Slalom in complex time - Supplementary information
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Closer view of the first soft recollision in Figure S3, showing in detail the topology of the quantum closest-approach surface. At $p_z>0$, shown in orange, the surface has three components which are mostly real-valued, while at $p_z<0$, in green, one component is real and two are imaginary. These components connect at the soft recollision, but which component connects to which depends on the value of $p_x$, which leads to the mixing behaviour shown in Figs. 11(b) and (c) of the main text.
Topologically, this means that what would otherwise be three separate components at high $p_x$, shown pale, are connected by 'bridges' of low transverse momentum. The surface thus consists of a single connected component, apart from the first few isolated roots with $\Re(\omega t)< 3 \pi/2$. Geometrically, this introduces two saddle points in the surface itself, which represent the degenerate points described at the end of Section IV.B, which give double roots of the closest-approach equation.
In trajectory terms, the real root at $p_z<0$ is an inward turning point. At high $p_x$, shown pale, it moves towards negative $\Re(\omega t)$ and becomes a closest-approach point, while at low $p_x$, shown solid, it becomes the central real root in orange, which is an outward turning point. Similarly, the imaginary solutions of $p_z<0$ will appear, at high $p_x$, as an alternating maximum and minimum on either side of the lobe, corresponding to an outward turning point and a closest-approach time. At low $p_x$, on the other hand, they both become closest-approach times on either side of the central turning point.