On Wigner Functions and Quantum Dynamics on a Circle

TL;DR

This work extends Wigner-function formalism to quantum dynamics on a circle by introducing two related two-parameter distributions, $W[\theta,p]$ and $W_{1/2}[\theta,p]$, defined for angular position $\theta$ and angular momentum $L$. By analyzing a $2\pi$-periodic pure state controlled by parameters $\lambda$ and the fractional part $\epsilon$ of $\bar{l}$, the authors derive how marginals relate to physical probabilities and demonstrate negativity for fractional $p$, while integer $p$ marginals recover Born-rule statistics. The study further shows that Fejér summation yields well-defined angle marginals, predicts distinctive large- and small-$\lambda$ asymptotics, and identifies a characteristic gap in $\Delta\theta$ when $\epsilon=1/2$, signaling the unique features of half-integer angular momentum. The results emphasize that the choice between $W[\theta,p]$ and $W_{1/2}[\theta,p]$ depends on experimental access, and together they illuminate how quantum phase on a circle can be probed via phase-space methods.

Abstract

In the present work we consider properly defined two-parameter Wigner functions that, e.g., can be used as witness for quantum behaviour for pure states as expressed in terms of a self-adjoint observable angular position and the corresponding angular momentum operator L. It is shown that negative values of the corresponding Wigner functions may reveal the quantum nature of superpositions of angular momentum eigenstates with a fractional mean value of L but in an ambiguous manner. For a suitable choice of parameters these Wigner functions are positive and are in accordance with Borns rule in quantum mechanics.

Figures (4)

• Figure 1: The Wigner distribution $W[\theta,p]$ according to Eq.(\ref{['eq:ourwigner_1']}) with $-\pi \le \theta \le \pi$ and $\bar{\theta}=0$ for the state $\psi(\theta)$ as given in Eq.(\ref{['eq:wigner2024_3']}). In general $W[\theta,p]$ is bounded by $|W[\theta,p]|\leq 1/2\pi$ . Here $q=\exp(-1/2\lambda)=0.5$ and integer-valued $\bar{l}= l$ as a function of the parameter $m =p-l$ in the range $-2\le m \le 2$. With this choice of parameters $W[\theta,p]$ is then symmetric with respect to a line with $m=0$. For half-integer values of $m$ the Wigner distribution $W[\theta,p]$ can clearly be negative at $|\theta| = {\cal O}(\pi)$, in which case the parameter $p$ is fractional.
• Figure 2: The Wigner distribution $W[\theta,p]$ according to Eq.(\ref{['eq:ourwigner_1']}) for $-\pi \le \theta \le \pi$ as in Fig.\ref{['fig:wignerplot_1']} now with $q=\exp(-1/2\lambda)=0.001$, $\epsilon=1/2$, and $\bar{l}= l +1/2$ in the range $-2\le m \le 2$ with $m=p-l$ for the state $\psi(\theta)$ as in Eq.(\ref{['eq:wigner2024_3']}). The symmetry with respect to the line with $m=0$ as in Fig.\ref{['fig:wignerplot_1']} is now not present. For half-integer values of $m$ the Wigner function $W[\theta,p]$ can be negative at $|\theta| = {\cal O}(\pi)$ and the parameter $p$ must then be fractional as in Fig.\ref{['fig:wignerplot_1']}.
• Figure 3: The Wigner distribution $W_{1/2}[\theta,p]$ according to Eq.(\ref{['eq:wigner_kastrup']}) for $-\pi \le \theta \le \pi$ for $-\pi \le \theta \le \pi$ with $q=0.001$, $\bar{l}= l +\epsilon$, and $\epsilon =1/2$ for the state $\psi(\theta)$ according to Eq.(\ref{['eq:wigner2024_3']}) as a function of the parameter $m =p-l$ in the range $-2\leq m \leq 2$. As in Fig.\ref{['fig:wignerplot_2']} the symmetry with respect to the line with $m=0$ is absent. Regions of negativity for $W_{1/2}[\theta,p]$ are in general not the same as in Fig.\ref{['fig:wignerplot_2']} for the same quantum state considered.
• Figure 4: The upper solid curve corresponds to the uncertainty $\Delta\theta$ as function of $\Delta L$ making use of the marginal Wigner distribution $W[\theta]$ according to Eq.(\ref{['eq:ourwigner_16']}) for a fractional value of $\bar{l} = l + \epsilon$ with $\epsilon = 1/2$ and using the state $\psi(\theta)$ in Eq.(\ref{['eq:wigner2024_3']}). The uncertainties $\Delta\theta$ and $\Delta L$ are independent of the integer $l$. The two upper horizontal dotted lines corresponds to the bounds $\pi/\sqrt{2}$ and $\pi/\sqrt{3}$, respectively. The lower solid curve corresponds to the marginal distribution $W_{1/2}[\theta]=|\psi(\theta)|^2$ as obtained from the Wigner function $W_{1/2}[\theta,p]$ in Eq.(\ref{['eq:wigner_kastrup']}). The lower dotted line corresponds to the upper bound $\Delta\theta=\sqrt{\pi^2/3 -2}$ in accordance with predictions of Ref.Skagerstam_2023.