Nonlinear optical analogues of quantum phase transitions in a squeezing-enhanced LMG model

Chon-Fai Kam

REVIEW 3 major objections 5 minor 1 cited by

Nonlinear optical analogues of quantum phase transitions in a squeezing-enhanced LMG model

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that a tetragonal-symmetry nonlinear optical fiber realizes a generalized Lipkin-Meshkov-Glick spin model whose extra $S_zS_x$ squeezing term drives a classical polarization bifurcation even without a rotor-like term…

desk verdict Useful symmetry catalog for fiber Hamiltonians, but the claimed new quantum phase transition is a basis artifact and the quantum spectrum is explicitly deferred. read the letter →

arxiv 2505.18618 v1 pith:QQYRTEWV submitted 2025-05-24 quant-ph

Chon-Fai Kam This is my paper · ORCID

classification quant-ph MSC 81Q8081V8037J2033E0578A60 PACS 42.65.-k42.81.Gs03.65.Vf05.70.Fh

keywords excited-statequantumphasetransitionLipkin-Meshkov-GlickmodelspinsqueezingnonlinearopticalfibertetragonalsymmetryPoincarésphereEulertoppolarizationbifurcation

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

This paper claims that light propagating in a nonlinear optical fiber with tetragonal crystal symmetry behaves like a generalized Lipkin-Meshkov-Glick spin model with an extra squeezing term, $S_zS_x$, that the standard LMG model does not contain. It argues that this extra term allows a classical bifurcation, the polarization energy surface switching from ellipsoidal to hyperbolic, to occur even when the linear rotor-like term is absent. If correct, this gives an experimentally accessible, room-temperature optical platform for studying excited-state quantum phase transitions and points toward new families of squeezed states. The paper develops the classical spin dynamics in detail, solving the trajectories with Jacobi elliptic functions, while deferring the quantum spectrum to future work.

What carries the argument

The machinery is the mapping of the two-mode polarization amplitudes $(u_x, u_y)$ to a spin vector $\mathbf{S}$ on the Poincaré sphere through $S_x = u_x^*u_y + u_y^*u_x$, $S_y = i(u_x^*u_y - u_y^*u_x)$, and $S_z = |u_x|^2 - |u_y|^2$, which turns the fiber Hamiltonian into a quadratic spin Hamiltonian. The paper then rotates the quadratic form to principal axes to write $H = M_x^2/(2I_x) + M_y^2/(2I_y)$, the asymmetric Euler top; when $\alpha\gamma < \beta^2$, one principal moment becomes negative, producing an inverted top with a hyperbolic energy surface. Jacobi elliptic functions and their $\theta$-function representations provide exact trajectories in every energy regime, and the separatrix or heteroclinic orbits locate the classical bifurcation.

What would settle it

Compute or measure the finite-spin quantum spectrum of $H = \Delta\beta\,S_z + c_0 S^2 + c_z S_z^2 + c_x S_x^2 + 2c\,S_zS_x$ with the rotor term $\Delta\beta\,S_z$ set to zero. If no sequence of excited-level crossings or singularities in the level density appears as $c$ is increased past the classical threshold, the claimed excited-state phase transition is absent. On the experimental side, a tetragonal fiber should show an abrupt change in the output Stokes vector and four heteroclinic orbits at a critical input power; their absence would falsify the optical analogue.

Watch

Extended reading notes

Core claim

The central claim is that tetragonal-symmetry fibers are the simplest nontrivial optical realization of the generalized LMG model: their polarization Hamiltonian contains the crossing term $2c\,S_zS_x$ alongside the usual single-axis and two-axis squeezing terms. In the large-spin classical limit this term flips the effective moments of inertia of the associated Euler top, so that the intersection of the energy surface with the unit sphere changes from an elliptic cylinder to a hyperbolic cylinder. The paper solves this classical model exactly, showing that for parameters satisfying $\alpha\gamma < \beta^2$ after a rotation that diagonalizes the quadratic form, the dynamics become those of an inverted asymmetric top: stable motion around the $S_x$ and $S_y$ axes, hyperbolic saddles at $S_z = \pm 1$, and four heteroclinic orbits at zero energy. The paper interprets this geometric change as the classical signature of a new excited-state quantum phase transition, distinct from the standard LMG transition that requires a rotor-like term, and states that the quantum spectrum itself is left for a subsequent paper.

