Revised Sauter-like Model: Theory & Applications

• Revised Sauter-like model is a family of adaptations of classic Sauter potential profiles tailored to specific phenomena in strong-field QED, condensed matter, and plasma physics.
• In strong-field QED, frequency modulation and chirping in Sauter wells enhance electron-positron pair production by broadening resonant channels.
• In lattice and plasma contexts, the model underpins effective particle-hole tunneling in Mott insulators and updated bootstrap-current closures for improved theoretical fits.

Searching arXiv for the cited papers to ground the article in the underlying literature. arXiv search: (Queisser et al., 2011) The revised Sauter-like model is not a single canonical construction but a family of domain-specific adaptations of Sauter-type profiles or Sauter-Schwinger analogies. In strong-field QED, it denotes time-dependent Sauter wells with frequency modulation and, separately, a finite Sauter-like pulse engineered so that the vector potential vanishes in the asymptotic past and future (Wang et al., 2020, Kamiński et al., 29 May 2026). In the tilted Bose-Hubbard model, it appears as a lattice realization of Sauter-Schwinger-like particle-hole creation in the Mott phase (Queisser et al., 2011). In magnetically confined plasmas, it denotes the revised bootstrap-current closure of Redl et al. (2021), implemented as a self-consistent source current in M3D-C1 and extended to quasisymmetric stellarators through the Landreman isomorphism (Saxena et al., 7 Jul 2025). The common feature is retention of a Sauter-derived structure together with a revision of temporal dependence, effective quasiparticle interpretation, or geometry-dependent coefficient fits.

1. Terminological scope

Across the cited literature, the phrase “revised Sauter-like model” is used for distinct constructions rather than for a universal model. In one usage, the revision is a change from a static Sauter well to a static-plus-oscillating or chirped Sauter well in the Dirac equation. In another, the revision is a change from the classic single Sauter pulse to a finite laser-pulse superposition with well-defined asymptotic in- and out-states. In condensed matter, the revision is an effective long-wavelength mapping from the tilted Bose-Hubbard model to a Klein-Gordon-like description of particle-hole tunneling. In plasma transport, the revision is a refitting of Sauter bootstrap coefficients to improve collisionality behavior while keeping the same gradient-driven closure structure.

Domain	Revised element	Principal consequence
Dirac pair creation	Static Sauter well replaced by static-plus-oscillating or chirped well	Pair production depends strongly on modulation amplitude and center frequency
Dynamical Sauter-Schwinger theory	Finite pulse with asymptotically vanishing vector potential; boundary- and initial-value formulations compared	Spin-summed agreement can coexist with spin- and helicity-resolved disagreement
Tilted Bose-Hubbard lattice	Effective particle/hole equations in a $1/Z$ expansion and long-wavelength limit	Nonperturbative particle-hole creation for small gradients; resonant tunneling for large tilts
Bootstrap current modeling	Redl et al. (2021) coefficients mapped to QS geometry	Self-consistent neoclassical current closure in M3D-C1

A recurrent misconception is that “revised” implies a single standardized extension of the Sauter potential. The literature instead uses the expression for several revisions that share a Sauter ancestry but differ in ontology, observables, and validity regime.

2. Frequency-modulated Sauter wells in Dirac pair creation

In the external-field Dirac framework, the revised Sauter-like model is a combined potential well

$$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$

with spatial profile

$$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$

and chirped instantaneous frequency

$$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$

The field operator evolves under

$$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$

and the number of created electrons is extracted from transition amplitudes via

$N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$

The model is “Sauter-like” because the spatial profile remains a smooth Sauter well, but it is revised by explicit time dependence and by frequency modulation of the oscillating component (Wang et al., 2020).

The static part has depth $V_1$ and is turned on and off smoothly by $f(t)$, while the oscillating part has amplitude $V_2$ and is active only during $t_0<t<t_0+t_1$. Representative values used in the main figures are $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$0, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$1, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$2, often $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$3, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$4, and $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$5.

