Tunneling driven by quantum light described via field Bohmian trajectories

Andrey S. Moskalenko; Denis V. Seletskiy; Sangwon Kim; Seongjin Ahn

REVIEW 3 major objections 5 minor 77 references

Tunneling driven by quantum light described via field Bohmian trajectories

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Squeezed vacuum drives tunneling through a bundle of classical fields.

desk verdict Fresh idea—Bohmian trajectories for the light field—but Eq. (12) is assumed, not derived, and the one-parameter field ensemble misses the multi-time correlations that control the electron's reduced dynamics. read the letter →

arxiv 2507.15972 v1 pith:KCDGLCSE submitted 2025-07-21 quant-ph physics.optics

Sangwon Kim , Seongjin Ahn , Denis V. Seletskiy , Andrey S. Moskalenko This is my paper · ORCID

classification quant-phphysics.optics

keywords brightsqueezedvacuumBohmiantrajectoriesquantumlighttunnelingKeldyshparametermultiphotonfieldemissiontip–surfacejunction

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 tries to establish that quantum light can be represented, without approximation, by a statistical ensemble of deterministic classical field trajectories, and that electron tunneling driven by such light can be computed as an ensemble average over those trajectories. For a one-sided tip–surface junction illuminated by bright squeezed vacuum, the total tunneling probability is shown to be the average of single-realization quasiclassical tunneling probabilities weighted by the Gaussian distribution of the initial Bohmian field quadrature. The payoff is a concrete picture of how quantum light statistics are imprinted on the measured electron current, and a demonstration that the standard Keldysh classification of multiphoton versus direct tunneling extends smoothly to quantum light as the squeezing parameter grows.

What carries the argument

The central object is the Bohmian trajectory of the field quadrature $X(t)$ for a single-mode squeezed vacuum, governed by $\dot{X} = \omega\,\mathrm{Im}[\partial_X\psi/\psi]$, with the electric field proportional to $P = \dot{X}/\omega$. Each trajectory is labeled by its initial value $X_i$, drawn from the Gaussian density $\rho(X_i)$, so the quantum state becomes a statistical ensemble of classical driving fields. These fields are fed into a generalized quasiclassical non-adiabatic tunneling theory, based on complex-time Hamilton–Jacobi dynamics, which yields a tunneling probability $P(X_i)$ for each realization. Averaging over the ensemble according to Eq. (12) imprints the quantum statistics of light onto the electron current, and the restriction to $X_i<0$ encodes the directional tip-to-surface geometry of the tunneling junction.

What would settle it

A decisive check is to solve the full light–matter Schrödinger equation for a single electron coupled to one squeezed vacuum mode without neglecting backaction, and compare the exact joint transition probability with the ensemble average in Eq. (12); a significant deviation for moderate squeezing (for example, $r\approx 1$–$3$) would show that the no-backaction ensemble average is not the full quantum probability. Experimentally, measuring the electron emission rate from a tip driven by bright squeezed vacuum as a function of $r$ and comparing the shape of $P_{\mathrm{tot}}(\gamma_{\mathrm{peak}})$ would settle the claim.

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Extended reading notes

Core claim

The central claim is that a single-mode squeezed vacuum state can be unraveled exactly into a bundle of Bohmian field trajectories, each carrying a definite classical electric-field profile, and that for a one-sided tip–surface junction the tunneling probability is the ensemble average $P_{\mathrm{tot}} = \int_{-\infty}^{0} P(X_i)\,\rho(X_i)\,dX_i$ (Eq. 12). Here $\rho(X_i)$ is the Gaussian probability density of the initial field quadrature and $P(X_i)$ is the quasiclassical non-adiabatic tunneling probability computed for the corresponding classical field realization. This trajectory-averaged probability produces a curve $P_{\mathrm{tot}}(\gamma_{\mathrm{peak}})$ that smoothly crosses from the multiphoton regime to the direct tunneling regime as the squeezing parameter $r$ increases, with the effective Keldysh parameter $\gamma_{\mathrm{peak}}$ passing through unity. The paper argues that this gives a rigorous and intuitive framework for light–matter interaction with non-classical light, avoiding the approximations of coherent-state expansions with positive $P$-distributions.

Load-bearing premise

The electron never reacts back on the light field, so each Bohmian field trajectory can be treated as an independent classical driving field and the single-electron probability is just the average of classical probabilities; if the emitted electron changes the field mode or entangles with it, the result is not the full quantum probability.

