Wigner Functions for Arbitrary Quantum Systems

The possibility of constructing a complete, continuous Wigner function for any quantum system has been a subject of investigation for over 50 years. A key system that has served to illustrate the difficulties of this problem has been an ensemble of spins. Here we present a general and consistent framework for constructing Wigner functions exploiting the underlying symmetries in the physical system at hand. The Wigner function can be used to fully describe any quantum system of arbitrary dimension or ensemble size.

Figure 1

(a)–(c) Polar plot of the Wigner function using ${\widehat{\mathrm{\Delta }}}^{[2j+1]}(\theta ,\varphi )$, as defined in Eq. (8) for high spin spaces of dimension $1/2$ (a), $3/2$ (b), and $7/2$ (c). Here we have used as examples normalized states of the form $|j,m=j⟩+|j,m=-j⟩$ (where we have labeled states in terms of the quantum numbers for ${\widehat{J}}^2$ and ${\widehat{J}}_z$). Note that there are $2j$ interference terms and that images are not to the same scale. (d)–(f) Mercator projection of the Wigner function using ${\widehat{\mathrm{\Delta }}}^{[2^k]}(\{{\theta }_i,{\varphi }_i\})$, as defined in Eq. (9), for a set of 1 (d), 2 (e), and 3 (f) spins and two-level atoms, where we have taken the slice ${\theta }_i=\theta$ (as the ordinate from 0 to $\pi /2$) and ${\varphi }_i=\varphi$ (as the abscissa for 0 to $\pi$). For all plots, blue is positive and red negative and black is the zero contour.

Figure 2

Comparison of the Wigner (left) and $Q$ (right) functions for states generated by one axis squeezing of a set of two-level systems. The initial state is $|{+}_6⟩≔\underset{i=1}{\overset{6}{\otimes }}({|j=\frac{1}{2},m=\frac{1}{2}⟩}_i+{|j=\frac{1}{2},m=-\frac{1}{2}⟩}_i)$ and the Hamiltonian is $\widehat{H}={[\underset{i}{⨁}({\widehat{\sigma }}_z{)}_i]}^2$. We calculate the Wigner function using Eq. (10) using ${\widehat{\mathrm{\Delta }}}^{[2^k]}(\{{\theta }_i,{\varphi }_i\})$, $\widehat{A}=|{+}_6⟩⟨{+}_6|$ and $\widehat{V}=\exp (-\mathrm{i}\widehat{H}t)$, where $t=\pi /125$. $Q$ is calculated in the usual way using the natural $\theta$, $\varphi$ parametrization of spin coherent states (see Ref. [50] for details). As in Figs. 1, we have taken the slice ${\theta }_i=\theta$ (as the ordinate) and ${\varphi }_i=\varphi$ (as the abscissa). In both figures we clearly see squeezing, but in the Wigner function we also see negative volume indicating the underlying quantum nature of the states. Note, unlike in Figs. 1, here $\varphi$ ranges from $-\pi /2$ to $\pi /2$. For all plots, blue is positive and red negative and black is the zero contour.