The principal control parameters are the modulation amplitude $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$6, the center frequency $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$7, and the chirp rate $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$8. The reported effect of $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$9 is not monotonic: increasing $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$0 initially enhances pair creation, but overly strong chirp can shorten the effective interaction time and reduce the yield. For the combined well at $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$1, the fixed-frequency case $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$2 gives $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$3, while the optimal value $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$4 gives $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$5. The abstract summarizes the gain more conservatively as “about two times.” The optimal modulation depends on $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$6: for low $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$7, the optimal modulation tends to satisfy $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$8, whereas at higher $$S(z)=\frac{1}{2}\left\{\tanh\left[\frac{z-D/2}{W}\right]-\tanh\left[\frac{z+D/2}{W}\right]\right\},$$9 excessive chirping reduces the effective interaction time.

The single frequency-modulated oscillating well, obtained by setting $$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$0, shows a qualitatively stronger low-frequency effect. In that regime, the optimized frequency-modulated yield can exceed the fixed-frequency yield by up to four orders of magnitude, and the paper states that the final number of created electrons can increase by $$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$1 compared with the fixed-frequency case. The proposed mechanism is that chirp broadens the spectrum, increases the number of resonant channels, and increases the number of times bound states dive into the Dirac sea. In the high-frequency regime, chirp also broadens the discrete peaks $$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$2 into intervals from $$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$3 to $$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$4, and some electrons escape the trap, reducing Pauli blocking.

3. Finite Sauter-like pulses and the comparison of boundary- and initial-value formulations

A second usage of the revised Sauter-like model arises in the study of dynamical Sauter-Schwinger pair creation from the vacuum by an electromagnetic background field. Here the revision is not a new field theory but a reorganization of the formulation: the same external-field Dirac problem is treated either by a Feynman-style boundary-value approach or by an initial-value approach obtained by replacing Feynman propagators with retarded propagators (Kamiński et al., 29 May 2026).

The boundary-value formulation starts from the exact Feynman propagator $$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$5, which satisfies

$$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$6

with the free Feynman propagator

$$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$7

In this picture, the amplitude is the exact sum of all Born terms in the background field. The associated Dirac solution is purely negative-energy in the remote past and decomposes into positive- and negative-energy components in the remote future. The pair-creation amplitude is the coefficient of the positive-energy outgoing term,

$$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$8

The initial-value formulation replaces the Feynman propagator by the retarded propagator

$$\omega(t)=\omega_0+\Delta\omega\,\sin[\Omega (t-t_0)].$$9

and evolves a normalized negative-energy state forward in time. In this picture, negative-energy states represent electrons filling the Dirac sea, and pair creation is described as excitation of one of these electrons into the positive-energy continuum. The amplitude is

$$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$0

The paper also shows that the quantum kinetic equation formalism is equivalent to this initial-value Dirac treatment for spatially homogeneous time-dependent fields.

The pulse used for numerical comparison is a homogeneous electric field formed by a superposition of three circularly polarized pulses with orthogonal polarization planes,

$$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$1

where each component has envelope

$$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$2

The vector potential is

$$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$3

so both $$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$4 and $$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$5 vanish outside the pulse for $$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$6. In the example shown, $$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$7, $$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$8, and $$i\frac{\partial \hat{\psi}(z,t)}{\partial t} = \left[c\,\alpha_z \hat{p}_z + \beta c^2 + V(z,t)\right]\hat{\psi}(z,t),$$9.

The central result is that the two formulations are exact within their own definitions but are not generally equivalent. For suitably weak fields, spin-summed momentum distributions can be nearly the same, yet spin-resolved, helicity-resolved, and entangled momentum distributions differ significantly. For homogeneous electric pulses, the scalar Bell-channel amplitude vanishes in both formulations,

$N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$0

but the remaining entangled channels remain nonzero and differ strongly between the two approaches. The paper’s interpretation is correspondingly sharp: the boundary-value/Feynman/S-matrix/worldline formulation is the correct one for relativistic vacuum pair creation with positrons as antiparticles, whereas the initial-value/retarded/QKE/DHW formulation is a mathematically consistent Dirac-sea excitation theory with a different physical meaning.

