Self-Intersecting Orbits in Strong Coulomb Fields

• Self-intersecting orbits are complex trajectories arising in intense Coulomb fields that blend quantum tunneling with nonlinear dynamics.
• They exhibit caustic formation, invariant manifold organization, and bifurcation phenomena, elucidating scaling laws and interference patterns.
• Research integrates semiclassical quantization with topological and relativistic corrections to explain recollision events and modified orbital structures.

Self-intersecting orbits in super strong Coulomb fields are a distinctive class of classical and quantum trajectories whose geometry, topology, and dynamical behavior are governed by the interplay of intense central potentials, external fields, and relativistic corrections. The physical realization and theoretical characterization of such orbits span atomic physics, strong-field ionization dynamics, semiclassical quantization, nonlinear dynamics, and topological analysis. Their paper provides insight into low-energy structures, periodic orbit theory, bifurcation phenomena, high-order processes in strong-field physics, and the frontier of superheavy element atomic structure.

1. Quantum and Semi-Classical Frameworks for Self-Intersecting Orbits

The emergence of self-intersecting orbits is most rigorously treated in the quantum-orbit theories that extend the Strong Field Approximation (SFA) to incorporate the long-range Coulomb potential. In the trajectory-based Coulomb-SFA (TC-SFA) and Coulomb Quantum Orbit Strong-Field Approximation (CQSFA) (Yan et al., 2010), electrons are liberated via tunneling in a strong laser field, and their subsequent evolution is governed by the coupled Newtonian equations: $$\dot{\mathbf{r}} = \mathbf{p} + \mathbf{A}(t),\qquad \dot{\mathbf{p}} = -\frac{\mathbf{r}}{|\mathbf{r}|^3}$$ where $\mathbf{A}(t)$ is the vector potential of the laser field, and the Coulomb force deflects the continuum electron, leading to revisiting of the ionic core. These revisitations manifest as "self-intersecting" trajectories in phase space and contribute to regions of high electron yield—specifically, caustics that underlie the low-energy structures (LES) observed experimentally.

In semiclassical terms, the ionization amplitude is evaluated via the saddle-point method, with the action containing both the laser and Coulomb contributions: $$S^{(C)}(t) = S^{(C\downarrow)}(t_s \to t_0) + S^{(C\rightarrow)}(t_0 \to T),\qquad S^{(C\rightarrow)} = \int_{t_0}^{T} \left[ \frac{v^2(t)}{2} - \frac{1}{|\mathbf{r}(t)|} \right] dt$$ The formation of caustics—points at which many semiclassical trajectories coalesce and the mapping from initial to final momenta becomes singular—is both an analytical and experimental signature of self-intersecting orbits.

2. Nonlinear Dynamics, Periodic Orbits, and Invariant Manifolds

Self-intersecting character can arise from nonlinear-dynamical organization by unstable periodic orbits and their invariant manifolds (Berman et al., 2015). In the presence of both laser and Coulomb fields, phase-space is structured by a small set of unstable periodic orbits (e.g., $\mathcal{O}$ and $\mathcal{O}_{\pm}$), whose stable and unstable manifolds "funnel" trajectories toward recollision events:

• Immediate recollisions: organized along continuous phase-space curves;
• Delayed recollisions: stratified into bands determined by manifold geometry, where the delayed returns can exceed the canonical energy limit ($3.17 U_p$) predicted by SFA.

The generic Hamiltonian is: $$\mathcal{H}(x,p,t) = \frac{1}{2}[p + \cos(t + \phi)]^2 - \frac{\varepsilon}{\sqrt{x^2 + a^2}}$$ where the gradient of the soft-Coulomb term produces focusing even when $\varepsilon/a \ll U_p$, thus establishing the persistence of Coulomb focusing and self-intersections at arbitrarily high laser intensity.

3. Topological and Bifurcation Analysis of Periodic Orbits

In classical and semiclassical models of three-body systems in Coulomb or "super strong" pairwise potentials, self-intersecting orbits are classified via topological sequences and bifurcation diagrams (Dmitrašinović et al., 2017, Šindik et al., 2018). For the $\sum_{i>j} -1/r_{ij}^2$ potential, periodic solutions with fixed hyper-radius and vanishing angular momentum form discrete sequences characterized by words in the free group $F_2$ (e.g., $a^n b^n$), corresponding to closed curves on the 'shape sphere' that puncture themselves at collision loci. This results in an accumulation of self-intersections as the index $n$ grows.