Editorial extensions

If this is right

The measured tunneling current from a tip driven by bright squeezed vacuum should follow the ensemble-averaged curve $P_{\mathrm{tot}}(\gamma_{\mathrm{peak}})$, so quantum light statistics are directly observable in the electron current.
As the squeezing parameter $r$ grows, the same junction crosses from multiphoton to direct tunneling, generalizing the Keldysh classification to quantum-light driving.
The optimal field realization $X_{\mathrm{peak}}$ shifts relative to the Gaussian width as $r$ changes, and the corresponding peak field strength $E_{\mathrm{peak}}$ grows exponentially with $r$, providing a quantitative handle on the crossover.
The framework gives a platform for treating optically induced tunneling with arbitrary quantum states of light and for bridging semiclassical electron-dynamics theories with fully quantum analogues.

Reading between the lines

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

Inference: The no-backaction assumption likely breaks down for strong coupling or high electron emission rates; a natural extension is to include the electron's reaction on the field mode and test whether the ensemble average in Eq. (12) remains the full quantum probability.
Inference: Extending the bundle picture to multimode squeezed vacuum may reveal intermode correlations as correlations in the tunneling current, not just in the mean rate.
Inference: A discriminating experiment could measure the full counting statistics of emitted electrons: a classical mixture per realization would produce a specific Poisson-mixture signature, whereas genuine quantum field entanglement would add extra correlations beyond the ensemble average.
Inference: The smooth crossover in Fig. 4 suggests a quantitative relation between squeezing and the effective Keldysh parameter that, if confirmed, could allow squeezing to be calibrated from tunneling-current measurements alone.

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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

The big new thing here is that the Bohmian trajectories are put on the light side, not the electron side. Prior Bohmian strong-field work kept the field classical and made the electron's position stochastic; this paper turns that around for a single-mode squeezed vacuum and feeds the resulting field histories into the quasiclassical tunneling machinery of Ref. [71]. That is a genuinely fresh combination, and it yields an easy-to-read picture: each field realization produces a tunneling probability P(Xi), and quantum statistics enter through the Gaussian weight ρ(Xi). Fig. 4 is physically plausible and the Keldysh-parameter presentation is nice.

The soft spot is Eq. (12). The paper simply assumes that the total tunneling probability is the ensemble average over the Bohmian field histories. The no-backaction assumption, stated after Eq. (4), is the only justification. Even granting that, the stress-test note is right: the Bohmian ensemble is a one-parameter family (initial Xi), while a single mode has two independent quadratures. The ensemble therefore cannot reproduce the full two-time correlation function of the field, which is what controls the electron's reduced dynamics under a Gaussian state. For a real Gaussian wavefunction the Bohmian momentum variance is zero while the quantum variance is positive; the quantum potential contribution is missing. So the distribution of E(t) in the bundle differs from the quantum field distribution, and Eq. (12) is an additional modeling assumption, not a derivation. It might be an excellent approximation for bright fields where backaction is negligible, but the paper gives no criterion and no benchmark against an exact no-backaction solution or against the classical limit.

That said, this is not a nonsensical paper. The formalism is coherent, the citation practice is honest (the tunneling theory is imported from their own Ref. [71], but used as a tool, not to define the target result), and the application to STM geometry is timely. The missing derivation is addressable: one could compute the reduced electron density matrix for a simple barrier with a Gaussian field state and compare with the ensemble average. Until that is done, the exactness claim is unsupported.

The paper deserves a serious referee—it is novel, readable, and the flaw is fixable. I would send it to review with the demand that the authors either derive Eq. (12) from the light-matter Schrödinger equation under controlled approximations or benchmark it against an exact calculation. A reading group would enjoy dissecting the correlation issue. I wouldn't cite it in my own work yet—I'd wait for the benchmark.

Referee Report

3 major / 5 minor

Summary. The paper proposes a framework for describing tunneling driven by quantum light, specifically bright squeezed vacuum (BSV), by representing the single-mode quantum field as an ensemble of deterministic Bohmian field trajectories. Each trajectory provides a classical driving field E(t) = sqrt(hbar omega / epsilon0 V) P(t), for which the electron tunneling probability is computed using the quasiclassical non-adiabatic theory of Ref. [71]. The total tunneling probability is then given by the ensemble average in Eq. (12), Ptot = integral over negative initial quadratures of P(X_i) rho(X_i) dX_i. The authors apply this to a gold tip-surface junction with a 5 eV barrier and a 0.775 eV mode, and obtain a smooth crossover from multiphoton to direct tunneling as the squeezing parameter r increases, expressed through the effective Keldysh parameter gamma_peak in Fig. 4. The central claim is that this provides an exact and intuitive unraveling of quantum light into classical field realizations for tunneling problems.

