{"id":"6f7c78ae-8b89-4169-8ceb-38a8080d3a21","arxiv_id":"2507.15972","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Bright squeezed vacuum is modeled as a bundle of Bohmian field trajectories, and electron tunneling probabilities are averaged over the bundle to predict the multiphoton-to-tunneling crossover in a tip-surface junction.","lead":"A theory paper treats a squeezed vacuum light pulse as a bundle of classical electric-field histories and computes electron tunneling for each one, then averages the results. This gives a way to predict how quantum light statistics appear in the current of a tunneling microscope, including the crossover from multiphoton to direct tunneling.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (12) averages over a one-parameter Bohmian field ensemble that does not reproduce the two-time field correlations controlling the electron action; the no-backaction assumption is not the only missing link.","rationale":"The reader correctly isolates the averaging step in Eq. (12) as the load-bearing point, but attributes it to backaction. The stronger difficulty is that the no-backaction replacement of the field by Bohmian c-number trajectories is itself an unproven averaging prescription. In standard Bohmian mechanics for a harmonic mode, only the position/quadrature distribution is reproduced by the trajectory ensemble; the local momentum is fixed by the wavefunction, so it cannot carry the independent quadrature fluctuations of the quantum field. Electron tunneling is a nonlinear functional of the field history, so multi-time correlations matter. The paper supplies no derivation of the factorization in Eq. (12) from the joint light-matter state and no comparison with an exact no-backaction calculation. I would therefore keep the CONDITIONAL verdict: the framework is coherent and the field-trajectory formulas (Eqs. 3-6) are internally consistent, but the central ensemble average needs a derivation or a quantitative benchmark. The proposed two-level test is a minimal, analytically solvable version that directly probes whether the Bohmian-field ensemble reproduces the field correlations needed by any electron dynamics. If the test passes for the relevant r, the concern is retired; if it fails, the Fig. 4 curve is not the quantum result.","tokens_in":11624,"tokens_out":23529,"duration_ms":298678,"concrete_test":"For the same single-mode BSV state and dipole coupling, take a two-level electron system with a short interaction window T comparable to the under-barrier time. In the weak-coupling limit the exact no-backaction excitation probability is P_ex ∝ ∫∫ g(t)g(t') ⟨E_hat(t)E_hat(t')⟩ dt dt', evaluated with the squeezed-vacuum two-time function. Compute the same P_ex by averaging the classical-field result over the Bohmian ensemble: integrate ρ(X_i) from Eq. (5) with E_B(t)=sqrt(ℏω/ε0V)P(t;X_i) from Eq. (3). If the two results differ for any r in the range 11–25, the one-parameter Bohmian ensemble does not carry the field correlations that drive the electron, and Eq. (12) needs either an independent derivation or a nontrivial correction before the Ptot(γpeak) curve in Fig. 4 can be trusted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central probability Ptot in Eq. (12) is obtained by replacing the BSV mode with classical driving fields E(t)=sqrt(ℏω/ε0V)P(t), where P(t) is the Bohmian momentum from Eqs. (3)-(4), and averaging over the Gaussian initial quadrature X_i. This is presented as an exact unraveling, but it is a one-parameter family: for a fixed X_i the entire history P(t;X_i) is fixed. A single mode has two independent quadratures, and the exact no-backaction electron dynamics depends on the field through time-ordered correlation functions (equivalently the characteristic functional of the Gaussian state). Those correlations are not rank-one functions f(t)f(t'). For example, for a Gaussian wavefunction with exponent C=a+ib the Bohmian ensemble variance of P is b^2/(2a), while the quantum variance is (a^2+b^2)/(2a): the 'quantum potential' part is missing. Thus even with strictly zero backaction from the electron, the ensemble in Eq. (12) is not guaranteed to equal the reduced electron probability; it is an additional modeling assumption. The paper gives no derivation of Eq. (12) from the light-matter Schrödinger equation and no benchmark against an exact no-backaction solution, so the multi-time correlation content of the BSV field is untested.