Similarly, in the nonrelativistic Coulomb three-body case, periodic orbits are classified by symmetry under reflections and topological word length $N$ (counting syzygies, or collinear events), and obey a linear scaling law,

$T |E|^{3/2} \sim N$

where $T$ is the period, $E$ the energy, and $N$ the topological complexity (Šindik et al., 2018). This scaling ties orbital topology directly to dynamical properties, providing predictive regularities for self-intersecting trajectories.

4. Influence of External Fields and Modified Potentials

The behavior of orbits in super strong fields is extensively modified by external fields (magnetic, electric) and by vacuum corrections to the central potential (Glazyrin et al., 2016, Schweiner et al., 2014, Leon et al., 2019). When a superstrong magnetic field is present, vacuum polarization alters the Coulomb potential: $$\Phi(\rho,z) = \frac{e}{\pi} \int dk_\parallel \int dk_\perp \frac{ k_\perp J_0(k_\perp\rho) e^{-ik_\parallel z} }{ k_\perp^2 + k_\parallel^2 + (2e^3 B / \pi) e^{-k_\perp^2/2eB} T(k_\parallel^2/4m^2) }$$ The resulting equipotential lines become "eye-shaped" in the mid-range, rather than purely elliptical—a geometric alteration that impacts orbit crossing and self-intersections in extended charge distributions.

For a hydrogen atom in crossed fields (Stark saddle problem), bifurcation analysis reveals transitions (pitchfork, cusp bifurcations) where the orbits evolve from simple elliptical to heart-shaped and ultimately develop self-intersections as energy and field strength increase (Schweiner et al., 2014). Quantum signatures of these classical structures appear in Husimi distributions and periodic oscillations in absorption spectra.

In planar two-body problems with both Coulomb and oscillator (magnetic) terms, bifurcation diagrams organized in elliptic coordinates show that a subset of orbits (e.g., certain satellite and deformed oscillatory types) become self-intersecting with increasing Coulomb dominance (Leon et al., 2019). The trajectory's repeated traversal through caustics is key to self-intersection.

5. Interference and Holography: Observable Consequences

In strong-field ionization, self-intersecting (recolliding, multipass) Coulomb-distorted quantum orbits give rise to experimentally observable interference carpets, spiral fringe patterns, and V-shaped structures in momentum distributions (Maxwell et al., 2020, Maxwell et al., 2018). Spiral interference arises from the interference of specific quantum orbits (labeled 3 and 4 in CQSFA) with saddle-point energetics: $$I + U + E_k = 2n\omega,\quad \text{with}\quad E_k = \frac{1}{2}p_\perp^2$$ This interference is a consequence of the orbits' half-cycle symmetry and close approach to the core, with the spiral itself directly reflective of the self-intersecting character of the underlying trajectory.

In photoelectron holography, the presence of orbital self-intersections enriches the interference landscape, yielding patterns that are highly sensitive to the binding potential and capable of probing structure-dependent phase information after ionization (Maxwell et al., 2018).

6. Relativistic Extensions and Superheavy Elements

The presence of self-intersecting orbits is not limited to nonrelativistic systems. In systems with high nuclear charge, relativistic corrections drive electron dynamics into regimes where the classical orbits no longer close after a single revolution. In the Bohr-Sommerfeld atomic model (Suslov, 15 Oct 2025), the quantized angular advance per cycle $\Delta$ leads to a "winding number" topological classification: $$N_{\rm winding} := 2\left(\frac{\Delta}{2\pi} - 1 \right)$$ For hydrogen-like ions of Oganesson ($Z=118$) and speculative $Z \lesssim 137$, electron orbits transition from open rosettes (winding number 1) to multiply self-intersecting (winding number $>1$), providing a historically significant and pedagogically intuitive measure of Coulomb field strength and orbital complexity.

7. Summary and Outlook

The paper of self-intersecting orbits in super strong Coulomb fields synthesizes methods from quantum orbit theory, periodic orbit topological analysis, nonlinear dynamics, bifurcation theory, and relativistic semiclassical quantization. Key mechanisms include:

• Caustic formation and saddle-point concentration of quantum trajectories.
• Organization by phase-space invariant manifolds and periodic orbits.
• Topological classification via word length and winding number.
• Modulation of orbit structure and intersection by field-induced potential modifications and relativistic dynamics.
• Manifestation in experimentally observable interference patterns and energy scaling relations.

The field continues to evolve through improved computational methods for trajectory identification, refined semiclassical and quantum models, and ongoing experimental verification in strong-field, high-charge, and multidimensional systems. Predictions for experimental signatures—such as spiral interference carpets, period-topology scaling, and caustic-induced low-energy structures—remain active topics at the intersection of atomic, molecular, and nonlinear physics.