Significance. If the central formula Eq. (12) were rigorously justified, the paper would give a computationally efficient and conceptually clear route to quantum-light-driven tunneling, potentially extendable to arbitrary quantum states and multimode fields. Strengths include the internal consistency of the Bohmian field derivation in Eqs. (1)-(6), the use of an established quasiclassical tunneling theory, and the absence of parameters fitted to the final Ptot curve. The paper also addresses an experimentally relevant geometry and connects to recent BSV experiments. However, the load-bearing step, namely the replacement of the quantum field by a one-parameter Bohmian trajectory ensemble and the averaging in Eq. (12), is assumed rather than derived, and no benchmark is provided against an exact no-backaction or full quantum calculation. Because this step is central to the quantitative predictions, the significance of the results as stated depends on an unproven equivalence; with appropriate derivation or numerical validation, the framework could become a valuable tool for the quantum-optics and strong-field communities.

major comments (3)

[Trajectory-averaged tunneling probability, Eq. (12)] Eq. (12) is the central result, but it is assumed rather than derived from the joint light-electron dynamics. The text after Eq. (4) states 'Assuming that there is no backaction of the electron to the field, we can select ti arbitrarily,' yet even with strictly zero backaction, the reduced electron probability is determined by the multi-time correlation functions of the field, equivalently by its characteristic functional. For a single mode, a Gaussian state has two independent quadratures, and its two-time correlations are not rank-one functions f(t)f(t'). The Bohmian ensemble in Eq. (12) is a one-parameter family: for a fixed X_i the entire history P(t; X_i) is fixed. To see the discrepancy, for a Gaussian wavefunction with exponent C = a + ib, the Bohmian ensemble variance of P is b^2/(2a), whereas the quantum variance is (a^2 + b^2)/(2a); the 'quantum potential' contribution (a^2/(2a) = a/2) is missing from the Bohmian variance. Thus Eq. (12) is not an exact unraveling but an additional modeling assumption. I request either a derivation of Eq. (12) from the light-matter Schrodinger equation under the no-backaction condition, or a numerical benchmark against an exact solution of the Schrodinger equation for the electron driven by the quantized single mode with backaction artificially neglected, for the parameters used in the paper. This is load-bearing because the quantitative Ptot(gamma_peak) curve in Fig. 4 rests entirely on Eq. (12).
[Bohmian description of quantum light] The paper gives no estimate of when the no-backaction assumption becomes significant. The parameters include a macroscopic photon number for large r and a field amplitude that grows exponentially with r, so the electron's emission could in principle modify the field mode or entangle it with the electron. The statement 'Assuming that there is no backaction' is presented without a criterion. An order-of-magnitude estimate based on the coupling strength, photon number, and interaction time would clarify the domain of validity. Without such an estimate, it is unclear whether the predicted crossover in Fig. 4 is robust for realistic BSV pulses or only for a hypothetical strictly no-backaction setting.
[Introduction and Conclusion] The manuscript repeatedly calls the trajectory decomposition 'exact' (e.g., 'an exact decomposition of quantum light into a bundle of deterministic field trajectories' in the Introduction and 'rigorous framework' in the Conclusion). In light of the missing two-time correlation content of the one-parameter Bohmian ensemble, this wording is too strong. Even if the no-backaction assumption holds, the equivalence between the quantum field and the trajectory ensemble for the specific tunneling observable must be proven or numerically demonstrated before calling the decomposition exact. I recommend softening the wording or providing the missing proof, since the current phrasing overstates what has been established.

minor comments (5)

[Bohmian description of quantum light, Eq. (3)] The symbol P(t) in Eq. (3) denotes the Bohmian momentum quadrature, while earlier in the same paragraph P is used for the operator quadrature (e.g., in [X,P] = i and in the definition of E). This dual use could confuse readers; please introduce a distinct notation, e.g., P_B(t) or a subscript, for the Bohmian trajectory value.
[Quasiclassical non-adiabatic tunneling theory, after Eq. (11)] The statement '0 < epsilon << 1 for all relevant realizations and considered values of r' would benefit from a quantitative bound or a figure showing epsilon as a function of r and X_i, since the real-time emergence time tau_0 is claimed to be very close to the vertical edges of the field profile.
[Fig. 2 caption] The caption says 'Red dotted line indicates t = tau_0 for all of the trajectories,' but tau_0 generally depends on the realization and on r. Please clarify whether the displayed line is for a representative trajectory or whether tau_0 is indeed common to all plotted curves.
[Reference list] Reference [49] contains a typographical error in the author list: 'O. Coehn' should be 'O. Cohen'.
[Introduction] The abstract contains the phrase 'the electron (under-) above-barrier dynamics', which reads awkwardly; consider rephrasing as 'under- and above-barrier dynamics'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Bohmian field ensemble and Eq. (12) are not a concealed restatement of the input; only a minor methodological self-citation is present.