Load-bearing premise

The load-bearing premise is that the fiber-derived Hamiltonian containing $2c\,S_zS_x$ reduces, by a rotation the paper does not display, to the classical spin Hamiltonian with $S_xS_y$ that is actually solved; a second premise, that the resulting classical bifurcation survives as an excited-state quantum phase transition, is asserted while the quantum spectrum is deferred.

Editorial extensions

If this is right

Tetragonal-symmetry fiber materials are identified as the simplest symmetry class that realizes a generalized LMG Hamiltonian, while isotropic, cubic, and most hexagonal or trigonal classes either reduce to the standard model or do not admit a Hamiltonian description.
A classical bifurcation in polarization dynamics occurs without the linear rotor term, so an optical experiment could in principle observe an LMG-type critical phenomenon without applying an external-field-like term.
The exact elliptic-function solutions give quantitative predictions for Stokes-vector trajectories near the separatrix and for the area enclosed by heteroclinic orbits, which can be compared directly with fiber experiments.
The paper's interpretation implies that the $S_zS_x$ squeezing term should generate a new family of squeezed states beyond one-axis and two-axis twisting; the squeezing properties are announced as future work.
If the classical bifurcation survives quantization, the corresponding excited-state quantum phase transition would appear as crossings of a sequence of excited levels rather than a single ground-state crossing.

Reading between the lines

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

If the missing rotation between the fiber-derived Hamiltonian with $S_zS_x$ and the studied classical form with $S_xS_y$ is supplied, the same inverted-top bifurcation should appear in the quantum spectrum as a sequence of level crossings; computing the finite-spin spectrum of Eq. (40) would settle this directly.
The heteroclinic-orbit area formula, $2\pi - 4\arctan\sqrt{J_x/J_y}$, offers a concrete experimental observable: the area of the separatrix on the Poincaré sphere could be measured from the Stokes trajectory near critical power and compared with material parameters.
The symmetry-class analysis could be extended to higher-order susceptibilities through the general polynomial Hamiltonian of Eq. (80), in which case the optical platform might realize phase diagrams beyond quadratic spin models, with bifurcations governed by discriminants of that polynomial.
A direct experimental test would use a polarization-maintaining fiber made of a tetragonal crystal, ramp the input power across the threshold $\alpha\gamma = \beta^2$, and look for an abrupt change or hysteresis in the output Stokes vector; the paper does not report such an experiment.

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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

Short version: the paper has two pieces. The first is a systematic derivation of coupled-mode Hamiltonians for nonlinear fibers in different crystal symmetry classes, plus the observation that only certain classes reduce to integrable spin Hamiltonians. That part is competent and, as a catalog, usable. The second is the claim that the tetragonal S_z S_x term produces a new quantum phase transition. That claim is not supported, and I think it is wrong as stated.

What is actually new: the classification of which fiber point groups yield Hamiltonian rather than dissipative polarization dynamics, with the tetragonal case isolated as the only nontrivial integrable one. The classical analysis of the asymmetric Euler top in Section IV is correct and detailed, with explicit elliptic-function solutions. Credit where due.

The problems: (1) The Hamiltonian actually analyzed in Section IV, Eq. (41), contains S_x S_y, not the S_z S_x term from Eq. (40). No basis change is given that connects them. (2) Even if one supplied the rotation, any traceless quadratic spin Hamiltonian is an SO(3) rotation of the standard LMG quadratic form; the five-dimensional space of such Hamiltonians is the l=2 representation, so the S_z S_x term is a coordinate artifact. The classical bifurcation from elliptic to hyperbolic energy surfaces is just the Euler top with one negative moment of inertia, known from rigid-body mechanics. (3) The paper explicitly defers the quantum spectrum, so the central claim of a QPT is not derived. The abstract and Section III.5 assert it anyway.

Proportion: the classical math is solid; the fiber derivation is standard but correctly done; the framing overreaches.