4. Lattice realization in the tilted Bose-Hubbard Mott phase

A condensed-matter counterpart appears in the tilted Bose-Hubbard model in the Mott phase. The Hamiltonian is

$N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$1

with hopping amplitude $N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$2, on-site interaction $N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$3, lattice adjacency matrix $N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$4, coordination number $N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$5, external on-site potential $N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$6, and $N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$7. The regime of interest is the Mott phase at unit filling,

$N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$8

with the tilt implemented by a spatially varying potential analogous to an electric field,

$N(t)=\sum_p\sum_n |U_{pn}(t)|^2.$9

The physical process is tunneling of a boson to a neighboring site, leaving behind a hole and thereby creating a particle-hole pair (Queisser et al., 2011).

The paper emphasizes that the tilt has no effect at the Gutzwiller mean-field level. In the $V_1$0 leading order,

$V_1$1

so the tilt contributes only a local phase and does not generate the intersite correlations required for a real particle-hole excitation. Pair creation is therefore invisible at mean field because it is fundamentally a quantum fluctuation effect.

Beyond mean field, a $V_1$2 hierarchy is derived for reduced density matrices. Local particle and hole operators are introduced as

$V_1$3

with correlators

$V_1$4

To first order in $V_1$5, these obey a closed linear system. The effective representation that reproduces the correlation dynamics is

$V_1$6

$V_1$7

In the long-wavelength limit, defining $V_1$8 yields

$V_1$9

and near the Mott transition,

$f(t)$0

Eliminating one field gives a Klein-Gordon-like equation,

$f(t)$1

with effective gap

$f(t)$2

and effective speed of light

$f(t)$3

This is the paper’s quantitative Sauter-Schwinger analogy: the Mott insulator plays the role of a relativistic vacuum, the particle-hole pair plays the role of an electron-positron pair, and the tilt plays the role of an electric field.

In Fourier space the tilt enters via a Peierls-type substitution, $f(t)$4, and positive/negative-frequency mixing gives

$f(t)$5

For small gradients, the pair-production probability obeys

$f(t)$6

For large tilts, the continuum approximation breaks down and the process becomes resonant. The first resonance occurs at

$f(t)$7

with growth such as

$f(t)$8

and higher-order resonances appear when

$f(t)$9

The resulting picture is explicitly two-regime: Schwinger-like tunneling for small gradients and resonant lattice-assisted pair production tied to Bloch oscillations for large gradients.

5. Bootstrap-current closure in M3D-C1

In plasma physics, the revised Sauter-like model denotes a bootstrap-current closure used in M3D-C1. The bootstrap current is treated as a neoclassical, internally driven parallel current and inserted as a source current $V_2$0 into Ohm’s law,

$V_2$1

Because the bootstrap current is an internal force, the opposite force is included in the ion momentum equation so that no spurious net force is added to the MHD force balance. The current is assumed to be purely parallel to $V_2$2, divergence-free on a flux surface, and representable by a flux-surface average of the parallel current drive (Saxena et al., 7 Jul 2025).

The closure is written in pressure-gradient form as

$V_2$3

or equivalently in density/temperature-gradient form as

$V_2$4

with

$V_2$5

The coefficients $V_2$6, $V_2$7, $V_2$8, and $V_2$9 depend on collisionality, trapped-particle fraction, effective charge, and geometry.

What is revised relative to the original generalized Sauter model is the fit, not the functional architecture. The original Sauter et al. (1999) model was fitted to axisymmetric CQL3D/CQLP calculations and becomes less reliable at higher electron collisionality, especially $t_0<t<t_0+t_1$0. The Redl et al. (2021) revision re-fits the formulas using NEO calculations, improves behavior over a broader collisionality range, and is more robust at high $t_0<t<t_0+t_1$1, while remaining tokamak-based in native form.

