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REVIEW 3 major objections 5 minor 73 references

Collective tunnel ionization in atomic systems

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Two electrons can tunnel out of an atom together, and this paper derives the rate and the observable signature.

desk verdict A serious attempt at a real two-electron tunneling theory with a new repulsion-induced action term and a plausible momentum signature, but the analytic rate is not quantitatively controlled at the fields where the TDSE evidence lives. read the letter →

arxiv 2608.03998 v1 pith:7GDS34S5 submitted 2026-08-04 physics.atom-ph

D.I. Tyurin , V.V. Strelkov , S.V. Popruzhenko This is my paper · ORCID
classification physics.atom-ph
keywords collectivetunnelingnonsequentialdoubleionizationtunnelelectron–electroncorrelationsstrong-fieldlaserphysicsimaginarytimemethodlateralmomentumdistributionnegativebromineion
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 argues that in strong low-frequency laser fields two electrons can tunnel out of an atom simultaneously, not one after the other, and that this collective channel leaves a measurable fingerprint. It develops an analytic semiclassical theory that includes electron–electron repulsion, which earlier collective-tunneling models neglected, and checks it against numerical solutions of the two-electron Schrödinger equation for negative bromine and neutral xenon. The most probable escape path keeps the two electrons at the same coordinate along the field while separating them in opposite lateral directions; the repulsion then imprints a characteristic lateral momentum on the pair. If the theory is right, collective tunneling becomes a quantitative, experimentally searchable process rather than a speculative correction to sequential ionization.

What carries the argument

The central object is the symmetric two-electron trajectory ansatz x1 = x2 ≡ x, y1 = −y2 ≡ y. On this slice the two-electron problem becomes a single effective particle with mass 2 in the potential V(x,y) = 2V_ei(x,y) + V_ee(x,y;x,−y) − 2E x. The semiclassical imaginary-time action separates into a longitudinal part Sx and a lateral part Sy; the lateral equation of motion y¨ = −1/(4y^2) encodes the electron–electron repulsion and yields both the sub-barrier action correction and, after real-time propagation, the asymptotic lateral momentum. The Coulomb attraction to the nucleus is then added perturbatively as SC through the standard matching procedure. The work this machinery does is to conv

What would settle it

Measure two-electron coincidence lateral momenta for nonsequential double ionization of xenon in a short circularly polarized pulse with recollisions suppressed. If the collective channel exists, the p1y = −p2y diagonal should show side peaks or shoulders near p_y = (π^2 E^2/(16 Ip))^(1/6); a distribution identical to the non-interacting sequential background at the same field would refute the prediction. Additionally, at fields E much below the barrier-suppression field, a direct numerical evaluation of Im S along the full two-particle trajectory should reproduce Eq. (48); a mismatch there wo

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Extended reading notes

Core claim

For strong fields below the barrier-suppression limit, the paper claims that nonsequential double ionization contains a collective tunneling channel in which both electrons pass under the barrier at once. Along the dominant trajectory the electron positions satisfy x1 ≈ x2 and y1 ≈ −y2, reducing the two-electron problem to a single particle of mass 2 moving in an effective potential. The electron–electron repulsion, kept as 1/(2|y|), dominates the nuclear attraction in the leading sub-barrier action and contributes the term Sy ≈ i 3π^(2/3)/(2^(4/3)) (Ip/E^2)^(1/6); adding the Coulomb factor gives the total action S_CT = S_0 + S_y + S_C and the rate w_CT ≃ exp(−2 Im S_CT). After the electrons

Load-bearing premise

The derivation assumes the most probable escape path is exactly the symmetric configuration x1 = x2, y1 = −y2, with the electron–electron repulsion term dominating the nuclear attraction inside the barrier; for regions where |y| ≥ |x|, no analytic correction is derived.

Editorial extensions

If this is right

  • The exponential rate of collective tunneling is specified by Eq. (48), so experiments and simulations can compare CT and sequential rates directly from ionization potentials, field amplitude, and atomic charge.
  • The lateral momentum prediction p_y ≈ (π^2 E^2/(16 Ip))^(1/6) gives an unambiguous signature: side peaks for Br− and shoulders for Xe in the p1y = −p2y diagonal of the two-electron momentum distribution.
  • Short circularly polarized or unipolar pulses that suppress recollision can expose CT, because the pair emerges with correlated off-plane momenta rather than in the polarization plane.
  • Including electron–electron repulsion removes the orders-of-magnitude overestimate of earlier quasiparticle CT models, bringing the CT rate to a level comparable to the sequential channel under short-pulse conditions.
  • The same imaginary-time action decomposition extends the standard single-electron tunnel-ionization formalism to a correlated two-particle problem, with potential applications to other two-electron escape processes.

