Semiclassical Propagator of the Wigner Function

Propagation of the Wigner function is studied on two levels of semiclassical propagation: one based on the Van Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a pair of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions.

Figure 1

The reduced action (shaded) of the Wigner propagator in the Van Vleck approximation, Eq. (5), is the symplectic area enclosed between the two classical trajectories ${\mathbf{r}}_{j\pm }(t)$ and the two transverse vectors ${\mathbf{r}}_{j+}^{\prime }-{\mathbf{r}}_{j-}^{\prime }$ and ${\mathbf{r}}_{j+}^{\prime \prime }-{\mathbf{r}}_{j-}^{\prime \prime }$ (schematic drawing). The solid central line is the classical trajectory ${\mathbf{r}}_{\mathrm{cl}}({\mathbf{r}}^{\prime },t)$; the broken line is the propagation path ${\overline{\mathbf{r}}}_j({\mathbf{r}}^{\prime },t)$.

Figure 2

(a) Classical building blocks entering the Wigner propagator according to Eqs. (4, 5), for a stable trajectory starting at ${\mathbf{r}}^{\prime }=(0.636,0)$ near the minimum of the cubic potential $V(q)=0.329q^3-0.69q$. (b) Classical trajectory ${\mathbf{r}}_{\mathrm{cl}}({\mathbf{r}}^{\prime },t)$ (black line), a pair of auxiliary trajectories ${\mathbf{r}}_{j\pm }(t)$ (green lines), and the corresponding propagation path ${\overline{\mathbf{r}}}_j({\mathbf{r}}^{\prime },t)$ (red dashed line). The grid of auxiliary initial points ${\mathbf{r}}_{j\pm }^{\prime }$ around ${\mathbf{r}}^{\prime }$ (yellow) is parametrized by polar coordinates. Propagated classically over time $t$, it deforms into the turquoise pattern around ${\mathbf{r}}^{\prime \prime }$. The red/blue cone is formed by midpoints ${\overline{\mathbf{r}}}_j^{\prime \prime }$ that correspond to extrema/saddles of the action. Its boundaries form caustics separating the region accessed by two midpoints ${\overline{\mathbf{r}}}^{\prime \prime }$ from the inaccessible rest. (c) Enlargement of the area around ${\mathbf{r}}^{\prime \prime }$, indicating the number of trajectory pairs that access each region.

Figure 3

Quantum spot replacing the classical delta function on a stable (elliptic) trajectory near the minimum of a cubic potential as shown in Fig. 2a, at $t=1.8$ ($\varphi \approx 2\pi /3$). (a) shows the exact quantum result for the Wigner propagator; (b) and (c) are semiclassical approximations based on Eqs. (4, 9), respectively, all for $\hslash =0.01$. Frames coincide with that in Fig. 2c. (d) Quantum spot, according to Eq. (4), for an unstable classical trajectory near the maximum of the same potential, at $t=1.0$. Crosses mark the classical trajectory. Color code ranges from red (negative) to blue (positive).