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11923,"tokens_out":3114,"duration_ms":37301,"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":[{"comment":"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).","section":"Trajectory-averaged tunneling probability, Eq. (12)"},{"comment":"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.","section":"Bohmian description of quantum light"},{"comment":"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.","section":"Introduction and Conclusion"}],"minor_comments":[{"comment":"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.","section":"Bohmian description of quantum light, Eq. (3)"},{"comment":"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.","section":"Quasiclassical non-adiabatic tunneling theory, after Eq. (11)"},{"comment":"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.","section":"Fig. 2 caption"},{"comment":"Reference [49] contains a typographical error in the author list: 'O. Coehn' should be 'O. Cohen'.","section":"Reference list"},{"comment":"The abstract contains the phrase 'the electron (under-) above-barrier dynamics', which reads awkwardly; consider rephrasing as 'under- and above-barrier dynamics'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper's reliance on the authors' own Ref. [71] is a legitimate import of an established quasiclassical tool, not circular reasoning. The main concern is the unproven status of Eq. (12); this is fixable by adding a derivation or a numerical benchmark. The manuscript fits the journal's scope and would be a useful contribution if the central averaging step is substantiated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThat 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.\n\nThe 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.","headline":"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.","tokens_in":12459,"tokens_out":8703,"would_cite":false,"duration_ms":101730,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Squeezed vacuum drives tunneling through a bundle of classical fields.","keywords":["bright squeezed vacuum","Bohmian trajectories","quantum light","tunneling","Keldysh parameter","multiphoton tunneling","field emission","tip–surface junction"],"falsifier":"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.","tokens_in":11434,"feed_emoji":"⚛️","tokens_out":5199,"duration_ms":57588,"temperature":0.7,"pith_summary":"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.","feed_headline":"Squeezed vacuum drives tunneling through a bundle of classical fields","feed_subtitle":"Electron current inherits the squeezed light statistics, crossing from multiphoton to direct tunneling as squeezing grows.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Bohmian mechanics formalism that converts the squeezed-vacuum wavefunction into deterministic field trajectories.","marker":"[57]"},{"why":"Provides the explicit squeezed-vacuum Bohmian trajectories for the field quadratures used in Eqs. (3)–(4).","marker":"[66]"},{"why":"Supplies the generalized quasiclassical non-adiabatic tunneling formulation that handles the asymmetric field profiles generated by the trajectories.","marker":"[71]"},{"why":"Defines the Keldysh parameter and its multiphoton-versus-direct-tunneling classification that the paper generalizes to quantum light.","marker":"[75–77]"},{"why":"Establishes the experimental availability of bright squeezed vacuum pulses with macroscopic photon numbers that motivate the scenario.","marker":"[42,43]"},{"why":"Provides the parameter choices for mode frequency and field amplitude used in the illustrative tip–surface calculation.","marker":"[45]"}],"fun_headline_variants":["Bohmian field trajectories translate squeezed vacuum into classical tunneling","Squeezed vacuum tunneling emerges from average over Bohmian fields","Quantum light tunneling unraveled: each field path is classical","Bohmian trajectories bridge multiphoton to direct tunneling under squeezed light","Squeezed vacuum's quantum stats arise from classical Bohmian field bundle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bohmian field trajectories translate squeezed vacuum into classical tunneling","Squeezed vacuum tunneling emerges from average over Bohmian fields","Quantum light tunneling unraveled: each field path is classical","Bohmian trajectories bridge multiphoton to direct tunneling under squeezed light","Squeezed vacuum's quantum stats arise from classical Bohmian field bundle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2872,"prompt_tokens":1044,"completion_tokens":1828,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":660,"completion_tokens_details":{"reasoning_tokens":1738}},"tokens_in":660,"tokens_out":1828,"duration_ms":15842,"temperature":1.0,"reasoning_tokens":1738,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:22:40.817047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"hidden” variables. I, Phys. Rev. 85, 166 (1952); A suggested interpretation of the quantum the- ory in terms of “hidden","cited_arxiv_id":null,"evidence_quote":"Supplies the Bohmian mechanics formalism that converts the squeezed-vacuum wavefunction into deterministic field trajectories."},{"cited_title":"What Bohmian mechanic says about arrival times of 1D vacuum squeezed states","cited_arxiv_id":"2502.05734","evidence_quote":"Provides the explicit squeezed-vacuum Bohmian trajectories for the field quadratures used in Eqs. (3)–(4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the generalized quasiclassical non-adiabatic tunneling formulation that handles the asymmetric field profiles generated by the trajectories."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the parameter choices for mode frequency and field amplitude used in the illustrative tip–surface calculation."}],"review_version":1}