116
4 Chaotic Scattering
x→ − x alone. As we shall see below, this gives rise to an asymmetry in the
scattering dynamics of this system, which depends on the initial phase φ 0 . The
fraction of incident particles that are transmitted and reflected will differ depending
on the incident direction.
In describing the behavior of this system, we use atomic units (see Appendix I).
Therefore, the electron mass is m = 1 a.u. and we choose V 0 = 0.27035 a.u., δ =
2 a.u., ω = 0.65 a.u. and α 0 = 0.9 a.u., values that have been shown to describe
the quantum behavior of a negative chlorine ion Cl − in the presence of a laser
field (Yao and Chu 1992; Marinescu and Gavrila 1996; Fearnside et al. 1995). The
classical scattering dynamics has been analyzed in great detail in Emmanouilidou
et al. (2003), Jung and Emmanouilidou (2005), Emmanouilidou and Jung (2006),
and Lin et al. (2011). We will mainly follow the approach used in Lin et al. (2011),
but comment on the other very interesting works as needed.
4.4.1 Scattering Map
The fractal structure of the scattering process can be seen in a Poincaré surface of
section (PSS) of the dynamics. We can solve Hamilton’s equations in the KH-frame
and plot p and x each time φ = π/2 + n2π , where n = 0, 1, 2, ... The inverted
Gaussian system, in the absence of radiation, has three primary periodic orbits (fixed
points of the PSS): a stable fixed point at (p = 0, x = 0), an unstable periodic orbit
(F L ) at (p = 0, x = −∞), and an unstable periodic orbit (F R ) at (p = 0, x =
+∞). We define a fundamental region R, in the PSS, whose boundaries are given
by segments of the invariant manifolds of the outer fixed points of the system. The
invariant manifolds and the fundamental area R are shown in Fig. 4.12a.
The invariant manifolds are located by taking a line of initial points with fixed
x but varying p > 0 near p = 0, which is the neighborhood of the separatrix in
the absence of radiation. Then evolve the points in time and determine which points
escape and which points do not escape. Points that do not escape are on the invariant
manifold. This process is repeated for different values of x until a segment of the
manifold is mapped out. The curves n–u and s–n, in Fig. 4.12a, are segments of the
unstable and stable manifolds, respectively, of the fixed point on the left, F L . The
curves r–p and p–v are segments of the stable and unstable manifolds, respectively,
to the fixed point on the right, F R . The fundamental area, in Fig. 4.12a, is defined
and enclosed by the curves n–q, q–p, n–m and m–p. Because the invariant manifolds
approach the p = 0 axis exponentially as x→±∞, it is sufficient to consider the
segments shown in the figure.
In Fig. 4.12b, we show four iterations backward in time of the segments r–p and
s–n of the stable manifolds. The stable manifolds cannot intersect themselves and
they cannot intersect each other. However, they can intersect the unstable manifolds.
The curves, as they are iterated backward in time, must enclose the same area
enclosed by the stable and unstable manifolds in Fig. 4.12a.
4 Chaotic Scattering
x→ − x alone. As we shall see below, this gives rise to an asymmetry in the
scattering dynamics of this system, which depends on the initial phase φ 0 . The
fraction of incident particles that are transmitted and reflected will differ depending
on the incident direction.
In describing the behavior of this system, we use atomic units (see Appendix I).
Therefore, the electron mass is m = 1 a.u. and we choose V 0 = 0.27035 a.u., δ =
2 a.u., ω = 0.65 a.u. and α 0 = 0.9 a.u., values that have been shown to describe
the quantum behavior of a negative chlorine ion Cl − in the presence of a laser
field (Yao and Chu 1992; Marinescu and Gavrila 1996; Fearnside et al. 1995). The
classical scattering dynamics has been analyzed in great detail in Emmanouilidou
et al. (2003), Jung and Emmanouilidou (2005), Emmanouilidou and Jung (2006),
and Lin et al. (2011). We will mainly follow the approach used in Lin et al. (2011),
but comment on the other very interesting works as needed.
4.4.1 Scattering Map
The fractal structure of the scattering process can be seen in a Poincaré surface of
section (PSS) of the dynamics. We can solve Hamilton’s equations in the KH-frame
and plot p and x each time φ = π/2 + n2π , where n = 0, 1, 2, ... The inverted
Gaussian system, in the absence of radiation, has three primary periodic orbits (fixed
points of the PSS): a stable fixed point at (p = 0, x = 0), an unstable periodic orbit
(F L ) at (p = 0, x = −∞), and an unstable periodic orbit (F R ) at (p = 0, x =
+∞). We define a fundamental region R, in the PSS, whose boundaries are given
by segments of the invariant manifolds of the outer fixed points of the system. The
invariant manifolds and the fundamental area R are shown in Fig. 4.12a.
The invariant manifolds are located by taking a line of initial points with fixed
x but varying p > 0 near p = 0, which is the neighborhood of the separatrix in
the absence of radiation. Then evolve the points in time and determine which points
escape and which points do not escape. Points that do not escape are on the invariant
manifold. This process is repeated for different values of x until a segment of the
manifold is mapped out. The curves n–u and s–n, in Fig. 4.12a, are segments of the
unstable and stable manifolds, respectively, of the fixed point on the left, F L . The
curves r–p and p–v are segments of the stable and unstable manifolds, respectively,
to the fixed point on the right, F R . The fundamental area, in Fig. 4.12a, is defined
and enclosed by the curves n–q, q–p, n–m and m–p. Because the invariant manifolds
approach the p = 0 axis exponentially as x→±∞, it is sufficient to consider the
segments shown in the figure.
In Fig. 4.12b, we show four iterations backward in time of the segments r–p and
s–n of the stable manifolds. The stable manifolds cannot intersect themselves and
they cannot intersect each other. However, they can intersect the unstable manifolds.
The curves, as they are iterated backward in time, must enclose the same area
enclosed by the stable and unstable manifolds in Fig. 4.12a.