 (2015)

2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

For the ground state, the maximum of the probability density is localized near the equilibrium position of the outer electron

of the lowest1S frozen planet states in theN= 4 series. For the ground state, the maximum of the probability density is localized near the equilibrium position of the outer electron. For excited FPS, the distribution shifts outward along the r1 axis while remaining centered inr 2, consistent with the structure of the classical configuration. of the electr...

work page

[2] [2]

Ossiander, F

M. Ossiander, F. Siegrist, V. Shirvanyan, R. Pazourek, A. Sommer, T. Latka, A. Guggenmos, S. Nagele, J. Feist, J. Burgd¨ orfer, et al., Nature Phys.13, 280 (2017)

work page 2017

[3] [3]

Busto, L

D. Busto, L. Barreau, M. Isinger, M. Turconi, C. Alexan- dridi, A. Harth, S. Zhong, R. J. Squibb, D. Kroon, S. Plogmaker, et al., J. Phys. B51, 044002 (2018)

work page 2018

[4] [4]

Beaulieu, A

S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. G´ eneaux, F. L´ egar´ e, S. Petit, et al., Science358, 1288 (2017)

work page 2017

[5] [5]

Y. Yang, R. E. Mainz, G. M. Rossi, F. Scheiba, M. A. Silva-Toledo, P. D. Keathley, G. Cirmi, and F. X. K¨ artner, Nature Communications12, 6641 (2021)

work page 2021

[6] [6]

Borrego-Varillas, M

R. Borrego-Varillas, M. Lucchini, and M. Nisoli, Reports on Progress in Physics85, 066401 (2022)

2022

[7] [7]

Nandi, E

S. Nandi, E. Olofsson, M. Bertolino, S. Carlstr¨ om, F. Za- pata, D. Busto, C. Callegari, M. D. Fraia, P. Eng- Johnsson, R. Feifel, et al., Nature608, 488 (2022)

work page 2022

[8] [8]

Richter and D

K. Richter and D. Wintgen, J. Phys. B26, 3719 (1993)

work page 1993