Pith. sign in

REVIEW 3 major objections 7 minor 36 references

Stokes phenomenon unifies pair generation in photonic time crystals

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · glm-5.2

2026-07-09 13:45 UTC pith:E4KZYW36

load-bearing objection Stokes/Airy mapping for PTC temporal boundaries is new and clean; quantum entanglement claims for the nonlinear regime are extrapolated from linear theory and not demonstrated. the 3 major comments →

arxiv 2607.07340 v1 pith:E4KZYW36 submitted 2026-07-08 physics.optics nlin.PS

Time domain Stokes mechanism of pair correlated k gap solitons in nonlinear photonic time crystal slabs

Jiaxiang Sun , Guowei Chen , Liang Zhang , Yiming Pan This is my paper
classification physics.optics nlin.PS
keywords photonic time crystalsStokes phenomenonk-gap solitonsphoton pair generationtime-varying mediaKerr nonlinearityBogoliubov transformationheralded quantum light
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that two processes previously treated separately in time-varying optical media — time reflection at temporal boundaries and exponential amplification inside momentum k-gaps — are both consequences of a single mechanism: the time-domain Stokes phenomenon. The authors map the wave equation near the switch-on and switch-off boundaries of a finite-duration photonic time crystal (PTC) slab onto the Airy equation, showing that each temporal boundary acts as a Stokes turning point. At the entry boundary, an incident mode (or vacuum fluctuation) crosses a Stokes line and activates an exponentially growing branch inside the k-gap. Kerr nonlinearity then arrests this unbounded growth and reshapes the frozen field into propagating k-gap solitons. At the exit boundary, a second Stokes transition converts these solitons back into oscillatory output modes, producing four spatially separated pulse branches. The paper claims these branches are entangled photon pairs, detectable via Hanbury Brown–Twiss bunching and Hong–Ou–Mandel anti-bunching measurements, and proposes a concrete four-port detection topology for experimental verification.

Core claim

The central discovery is the identification of a common mathematical structure — the Stokes phenomenon from asymptotic analysis — underlying both temporal-boundary scattering and k-gap amplification in photonic time crystals. By showing that the slowly varying envelope of the PTC wave equation reduces to the Airy equation near each temporal boundary, the authors demonstrate that mode conversion at temporal interfaces is governed by the same oscillatory-to-exponential branching that defines Stokes lines in classical asymptotics. This unification means that pair generation at the entry boundary, parametric amplification inside the k-gap, and secondary pair generation at the exit boundary are三个

What carries the argument

The Airy/Stokes turning-point mapping applied to temporal boundaries of a finite-duration PTC slab, combined with Kerr nonlinearity that saturates k-gap amplification into stable soliton branches.

Load-bearing premise

The claim that the four output branches are entangled photon pairs rests on applying linear quantum theory — the Bogoliubov transformation and two-mode squeezed states — to a regime where Kerr nonlinearity is essential. The classical FDTD simulations confirm the soliton dynamics, but the quantum treatment of the nonlinear regime is deferred to a companion manuscript that is not yet available.

