Coherent States of Non-Null Torus Knots

Abstract: We construct coherent states for the quantized electromagnetic field that correspond to the classical non-null torus knot solutions of Maxwell's equations in vacuum. We derive the displacement operators from the general relation between classical fields and coherent state amplitudes and verify the defining properties of coherent states through direct computation. We determine the observables of the model: field expectation values, energy density, Poynting vector, helicity, photon number, quadrature uncertainties, and correlation functions, and calculate their expectation values in the knotted coherent states in terms of the integer parameters $(n,m,l,s)$ of the classical solutions. As an example, we particularize the construction in the case of the Hopfion coherent state. These results establish the quantum-classical correspondence for this type of vacuum topological electromagnetic systems.

• The paper introduces a coherent state quantization method that reproduces classical non-null torus knot electromagnetic field configurations using canonical quantization in the Coulomb gauge.
• It provides explicit formulas for observables like energy, photon number, and helicity, directly determined by the knot’s integer parameters.
• The construction rigorously verifies quantum-classical correspondence, paving the way for exploring topological effects in quantum optics and field theory simulation.

Coherent State Quantization of Non-Null Torus Knots in Electromagnetism

Introduction

This work, "Coherent States of Non-Null Torus Knots" (2605.15420), presents a rigorous construction of coherent quantum states associated with topologically nontrivial vacuum solutions of Maxwell's equations, specifically the non-null torus knot family. The framework bridges classical topological electromagnetic field configurations described by parameters $(n,m,l,s)$ with their quantum coherent state counterparts, providing a comprehensive quantum-classical correspondence. The analysis encompasses explicit displacement operators, quantization prescriptions, observable calculations, and explicit expressions for photon-number distributions, helicity, and correlation functions.

Construction of Coherent States for Topological Fields

The authors operate within the Coulomb gauge, leveraging the canonical quantization of the electromagnetic field in terms of its oscillator modes. Coherent states, defined as eigenstates of annihilation operators in this mode expansion, are constructed via displacement operators parameterized by complex amplitudes $\alpha_{\mathbf{k}\lambda}$. These amplitudes are determined through the initial classical field data using explicit Fourier transform relations.

For general Ra~{n}ada-Hopf solutions, the mapping between classical vector potentials/electric fields and the quantum coherent state amplitudes is established. The displacement operator generating the coherent state is given as

$$\hat{D}(\{\alpha\}) = \exp\left[ \sum_{\lambda} \int \frac{d^3k}{(2\pi)^3} \left( \alpha_{\mathbf{k}\lambda} \hat{a}^{\dagger}_{\mathbf{k}\lambda} - \bar{\alpha}_{\mathbf{k}\lambda} \hat{a}_{\mathbf{k}\lambda} \right) \right].$$

The knotted field configurations are determined by nontrivial mappings $S^3 \to S^2$ encoding Hopf fibrations, and the explicit mode amplitudes $\alpha_{\mathbf{k}\lambda}$ are derived for these spatially-localized, finite-energy configurations.

Non-Null Torus Knot Coherent States

The central objects in the analysis are the non-null torus knot solutions, whose field lines correspond to torus knots classified by coprime integers $(n,m)$ for magnetic field topology and $(l,s)$ for electric field topology. The explicit construction of time-dependent solutions is reviewed, and their quantization is addressed by computing the corresponding coherent state amplitudes. The Fourier-transformed amplitudes feature exponential spatial cut-offs, directly linked to the physical knot size.

The displacement operator for these coherent states incorporates the knot parameters, field strength scale $a$, and characteristic length $L_0$:

$$\alpha_{\mathbf{k}\lambda} = \frac{\sqrt{a} L_0}{2^{3/2}\pi^2 \sqrt{2\omega_{\mathbf{k}}}} e^{-K} \boldsymbol{\epsilon}_{\mathbf{k}\lambda}^* \cdot \mathbf{W}(\mathbf{K})$$

where $\alpha_{\mathbf{k}\lambda}$0 encodes all topological and geometrical information. The convergent nature of the construction is ensured by the physically motivated exponential cutoff derived from the spatial localization of the classical field.

Calculation of Observables

A detailed analysis of quantum observables in these coherent states reveals how topological information translates into quantum expectation values:

• Energy: The normal-ordered Hamiltonian expectation value in the non-null torus knot coherent state is

$\alpha_{\mathbf{k}\lambda}$1

indicating degeneracy in energy among different toroidal topologies with the same sum of parameter squares.

• Photon Number: The expected photon number is similarly linked to the knot parameters:

$\alpha_{\mathbf{k}\lambda}$2

with the mean photon energy fixed at $\alpha_{\mathbf{k}\lambda}$3 (in units where $\alpha_{\mathbf{k}\lambda}$4), set by the spectral cutoff originating from the knot size.

• Momentum Density: The Poynting vector expectation value exhibits a degenerate structure in the knot parameters, with explicit time- and space-dependent toroidal patterns.
• Helicity: Both spin (optical) helicity and magnetic helicity expectation values are determined solely by the linking invariants $\alpha_{\mathbf{k}\lambda}$5 and $\alpha_{\mathbf{k}\lambda}$6:

$\alpha_{\mathbf{k}\lambda}$7

confirming the quantum preservation of topological invariants of the classical field.

• Correlations: First- and second-order correlation functions reveal nontrivial spatial variation in $\alpha_{\mathbf{k}\lambda}$8, attributable to spatial polarization structure, while $\alpha_{\mathbf{k}\lambda}$9 saturates the value 1, confirming full quantum coherence (minimum-uncertainty, Poissonian statistics).

Hopfion as a Special Case

The Hopfion field, corresponding to the minimal case $$\hat{D}(\{\alpha\}) = \exp\left[ \sum_{\lambda} \int \frac{d^3k}{(2\pi)^3} \left( \alpha_{\mathbf{k}\lambda} \hat{a}^{\dagger}_{\mathbf{k}\lambda} - \bar{\alpha}_{\mathbf{k}\lambda} \hat{a}_{\mathbf{k}\lambda} \right) \right].$$0, is treated explicitly. The observables for this configuration are obtained directly from the general results, illustrating the inheritance of topological and energetic properties by the quantum state. The spatial structure and helicity expectation clearly track the underlying Hopf link topology.

Theoretical and Practical Implications

The results establish an unambiguous quantum-classical correspondence for topologically nontrivial electromagnetic fields in vacuum, providing a solvable model where quantization preserves knot invariants and field topology at the level of quantum expectation values. This formalism opens the following research directions:

• Generalization to Nonclassical States: Extension to squeezed and entangled multi-mode states could unveil non-classical topological effects and potential topological robustness under quantum noise.
• Matter Interaction and Decoherence: The model provides a starting point for exploring the interaction of topological light states with quantum matter, as well as the robustness of topological invariants under decoherence.
• Quantum Information and Photonic Devices: The explicit linkage of quantum observables to topological indices could have implications for the generation of robust quantum states for information processing or for engineering topologically-structured light in quantum optical devices.
• Simulating Topological Field Theories: Coherent topological states provide testbeds for simulating aspects of topological field theory and exploring their consequences in controlled laboratory environments.

Conclusion

The paper presents a robust quantization framework for topological electromagnetic fields, explicitly constructing coherent state representations for non-null torus knots and Ra~{n}ada-Hopf solutions. All field observables are explicitly linked to the topological invariants of the underlying classical configuration, and the quantum-classical correspondence is realized in a physically transparent and computationally tractable way. This work lays foundational groundwork for further investigation of topological phenomena in quantum optics and the interplay between geometry, topology, and quantum field theory.