Isotopic variations and Zeeman-like splitting in the spectra of nonlinear photonic meta-atoms

Abstract: We study photonic meta-atoms, a unique class of composite solitary wave, supported in nonlinear waveguides. We establish an analogy to one-dimensional soft-core atoms, allowing to describe the complex dynamics via concepts from atomic physics. Higher-order dispersive effects cause specific spectral resonances characteristic for the eigenspectrum of a meta-atom. We demonstrate that subtle changes in this level spectrum causes frequency shifts of the resonances. These shifts consist of isotopic and isomeric contributions that can be distinguished in terms of a simple model. We further demonstrate a generic mechanism that causes a Zeeman-like splitting of resonance lines.

• The paper develops a quantitative spectroscopy framework in which trapped states of nonlinear photonic meta-atoms emit phase-matched resonant radiation that fingerprints their eigenspectrum.
• The paper shows that changing soliton duration shifts resonances by about 0.026 and 0.019 frequency units for the two lowest states, while an analytical formula separates isotopic and isomeric effects.
• The paper demonstrates that periodic soliton breathing splits resonance lines according to D(Ω) = −κₙ + mKₛ, providing a field-free optical analogue of Zeeman splitting and a probe of internal dynamics.

Overview

This paper develops the analogy between photonic meta-atoms—composite solitary waves in nonlinear waveguides—and one-dimensional (1D) soft-core atoms from strong-field physics. The authors show that higher-order dispersive effects couple the discrete trapped states of a meta-atom to dispersive-wave continua, producing resonance lines that act as a spectral fingerprint. Subtle changes in the trapped-state eigenspectrum shift these resonances, and the shifts decompose into isotopic and isomeric contributions analogous to atomic spectroscopy. A further mechanism, arising from $z$-periodic oscillation of the soliton component, produces a Zeeman-like splitting of the resonance lines (2603.17748).

Model and meta-atomic eigenproblem

The governing equation is a higher-order nonlinear Schrödinger equation (HONSE) with quadratic dispersion $\beta_2>0$ and quartic dispersion $\beta_4<0$, supporting two anomalous-dispersion domains separated by a normal-dispersion region with zero-dispersion points at $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$. Meta-atoms are described by a two-pulse ansatz: a strong soliton $A_{\rm S}$ at detuning $\Omega_{\rm S}$ and a weak pulse $A_{\rm G}$ at $\Omega_{\rm G}$, group-velocity matched across the frequency gap. In the limit $\max(|A_{\rm G}|)\ll\max(|A_{\rm S}|)$, the weak pulse obeys a Schrödinger equation with a soliton-induced Pöschl–Teller-type trapping potential $V_{\rm T}(t)=-2\gamma P_0\,{\rm sech}^2(t/t_0)$, solvable exactly with $\beta_2>0$0 trapped states at eigenvalues $\beta_2>0$1, where $\beta_2>0$2 is an effective charge determined by the ratio $\beta_2>0$3.

The analogy to 1D soft-core Coulomb potentials is established by matching well depths rather than ground-state energies: associating the bound-state count $\beta_2>0$4 with atomic number and the soliton duration $\beta_2>0$5 with nuclear extent (mass number). Nuclides with equal $\beta_2>0$6 but different $\beta_2>0$7 are termed "isotopes"; those with equal $\beta_2>0$8 but different $\beta_2>0$9 (without changing $\beta_4<0$0) are "isomers." Higher-order dispersion retained about $\beta_4<0$1 yields a Gross-Pitaevskii-type equation whose phase-matching condition $\beta_4<0$2 predicts resonant radiation frequencies—a decay channel absent in quantum mechanics, since here the "nucleus" (the soliton) is itself dynamical and radiative.