What would settle it

If HBT and HOM measurements on the four output branches fail to show the predicted bunching peak and coincidence dip — or show them with visibility inconsistent with the squeezing parameter — the entanglement claim for the nonlinear soliton regime would not hold.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the Stokes unification is correct, temporal-boundary pair generation and k-gap amplification can be treated within a single analytical framework, simplifying the design of time-varying quantum light sources.
  • The four-port emission topology provides a directly testable prediction: HBT bunching peaks and HOM coincidence dips should appear in specific output branches, with visibility depending on the squeezing parameter set by modulation depth.
  • The amplitude-dependent scaling from single pairs to multiplexed entangled pulse trains (via higher-order k-gap solitons) suggests a tunable repetition-rate source without changing the modulation frequency.
  • The Stokes framework could extend to other time-varying platforms — epsilon-near-zero media, silicon waveguides, or time-modulated metasurfaces — wherever abrupt temporal boundaries produce mode conversion.
  • If experimentally confirmed, heralded detection of backward-propagating pulses would enable ultrafast quantum communication protocols where the idler arrival time is set by femtosecond-scale temporal switching.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Stokes mapping is derived in the linear regime; whether the Airy turning-point structure survives intact under strong Kerr nonlinearity — where the soliton itself modifies the local dispersion — is not explicitly verified in the paper's analytical treatment.
  • The four-branch entanglement claim inherits the structure of two sequential Bogoliubov transformations (one per boundary), but the cross-correlations between branches from different boundaries are not computed; only within-pair correlations are analyzed.
  • If the Stokes framework generalizes, one could predict pair-generation efficiencies directly from Airy connection formulas without full Floquet analysis, potentially offering a shortcut for engineering temporal boundaries in other time-varying media.
  • The proposal to use higher-order solitons for multiplexed entangled trains raises an unaddressed question: whether inter-soliton interactions in the multi-peak regime preserve the pairwise entanglement structure or introduce cross-talk between pulse pairs.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This manuscript proposes that pair generation via time reflection and k-gap amplification in photonic time crystals (PTCs) arises from a common mechanism: the time-domain Stokes phenomenon. The authors map the wave equation near temporal boundaries to an Airy/Stokes turning-point problem (Eqs. 3–6), perform classical FDTD simulations of a nonlinear PTC slab showing k-gap soliton formation and four-branch output (Fig. 2), and propose HBT/HOM measurements to verify nonclassical correlations (Eq. 8). The Airy/Stokes mapping in the linear regime is mathematically clean and the FDTD simulations are internally consistent. However, the central quantum claim — that the four output branches are entangled photon pairs — relies on extrapolating linear-limit two-mode squeezing formulas (U→0) to the nonlinear soliton regime without computing the nonlinear quantum dynamics. The companion manuscript [34] containing the full nonlinear quantum treatment is unavailable.

Significance. The paper identifies a potentially valuable connection between asymptotic Stokes physics and temporal scattering in PTCs. The linear Airy/Stokes mapping (Eqs. 3–6, M1–M3) is a legitimate asymptotic analysis with a parameter-free connection formula. The FDTD simulations (Methods M2) demonstrate classical field dynamics consistent with soliton formation and four-branch splitting. The proposed HBT/HOM detection scheme (Eq. 8, Figs. 2c–d) provides a falsifiable experimental protocol. These are genuine strengths. However, the significance is substantially reduced by the gap between the classical simulations and the quantum entanglement claims, which are not bridged within this manuscript.