full rationale

The paper's central chain is not circular. The field Bohmian trajectories are obtained from the squeezed-vacuum wavefunction via the continuity equation (Eqs. (2)-(6)), and the ensemble distribution rho(X_i) is the modulus squared of that wavefunction, not a quantity fitted to the final tunneling curve. For each fixed initial quadrature X_i the field realization E(t)=sqrt(hbar omega/eps0 V) P(t) is deterministic, and P(X_i) is computed by solving the quasiclassical action equations, with no parameter adjusted to reproduce the target Ptot. Eq. (12) is the Bohmian ensemble-averaging rule applied to the one-sided geometry, not an equation that defines P(X_i) in terms of Ptot. The effective Keldysh parameter in Eq. (13) is a derived plotting variable: Ptot is computed first and gamma_peak is then introduced as a function of the maximizing realization, so the Fig. 4 curve is not a fit dressed as a prediction. The only author-overlapping citation that carries computational weight is Ref. [71], which supplies the quasiclassical non-adiabatic tunneling machinery; this is a methodological import rather than a self-referential uniqueness or ansatz claim, and the central Bohmian-field premise is derived in the present paper rather than taken from that citation. The explicit no-backaction assumption is a validity limitation, and the concern that a single-parameter Bohmian ensemble may not reproduce the two-time field correlations seen by the electron is a correctness question about the modeling step, not an instance of an output being equivalent to its input by construction. The score of 2 reflects only the presence of the minor self-citation, not any demonstrated circular reduction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on reducing single-mode squeezed light to a bundle of Bohmian trajectories and on applying a quasiclassical tunneling calculation to each. No parameters are fitted to the output curve: ΔU, ω, and the field amplitude are experimental inputs, and r is scanned. The least supported steps are the single-mode truncation, no-backaction factorization, rectangular step barrier, and exponential-accuracy quasiclassics. No new physical entities are introduced.

free parameters (4)

Barrier height ΔU = 5 eV
Sets the tunneling exponent scale; chosen as a representative gold tip-surface work function, not fitted to the final result.
Single-mode frequency ω = 0.775 eV (0.0285 a.u.)
Central frequency of the squeezed vacuum mode, tied to Ref. [67] parameters; enters the effective Keldysh parameter.
Field amplitude prefactor sqrt(hbar omega / epsilon0 V) = sqrt(2) x 10^-8 a.u.
Sets the absolute electric-field scale for each trajectory; value chosen similar to Ref. [45], not optimized.
Squeezing parameter r = scanned from 11 to 25
Controls the quantum statistics and field amplitude; varied to display the crossover, not fitted to data.

assumptions (5)

domain assumption The BSV pulse is represented by a single optical mode with fixed linear polarization.
Introduced in 'For simplicity, we restrict our consideration to a single-frequency mode...' Real BSV pulses are multi-mode; this restricts the exactness claim.
domain assumption The electron exerts no backaction on the light field.
Stated after Eq. (4). It justifies choosing ti arbitrarily and averaging independent field realizations; if backaction matters, Eq. (12) is not the full joint probability.
domain assumption The static potential is U(x) = -Delta U Theta(-x), a constant step at the emission point.
Assumed in the quasiclassical section. Ignores image potentials, gap geometry, and barrier curvature; affects quantitative tunneling probabilities.
domain assumption The quasiclassical non-adiabatic tunneling theory of Ref. [71] is valid for each field realization.
Used to compute P(Xi). It gives exponential accuracy only and omits prefactors and multi-path interference.
domain assumption The electron starts under the barrier with H = -Delta U and complex velocity i sqrt(2 Delta U / m) at complex time t0.
Boundary conditions in the quasiclassical method; assumes no initial spread or field-induced modification before the tunneling point.

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Cite this review

Pith. "Pith review of Tunneling driven by quantum light described via field Bohmian trajectories." pith.science (2026). https://pith.science/paper/KCDGLCSE

@misc{pith2026250715972,
  author       = {Pith},
  title        = {Pith review of: Tunneling driven by quantum light described via field Bohmian trajectories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCDGLCSE}},
  note         = {Machine review of arXiv:2507.15972}
}