For quasisymmetric stellarators, M3D-C1 uses the Landreman–Buller–Drevlak isomorphism. The bootstrap-current drive is replaced by a QS expression involving $t_0<t<t_0+t_1$2, derivatives with respect to $t_0<t<t_0+t_1$3, and the quasisymmetric field form $t_0<t<t_0+t_1$4. The conversion between toroidal-flux and poloidal-flux derivatives is

$t_0<t<t_0+t_1$5

and for axisymmetry $t_0<t<t_0+t_1$6 and $t_0<t<t_0+t_1$7, recovering the tokamak formula. This is the key mapping that allows the revised Sauter-like model to be used in quasi-axisymmetric stellarators.

The implementation is extended consistently to the energy equation, where resistive heating becomes

$t_0<t<t_0+t_1$8

rather than the standard $t_0<t<t_0+t_1$9. Since M3D-C1 does not assume flux surfaces a priori, an approximate magnetic coordinate system is constructed from M3D-C1 output using Fusion-IO, with electron-temperature isotherms used as proxies for magnetic surfaces. A normalized temperature coordinate,

$$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$00

is used to tabulate $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$01, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$02, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$03, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$04, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$05, and $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$06; the coefficients $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$07, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$08, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$09, and $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$10 are then evaluated locally during time evolution.

Benchmarking is reported against NEO, XGCa, and SFINCS. In the low-aspect-ratio circular tokamak case CIRC1, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$11 at the outer boundary, the peak bootstrap current occurs near $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$12, and local parameters there are $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$13 and $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$14. At the bootstrap-current peak, the reported differences are $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$15 for M3D-C1 + Sauter vs NEO Sauter, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$16 for M3D-C1 vs XGCa, $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$17 for M3D-C1 + Redl vs XGCa, and $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$18 for M3D-C1 + Redl vs SFINCS. In QS stellarators, the Redl-plus-Landreman implementation agrees closely with SFINCS and with Redl-based results from Landreman et al. Minor discrepancies are attributed to numerical treatment differences. The explicit caveat is that the Sauter-Redl-Landreman model is strictly valid only for axisymmetric or quasisymmetric magnetic geometries.

6. Unifying structure, domain limits, and interpretive cautions

The family resemblance among these models lies in the retention of a Sauter-shaped potential, a Sauter-Schwinger tunneling structure, or a Sauter-derived closure while revising the object that carries the physics. In the frequency-modulated well, the revision is spectral: the field contains a range of frequencies rather than a single monochromatic component. In the finite-pulse comparison, the revision is asymptotic and interpretive: the pulse is chosen so that both $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$19 and $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$20 vanish outside the interaction region, and the same Dirac equation is assigned two different propagator-based meanings. In the Bose-Hubbard problem, the revision is emergent: a lattice Mott insulator acquires an effective relativistic description only in the long-wavelength, small-gradient limit. In bootstrap-current modeling, the revision is closure-theoretic: the Sauter functional form is retained, but the coefficient fits and collisionality dependence are updated.

The limitations are correspondingly different. The chirped-well calculations are tied to the chosen temporal protocol, and the enhancement with $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$21 is explicitly non-monotonic. The boundary- and initial-value formulations are not generally physically equivalent, even where spin-summed spectra nearly coincide; their differences survive in spin-sensitive observables. The Bose-Hubbard analogy requires unit filling, the Mott phase, a controlled $$V(z,t)=V_1 S(z) f(t)+V_2 \sin[\omega(t)t]\,S(z)\,\theta(t;t_0,t_0+t_1),$$22 expansion, and the small-gradient/long-wavelength regime for the Klein-Gordon-like reduction; at large tilt the correct description is resonant tunneling related to Bloch oscillations rather than smooth-barrier Schwinger tunneling. The plasma closure is approximate away from axisymmetry or quasisymmetry, relies on the existence of usable surface proxies, and does not capture dynamic neoclassical response on Alfvénic timescales.

A plausible implication is that the term “revised Sauter-like model” should always be read locally, with the revision specified by context. In one literature it names a chirped external potential, in another a finite-pulse formulation of vacuum pair creation, in a third an effective lattice analogue of Sauter-Schwinger tunneling, and in a fourth an updated neoclassical bootstrap-current fit. The shared label indicates continuity with Sauter-derived reasoning, not identity of mathematical object or physical interpretation.