major comments (3)
  1. The central quantum claim — that the four output branches are entangled photon pairs — rests on two disconnected pieces of evidence: (1) classical FDTD simulations of the nonlinear PTC slab (Eqs. M4–M10) showing pulse splitting, and (2) analytical HBT/HOM correlation functions (Eq. 8 / M20–M22) derived exclusively in the linear limit U→0. The Hamiltonian in Eq. (7)/M18 includes the Kerr term (ℏU), but the state evolution, photon statistics, and correlation functions for finite U are never computed. The statement on p.7 (and repeated in M4) that 'the same observables are evaluated using the paired state obtained from the nonlinear Hamiltonian in M3' describes a computation that is not actually performed — only the linear-limit formulas are shown. Without the nonlinear quantum treatment, the entanglement claim for the soliton regime is an extrapolation from the linear limit, not a result.
  2. The FDTD simulations (M2) initialize a coherent seed pulse (Eq. M7), not vacuum. Classical simulations of a coherent seed undergoing nonlinear dynamics cannot, even in principle, demonstrate quantum pair correlations or entanglement. The claim on p.6 that 'these outputs should be viewed as entangled soliton-pair branches rather than classical wave fragments' is therefore not supported by the simulations shown. The paper should either clearly state that the FDTD results demonstrate only the classical wave dynamics (soliton formation and splitting) while the quantum correlations are predicted from the linear theory, or provide the nonlinear quantum analysis.
  3. p.4: The assertion that Kerr nonlinearity 'does not alter the local soft-boundary structure' extends the linear Airy/Stokes mapping to the nonlinear regime without proof. This claim is load-bearing because the entire mechanism for secondary pair generation at the exit boundary depends on the Stokes mapping remaining valid in the nonlinear regime. A perturbative argument or reference to where this is justified would strengthen this claim. Without it, the extension from linear to nonlinear Stokes mapping is an assumption.
minor comments (7)
  1. Abstract: 'entangled pulse branches' is stated as an established result. Given that the quantum entanglement is not demonstrated within this manuscript, the abstract should distinguish between what is shown (classical soliton dynamics, linear-limit quantum predictions) and what is conjectured (nonlinear-regime entanglement).
  2. Eq. (2): The notation switches between Ẽ_k and Ẽ_k without clear definition of the tilde vs. double-tilde convention. Clarifying this would improve readability.
  3. p.5, Fig. 1 caption: The term 'antiphoton' is used without definition. If this refers to the backward-propagating partner mode, a brief clarification would help readers unfamiliar with this terminology.
  4. p.7: The squeezing parameter r = κt = (Δn/2n₀)ω₀t is defined, but the connection between the FDTD simulation parameters (δ₁, β, Ω) and the squeezing parameter r used in the HBT/HOM plots (Fig. 2d) is not specified. Readers cannot verify that the r values shown (0.3, 1.5) correspond to the simulation results.
  5. Fig. 2d: The HBT and HOM curves appear to be analytical plots of Eq. (8), not simulation outputs. The caption should clarify whether these are theoretical predictions or data extracted from the FDTD simulations.
  6. p.8: 'requires optical-frequency quantum platforms' — the experimental parameters mentioned (Δn/n₀ ≈ 0.2, femtosecond switching) are demanding. A brief discussion of whether ENZ or silicon waveguide platforms can realistically achieve these simultaneously with Kerr nonlinearity would strengthen the experimental feasibility discussion.
  7. Reference [34] is cited as 'manuscript in submission.' Since key quantum claims depend on [34], the authors should either incorporate the essential results here or clearly flag that the nonlinear quantum treatment is forthcoming.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive reading of our manuscript. The referee correctly identifies the linear Airy/Stokes mapping and the FDTD simulations as genuine strengths, and raises three substantive concerns about the gap between our classical simulations and our quantum entanglement claims. We agree with the core of each concern and will revise the manuscript accordingly. Below we address each major comment in turn.

read point-by-point responses

  1. Referee: The central quantum claim — that the four output branches are entangled photon pairs — rests on disconnected evidence: classical FDTD simulations and analytical HBT/HOM formulas derived exclusively in the linear limit U→0. The nonlinear quantum dynamics are never computed, and the companion manuscript [34] is unavailable.

    Authors: The referee is correct. The manuscript as written does not compute the nonlinear quantum dynamics at finite U. The HBT and HOM correlation functions in Eq. (8) / M20–M22 are derived in the linear limit (U → 0), and the statement on p. 7 and in M4 that 'the same observables are evaluated using the paired state obtained from the nonlinear Hamiltonian in M3' describes a computation that is not actually performed in this manuscript. We will revise the manuscript to make this distinction explicit: (i) the FDTD simulations demonstrate classical nonlinear wave dynamics (soliton formation and four-branch splitting); (ii) the HBT/HOM correlation functions are predictions from the linear-limit quantum theory; and (iii) the extension to the nonlinear soliton regime is a conjecture supported by the physical argument that the Bogoliubov pair-generation channel persists under weak Kerr nonlinearity, but is not rigorously established here. We will remove or substantially soften the language asserting entanglement of the soliton branches as an established result, and reframe it as a prediction to be tested by the proposed experiments and by the full nonlinear quantum treatment in [34]. We will also add an explicit caveat that [34] is not yet available and that the nonlinear quantum claims are therefore provisional. revision: yes

  2. Referee: The FDTD simulations initialize a coherent seed pulse, not vacuum. Classical simulations of a coherent seed cannot demonstrate quantum pair correlations or entanglement. The claim on p.6 that 'these outputs should be viewed as entangled soliton-pair branches rather than classical wave fragments' is not supported by the simulations shown.