read the original abstract

Recent realization of an intense quantum light, namely bright squeezed vacuum, opened a new perspective on quantum light-matter interaction. Several theoretical works have appeared based on coherent state expansions of quantum state of light to investigate non-classical driving of high-harmonic generation in atomic gases and solids, or free-electron dynamics, but their predictions surprisingly coincide with what one could expect from essentially classical interpretations of the light statistics. A deeper theoretical insight into the underlying physics is necessary for understanding of observed experimental findings and predicting emerging effects relying on this new configuration. Here we present a theoretical framework to describe tunneling driven by quantum light, where the properties of such light are captured by a statistical ensemble of classical fields via a hydrodynamic, also referred to as Bohmian, formulation. Generalizing the quasiclassical theory of non-adiabatic tunneling driven by classical light, a single tunneling event is described by a bundle of tunneling solutions, each driven by a classical field corresponding to one realization in the ensemble. Quantum statistics of light are thus imprinted on the measured current. Fully quantum description of light via the Bohmian trajectories of its field provides a perfect fit to the description of the electron (under-) above-barrier dynamics in terms of (complex quasiclassical) real classical trajectories, resulting in a consistent and elegant theoretical approach. To illustrate this, we consider BSV-induced electron transport from the tip to the surface in the tunneling microscope configuration demonstrating the transition from the multiphoton to the direct tunneling regime.

Figures

Figures reproduced from arXiv: 2507.15972 by the authors.

**Figure 1.** FIG. 1. (a) Schematic diagram of the setup. A squeezed vacuum state of light drives electronic transport in a metal tip–surface [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Trajectories of the released electrons in the classically [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4. Dependence of the total tunneling probability of the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Reference graph

Works this paper leans on

77 extracted references · 65 canonical work pages

[71]

S. Kim, T. Schmude, G. Burkard, and A. S. Moskalenko, Quasiclassical theory of non-adiabatic tunneling in nanocontacts induced by phase-controlled ultrashort light pulses, New J. Phys. 23, 083006 (2021)

work page 2021
[1]

McPherson, G

A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases, J. Opt. Soc. Am. B 4, 595 (1987)

1987
[2]

Ferray, A

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompre, G. Main- fray, and C. Manus, Multiple-harmonic conversion of 1064 nm radiation in rare gases, J. Phys. B 21, L31 (1988)

work page 1988
[3]

Ghimire, A

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nat. Phys. 7, 138 (2011)

work page 2011
[4]

T. T. Luu, Z. Yin, A. Jain, T. Gaumnitz, Y. Pertot, J. Ma, and H. J. W¨ orner, Extreme–ultraviolet high– harmonic generation in liquids, Nat. Commun. 9, 3723 6 (2018)

work page 2018
[5]

S. Han, H. Kim, Y. W. Kim, Y.-J. Kim, S. Kim, I.-Y. Park, and S.-W. Kim, High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanos- tructure, Nat. Commun. 7, 13105 (2016)

work page 2016
[6]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Aug´ e, P. Balcou, H. G. Muller, and P. Agostini, Observation of a train of attosecond pulses from high harmonic generation, Science 292, 1689 (2001)

work page 2001
[7]

Hentschel, R

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Rei- der, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Attosecond metrology, Na- ture (London) 414, 509 (2001)

work page 2001

Show all 77 references

[8]

Krausz and M

F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys. 81, 163 (2009)

2009
[9]

P. B. Corkum and F. Krausz, Attosecond science, Nat. Phys. 3, 381 (2007)

2007
[10]

D. M. Villeneuve, Attosecond science, Contemp. Phys. 59, 47 (2018)

2018
[11]

Goulielmakis, V

E. Goulielmakis, V. S. Yakovlev, A. L. Cavalieri, M. Uiberacker, V. Pervak, A. Apolonski, R. Kienberger, U. Kleineberg, and F. Krausz, Attosecond control and measurement: Lightwave electronics, Science 317, 769 (2007)

2007
[12]

Eckle, M

P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Sch¨ offler, H. G. Muller, R. D¨ orner, and U. Keller, Attosecond angular streaking, Nat. Phys. 4, 565 (2008)

2008
[13]

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Extreme ultraviolet high- harmonic spectroscopy of solids, Nature (London) 521, 498 (2015)

2015
[14]

J. L. Krause, K. J. Schafer, and K. C. Kulander, Cal- culation of photoemission from atoms subject to intense laser fields, Phys. Rev. A 45, 4998 (1992)

1992
[15]

J. L. Krause, K. J. Schafer, and K. C. Kulander, High- order harmonic generation from atoms and ions in the high intensity regime, Phys. Rev. Lett. 68, 3535 (1992)

1992
[16]

P. B. Corkum, Plasma perspective on strong field multi- photon ionization, Phys. Rev. Lett. 71, 1994 (1993)

1993
[17]

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Ku- lander, Above threshold ionization beyond the high har- monic cutoff, Phys. Rev. Lett. 70, 1599 (1993)

1993
[18]

L’Huillier, M

A. L’Huillier, M. Lewenstein, P. Sali` eres, P. Balcou, M. Y. Ivanov, J. Larsson, and C. G. Wahlstr¨ om, High- order harmonic-generation cutoff, Phys. Rev. A 48, R3433 (1993)

1993
[19]