Reading between the lines

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

  • The scaling p_y ∝ E^(1/3) Ip^(−1/6) is a clean cross-species test: measuring the side-peak position at several field amplitudes and atomic targets would directly check the symmetric-path assumption on which the whole derivation rests.
  • In a real three-dimensional atom, CT should emit the two electrons along a cone around the field axis rather than exactly in a plane; a coincidence experiment seeking pairs with nearly equal energy and opposite polar angles could detect CT even at low yield.
  • Because the theory splits the action into short-range (electron–electron repulsion) and Coulomb parts, the same decomposition may transfer to other correlated tunneling problems in which a repulsive interaction dominates along one coordinate.
  • The authors note that 1D model atoms show no collective channel at all; this implies that any simulation or experiment searching for CT must preserve at least two-dimensional geometry for the electron motion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents a combined TDSE and semiclassical study of simultaneous two-electron tunnel ionization (collective tunneling) in strong low-frequency laser fields. Two-dimensional TDSE simulations for Br^- and Xe in short unipolar pulses show a two-electron flux along the diagonal x1≈x2 with lateral anti-correlation y1≈-y2, which is absent in 1D and disappears at longer pulse durations. The authors interpret this as tunneling through a two-dimensional barrier and develop an imaginary-time semiclassical theory for the symmetric trajectory. The main analytic results are the collective-tunneling action S_CT = S_0 + S_y + S_C (Eq. 48), with S_y arising from electron-electron repulsion and S_C from the nuclear attraction, and the asymptotic lateral momentum prediction p_y ≈ (π^2 E^2/(16 I_p))^{1/6} (Eq. 59). The paper further proposes experimental signatures of collective tunneling in circularly polarized or short pulses.

Significance. If the analytic theory were quantitatively reliable, this would be an important step: it gives a parameter-free semiclassical description of a correlated two-electron tunneling channel, corrects the earlier quasiparticle models by including electron-electron repulsion, and produces a falsifiable prediction for lateral momentum distributions. The TDSE evidence for the collective channel and for the lateral shoulder is direct and not fitted to the analytic model; the lateral-momentum comparison is a genuine consistency check. The limitations of the semiclassical expansion are acknowledged in the text, but the quantitative rate formula is used for experimental estimates, so the status of the analytic predictions is a key issue.

major comments (3)
  1. [Sec. V, Eq. (51) and Fig. 7] The quantitative rate formula is not supported at the field strengths used in the TDSE demonstration. For Br^- at E=0.035, the paper's own numerical minimization of the full action gives ImS_min^(1)=1.18 (and ImS_min^(2)=1.228 with the unsmoothed potential), while Eq. (51) gives ImS_CT ≈ 4.7. At E=0.02 the corresponding numbers are 7.24 and ≈10.6. Thus the analytic exponent is larger by factors of about 4 and 1.5, corresponding to many orders of magnitude in rate. This contradicts the statement in Sec. V (note after Eq. 49) that the perturbative results "demonstrate a good agreement with those obtained without perturbative expansions as well as with the TDSE solutions." Since Eq. (48) is used in Sec. VI.A to compare CT and sequential rates and to estimate experimental feasibility, the central quantitative claim is not established in the demonstrated regime.
  2. [Sec. V.A and V.B, Eqs. (28), (42), (47)] The derivation rests on a perturbative treatment of the nuclear attraction that is not controlled at the parameters of interest. The term -2Z/sqrt(x^2+y^2) is dropped in Eq. (28) using y0/x0~sqrt(mu)<<1 (Eq. 45), but for Br^- at E=0.02 the paper's own Eq. (43) gives mu=0.61, so sqrt(mu)~0.8, not <<1. The resulting short-range action (42) at E=0.035 is about 15, whereas the full numerical action is 1.18; the first-order Coulomb correction (47) reduces this only to about 4.7. The near-nucleus region, where |y|>=|x| and the integrand in Eq. (46) should be 1/|y| rather than 1/|x|, contributes nonperturbatively, as the paper partly acknowledges in comment 2 of Sec. V.B. This is not a small correction and requires either a nonperturbative treatment or a strictly asymptotic statement of validity.
  3. [Sec. VI.B, Eq. (59)] The lateral momentum prediction p_y=(π^2 E^2/(16 I_p))^{1/6} is derived from the asymptotic exit coordinate y0 of Eq. (37), i.e. from the short-range approximation without the nuclear attraction. The TDSE position of the side maxima agrees only to about 20%, and the improvement is obtained by solving the full equations of motion (58) with a numerical initial condition. The paper should state more clearly that Eq. (59) is a leading-order estimate, not a quantitative prediction at E/E_BS~0.8, and should quantify the expected error or give a parameter scan before using it as an experimental calibration.
minor comments (5)
  1. [Sec. IV] The text refers to "Fig.2(c.1–c.3)" for the xenon panels, but the figure uses panels (g)-(i).
  2. [Sec. III] Typo: "anlyze" should be "analyze".
  3. [Sec. V] Typo: "the very fact the the action" should read "the very fact that the action".
  4. [Sec. VI.C] "Not that in the 3D geometry" should be "Note that in the 3D geometry."
  5. [Sec. VI.C, Fig. 11] The text around Fig. 11 states the field propagates along the y-axis, while the rest of the paper uses x as the polarization direction; please clarify the coordinate convention in the figure caption.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the analytic rate and lateral-momentum predictions are parameter-free functions of I_p and E and are compared with independent TDSE solutions; self-citations to prior work are not load-bearing.

