competes with the quadratic Zeeman field, giving rise to interesting dynamics [@sadler].

In the condensed gas, dipolar interactions may also be important [@mukund; @mukund2]. At the lower densities found in a normal gas these interactions, which fall as $1/r^{3}$, may be neglected.

Following standard arguments [@kadanoff] we obtain the equations of motion for the Wigner density matrix $\textbf{F}_{ab}(\vec{p}, \vec{R}, t)$, whose elements are $f_{ab}(\vec{p}, \vec{R}, t)= \int d \vec{r} e^{-i\vec{p}\cdotp \vec{r}} \langle \psi^{\dagger}_{a}(\vec{R} - \frac{\vec{r}}{2}, t) \psi_{b}(\vec{R} + \frac{\vec{r}}{2}, t) \rangle$. The diagonal components of the spin density matrix, when integrated in momentum, give the densities of each of the spin species. The off diagonal terms, often referred to as *coherences*, are responsible for spin dynamics. Here $\vec{p}$ represents momentum, $\vec{R}$ and $\vec{r}$ denote the center of mass and relative coordinates. The Wigner function is the quantum analog of the classical distribution function. By taking moments of the Wigner function, we obtain physical observables such as the density $\textbf{n}(\vec{R}, t) = \int \frac{d\vec{p}}{(2 \pi)^{3}}~\textbf{F}(\vec{p}, \vec{R}, t)$ and spin current $\vec{\textbf{j}}(\vec{R}, t) = \int \frac{d\vec{p}}{(2 \pi)^{3}}~\vec{p}~\textbf{F}(\vec{p}, \vec{R}, t)$.

The equation of motion takes the standard form [@bashkin]: $$\label{eq:2}
\frac{\partial}{\partial t}\textbf{F} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}\textbf{F}= i\Big[\textbf{V}, \textbf{F}\Big] + \frac{1}{2}\Big\{\vec{\nabla}_{R}\textbf{V},\vec{\nabla}_{p}\textbf{F}\Big\} + \textbf{I}_{c}$$ where $\textbf I_{c}$ 