    Authors: We agree. Classical FDTD simulations with a coherent seed demonstrate the classical nonlinear wave dynamics — amplification, Kerr arrest, soliton formation, and Stokes splitting at the exit boundary — but cannot, even in principle, establish quantum pair correlations or entanglement. The sentence on p. 6 asserting that the outputs 'should be viewed as entangled soliton-pair branches rather than classical wave fragments' overstates what the simulations show. We will revise this passage to clearly state that the FDTD results demonstrate the classical wave dynamics (soliton formation and four-branch splitting), and that the quantum correlations are predicted from the linear-limit Bogoliubov theory and remain to be verified experimentally. We will also add a sentence noting that the coherent seed in the simulations corresponds to the stimulated (classical) regime, while the quantum pair-generation channel operates on vacuum fluctuations and is not captured by the classical simulation. revision: yes

  3. Referee: The assertion that Kerr nonlinearity 'does not alter the local soft-boundary structure' extends the linear Airy/Stokes mapping to the nonlinear regime without proof. This claim is load-bearing for the secondary pair generation mechanism.

    Authors: The referee raises a valid concern. The statement on p. 4 that Kerr nonlinearity 'does not alter the local soft-boundary structure' is presented without justification. We can offer the following perturbative argument, which we will add to the manuscript: the Airy/Stokes mapping is derived in a narrow transition region near the temporal boundary where the modulation depth ramps through zero. In this region, the field amplitude is still small (the k-gap amplification has not yet saturated), so the Kerr term β|Ẽ|² is perturbative relative to the linear terms and does not modify the leading-order Airy structure. At the exit boundary, the soliton amplitude is finite but the transition is rapid (controlled by the ramp rate γ), so the local Stokes connection is governed by the linear turning-point physics on timescales short compared to the nonlinear evolution. This is a perturbative argument, not a rigorous proof, and we will present it as such. We will also acknowledge that the validity of the Stokes mapping in the nonlinear regime is an assumption that ultimately requires verification through the full nonlinear quantum treatment, and we will note this explicitly as a limitation. revision: partial

Circularity Check

0 steps flagged

No significant circularity found; derivation chain is self-contained for the linear-limit results actually derived.

full rationale

The paper's derivation chain proceeds as follows: (1) the nonlinear wave equation (Eq. 1) is reduced to a Mathieu-type equation (Eq. 2); (2) in the linear limit, the envelope maps to an inverted oscillator (Eq. 3), and a linear-ramp approximation near the band edge yields the Airy equation (Eq. 4), whose standard asymptotics (Eqs. 5–6) give the Stokes connection — this is a legitimate asymptotic technique with no definitional circularity; (3) the HBT/HOM correlation functions (Eq. 8 / M20–M22) are derived from the two-mode squeezed state obtained via the Bogoliubov transformation in the linear limit (U→0), and these are standard quantum-optics textbook results [18–22], not circularly defined; (4) the FDTD simulations (Methods M2) are classical and independent of the analytical Stokes mapping — they serve as verification, not as input. The squeezing parameter r = κt is set by physical parameters (modulation depth, frequency, time), not fitted to the HBT/HOM data. The self-citations to [16] (Pan et al., PRL 2023) are to a published, independently verifiable paper for the k-gap soliton concept and displacement-field formulation. The companion manuscript [34] (Zhang, Pan, Pan, in submission) is referenced for the nonlinear quantum treatment, but the paper is explicit that the formulas shown are the 'linear reference limit' and that the nonlinear treatment is deferred — this is an extrapolation gap (correctness risk), not circularity. No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 2 invented entities

The paper introduces one genuinely new conceptual entity (time-domain Stokes mapping for PTCs) that is supported by parameter-free asymptotic analysis. The quantum entanglement claims for the nonlinear regime rely on an ad-hoc axiom (extrapolation from linear theory) and an unavailable companion manuscript. The free parameters are simulation/experimental choices, not fitted constants, but several lack stated numerical values.