full rationale

The central analytic derivation is not circular. The symmetry condition (16) is introduced after being observed in the TDSE solutions (Section IV, Figs. 2-3), but the quantitative results — the action S_CT in Eq. (48) and the lateral momentum prediction in Eq. (59) — are derived from the semiclassical action integral with only I_p and E (plus the standard Coulomb matching) as inputs. No parameter is fitted to the TDSE output to produce those predictions; the comparison in Figs. 7, 9, and 10 is a genuine consistency check, and the paper reports the residual discrepancies (e.g., about 20% in the side-peak positions) as physical approximation errors rather than as a sign that the theory was adjusted to reproduce the numerics. The self-citations to [32] supply background: the earlier numerical demonstration of a collective flux, the statement that the 1D channel is blocked, and the construction of the non-interacting-electron reference system. These are not load-bearing for the analytic derivation, which stands on the present paper's own TDSE solutions and on the standard imaginary-time-method formalism. The paper is also explicit about its limitations: Section V states that the semiclassical expansion is "on the edge of its applicability" for the parameters used, that a correction from the region |y| >= |x| "cannot be analytically derived," and the Conclusions note that the pre-factor remains to be found. These are correctness/accuracy caveats, not circularity. I therefore find no circular step that reduces a prediction to its own input; the minor self-citations do not warrant more than a very low score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The main free parameters are the soft-core potential smoothing parameters, which calibrate the model TDSE to the target ionization potentials. The analytic action formula depends only on physical inputs (I_p, E, Z) plus the stated semiclassical approximations. No new entities are introduced.

free parameters (3)
  • a (electron-ion smoothing parameter) = 1.66 (1D Br-), 1 (2D Br-), 1.546 (2D Xe)
    Adjusted so that the soft-core potential (9) reproduces the second ionization potential I_p2 of the target.
  • b (electron-electron smoothing parameter) = 2.6 (1D Br-), 2.2 (2D Br-), 2.1 (2D Xe)
    Adjusted so that the soft-core potential (10) reproduces the first ionization potential I_p1.
  • rcut (momentum filter cutoff) = 15 a.u.
    Defines the spatial filter (52) that separates double ionization from bound and single-ionization contributions; a technical choice for the momentum analysis.
assumptions (6)
  • domain assumption Semiclassical approximation / Imaginary Time Method (ITM) for tunneling
    The rate formula (24) and action (25) rely on ITM, standard in strong-field ionization; the paper applies it to the two-electron problem.
  • domain assumption Most probable trajectory is symmetric: x1=x2=X, y1=-y2=y (Eq. 16)
    Section V restricts the action (25) to these trajectories; the authors argue from symmetry and verify numerically in Fig.3, but this is an assumption for the analytic derivation.
  • domain assumption Electron-electron repulsion dominates over nuclear attraction along the sub-barrier path, so the short-range potential V_sr = 1/(2|y|) - 2Ex is used in the leading action (28)
    Used to split the action into longitudinal and lateral parts (29). The paper argues y0/x0 << 1, but mu is only ~0.5 for the studied parameters.
  • domain assumption Static field limit (Keldysh parameter gamma -> 0) for the action
    Section V sets Re(t)=0 and treats E as constant; for T=2 fs, gamma1=0.54, which is not deeply in the static regime.
  • domain assumption Soft-core potentials (9)-(10) model the real atom
    The 2D TDSE uses smoothed Coulomb potentials with parameters a,b fitted to I_p1, I_p2. The claim that this captures the real atom's collective tunneling is an assumption.
  • domain assumption Coulomb correction (47) can be borrowed from single-electron theory with the same matching procedure
    Section V.B adopts the single-electron logarithmic factor; the paper estimates the near-nucleus correction is small but does not derive it analytically.