Who this is for: someone wanting a clear derivation of spin Hamiltonians from fiber susceptibilities might use the catalog. Anyone looking for a new QPT will be disappointed. I would not cite this as evidence of a new phase transition.

Recommendation: the paper deserves a serious referee only if the authors reframe it as a classical bifurcation study, or actually compute the ESQPT spectrum. As is, it is borderline reversible, but I would not send it to review under its current title and abstract.

Referee Report

3 major / 5 minor

Summary. The paper derives coupled-mode equations for polarization propagation in weakly guiding nonlinear fibers and classifies the resulting Hamiltonians for isotropic, cubic, hexagonal, trigonal, tetragonal, monoclinic, and triclinic symmetries. For tetragonal fibers satisfying d=-c, it writes a Hamiltonian (Eq. (40)) whose quadratic spin part contains a term 2c S_z S_x, which it identifies as an unconventional squeezing term giving rise to a novel quantum phase transition. Section IV then studies a classical spin Hamiltonian H = α/2 S_x^2 + β S_x S_y + γ/2 S_y^2, solving the equations of motion with Jacobi elliptic functions and describing the change from an elliptic to a hyperbolic intersection of the energy surface with the sphere. The paper concludes that the corresponding classical bifurcation may signify a new type of excited-state quantum phase transition, with spectral and squeezing properties deferred to a future paper.

Significance. Credit is due for the self-contained derivation of the fiber coupled-mode equations, for the explicit symmetry classification that identifies when the dynamics are Hamiltonian, and for the detailed classical solution of the asymmetric Euler top with elliptic functions. The classical analysis in Section IV is internally consistent. However, the advertised central result, namely a novel quantum phase transition induced by the S_z S_x term, is not established. The S_z S_x term is equivalent under an SO(3) spin rotation to the conventional quadratic anisotropies of the LMG model, the analysis in Section IV is not connected to Eq. (40) by any displayed transformation, and the quantum spectral statements are explicitly deferred. The manuscript therefore does not support its title or abstract claims as they stand.

major comments (3)

[Section III.5, Eqs. (40)-(41)] The paper's central claim rests on the statement that the 2c S_z S_x term in Eq. (40) is an unconventional squeezing term absent from the conventional LMG model. Yet the classical model actually analyzed in Section IV, Eq. (41), contains S_x S_y but no S_z S_x term, and no rotation or basis change connecting Eq. (40) to Eq. (41) is given. The missing connection matters: for a traceless quadratic spin Hamiltonian, any symmetric 3x3 coupling matrix can be diagonalized by an SO(3) rotation, and the space of such Hamiltonians is the l=2 representation of SO(3). Hence S_z S_x is unitarily equivalent to a linear combination of S_x^2, S_y^2, and S_z^2 up to the conserved S^2, so it does not by itself establish a new type of quantum phase transition.
[Section III.5 and Section V] The manuscript asserts that the transition 'manifests as an excited-state quantum phase transition' and that at the transition point a series of excited states cross, but then states that the spectral properties and squeezing phenomena are deferred to a forthcoming paper. No finite-N spectrum, level density, fidelity susceptibility, or level-crossing calculation is presented. A classical bifurcation in the energy surface is a necessary analogue, not a sufficient demonstration of an ESQPT; the central quantum claim is therefore explicitly unsupported.
[Abstract, Section III.5, and Section V] The advertised 'geometric gauge structures akin to Berry-like phases' are not derived anywhere in the manuscript. Section III.5 contains only a qualitative statement that the geometric phase is generated by a singular gauge field and takes the form of a conical cosmological singularity in de Sitter space; no Berry connection, holonomy, curvature, or adiabatic evolution is computed. This is a load-bearing advertised finding and should be either derived or removed from the abstract and conclusions.

minor comments (5)

[Section II, after Eq. (26)] 'Where where each point on S^2' contains a duplicated 'where'.
[Section II, text around Eq. (26)] The paragraph says the Hamiltonian contains an Euler-top-like term proportional to S_y^2, but Eq. (26) displays S_z^2 + S_x^2, which equals S^2 - S_y^2 up to the conserved S^2; the wording should be reconciled.
[Section IV, Eq. (44)] The angle θ defined by tan 2θ = 2β/(α-γ) is undefined when α=γ and β=0; the isotropic limit should be discussed separately.
[Section III.5, Eq. (37)] The statement that Eq. (37) is integrable only when d=-c would benefit from an explicit reference to the general conditions bx=by, dy=cx, and dx=cy derived in Appendix B.
[References] Reference [16] is an unpublished arXiv preprint and is cited for the framework of optical analogues and geometric gauge structures; the dependence on it for key motivational claims should be reduced or the reference updated.