Winterfeldt, C

C. Winterfeldt, C. Spielmann, and G. Gerber, Collo- quium: Optimal control of high-harmonic generation, Rev. Mod. Phys. 80, 117 (2008)

2008
[20]

Lewenstein, P

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, Theory of high-harmonic generation by low-frequency laser fields, Phys. Rev. A 49, 2117 (1994)

1994
[21]

Becker, A

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, A unified theory of high-harmonic generation: Application to polarization properties of the harmonics, Phys. Rev. A 56, 645 (1997)

1997
[22]

Y. I. Salamin, S. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, Relativistic high-power laser–matter interactions, Phys. Rep. 427, 41 (2006)

2006
[23]

O. E. Alon, V. Averbukh, and N. Moiseyev, Selection rules for the high harmonic generation spectra, Phys. Rev. Lett. 80, 3743 (1998)

1998
[24]

Hern´ andez-Garc ´ ıa, J

C. Hern´ andez-Garc ´ ıa, J. A. P´ erez-Hern´ andez, T. Pop- mintchev, M. M. Murnane, H. C. Kapteyn, A. Jaron- Becker, A. Becker, and L. Plaja, Zeptosecond high har- monic keV X-ray waveforms driven by midinfrared laser pulses, Phys. Rev. Lett. 111, 033002 (2013)

2013
[25]

Vampa, C

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, Theoretical analysis of high-harmonic generation in solids, Phys. Rev. Lett.113, 073901 (2014)

2014
[26]

Schubert, M

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Sub-cycle control of tera- hertz high-harmonic generation by dynamical bloch os- cillations, Nat. Photon. 8, 119 (2014)

2014
[27]

Ndabashimiye, S

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Solid-state harmonics beyond the atomic limit, Nature (London) 534, 520 (2016)

2016
[28]

M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, Multilevel perspective on high-order harmonic generation in solids, Phys. Rev. A 94, 063403 (2016)

2016
[29]

Ghimire and D

S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nat. Phys. 15, 10 (2019)

2019
[30]

Yue and M

L. Yue and M. B. Gaarde, Introduction to theory of high- harmonic generation in solids: tutorial, J. Opt. Soc. Am. B 39, 535 (2022)

2022
[31]

R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963)

1963
[32]

R. J. Glauber, Coherent and incoherent states of the ra- diation field, Phys. Rev. 131, 2766 (1963)

1963
[33]

Davidovich, Sub-Poissonian processes in quantum op- tics, Rev

L. Davidovich, Sub-Poissonian processes in quantum op- tics, Rev. Mod. Phys. 68, 127 (1996)

1996
[34]

Chekhova, G

M. Chekhova, G. Leuchs, and M. ˙Zukowski, Bright squeezed vacuum: Entanglement of macroscopic light beams, Opt. Commun. 337, 27 (2015)

2015
[35]

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, 30 years of squeezed light generation, Physica Scripta 91, 053001 (2016)

2016
[36]

Cruz-Rodriguez, D

L. Cruz-Rodriguez, D. Dey, A. Freibert, and P. Stammer, Quantum phenomena in attosecond science, Nat. Rev. Phys. 6, 691 (2024)

2024
[37]

Gorlach, O

A. Gorlach, O. Neufeld, N. Rivera, O. Cohen, and I. Kaminer, The quantum-optical nature of high har- monic generation, Nat. Commun. 11, 4598 (2020)

2020
[38]

Stammer, J

P. Stammer, J. Rivera-Dean, A. S. Maxwell, T. Lam- prou, J. Arg¨ uello-Luengo, P. Tzallas, M. F. Ciappina, and M. Lewenstein, Entanglement and squeezing of the optical field modes in high harmonic generation, Phys. Rev. Lett. 132, 143603 (2024)

2024
[39]

Sloan, A

J. Sloan, A. Gorlach, M. E. Tzur, N. Rivera, O. Cohen, I. Kaminer, and M. Soljaˇ ci´ c, Entangling extreme ultravi- olet photons through strong field pair generation (2023), arXiv:2309.16466 [quant-ph]

2023 arXiv
[40]

C. S. Lange, T. Hansen, and L. B. Madsen, Electron- correlation-induced nonclassicality of light from high- order harmonic generation, Phys. Rev. A 109, 033110 (2024)

2024
[41]

Lewenstein, M

M. Lewenstein, M. F. Ciappina, E. Pisanty, J. Rivera- Dean, P. Stammer, T. Lamprou, and P. Tzallas, Gener- ation of optical Schr¨ odinger cat states in intense laser– matter interactions, Nat. Phys. 17, 1104 (2021)

2021
[42]

