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Structured Gaussians From Geometric Quantisation

Kerr Maxwell

TL;DR

The paper develops a geometric framework for structured Gaussian beams by applying geometric quantisation and symplectic reduction to the 2D harmonic oscillator, showing the reduced phase space coincides with the modal Poincaré sphere and that interbasis expansions of Generalised Hermite-Laguerre-Gauss modes correspond to $SU(2)$ rotations. It first quantises the 2DSHO directly and via reduction, demonstrating that quantisation commutes with reduction in this setting, and then reduces the system to a sphere $S^2$ whose action-angle (and Kähler) quantisations yield a finite-dimensional spin-$N/2$ Hilbert space with $2N+1$ states. Rotations generated by the $\mathfrak{su}(2)$ invariants are shown to enact $SU(2)$ rotations on the quantum space, with interbasis expansions governed by Wigner $D$-matrices, and the rotated GG modes are identified with points on the modal Poincaré sphere via the Segal–Bargmann transform. Overall, the work provides a rigorous geometric interpretation of structured light symmetries and presents a pathway to extend the formalism to other symmetry groups and optical degrees of freedom.

Abstract

We develop a geometric description of structured Gaussian beams, a form a structured light, by applying geometric quantisation and symplectic reduction to the 2D harmonic oscillator. Our results show that the geometric quantisation of the oscillator's reduced phase space coincides with the modal Poincaré sphere in optics. We explicitly consider the case of the Generalised Hermite-Laguerre-Gauss modes, identifying their interbasis expansions with rotations of the reduced phase space and the geometric data accompanying the quantisation. This description simplifies the presentation of $SU(2)$ symmetries in structured light beams and is extensible to other symmetry groups.

Structured Gaussians From Geometric Quantisation

TL;DR

The paper develops a geometric framework for structured Gaussian beams by applying geometric quantisation and symplectic reduction to the 2D harmonic oscillator, showing the reduced phase space coincides with the modal Poincaré sphere and that interbasis expansions of Generalised Hermite-Laguerre-Gauss modes correspond to rotations. It first quantises the 2DSHO directly and via reduction, demonstrating that quantisation commutes with reduction in this setting, and then reduces the system to a sphere whose action-angle (and Kähler) quantisations yield a finite-dimensional spin- Hilbert space with states. Rotations generated by the invariants are shown to enact rotations on the quantum space, with interbasis expansions governed by Wigner -matrices, and the rotated GG modes are identified with points on the modal Poincaré sphere via the Segal–Bargmann transform. Overall, the work provides a rigorous geometric interpretation of structured light symmetries and presents a pathway to extend the formalism to other symmetry groups and optical degrees of freedom.

Abstract

We develop a geometric description of structured Gaussian beams, a form a structured light, by applying geometric quantisation and symplectic reduction to the 2D harmonic oscillator. Our results show that the geometric quantisation of the oscillator's reduced phase space coincides with the modal Poincaré sphere in optics. We explicitly consider the case of the Generalised Hermite-Laguerre-Gauss modes, identifying their interbasis expansions with rotations of the reduced phase space and the geometric data accompanying the quantisation. This description simplifies the presentation of symmetries in structured light beams and is extensible to other symmetry groups.

Paper Structure

This paper contains 15 sections, 3 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Symplectic Reduction
  4. Geometric Quantisation
  5. Kähler Quantisation of the 2DSHO and the Jordan-Schwinger Map
  6. Direct Quantisation
  7. Quantisation via Reduction
  8. Reduced Quantisations of the 2DSHO
  9. The Reduction $T^*\mathbb{R}^2\rightarrow S^2$
  10. Quantisation From Action-Angle Foliation
  11. Kähler Quantisation
  12. Applications
  13. Quantisation of Rotations
  14. Rotating the Modal Poincaré Sphere and the Generalised Hermite-Laguerre-Gauss Modes
  15. Conclusion

Key Result

Theorem 2.4

Let $\undefined$ be a free and proper Hamiltonian $G$ action on $M$ with moment map $\undefined$. Let $\undefined$ act freely and properly on $\undefined^{-1}(\mathcal{O})$ where $\mathcal{O}\subset \mathfrak{g}^*$ is a coadjoint orbit and every point in $\mathcal{O}$ is a regular value of $\undefin is a symplectic manifold, called the reduced phase space. The symplectic structure $\overline{\unde

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Marsden-Weinstein Reduction, 5.4.15 in mcduff2017introduction
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 9 more