Circularity Check

1 steps flagged · score 6.0 of 10

Partial circularity: the claimed novel S_zS_x term is a coordinate artifact that reduces to the standard LMG/asymmetric-top family under SO(3) rotation, so the 'novel QPT' is not an independent result.

renaming known result [Section III.5 (Eq. (40)) and Section IV (Eqs. (41), (44)-(46))]
"In Eq. (40), the Hamiltonian features a quadratic crossing term, S_zS_x, which is absent from the conventional LMG model. In addition to the standard single-axis squeezing term, S^2−S_y^2, and the two-axis squeezing term, S_z^2−S_x^2, the appearance of this unconventional squeezing term, S_zS_x, gives rise to a novel quantum phase transition in the generalized LMG model."

The 'unconventional' term is defined by its absence from the conventional LMG model in one fixed spin basis, but the model is rotationally covariant. The paper's own Eq. (44) transformation turns the analyzed cross-term Hamiltonian Eq. (41) into the standard asymmetric Euler top Eqs. (45)-(46); every traceless quadratic spin coupling, including the S_zS_x term of Eq. (40), is an SO(3) rotation away from a diagonal LMG-type anisotropy. The claimed novel QPT is therefore not an independent output of the derivation; it is the known LMG/asymmetric-top bifurcation expressed in rotated coordinates, so the 'novelty' is equivalent by construction to the coordinate choice used to define it.

full rationale

The central circular step is the claim that the S_zS_x term in Eq. (40) produces a novel quantum phase transition absent from the conventional LMG model. The paper's own Section IV analysis shows that the cross-term Hamiltonian Eq. (41) is transformed by the rotation Eq. (44) into the standard asymmetric Euler top, Eqs. (45)-(46); since a general traceless quadratic spin Hamiltonian is equivalent under SO(3) to a diagonal LMG-type anisotropy, the S_zS_x term is a basis-dependent label rather than a new interaction. The subsequent classical bifurcation from an ellipsoidal to a hyperbolic energy surface is therefore a known asymmetric-top phenomenon presented in rotated coordinates, and the 'novel QPT' is not an independent result of the derivation. The missing explicit map from Eq. (40) to Eq. (41) and the deferred quantum spectrum ('We defer the discussion of the spectral properties of the excited-state quantum phase transition and the associated squeezing phenomena to a forthcoming paper') further weaken the central claim, but these are support gaps rather than additional circularities. The self-citations [14,16] for the optical-quantum correspondence are not the core of the circularity here. Score 6 reflects a central novelty claim that reduces by construction to a known model in a rotated basis.

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

All parameters are material coefficients or generic model constants, not fitted to data. The derivation rests on standard fiber-optics approximations and on a semiclassical correspondence that is asserted rather than derived. The most significant unstated load is the assumed equivalence between the tetragonal fiber Hamiltonian and the classical spin model analyzed.

assumptions (5)

domain assumption Weakly guiding approximation (Δ << 1) so the fiber supports the fundamental HE11 mode with two linear polarizations.
Used throughout Sections II and III to derive the coupled-mode equations from Maxwell's equations.
domain assumption Instantaneous third-order nonlinear response and neglect of frequency tripling.
Justified in Appendix A for pulses longer than about 5 ps; standard in fiber optics.
domain assumption The classical Poisson-bracket dynamics on the unit sphere correctly represents the large-spin limit of the quantum LMG model.
Standard semiclassical correspondence, cited to Refs. [4,5,14], but not derived in this paper.
ad hoc to paper Tetragonal fiber coefficients satisfy d = -c so that the coupled-mode equations are integrable.
Stated in Section III.5; this condition selects the integrable subclass, excluding the general tetragonal case.
ad hoc to paper The Hamiltonian H = α/2 S_x^2 + β S_x S_y + γ/2 S_y^2 captures the unconventional squeezing of the tetragonal fiber case.
Assumed at the start of Section IV; no explicit transformation from Eq. (40), which contains S_z S_x, is provided.

