Wigner distribution of Sine Gordon and Kink solitons

Technical Summary: Wigner Distribution of Sine-Gordon and Kink Solitons

Problem Statement
The paper addresses the challenge of formulating a phase-space analogue for quantum mechanics regarding classical solitons, specifically the Kink and Sine-Gordon solitons. While the Wigner distribution is a standard tool for studying the quantum-classical interface and calculating properties like charge distributions and quantum speed limits, it requires a well-behaved quantum wave function (Schrodinger wave-functional) as an input. Classical soliton solutions, which satisfy non-linear hyperbolic differential equations, do not inherently possess the properties of a quantum wave function and cannot be directly used to evaluate the Wigner distribution. The authors aim to bridge this gap by deriving the necessary quantum wave functionals for these solitons to compute their Wigner distributions and subsequent physical observables.

Methodology
The authors employ a semi-classical approach rooted in the "shifted Hamiltonian" framework proposed in reference [36]. The methodology proceeds through the following steps:

1. Hamiltonian Transformation: The authors define a new Hamiltonian $H'$ related to the original Hamiltonian $H$ via a similarity transformation $X = \exp(-i \int dx \phi_{cl} \pi(x))$, where $\phi_{cl}$ is the classical soliton solution. This transformation isolates the oscillations in the soliton configuration from the classical background.
2. Quantization and Mode Decomposition: The field $\phi(x)$ and its conjugate momentum $\pi(x)$ are quantized by decomposing them into oscillator modes. This involves identifying continuum modes ($k$) and bound states (Goldstone modes) associated with the soliton. The transformed Hamiltonian is expressed in terms of creation and annihilation operators for these modes.
3. Derivation of Wave Functionals: Using the ground state conditions of the transformed Hamiltonian ($b_k |0\rangle = 0$), the authors derive the Schrodinger wave-functional ($\Psi$) for the soliton states.
   • For the Sine-Gordon soliton, the wave functional is derived by evaluating expectation values of the field operators in the ground state sector, resulting in an analytical expression for the wave function $\psi_{SG}(x)$.
   • For the Kink soliton (a double-well potential solution), a similar procedure is followed, accounting for additional bound states ($g_B$ and $g_{BO}$) and the specific form of the potential $V(\phi)$. This yields the wave function $\psi_K(x)$.
4. Wigner Distribution Calculation: The derived wave functions are substituted into the standard definition of the Wigner distribution:
   $$W(x, k) = \frac{1}{\pi\hbar} \int_{-\infty}^{\infty} dy \, \psi^*(x+y)\psi(x-y)e^{\frac{2iky}{\hbar}}$$
   Analytical integration is performed for the Sine-Gordon case, while numerical integration (with a cutoff for the $y$-variable) is utilized for the Kink soliton due to the complexity of the analytical form.
5. Derivation of Observables: Using the calculated Wigner distributions, the authors derive expressions for charge density ($Q(x) = \int dk W(x,k)$) and current density ($J(x) = \int dk \, k \, W(x,k)$).

Key Results

• Analytical Expressions: The paper provides explicit analytical expressions for the Wigner distributions of both the Sine-Gordon and Kink solitons.
  • The Sine-Gordon Wigner distribution is found to be symmetric in momentum space, with its magnitude increasing as one moves away from the origin in position space (under the specific parameter choices made).
  • The Kink soliton Wigner distribution is also symmetric in momentum space, with a maximum magnitude at zero momentum.
• Charge and Current Densities:
  • For both soliton types, the calculated current density $J(x)$ is found to be zero, consistent with the static nature of the soliton solutions in the chosen frame.
  • The charge density $Q(x)$ is shown to be approximately proportional to the square of the wave function amplitude ($|\psi(x)|^2$).
  • The charge density for the Sine-Gordon soliton is minimum at the origin and increases exponentially away from it.
  • The charge density for the Kink soliton exhibits a peak near the origin. It decreases sharply to zero on the positive $x$-axis but shows a damped oscillation behavior on the negative $x$-axis before becoming constant.
• Symmetry: The 3D plots of the Wigner distributions confirm their symmetry in momentum space.

Significance and Claims
The authors claim that their work successfully extends the application of Wigner distributions (quasi-probability distributions) to classical solitons by constructing the requisite quantum wave functionals. The significance of this derivation lies in its utility for:

• Calculating charge distributions and current densities for solitons in a phase-space framework.
• Providing a foundation for calculating the Quantum Speed Limit (QSL) time. The authors note that the QSL time, which defines the minimum time required for a quantum state to evolve to an orthogonal state, is fundamental to quantum information theory. They propose that the derived Wigner distributions can be used to calculate the quantum fidelity ($F(t)$) between initial and time-dependent states, thereby determining the QSL time for these soliton systems.
• Offering a tool to study the quantum analogue of the classical phase space approach for solitons, which have applications in condensed matter physics and quantum computing.

The paper concludes by stating that these results are derived for pure states and suggests that future work could extend this methodology to tunnelling instantons to further explore the rate of evolution of quantum states in those contexts.