All-optical analogue of anti-skyrmions and anti-merons in difference-frequency-generation process

Technical Summary: All-Optical Analogue of Anti-Skyrmions and Anti-Merons in Difference-Frequency-Generation Process

Problem Statement
Topological quasi-particles, such as Skyrmions, have garnered significant attention in spintronics for their potential in energy-efficient devices like racetrack memory and neuromorphic computing. While optical analogues of these textures have been explored in polarization optics and frequency upconversion, existing methods often face complex parameter constraints that render experimental realization impractical. Furthermore, there is a need to explore these topological states within the context of non-Hermitian dynamics and 'anti' parity-time (anti-PT) symmetry in classical optical interactions. This paper addresses the challenge of identifying a practical, experimentally feasible route to generate and observe topologically non-trivial optical analogues of anti-Skyrmions and anti-merons using difference-frequency generation (DFG).

Methodology
The authors propose a theoretical and analytical framework for a paraxial difference-frequency-generation (PDFG) process within a periodically-poled lithium tantalate (PPLT) crystal. The methodology involves the following key steps:

1. Hamiltonian Formulation: Under the undepleted pump approximation, the interaction Hamiltonian ($\hat{H}$) for the DFG process (involving signal and idler modes) is mapped onto a quantum mechanical 'two-level' atomic system. This system is analogous to a spin-1/2 particle in a magnetic field, where the signal and idler modes constitute the basis.
2. Non-Hermitian Dynamics: The system is analyzed in the spatial frequency (SF) domain. The Hamiltonian is shown to be non-Hermitian ($\hat{H} \neq \hat{H}^\dagger$), introducing singularities known as 'exceptional rings' in the SF space. The dynamics are examined under the condition of anti-PT symmetry, defined by the relation $(PT)\hat{H}(PT)^{-1} = -\hat{H}$.
3. Experimental Configuration: A practical setup is modeled using a 30-mm-long PPLT crystal pumped by a visible (532 nm) Gaussian laser beam. The study considers a variable grating period (fanned-out grating) to tune the phase-mismatch factor ($\Delta k'$) across three specific regimes at a crystal temperature of 200°C:
   • Negative phase-mismatch ($\Delta k' = -0.004$)
   • Perfect phase-matching ($\Delta k' = 0$)
   • Positive phase-mismatch ($\Delta k' = +0.002$)
4. Topological Analysis: The authors define a pseudo-magnetization vector ($\hat{h}$) in the parameter space derived from the Hamiltonian. They calculate the Skyrmion number ($N_{sk}$) as a topological invariant to characterize the texture of the signal and idler fields. The analysis decomposes $N_{sk}$ into polarity ($p$) and vorticity ($v$) components to distinguish between different topological textures (Skyrmions, anti-Skyrmions, merons, and anti-merons).

Key Results
The analysis yields distinct topological phases depending on the phase-mismatch condition:

• Negative Phase-Mismatch ($\Delta k' < 0$): In this regime, the system enters an anti-PT symmetry-broken phase. The eigenvalue spectrum is real in specific regions of the SF space. The resulting pseudo-magnetization texture exhibits a Skyrmion number of $N_{sk} = 1$. Detailed analysis of polarity and vorticity reveals $p = -1$ and $v = -1$, identifying the texture as an anti-Skyrmion. The vector field flips adiabatically from the beam center to infinity, signifying a topologically non-trivial state.
• Perfect Phase-Matching ($\Delta k' = 0$): This condition separates the trivial and non-trivial phases. The Skyrmion number is calculated as $N_{sk} = 1/2$. The pseudo-magnetization vector remains non-zero up to the equatorial plane of the parameter space, corresponding to an anti-meron quasi-particle.
• Positive Phase-Mismatch ($\Delta k' > 0$): In this regime, the system exhibits a topologically trivial interaction dynamics with $N_{sk} = 0$. The pseudo-magnetization vector is confined to the northern hemisphere of the parameter space without reaching the equatorial plane.

Significance and Claims
The paper claims to demonstrate the existence of topologically non-trivial optical analogues of anti-Skyrmions and anti-merons within a classical, all-optical DFG process. By mapping the DFG dynamics onto a non-Hermitian two-level system, the authors show that the process naturally supports anti-PT symmetry breaking.

The primary significance lies in providing an experimentally feasible platform (using standard PPLT crystals and visible pumps) to generate these complex topological textures without the impractical constraints found in previous upconversion proposals. The work establishes that negatively phase-mismatched DFG processes can host anti-Skyrmionic quasi-particles in the signal and idler modes, while phase-matched processes host anti-merons.

The authors conclude that these findings pave the way for developing anti-Skyrmion-encoded parametric devices. Such devices could facilitate reliable spatial-mode manipulation and topologically safeguarded beam-shaping in nonlinear optics, leveraging the broad wavelength scalability of the DFG process. The study aims to unify diverse domains of physical sciences by demonstrating how topological invariants govern the dynamics of nonlinear optical interactions.