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Pith. "Pith review of Nonlinear optical analogues of quantum phase transitions in a squeezing-enhanced LMG model." pith.science (2026). https://pith.science/paper/QQYRTEWV

@misc{pith2026250518618,
  author       = {Pith},
  title        = {Pith review of: Nonlinear optical analogues of quantum phase transitions in a squeezing-enhanced LMG model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQYRTEWV}},
  note         = {Machine review of arXiv:2505.18618}
}

read the original abstract

We investigate nonlinear optical analogues of quantum phase transitions within a squeezing-enhanced generalized Lipkin-Meshkov-Glick (LMG) model, focusing on excited-state quantum phase transitions in optical fibers with tetragonal symmetry. Our analysis reveals a novel squeezing effect that induces classical bifurcations in polarization dynamics, even without a linear rotor-like term. By mapping the nonlinear polarization dynamics to the generalized LMG model, we establish a direct correspondence between optical bifurcations and quantum critical phenomena, uncovering geometric gauge structures akin to Berry-like phases. These findings highlight the interplay between classical and quantum behaviors in optical systems, offering a versatile platform for studying quantum many-body physics with applications in quantum metrology and simulation.

Figures

Figures reproduced from arXiv: 2505.18618 by the authors.

**Figure 2.** FIG. 2. Intersection of the hyperbolic cylinder [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Nonlinear optical realization of non-integrable phases accompanying quantum phase transitions
quant-ph 2025-05 conditional novelty 5.0 of 10

A tetragonal-symmetry nonlinear fiber is shown to accumulate a Hannay angle whose conic singularity matches a BEC quantum phase transition, enabling an optical analog simulator.

Reference graph

Works this paper leans on

77 extracted references · 72 canonical work pages · cited by 1 Pith paper

[1]

(20) Hence, we have a j = 3b j, c j = d j = 0 and ax = ay = γ in Eq

Isotropic Fibers For isotropic fibers, the third-order nonlinear susceptibility tensorχ(3) has eight nonzero elements, of which only three are independent, and its elements obey [23] χ(3) xxyy =χ(3) yyxx, χ(3) xyxy =χ(3) yxyx, χ(3) xyyx =χ(3) yxxy, χ(3) xxxx =χ(3) yyyy =χ(3) xxyy +χ(3) xyxy +χ(3) xyyx. (20) Hence, we have a j = 3b j, c j = d j = 0 and ax ...

work page
[2]

The cubic crystal system has five point groups, 23, m3, 432, ¯43m and m3m [31]

Cubic Symmetry We now discuss anisotropic fibers with cubic symme- try. The cubic crystal system has five point groups, 23, m3, 432, ¯43m and m3m [31]. For the two classes 23 and m3, the third-order nonlinear susceptibility tensor χ(3) has eight nonzero elements and seven of them are indepen- dent. Those independent elements are given by χ(3) xxxx = χ(3) ...

work page
[3]

The hexagonal crystal system has seven point groups, 6, ¯6, 6/m, 622, 6mm, ¯62m and 6/mmm [31]

Hexagonal Symmetry We now discuss anisotropic fibers with hexagonal symme- try. The hexagonal crystal system has seven point groups, 6, ¯6, 6/m, 622, 6mm, ¯62m and 6/mmm [31]. For the three crystal classes 6, ¯6 and 6/m, the third-order nonlinear susceptibility tensorχ(3) has sixteen nonzero elements, of which only six are independent, and its elements ob...

work page
[4]

The trigonal crystal system encompasses five point groups: 3, ¯3, 32, 3 m, and ¯3m [31]

Trigonal Symmetry We now turn our attention to anisotropic fibers exhibiting trigonal symmetry. The trigonal crystal system encompasses five point groups: 3, ¯3, 32, 3 m, and ¯3m [31]. In the crystal classes 3 and ¯3, the third-order nonlinear susceptibility tensor χ(3) contains sixteen nonzero elements, of which only six are independent. These nonzero el...