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Pith. "Pith review of Collective tunnel ionization in atomic systems." pith.science (2026). https://pith.science/paper/7GDS34S5

@misc{pith2026260803998,
  author       = {Pith},
  title        = {Pith review of: Collective tunnel ionization in atomic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GDS34S5}},
  note         = {Machine review of arXiv:2608.03998}
}
read the original abstract

We present a theory of the collective tunnel effect in atomic systems subject to strong low-frequency laser fields. Using an analytic semi-classical method and numerical solution to the time-dependent Schrodinger equation we demonstrate the collective channel of the nonsequential double ionization in neutral xenon and in the negative bromine ion. Collective tunneling in the presence of the electron-electron repulsion comprises the joint sub-barrier motion of the two electrons along quantum trajectories transversely shifted in opposite directions with respect to the direction of the external electric field. Momentum distributions of the electron pairs carry clear signatures of the collective channel in the form of shoulders in the direction lateral to the field polarization. We propose and discuss potential experimental approaches to a search for collective tunneling in atomic systems using short circularly polarized laser pulses.

Figures

Figures reproduced from arXiv: 2608.03998 by the authors.

Figure 1
Figure 1. FIG. 1: Single- and two-electron potential energies [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Probability density [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Lateral distributions [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Panel (a): the potential surface (18) for Br [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Potential barriers (18) for the negative bromine [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: shows the trajectory (33) with y0 given by (37) (green solid line) and that obtained numerically by solv￾ing the exact equation (35) (green dashed line). Also, the trajectory found by direct numerical minimization of the action (25) with the potential energy (26), whic…
Figure 7
Figure 7. Figure 7: FIG. 7: Logarithm of the CT ionization rate, [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Lateral momentum distributions along the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Lateral momentum distributions (54) for the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Left column: lateral momentum distributions [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: On the experimental observation of the [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

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Works this paper leans on

73 extracted references · 72 canonical work pages

  1. [1]

    This is achieved by ap- plying a unipolar pulse, which does not allow the outer electron returning back to its parent ion

    Recollision is eliminated. This is achieved by ap- plying a unipolar pulse, which does not allow the outer electron returning back to its parent ion

    work page

  2. [2]

    Therefore, we consider the short pulses, where sig- nificant deviations from SI can be expected

    In the recollision-suppressed regime, double ioniza- tion was shown perfectly sequential in relatively long pulses of durationτ≈20 fs and more for systems with ionization potentialsI p ≃10eV [32]. Therefore, we consider the short pulses, where sig- nificant deviations from SI can be expected

    work page

  3. [3]

    Instead, in 1D the collective chan- nel is entirely blocked [32]

    Numerical TDSE solution for two electrons in 3D requires, for the time intervals necessary to reveal the collective effect, an extremely large computa- 4 tional capacity. Instead, in 1D the collective chan- nel is entirely blocked [32]. Thus, our study is fo- cused mostly on a model 2D two-electron atom, al- though we also use its 1D analog when appropria...

    work page

  4. [4]

    These results require two comments

    =−0.35. These results require two comments

    work page

  5. [5]

    Under these conditions,µ→0 (43), and|S 0| ≫ |Sy| ≫ |SC| ≫1

    The semiclassical theory of tunneling is asymptot- ically exact forE→0 [33, 55, 56], and the cor- responding formulas for tunneling rates give quan- titatively correct results atE≪E BS, Ech. Under these conditions,µ→0 (43), and|S 0| ≫ |Sy| ≫ |SC| ≫1. However, for parameters we consider hereµ≈0.5, and Λ CT = 4..5 so that all three ac- tions appear of compa...