Resonant radiation as spectral fingerprint

Propagation simulations confirm that energy captured by the trapped states is re-emitted as phase-matched resonant radiation. For the $\beta_4<0$3-nuclide, the graphical solution of the phase-matching condition predicts $\beta_4<0$4 ($\beta_4<0$5) and $\beta_4<0$6 ($\beta_4<0$7), which the observed spectra match closely; the $\beta_4<0$8 peaks overlap due to radiative broadening—higher states decay faster—but a Gaussian peak-fitting analysis resolves all five components, with widths increasing monotonically with $\beta_4<0$9. Resonances are restricted to frequencies where $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$0, producing a sharp cutoff at $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$1. The appendix analysis also identifies a far resonance at $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$2 whose driving is negligible because it occurs far from the localized trapped state.

Isotopic and isomeric shifts

Increasing the soliton duration from $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$3 to $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$4 shifts the resolvable lines to higher frequencies: numerically, $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$5 and $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$6. A linearized dispersion model near $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$7 yields the closed-form estimate

$\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$8

which predicts $\Omega_{\rm Z1,Z2}=\mp\sqrt{2}$9 and $A_{\rm S}$0 for $A_{\rm S}$1—excellent agreement despite the approximations involved. The two terms correspond to the isotopic contribution ($A_{\rm S}$2) and the isomeric contribution ($A_{\rm S}$3), mirroring volume and charge-distribution effects in atomic isotope-shift spectroscopy. Notably, the cutoff frequency remains fixed at $A_{\rm S}$4 across isotopes, confirming that the shift originates in the level spectrum rather than the continuum boundary. Appendix results demonstrate the isomeric variant by varying $A_{\rm S}$5 over $A_{\rm S}$6 while holding $A_{\rm S}$7. These results imply that meta-atom spectra can serve as sensitive probes of their own soliton parameters, in direct analogy to precision isotope-shift spectroscopy.

Zeeman-like splitting for vibrating meta-atoms

For soliton order $A_{\rm S}$8, the soliton component undergoes $A_{\rm S}$9-periodic amplitude and width oscillations, making the trapping potential time-periodic in propagation distance. Transferring a known radiation mechanism for oscillating solitary waves, the modified wavenumber-matching condition becomes

$\Omega_{\rm S}$0

with $\Omega_{\rm S}$1 the angular wavenumber of the oscillation and $\Omega_{\rm S}$2 labeling its spatial harmonics; the range of $\Omega_{\rm S}$3 is set by the anharmonicity of the oscillation. Simulations show both shifting and splitting of the groundstate resonance in excellent agreement with this prediction. The analogy to the Zeeman effect is structural: electron energy shifts $\Omega_{\rm S}$4 map onto wavenumber shifts $\Omega_{\rm S}$5. This establishes line splitting as a generic signature of internal soliton dynamics, observable without external fields.

Limitations and open questions

The analysis rests on several assumptions stated by the authors. The coupled-system reduction neglects higher-order dispersion locally at $\Omega_{\rm S}$6 and linearizes about the weak pulse; retaining the nonlinear term would couple the trapped states to each other, and the effect of such coupling on the spectral signature in a "mildly" nonlinear regime remains unexplored. The shift estimate relies on a linear approximation of the dispersion near the resonances, valid only for small parameter variations. The far resonance is discarded on the grounds of weak driving, and the role of reflectionless potentials at integer $\Omega_{\rm S}$7—whose highest-order state at $\Omega_{\rm S}$8 constitutes an optical analog of a half-bound state—is left open. The authors also anticipate additional broadening and splitting mechanisms in collisions between meta-atoms and dispersive waves, which are not treated here.

Conclusion

The paper substantiates the meta-atom/soft-core-atom analogy into a quantitative spectroscopic framework: trapped-state eigenspectra produce predictable resonance lines, parameter changes yield analytically estimable isotopic and isomeric shifts, and soliton breathing produces Zeeman-like multiplets via harmonics of the oscillation wavenumber. Together these results provide a diagnostic toolkit for reading soliton parameters off optical spectra, and they frame specific open problems concerning nonlinearly coupled trapped states and half-bound-state dynamics in HONSE-guided waveguides.