work page
[5]

Tetragonal Symmetry The tetragonal crystal system has seven point groups, 4, ¯4, 4/m, 422, 4 mm, ¯42m and 4/mmm [31]. For the three crystal classes 4, ¯4 and 4/m, the third-order nonlinear susceptibility tensorχ(3) has sixteen nonzero elements, of which only eight are independent, and its elements obey [23]  χ(3) xxyy =χ(3) yyxx, χ(3) xyxy ...

work page
[6]

The coupled-mode equations Eq

Monoclinic and Triclinic Symmetries Last but not least, the remaining crystal systems are: the monoclinic crystal system, which has three point groups 2, m and 2/m; the orthorhombic crystal system, which has three point groups 222, mm2 and mmm; and the triclinic crys- tal system, which has two point groups 1 and ¯1 [31]. The coupled-mode equations Eq. (19...

work page
[7]

dn(κt, k1), (61a) My = q Iy(1− 2HI x)/(Iy− Ix) sn(κt, k1) = k1 nd(ka, k′

work page
[8]

sn(κt, k1), (61b) Mz = p 1− 2HI x cn(κt, k1) = k1 sd(ka, k′

work page

Show all 77 references

[9]

(62) Here, cd u, nd u and sd u are auxiliary elliptic functions de- fined by cd u ≡ cn u/ dn u, nd u ≡ 1/ dn u, and sd u ≡ sn u/ dn u

cn(κt, k1), (61c) where we have used the identity cd( ka,−ik′/k) = cn(a, k′), and κ≡ω q 2H(Iy− Ix), k1≡ 1 k = s 1− 2HI x 2H(Iy− Ix), k′ 1≡− ik′ k = s 2HIy− 1 2H(Iy− Ix). (62) Here, cd u, nd u and sd u are auxiliary elliptic functions de- fined by cd u ≡ cn u/ dn u, nd u ≡ 1/ d...
[10]

dn(κt, k1) + k1 sinθ nd(ka, k′
[11]

sn(κt, k1), (64a) S y = k1 cosθ nd(ka, k′
[12]

sn(κt, k1)− cosθ cd(ka, k′
[13]

dn(κt, k1), (64b) S z = k1 sd(ka, k′
[14]

cn(κt, k1). (64c) As a remark, the energies of the classical spin at the fixed points Mx = ±1 and My = ±1 are Hmax ≡ 1/(2Ix) and Hsep ≡ 1/(2Iy) respectively, where Hmax denotes the maxi- mum energy of the classical spin for fixed parameters, and 10 Hsep represents the energy f...
[15]

cn(ρt, k1), (74a) My = s Jy(1− 2HJ x) Jx + Jy dn(ρt, k1) = sn(ϑ1, k′
[16]

dn(ρt, k1), (74b) Mz =− q 1 + 2HJ y sn(ρt, k1) = dn(ϑ1, k′
[17]

In par- ticular, for H = 0 and k = ˜k = 1, the Jacobi elliptic func- tions degenerate into the hyperbolic functions: snτ→ tanhτ, cnτ→ sechτ, dnτ→ sechτ

sn(ρt, k1), (74c) where ρ≡ Ω√1− 2HJ x, k1 ≡ 1/k, k′ 1 ≡ q 1− k2 1, and ϑ1 is a real number defined by dn(ϑ1, k′ 1)≡ p1 + 2HIy. In par- ticular, for H = 0 and k = ˜k = 1, the Jacobi elliptic func- tions degenerate into the hyperbolic functions: snτ→ tanhτ, cnτ→ sechτ, dnτ→ sech...
[18]

cn(ρt, k1) + sinθ sn(ϑ1, k′
[19]

dn(ρt, k1), (77a) S y =− sinθ cn(ϑ1, k′
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cn(ρt, k1) + cosθ sn(ϑ1, k′
[21]

dn(ρt, k1), (77b) S z =− dn(ϑ1, k′
[22]

