The Dirac sea of phase: Unifying phase paradoxes and Talbot revivals in multimode waveguides

Abstract

The quantum mechanical description of phase remains a fundamental challenge, with theoretical efforts tracing from the early works of London and Dirac to discrete formalisms. In this work, we extend the action-angle formalism to the Helmholtz-Schrödinger equation by introducing a phase-dependent wavefunction $φ(θ, t)$ residing in the Hardy space $H^2(\mathbb{D})$. This mathematical structure, defined by functions analytic on the unit disk with square-integrable boundary values, naturally ensures the positivity of the energy spectrum while providing a rigorous framework for wave dynamics in photonic systems. We demonstrate that establishing a self-adjoint phase operator requires extending the Hilbert space to $L^2$, a procedure that necessitates the admission of negative energy states. We interpret these states through an analogy with the Dirac sea, where the existence of antiphase or antiphoton modes provides a conceptual framework for understanding the fundamental limits of phase localization and quantum uncertainty. This formalism is applied to light propagation in multimode waveguides characterized by anharmonic refractive index profiles. By mapping modal dispersion to our phase representation, we show that the deviation of propagation constants from linear spacing governs the spatial evolution of the optical field. This approach offers a clear mechanism for the emergence of periodic self-imaging known as the Talbot effect, the generation of fractional revivals, and the formation of complex fractal interference patterns, providing a robust toolkit for the characterization and design of multimode interference devices.

Figures (5)

• Figure 1: Spectral evolution of the anharmonic oscillator. The plot displays the first twelve eigenvalues $\mathcal{E}_k$ as a function of the anharmonicity parameter $\lambda$. The deviation from the equidistant harmonic ladder increases with both coupling strength $\lambda$ and mode index $k$, indicating the onset of strong nonlinear dispersion required for revivals.
• Figure 2: Eigenfunctions of the anharmonic oscillator in dual configuration and phase-space representations. The upper panels show the spatial wavefunctions $\varphi_k(x)$ for two distinct anharmonicity strengths, $\lambda=0.05$ and $\lambda=0.50$, where the steeper quartic potential leads to increased confinement and a characteristic deformation of the nodal structure compared to the ideal harmonic case. The lower panels display the corresponding eigenfunctions projected onto the phase basis, $\varphi_k(\theta)$, separated into their real ($\text{Re}[\varphi_k(\theta)]$) and imaginary ($\text{Im}[\varphi_k(\theta)]$) components. This angular representation reveals the underlying phase topology and localization properties governed by the Hardy space $H^2(\mathbb{D})$ structure, highlighting how the anharmonicity distorts the phase-space symmetry and encodes the nonlinear frequency shifts that drive the Talbot-revival dynamics.
• Figure 3: Spatiotemporal evolution of the transverse probability density $|\psi(x,t)|^2$ in an anharmonic waveguide with quartic potential. The density maps illustrate the longitudinal propagation ($t=z$) for two coupling strengths: $(\mathrm{a}_j)$$\lambda=0.01$ and $(\mathrm{b}_j)$$\lambda=0.02$, where the index $j=1,2,3$ denotes selected temporal windows corresponding to distinct dynamical regimes. The initial configuration is a coherent state with $\alpha=4i$. Higher anharmonicity is shown to accelerate the collapse-revival cycle, inducing faster modal dephasing and a reduction in the fundamental revival period. These spatial carpets visualize the transition from near-field coherence to complex interference and eventual self-imaging, encoding the underlying modal dispersion of the system.
• Figure 4: Spatiotemporal evolution of the phase probability density $|\phi(\theta,t)|^2$ within a polar phase-space representation. The density is mapped into a polar coordinate system $(t, \theta)$, where the radial distance represents the propagation axis $t=z$ and the angular coordinate corresponds to the phase $\theta \in [0, 2\pi)$. Panels correspond to coupling strengths $(\mathrm{a}_j)$$\lambda=0.01$ and $(\mathrm{b}_j)$$\lambda=0.02$, initialized with a coherent state $\alpha=4i$. This visualization exposes the analytic structure of the state in the Hardy space $H^2(\mathbb{D})$, revealing the formation of fractal Talbot carpets and substructures at fractional revival times. The interplay between the linear rotation and quadratic de-alignment driven by the spectrum $\mathcal{E}_k$ results in the shredding and subsequent periodic reconstruction of the phase distribution.
• Figure 5: Temporal evolution of the position expectation value $\langle \hat{x}(t) \rangle$ in the anharmonic waveguide, demonstrating Talbot-type collapse and revival dynamics. The state undergoes periodic dephasing and rephasing governed by quadratic spectral dispersion characterized by $a_2=\frac{3}{2}\lambda$. Upper panel: $\lambda=0.01$ ($a_2=0.015$) exhibits a longer revival period $T_{\text{rev}} \approx [300-400]$ with high-fidelity reconstruction. Lower panel: $\lambda=0.02$ ($a_2=0.03$) accelerates the cycle, reducing $T_{\text{rev}}$ by a factor of two while introducing degradation in subsequent revivals. The reconstitution of $\langle \hat{x}(t) \rangle$ at integer multiples of $T_{\text{rev}}$ is the temporal signature of the Talbot self-imaging effect.