free parameters (6)
  • δ₁ (modulation depth) = ~0.2 (Δn/n₀ ≈ 0.2, stated in experimental discussion)
    Controls the k-gap width and amplification rate; chosen for simulation and proposed experiment.
  • β (effective Kerr coefficient) = not specified numerically
    Determines soliton stabilization; stated as β ≪ 1 but exact value used in FDTD not given.
  • Ω (modulation frequency) = 2ω₀, with ω₀ corresponding to λ₀ ≈ 800 nm
    Set by the carrier frequency of the seed pulse; chosen for near-infrared implementation.
  • α (boundary sharpness in tanh profile) = not specified
    Controls the softness of the temporal boundary; affects the Airy mapping validity but exact value not stated.
  • r (squeezing parameter) = 0.3 (weak) and 1.5 (strong), used for HBT/HOM curves
    Sets the quantum correlation strength in the analytical expressions; chosen to illustrate regimes, not fitted to data.
  • τ_c (temporal coherence time) = not specified
    Sets the Gaussian envelope width in HBT/HOM correlation functions; assumed but not derived from simulation parameters.
axioms (5)
  • domain assumption Weak-modulation and weak-nonlinearity approximation: δ₁ ≪ 1 and β ≪ 1, allowing inversion of constitutive relations to obtain Eq. 1.
    Stated in the Modelling section; necessary for the displacement-field formulation and the reduction to the nonlinear Mathieu equation (Eq. 2).
  • domain assumption Linear ramp approximation: δ̃(τ) ≈ δ̃₀ + γτ near temporal boundaries, enabling the Airy mapping.
    Stated in the Stokes phenomenon section; the validity of the Airy reduction depends on the modulation profile being locally linear near the turning point.
  • domain assumption Rotating-wave approximation (RWA) for the quantum Hamiltonian, dropping fast-oscillating terms.
    Invoked in Methods M3 to derive the effective parametric Hamiltonian (Eq. M12→M18); standard in quantum optics but limits validity to near-resonant, weak-modulation regimes.
  • ad hoc to paper The quantum correlations of the nonlinear soliton regime are adequately approximated by the linear two-mode squeezed state expressions.
    The HBT/HOM expressions (Eq. 8) are derived for the linear limit (U→0). The paper applies them to the nonlinear regime without derivation, deferring the full treatment to [34].
  • domain assumption Spatial homogeneity of the modulation (ε(t,x) = ε(t)), ensuring momentum conservation.
    Stated in Methods M2; necessary for the momentum-space reduction and the pairing of ±k modes.
invented entities (2)
  • Time-domain Stokes phenomenon (as applied to PTC temporal boundaries) independent evidence
    purpose: Provides the mathematical framework unifying time-reflection pair generation and k-gap amplification.
    The Airy mapping (Eq. 4) is a parameter-free asymptotic reduction with falsifiable predictions (Stokes phase π/4 in oscillatory tails, Eq. 6). The FDTD simulations provide classical verification of the boundary splitting.
  • Four-port entangled soliton-pair emission no independent evidence
    purpose: Proposed experimental signature of secondary Stokes-induced pair generation at the exit boundary.
    The four branches are observed in classical FDTD, but their entanglement is predicted from the linear quantum theory without a full nonlinear quantum treatment in this paper.

pith-pipeline@v1.1.0-glm · 17229 in / 3665 out tokens · 339787 ms · 2026-07-09T13:45:36.217789+00:00 · methodology