(77c) In particular, for H = 0, we obtain S x = sin(ϑ +θ) sechΩt, (78a) S y = cos(ϑ +θ) sechΩt, (78b) S z =− tanh Ωt

sn(ρt, k1). (77c) In particular, for H = 0, we obtain S x = sin(ϑ +θ) sechΩt, (78a) S y = cos(ϑ +θ) sechΩt, (78b) S z =− tanh Ωt. (78c) The area enclosed by the heteroclinic orbits passing through the hyperbolic saddles at S z = ±1 may be calculated from elementary geometry. F...
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(B2a) - (B2b), and Eq

(B3) Now, if we assume the coupled-mode equations can be written in the form of nonlinear Schrödinger equations, Eqs. (B2a) - (B2b), and Eq. (B3) immediately yield i˙ux = ∂H ∂u∗x = Aux + C∗uy + 2E1|ux|2ux + F1u2 xu∗ y + F2|uy|2uy + 2F∗ 1|ux|2uy + H|uy|2ux + 2I∗ 1u2 yu∗ x, (B4a...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

(20) Hence, we have a j = 3b j, c j = d j = 0 and ax = ay = γ in Eq

Isotropic Fibers For isotropic fibers, the third-order nonlinear susceptibility tensorχ(3) has eight nonzero elements, of which only three are independent, and its elements obey [23] χ(3) xxyy =χ(3) yyxx, χ(3) xyxy =χ(3) yxyx, χ(3) xyyx =χ(3) yxxy, χ(3) xxxx =χ(3) yyyy =χ(3) xxyy +χ(3) xyxy +χ(3) xyyx. (20) Hence, we have a j = 3b j, c j = d j = 0 and ax ...

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[2] [2]

The cubic crystal system has five point groups, 23, m3, 432, ¯43m and m3m [31]

Cubic Symmetry We now discuss anisotropic fibers with cubic symme- try. The cubic crystal system has five point groups, 23, m3, 432, ¯43m and m3m [31]. For the two classes 23 and m3, the third-order nonlinear susceptibility tensor χ(3) has eight nonzero elements and seven of them are indepen- dent. Those independent elements are given by χ(3) xxxx = χ(3) ...

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[3] [3]

The hexagonal crystal system has seven point groups, 6, ¯6, 6/m, 622, 6mm, ¯62m and 6/mmm [31]

Hexagonal Symmetry We now discuss anisotropic fibers with hexagonal symme- try. The hexagonal crystal system has seven point groups, 6, ¯6, 6/m, 622, 6mm, ¯62m and 6/mmm [31]. For the three crystal classes 6, ¯6 and 6/m, the third-order nonlinear susceptibility tensorχ(3) has sixteen nonzero elements, of which only six are independent, and its elements ob...

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[4] [4]

The trigonal crystal system encompasses five point groups: 3, ¯3, 32, 3 m, and ¯3m [31]

Trigonal Symmetry We now turn our attention to anisotropic fibers exhibiting trigonal symmetry. The trigonal crystal system encompasses five point groups: 3, ¯3, 32, 3 m, and ¯3m [31]. In the crystal classes 3 and ¯3, the third-order nonlinear susceptibility tensor χ(3) contains sixteen nonzero elements, of which only six are independent. These nonzero el...

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[5] [5]

Tetragonal Symmetry The tetragonal crystal system has seven point groups, 4, ¯4, 4/m, 422, 4 mm, ¯42m and 4/mmm [31]. For the three crystal classes 4, ¯4 and 4/m, the third-order nonlinear susceptibility tensorχ(3) has sixteen nonzero elements, of which only eight are independent, and its elements obey [23]  χ(3) xxyy =χ(3) yyxx, χ(3) xyxy ...

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[6] [6]

The coupled-mode equations Eq

Monoclinic and Triclinic Symmetries Last but not least, the remaining crystal systems are: the monoclinic crystal system, which has three point groups 2, m and 2/m; the orthorhombic crystal system, which has three point groups 222, mm2 and mmm; and the triclinic crys- tal system, which has two point groups 1 and ¯1 [31]. The coupled-mode equations Eq. (19...

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[7] [7]

dn(κt, k1), (61a) My = q Iy(1− 2HI x)/(Iy− Ix) sn(κt, k1) = k1 nd(ka, k′

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[8] [8]

sn(κt, k1), (61b) Mz = p 1− 2HI x cn(κt, k1) = k1 sd(ka, k′

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