    work page

  6. [6]

    The two asymptotics do not match, but the contribution of that part of space where|y| ≥ |x|into the can be estimated by inte- grating 1/|y(t)|fromτ 0 to ˜τsuch thaty(˜τ) =x(˜τ)

    Close to the nucleus, p x2 +y 2 ≪x 0, the condition |x| ≫ |y|inverts and the integrand in (46) should be replaced by 1/|y(τ)|. The two asymptotics do not match, but the contribution of that part of space where|y| ≥ |x|into the can be estimated by inte- grating 1/|y(t)|fromτ 0 to ˜τsuch thaty(˜τ) =x(˜τ). This estimation gives for the contribution of this p...

    work page

  7. [7]

    The CT rate has to be at least comparable in abso- lute value to that of the other competing channels

    work page

  8. [8]

    In the following Subsections, we separately address these two points

    Momentum distribution of the electron pairs or the double charged ions have to bear some unambigu- ous signatures of the ionization mechanism. In the following Subsections, we separately address these two points. 12 A. Comparison of the sequential single-electron and collective ionization rates The spatial distribution of the two-electron probabil- ity de...

    work page

Show all 73 references
  1. [9]

    Eq.(55) neglects the effect of the ion Coulomb po- tential on the electron dynamics after ionization. 14

  2. [10]

    As a result, the asymptotic formula (37) is not expected to be sufficiently accurate

    Although the field amplitudeE 0 = 0.035 satisfies the condition (20), it does not fall deeply into the regime of tunneling,E 0 ≪E BS, Ech. As a result, the asymptotic formula (37) is not expected to be sufficiently accurate

  3. [11]

    The analytics above is assumes a constant field, while the slow time dependence of the electric field (11) used for the TDSE numerical solution can also affect the momentum distributions

  4. [12]

    The first factor connected with the Coulomb focusing effect [59, 60] can be easily accounted for

    Finally, the tails of the central peak originating from the sequential ionization channel, overlap with the side structures and effectively pull the lateral maxima toward the origin. The first factor connected with the Coulomb focusing effect [59, 60] can be easily accounted f...

  5. [13]

    Suran and I

    V. Suran and I. Zapesochny, Soviet Technical Physics Letters1, 2 (1975)

    1975

  6. [14]

    l’Huillier, L

    A. l’Huillier, L. Lompre, G. Mainfray, and C. Manus, Physical Review A27, 2503 (1983)

    1983

  7. [15]

    D. N. Fittinghoff, P. R. Bolton, B. Chang, and K. C. Kulander, Physical review letters69, 2642 (1992)

    1992

  8. [16]

    Walker, B

    B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, Physical review letters73, 1227 (1994)

    1994

  9. [17]

    Kondo, A

    K. Kondo, A. Sagisaka, T. Tamida, Y. Nabekawa, and S. Watanabe, Physical Review A48, R2531 (1993)

    1993

  10. [18]

    Cornaggia and P

    C. Cornaggia and P. Hering, Journal of Physics B: Atomic, Molecular and Optical Physics31, L503 (1998)

    1998

  11. [19]

    Weber, M

    T. Weber, M. Weckenbrock, A. Staudte, L. Spiel- berger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen,et al., Physical review letters84, 443 (2000)

    2000

  12. [20]

    Moshammer, B

    R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schr¨ oter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn,et al., Physical review letters 84, 447 (2000)

    2000

  13. [21]

    Rudenko, T

    A. Rudenko, T. Ergler, K. Zrost, B. Feuerstein, V. De Je- sus, C. Schr¨ oter, R. Moshammer, and J. Ullrich, Journal of Physics B: Atomic, Molecular and Optical Physics41, 081006 (2008)

    2008

  14. [22]

    Becker, X

    W. Becker, X. Liu, P. J. Ho, and J. H. Eberly, Rev. Mod. Phys.84, 1011 (2012)

    2012

  15. [23]

    Larochelle, A

    S. Larochelle, A. Talebpour, and S.-L. Chin, Journal of Physics B: Atomic, Molecular and Optical Physics31, 1201 (1998)

    1998

  16. [24]

    M. Y. Kuchiev, JETP Lett45, 404 (1987)

    1987

  17. [25]

    P. B. Corkum, Phys. Rev. Lett.71, 1994 (1993)

    1994

  18. [26]