0 comments
read the original abstract

Pair generation in time-varying media is commonly attributed to time reflection at temporal boundaries or to amplification inside momentum k gaps. Here we show that these two processes are connected by the time domain Stokes phenomenon. A finite duration photonic time crystal (PTC) slab provides the necessary Stokes connection between the incident vacuum mode, transient k gap amplification, and time boundary scattering. With Kerr nonlinearity, the otherwise unbounded amplification is arrested, spawning Kerr stabilized k gap solitons. When these solitons cross the exit boundary of the time slab, Stokes induced mode conversion produces a secondary pair generation process, yielding four spatially separated and entangled pulse branches. Detection of a backward propagating light pulse therefore heralds its forward propagating partner. We further propose combined Hanbury Brown Twiss and Hong Ou Mandel measurements to test their nonclassical correlations. These results reveal a link between asymptotic Stokes physics and quantum temporal scattering in PTCs, and suggest a route toward ultrafast heralded quantum light sources.

Figures

Figures reproduced from arXiv: 2607.07340 by Guowei Chen, Jiaxiang Sun, Liang Zhang, Yiming Pan.

Figure 1
Figure 1. Figure 1: FIG. 1. The time [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    , Bi(𝜁) ∼ 1 √𝜋 |𝜁|−1/4 cos (2 3 |𝜁|3/2 + 𝜋

    work page

  2. [2]

    Maxwell’s equations tell a quantum story

    (6) Note that the Stokes phase 𝜋/4 in the oscillatory tails is the asymptotic signature of temporal - boundary mode conversion [7-9] and fixes the phase relation between the incident and generated branches [7-9]. By symmetry, the same Airy mapping applies to the switch-off temporal boundary at 𝜏 = 𝜏2, as shown in Fig. 1a. Kerr nonlinearity saturates the 𝑘...

    work page

  3. [3]

    Velocity modulation of electromagnetic waves,

    F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Transactions on Microwave Theory and Techniques 6, 167–172 (1958). DOI: 10.1109/TMTT.1958.1124533

    work page doi:10.1109/tmtt.1958.1124533 1958

  4. [4]

    Parametric properties of fields in a slab of time-varying permittivity,

    D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Transactions on Antennas and Propagation 14, 183–194 (1966). DOI: 10.1109/TAP.1966.1138637

    work page doi:10.1109/tap.1966.1138637 1966

  5. [5]

    Time refraction and time reflection: two basic concepts,

    J. T. Mendonça and P. K. Shukla, “Time refraction and time reflection: two basic concepts,” Physica Scripta 65, 160–163 (2002). DOI: 10.1238/Physica.Regular.065a00160

    work page doi:10.1238/physica.regular.065a00160 2002

  6. [6]

    Quantum theory of time refraction,

    J. T. Mendonça, A. Guerreiro, and A. M. Martins, “Quantum theory of time refraction,” Physical Review A 62, 033805 (2000). DOI: 10.1103/PhysRevA.62.033805

    work page doi:10.1103/physreva.62.033805 2000

  7. [7]

    Time refraction and the quantum properties of vacuum,

    J. T. Mendonça and A. Guerreiro, “Time refraction and the quantum properties of vacuum,” Physical Review A 72, 063805 (2005). DOI: 10.1103/PhysRevA.72.063805

    work page doi:10.1103/physreva.72.063805 2005

  8. [8]

    Photonics of time-varying media,

    E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, R. Sapienza, A. Alù, and J. B. Pendry, “Photonics of time-varying media,” Advanced Photonics 4, 014002 (2022). DOI: 10.1117/1.AP.4.1.014002

    work page doi:10.1117/1.ap.4.1.014002 2022

  9. [9]

    Stokes’ phenomenon; smoothing a Victorian discontinuity,

    M. V. Berry, “Stokes’ phenomenon; smoothing a Victorian discontinuity,” Publications Mathématiques de l’IHÉS 68, 211–221 (1988). DOI: 10.1007/BF02698550

    work page doi:10.1007/bf02698550 1988

  10. [10]

    Waves near Stokes lines,

    M. V. Berry, “Waves near Stokes lines,” Proceedings of the Royal Society A 427, 265–280 (1990). DOI: 10.1098/rspa.1990.0013

    work page doi:10.1098/rspa.1990.0013 1990

  11. [11]

