Wigner distribution of Sine Gordon and Kink solitons

TL;DR

This work develops phase-space descriptions of topological solitons by deriving Wigner distributions $\mathcal{W}(x,k)$ from Schrödinger wave-functionals obtained through a shifted-Hamiltonian transformation. The Sine-Gordon soliton yields a closed analytic form for $\mathcal{W}(x,k)$, while the kink soliton is treated with its oscillator-mode decomposition to produce a numerically evaluated $\mathcal{W}(x,k)$. From these distributions, the authors compute charge densities $\mathcal{Q}(x)$ and current densities $\mathcal{J}(x)$, finding momentum-space symmetry and vanishing currents under symmetric integration, with charge densities closely matching $|\psi(x)|^{2}$. The results lay a foundation for applying Wigner-phase-space methods to solitons and enable connections to quantum-speed-limit quantities via fidelity between Wigner states, suggesting further exploration of instantons and semi-classical dynamics.

Abstract

Wigner distributions play a significant role in formulating the phase space analogue of quantum mechanics. The Schrodinger wave-functional for solitons is needed to derive it for solitons. The Wigner distribution derived can further be used for calculating the charge distributions, current densities and wave function amplitude in position or momentum space. It can be also used to calculate the upper bound of the quantum speed limit time. We derive and analyze the Wigner distributions for Kink and Sine-Gordon solitons by evaluating the Schrodinger wave-functional for both solitons. The charge, current density, and quantum speed limit for solitons are also discussed which we obtain from the derived analytical expression of Wigner distributions.

Figures (4)

• Figure 1: 3D plot of Wigner distribution of Sine - Gordon soliton for $k_{0}\rightarrow 1, \hbar\rightarrow 1, n_{k} \rightarrow 0, n_{-k} \rightarrow 0, m\rightarrow 1.$
• Figure 2: Plot of charge density of Sine- Gordon soliton for $k_{0}\rightarrow 1, \hbar\rightarrow 1, n_{k} \rightarrow 0, n_{-k} \rightarrow 0, m\rightarrow 0.3.$
• Figure 3: 3D plot of Wigner distribution for $k_{0}\rightarrow 1, \hbar\rightarrow 1, n_{BO}^{\prime} \rightarrow 0, n_{k}^{\prime\prime} \rightarrow 0, n_{-k}^{\prime\prime} \rightarrow 0 , m\rightarrow 1, \beta \rightarrow 1, \lambda \rightarrow 1.$
• Figure 4: Plot of charge density of Kink soliton for $k_{0}\rightarrow 1, \hbar\rightarrow 1, n_{BO}^{\prime} \rightarrow 0, n_{k}^{\prime\prime} \rightarrow 0, n_{-k}^{\prime\prime} \rightarrow 0, m\rightarrow 1, \beta \rightarrow 1, \lambda \rightarrow 1.$