    Kopold, W

    R. Kopold, W. Becker, H. Rottke, and W. Sandner, Physical review letters85, 3781 (2000)

    2000

  19. [27]

    Agostini, Rev

    P. Agostini, Rev. Mod. Phys.96, 030501 (2024)

    2024

  20. [28]

    L’Huillier, Rev

    A. L’Huillier, Rev. Mod. Phys.96, 030503 (2024)

    2024

  21. [29]

    Krausz, Rev

    F. Krausz, Rev. Mod. Phys.96, 030502 (2024)

    2024

  22. [30]

    Dimauro and P

    L. Dimauro and P. Agostini, inAdvances in Atomic, Molecular, and Optical Physics, Vol. 35 (Elsevier, 1995) pp. 79–120

    1995

  23. [31]

    Zon, Journal of Experimental and Theoretical Physics 89, 219 (1999)

    B. Zon, Journal of Experimental and Theoretical Physics 89, 219 (1999)

    1999

  24. [32]

    Eichmann, M

    U. Eichmann, M. D¨ orr, H. Maeda, W. Becker, and W. Sandner, Phys. Rev. Lett.84, 3550 (2000)

    2000

  25. [33]

    Popruzhenko and D

    S. Popruzhenko and D. Tyurin, Bulletin of the Lebedev Physics Institute50, S922 (2023)

    2023

  26. [34]

    M. Lein, E. K. Gross, and V. Engel, Phys. Rev. Lett. 85, 4707 (2000)

    2000

  27. [35]

    Klaiber, K

    M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, Phys- ical Review A—Atomic, Molecular, and Optical Physics 75, 063413 (2007)

    2007

  28. [36]

    Shvetsov-Shilovski, S

    N. Shvetsov-Shilovski, S. Goreslavski, S. Popruzhenko, and W. Becker, Physical Review A—Atomic, Molecular, and Optical Physics77, 063405 (2008)

    2008

  29. [37]

    Liu and C

    X. Liu and C. Figueira de Morisson Faria, Physical review letters92, 133006 (2004)

    2004

  30. [38]

    Miloˇ sevi´ c, G

    D. Miloˇ sevi´ c, G. Paulus, D. Bauer, and W. Becker, Jour- nal of Physics B: Atomic, Molecular and Optical Physics 39, R203 (2006)

    2006

  31. [40]

    K¨ ubel, N

    M. K¨ ubel, N. G. Kling, K. J. Betsch, N. Camus, A. Kaldun, U. Kleineberg, I. Ben-Itzhak, R. Jones, G. Paulus, T. Pfeifer,et al., Physical Review A—Atomic, Molecular, and Optical Physics88, 023418 (2013)

    2013

  32. [41]

    S. Ben, T. Wang, T. Xu, J. Guo, and X. Liu, Optics Express24, 7525 (2016)

    2016

  33. [42]

    Becker, S

    W. Becker, S. Goreslavski, D. Miloˇ sevi´ c, and G. Paulus, Journal of Physics B: Atomic, Molecular and Optical Physics51, 162002 (2018)

    2018

  34. [43]

    S. Li, D. Sierra-Costa, M. J. Michie, I. Ben-Itzhak, and M. Dantus, Communications Physics3, 35 (2020)

    2020

  35. [44]

    Tyurin, V

    D. Tyurin, V. Strelkov, and S. Popruzhenko, arXiv preprint arXiv:2504.20583 (2025)

    2025 arXiv

  36. [45]

    J. R. Oppenheimer, Phys. Rev.31, 66 (1928)

    1928

  37. [46]

    Keldysh, Soviet Phys

    L. Keldysh, Soviet Phys. JETP20(1965)

    1965

  38. [47]

    Perelomov, V

    A. Perelomov, V. Popov, and M. Terent’ev, Zh. Eksp. Teor. Fiz50, 1393 (1966)

    1966

  39. [48]

    Y. B. Zeldovich, Sov. Phys. JETP38, 1123 (1960)

    1960

  40. [49]

    Galitsky and V

    V. Galitsky and V. Cheltsov, Nuclear Physics56, 86 (1964)

    1964

  41. [50]

    Blank and M

    B. Blank and M. P loszajczak, Reports on Progress in Physics71, 046301 (2008)

    2008

  42. [51]

    V. S. Popov, Physics-Uspekhi47, 855 (2004)

    2004

  43. [52]

    Popov, Physics of Atomic Nuclei68, 686 (2005)