    F. W. J. Olver, Asymptotics and Special Functions. Academic Press, New York (1974)

    work page 1974

  12. [12]

    Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect,

    E. Yablonovitch, “Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-DeWitt radiation and the nonadiabatic Casimir effect,” Physical Review Letters 62, 1742–1745 (1989). DOI: 10.1103/PhysRevLett.62.1742

    work page doi:10.1103/physrevlett.62.1742 1989

  13. [13]

    Observation of the dynamical Casimir effect in a superconducting circuit

    C. M. Wilson, G. Johansson, A. Pourkabirian, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, “Observation of the dynamical Casimir effect in a superconducting circuit,” Nature 479, 376–379 (2011). DOI: 10.1038/nature10561

    work page doi:10.1038/nature10561 2011

  14. [14]

    Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function 𝜖(𝑡),

    J. R. Zurita-Sá nchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function 𝜖(𝑡),” Physical Review A 79, 053821 (2009). DOI: 10.1103/PhysRevA.79.053821. 16

    work page doi:10.1103/physreva.79.053821 2009

  15. [15]

    Observation of genuine wave vector (𝑘 or 𝛽)gap in a dynamic transmission line and temporal photonic crystals,

    J. R. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (𝑘 or 𝛽)gap in a dynamic transmission line and temporal photonic crystals,” Applied Physics Letters 107, 074101 (2015). DOI: 10.1063/1.4928659

    work page doi:10.1063/1.4928659 2015

  16. [16]

    Topological aspects of photonic time crystals,

    E. Lustig, Y. Sharabi, and M. Segev, “Topological aspects of photonic time crystals,” Optica 5, 1390–1395 (2018). DOI: 10.1364/OPTICA.5.001390

    work page doi:10.1364/optica.5.001390 2018

  17. [17]

    Amplified emission and lasing in photonic time crystals,

    M. Lyubarov, Y. Lumer, A. Dikopoltsev, E. Lustig, Y. Sharabi, and M. Segev, “Amplified emission and lasing in photonic time crystals,” Science 377, 425–428 (2022). DOI: 10.1126/science.abo3324

    work page doi:10.1126/science.abo3324 2022

  18. [18]

    Superluminal k-gap solitons in nonlinear photonic time crystals,

    Y. Pan, M.-I. Cohen, and M. Segev, “Superluminal k-gap solitons in nonlinear photonic time crystals,” Physical Review Letters 130, 233801 (2023). DOI: 10.1103/PhysRevLett.130.233801

    work page doi:10.1103/physrevlett.130.233801 2023

  19. [19]

    Why you should not use the electric field to quantize in nonlinear optics,

    N. Quesada and J. E. Sipe, “Why you should not use the electric field to quantize in nonlinear optics,” Optics Letters 42, 3443–3446 (2017). DOI: 10.1364/OL.42.003443

    work page doi:10.1364/ol.42.003443 2017

  20. [20]

    P. D. Drummond and M. Hillery, The Quantum Theory of Nonlinear Optics. Cambridge University Press, Cambridge (2014)

    work page 2014

  21. [21]

    D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. Springer, Berlin (2008)

    work page 2008

  22. [22]

    The quantum theory of optical coherence,

    R. J. Glauber, “The quantum theory of optical coherence,” Physical Review 130, 2529–2539 (1963). DOI: 10.1103/PhysRev.130.2529

    work page doi:10.1103/physrev.130.2529 1963

  23. [23]

    Correlation between Photons in two Coherent Beams of Light,

    R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). DOI: 10.1038/177027a0

    work page doi:10.1038/177027a0 1956

  24. [24]

    Measurement of subpicosecond time intervals between two photons by interference,

    C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Physical Review Letters 59, 2044–2046 (1987). DOI: 10.1103/PhysRevLett.59.2044

    work page doi:10.1103/physrevlett.59.2044 2044

  25. [25]

    Über die Fortpflanzung des Lichtes in dispergierenden Medien,

    A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Annalen der Physik 349, 177–202 (1914). DOI: 10.1002/andp.19143491002

    work page doi:10.1002/andp.19143491002 1914

  26. [26]