K. Y. Spasibko, D. A. Kopylov, V. L. Krutyanskiy, T. V. Murzina, G. Leuchs, and M. V. Chekhova, Multiphoton effects enhanced due to ultrafast photon-number fluctu- 7 ations, Phys. Rev. Lett. 119, 223603 (2017)

2017
[43]

Manceau, K

M. Manceau, K. Y. Spasibko, G. Leuchs, R. Filip, and M. V. Chekhova, Indefinite-mean pareto photon distri- bution from amplified quantum noise, Phys. Rev. Lett. 123, 123606 (2019)

2019
[44]

M. E. Tzur, M. Birk, A. Gorlach, M. Kr¨ uger, I. Kaminer, and O. Cohen, Photon-statistics force in ultrafast elec- tron dynamics, Nat. Photonics 17, 501 (2023)

2023
[45]

M. E. Tzur and O. Cohen, Motion of charged particles in bright squeezed vacuum, Light Sci. Appl. 13, 41 (2024)

2024
[46]

Gorlach, M

A. Gorlach, M. E. Tzur, M. Birk, M. Kr¨ uger, N. Rivera, O. Cohen, and I. Kaminer, High-harmonic generation driven by quantum light, Nat. Phys. 19, 1689 (2023)

2023
[47]

M. E. Tzur, M. Birk, A. Gorlach, I. Kaminer, M. Kr¨ uger, and O. Cohen, Generation of squeezed high-order har- monics, Phys. Rev. Res. 6, 033079 (2024)

2024
[48]

Stammer, Absence of quantum optical coherence in high harmonic generation, Phys

P. Stammer, Absence of quantum optical coherence in high harmonic generation, Phys. Rev. Res. 6, L032033 (2024)

2024
[49]

Rasputnyi, Z

A. Rasputnyi, Z. Chen, M. Birk, O. Coehn, , I. Kaminer, M. Kr¨ uger, D. Seletskiy, M. V. Chekhova, and F. Tani, High-harmonic generation by a bright squeezed vacuum, Nat. Phys. 20, 1960 (2024)

2024
[50]

W. P. Schleich, Quantum Optics in Phase Space (Wiley- VCH Verlag, Weinheim, Germany, 2001)

2001
[51]

Ma and M

B. Ma and M. Kr¨ uger, Strong-field theory of attosec- ond tunneling microscopy, Phys. Rev. Lett. 133, 236901 (2024)

2024
[52]

H. Y. Kim, M. Garg, S. Mandal, L. Seiffert, T. Fennel, and E. Goulielmakis, Attosecond field emission, Nature (London) 613, 662 (2023)

2023
[53]

Heimerl, A

J. Heimerl, A. Mikhaylov, S. Meier, H. H¨ ollerer, I. Kaminer, M. Chekhova, and P. Hommelhoff, Multi- photon electron emission with non-classical light, Nat. Phys. 20, 945 (2024)

2024
[54]

Siday, J

T. Siday, J. Hayes, F. Schiegl, F. Sandner, P. Menden, V. Bergbauer, M. Zizlsperger, S. Nerreter, S. Lingl, J. Repp, J. Wilhelm, M. A. Huber, Y. A. Gerasimenko, and R. Huber, All-optical subcycle microscopy on atomic length scales, Nature (London) 629, 329 (2024)

2024
[55]

G. Yang, M. Kizmann, A. Leitenstorfer, and A. S. Moskalenko, Subcycle tomography of quantum light (2023), arXiv:2307.12812 [quant-ph]

2023
[56]

G. Yang, S. Sharma, and A. S. Moskalenko, Electro- optic sampling of the electric-field operator for ultra- broadband pulses of gaussian quantum light (2025), arXiv:2506.01730 [quant-ph]

2025
[57]

hidden” variables. I, Phys. Rev. 85, 166 (1952); A suggested interpretation of the quantum the- ory in terms of “hidden

D. Bohm, A suggested interpretation of the quantum the- ory in terms of “hidden” variables. I, Phys. Rev. 85, 166 (1952); A suggested interpretation of the quantum the- ory in terms of “hidden” variables. II, 85, 180 (1952)

1952
[58]

H. Z. Jooya, D. A. Telnov, P.-C. Li, and S.-I. Chu, Explo- ration of the subcycle multiphoton ionization dynamics and transient electron density structures with Bohmian trajectories, Phys. Rev. A 91, 063412 (2015)

2015
[59]

H. Z. Jooya, D. A. Telnov, and S.-I. Chu, Exploration of the electron multiple recollision dynamics in intense laser fields with Bohmian trajectories, Phys. Rev. A 93, 063405 (2016)

2016
[60]

Li, Y.-L

P.-C. Li, Y.-L. Sheu, H. Z. Jooya, X.-X. Zhou, and S.-I. Chu, Exploration of laser-driven electron- multirescattering dynamics in high-order harmonic gen- eration, Sci. Rep. 6, 32763 (2016)