    V. Popov, Physics of Atomic Nuclei68, 686 (2005)

    2005

  44. [53]

    Salieres, B

    P. Salieres, B. Carr´ e, L. Le D´ eroff, F. Grasbon, G. Paulus, H. Walther, R. Kopold, W. Becker, D. Milosevic, A. San- pera,et al., Science292, 902 (2001)

    2001

  45. [54]

    Eckhardt and K

    B. Eckhardt and K. Sacha, EPL (Europhysics Letters) 56, 651 (2001)

    2001

  46. [55]

    Sacha and B

    K. Sacha and B. Eckhardt, Physical Review A63, 043414 (2001)

    2001

  47. [56]

    Eckhardt and K

    B. Eckhardt and K. Sacha, Journal of Physics B: Atomic, Molecular and Optical Physics39, 3865 (2006)

    2006

  48. [57]

    Bader, S

    P. Bader, S. Blanes, and F. Casas, The Journal of chem- ical physics139(2013)

    2013

  49. [58]

    Neuhasuer and M

    D. Neuhasuer and M. Baer, The Journal of chemical physics90, 4351 (1989)

    1989

  50. [59]

    Smirnov and M

    B. Smirnov and M. Chibisov, Sov. Phys. JETP22, 23 (1966). 17

    1966

  51. [60]

    Tong and C

    X. Tong and C. Lin, Journal of Physics B: Atomic, Molec- ular and Optical Physics38, 2593 (2005)

    2005

  52. [61]

    I. Y. Kostyukov and A. Golovanov, Physical Review A 98, 043407 (2018)

    2018

  53. [62]

    Popruzhenko, Journal of Physics B: Atomic, Molecular and Optical Physics47, 204001 (2014)

    S. Popruzhenko, Journal of Physics B: Atomic, Molecular and Optical Physics47, 204001 (2014)

    2014

  54. [63]

    Bauer,Computational strong-field quantum dynamics (de Gruyter Berlin, 2017)

    D. Bauer,Computational strong-field quantum dynamics (de Gruyter Berlin, 2017)

    2017

  55. [64]

    Shvetsov-Shilovski, Laser Physics35, 123001 (2025)

    N. Shvetsov-Shilovski, Laser Physics35, 123001 (2025)

    2025

  56. [65]

    Perelomov, V

    A. Perelomov, V. Popov, and M. Terent’ev, Zh. Eksp. Teor. Fiz52, 514 (1967)

    1967

  57. [66]

    Popruzhenko, V

    S. Popruzhenko, V. Mur, V. Popov, and D. Bauer, Phys- ical review letters101, 193003 (2008)

    2008

  58. [67]

    Ankerhold,Quantum tunneling in complex systems: the semiclassical approach, Vol

    J. Ankerhold,Quantum tunneling in complex systems: the semiclassical approach, Vol. 224 (Springer, 2007)

    2007

  59. [68]

    Razavy,Quantum theory of tunneling(World Scien- tific, 2013)

    M. Razavy,Quantum theory of tunneling(World Scien- tific, 2013)

    2013

  60. [69]

    C. F. de Morisson Faria and X. Liu, Journal of Modern Optics58, 1076 (2011)

    2011

  61. [70]

    Bergues, M

    B. Bergues, M. K¨ ubel, N. G. Johnson, B. Fischer, N. Ca- mus, K. J. Betsch, O. Herrwerth, A. Senftleben, A. M. Sayler, T. Rathje,et al., Nature communications3, 813 (2012)

    2012

  62. [71]

    Brabec, M

    T. Brabec, M. Y. Ivanov, and P. B. Corkum, Physical Review A54, R2551 (1996)

    1996

  63. [72]

    Comtois, D

    D. Comtois, D. Zeidler, H. P´ epin, J. Kieffer, D. Vil- leneuve, and P. Corkum, Journal of Physics B: Atomic, Molecular and Optical Physics38, 1923 (2005)

    1923

  64. [73]

    R. M. Arkhipov, M. V. Arkhipov, A. V. Pakhomov, P. A. Obraztsov, and N. N. Rosanov, JETP Letters117, 8 (2023)

    2023

  65. [74]

    U. S. Sainadh, H. Xu, X. Wang, A. Atia-Tul-Noor, W. C. Wallace, N. Douguet, A. Bray, I. Ivanov, K. Bartschat, A. Kheifets,et al., Nature568, 75 (2019)

    2019

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