    Brillouin, Wave Propagation and Group Velocity

    L. Brillouin, Wave Propagation and Group Velocity. Academic Press, New York (1960)

    work page 1960

  27. [27]

    Signal velocity, causality, and quantum noise in superluminal light pulse propagation,

    A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Physical Review Letters 86, 3925–3929 (2001). DOI: 10.1103/PhysRevLett.86.3925. 17

    work page doi:10.1103/physrevlett.86.3925 2001

  28. [28]

    Broadband frequency translation through time refraction in an epsilon-near-zero material,

    Y. Zhou, M. Z. Alam, M. Karimi, J. Upham, O. Reshef, C. Liu, A. E. Willner, and R. W. Boyd, “Broadband frequency translation through time refraction in an epsilon-near-zero material,” Nature Communications 11, 2180 (2020). DOI: 10.1038/s41467-020-15682-2

    work page doi:10.1038/s41467-020-15682-2 2020

  29. [29]

    Saturable time-varying mirror based on an epsilon-near-zero material,

    R. Tirole, E. Galiffi, J. Dranczewski, T. Attavar, B. Tilmann, Y.-T. Wang, P. A. Huidobro, A. Alù, J. B. Pendry, S. A. Maier, S. Vezzoli, and R. Sapienza, “Saturable time-varying mirror based on an epsilon-near-zero material,” Physical Review Applied 18, 054067 (2022). DOI: 10.1103/PhysRevApplied.18.054067

    work page doi:10.1103/physrevapplied.18.054067 2022

  30. [30]

    Metasurface-based realization of photonic time crystals,

    X. Wang, M. S. Mirmoosa, V. S. Asadchy, C. Rockstuhl, S. Fan, and S. A. Tretyakov, “Metasurface-based realization of photonic time crystals,” Science Advances 9, eadg7541 (2023). DOI: 10.1126/sciadv.adg7541

    work page doi:10.1126/sciadv.adg7541 2023

  31. [31]

    Quantum electrodynamics of time-varying gratings,

    S. A. R. Horsley and J. B. Pendry, “Quantum electrodynamics of time-varying gratings,” Proceedings of the National Academy of Sciences 120, e2302652120 (2023). DOI: 10.1073/pnas.2302652120

    work page doi:10.1073/pnas.2302652120 2023

  32. [32]

    Slow Bragg solitons in nonlinear periodic structures,

    D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Physical Review Letters 62, 1746–1749 (1989). DOI: 10.1103/PhysRevLett.62.1746

    work page doi:10.1103/physrevlett.62.1746 1989

  33. [33]

    Time reversal and holography with spacetime transformations,

    V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nature Physics 12, 972–977 (2016). DOI: 10.1038/nphys3810

    work page doi:10.1038/nphys3810 2016

  34. [34]

    Time-reflection of microwaves by a fast optically-controlled time-boundary,

    T. R. Jones, A. V. Kildishev, M. Segev, and D. Peroulis, “Time-reflection of microwaves by a fast optically-controlled time-boundary,” Nature Communications 15, 6786 (2024). DOI: 10.1038/s41467-024-51171-6

    work page doi:10.1038/s41467-024-51171-6 2024

  35. [35]

    Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,

    R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Fö rst, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Optics Express 14, 8336–8346 (2006). DOI: 10.1364/OE.14.008336

    work page doi:10.1364/oe.14.008336 2006

  36. [36]

    Quantization and Biphoton Statistics of k-Gap Solitons in Nonlinear Photonic Time Crystals,

    L. Zhang, C. Pan, and Y. Pan, “Quantization and Biphoton Statistics of k-Gap Solitons in Nonlinear Photonic Time Crystals,” manuscript in submission. 18 Figures: FIG. 1. The time -domain Stokes phenomenon and photon pair generation in a finite -time photonic time-crystal (PTC) slab. (a) Macroscopic temporal scattering dynamics. An incident seed pulse unde...

    work page