2016
[61]

I. A. Ivanov, C. H. Nam, and K. T. Kim, Exit point in the strong field ionization process, Sci. Rep. 7, 39919 (2017)

2017
[62]

Douguet and K

N. Douguet and K. Bartschat, Dynamics of tunneling ionization using Bohmian mechanics, Phys. Rev. A 97, 013402 (2018)

2018
[63]

T. Moon, K. Bartschat, and N. Douguet, Strong-field ionization phenomena revealed by quantum trajectories, Phys. Rev. Lett. 133, 073201 (2024)

2024
[64]

Sharoglazova, M

V. Sharoglazova, M. Puplauskis, C. Mattschas, C. Toebes, and J. Klaers, Energy–speed relationship of quantum particles challenges Bohmian mechanics, Nature (London) 643, 67 (2025)

2025
[65]

Madelung, Quantentheorie in hydrodynamischer form, Z

E. Madelung, Quantentheorie in hydrodynamischer form, Z. Phys. 40, 322 (1927)

1927
[66]

Garcia-Chung and H

A. Garcia-Chung and H. G. Laguna, What Bohmian me- chanic says about arrival times of 1d vacuum squeezed states (2025), arXiv:2502.05734 [quant-ph]

2025 arXiv
[67]

C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer, Direct sampling of electric-field vacuum fluctuations, Science 350, 420 (2015)

2015
[68]

L. D. Landau and E. M. Lifshitz, Quantum Mechan- ics: Non-Relativistic Theory , 3rd ed. (Pergamon, Oxford, 1977)

1977
[69]

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Ionization of atoms in an alternating electric field, Sov. Phys. JETP 23, 924 (1966)

1966
[70]

V. S. Popov, Imaginary-time method in quantum me- chanics and field theory, Phys. At. Nucl. 68, 686 (2005)

2005
[72]

S. T. Epstein, General solutions of the Hamilton-Jacobi equation, Am. J. Phys. 32, 688 (1964)

1964
[73]

A. S. Moskalenko, S. D. Ganichev, V. I. Perel’, and I. N. Yassievich, Magnetic field effect on tunnel ionization of deep impurities by far-infrared radiation, Physica B 273, 1007 (1999)

1999
[74]

A. S. Moskalenko, V. I. Perel’, and I. N. Yassievich, Effect of a magnetic field on thermally stimulated ionization of impurity centers in semiconductors by submillimeter radiation, JETP 90, 217 (2000)

2000
[75]

L. V. Keldysh, Ionization in the field of a strong electro- magnetic wave, Sov. Phys. JETP 20, 1307 (1965)

1965
[76]

V. S. Popov, Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory), Phys. Usp. 47, 855 (2004)

2004
[77]

V Popruzhenko, Keldysh theory of strong field ioniza- tion: history, applications, difficulties and perspectives, J

S. V Popruzhenko, Keldysh theory of strong field ioniza- tion: history, applications, difficulties and perspectives, J. Phys. B: At. Mol. Opt. Phys. 47, 204001 (2014)

2014

Pith tools

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

[1] [71]

S. Kim, T. Schmude, G. Burkard, and A. S. Moskalenko, Quasiclassical theory of non-adiabatic tunneling in nanocontacts induced by phase-controlled ultrashort light pulses, New J. Phys. 23, 083006 (2021)

work page 2021

[2] [1]

McPherson, G

A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases, J. Opt. Soc. Am. B 4, 595 (1987)

1987

[3] [2]

Ferray, A

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompre, G. Main- fray, and C. Manus, Multiple-harmonic conversion of 1064 nm radiation in rare gases, J. Phys. B 21, L31 (1988)

work page 1988

[4] [3]

Ghimire, A

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nat. Phys. 7, 138 (2011)

work page 2011

[5] [4]

T. T. Luu, Z. Yin, A. Jain, T. Gaumnitz, Y. Pertot, J. Ma, and H. J. W¨ orner, Extreme–ultraviolet high– harmonic generation in liquids, Nat. Commun. 9, 3723 6 (2018)

work page 2018

[6] [5]

S. Han, H. Kim, Y. W. Kim, Y.-J. Kim, S. Kim, I.-Y. Park, and S.-W. Kim, High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanos- tructure, Nat. Commun. 7, 13105 (2016)

work page 2016

[7] [6]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Aug´ e, P. Balcou, H. G. Muller, and P. Agostini, Observation of a train of attosecond pulses from high harmonic generation, Science 292, 1689 (2001)

work page 2001

[8] [7]

Hentschel, R

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Rei- der, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Attosecond metrology, Na- ture (London) 414, 509 (2001)

work page 2001