$$ \newcommand{\vb}[1]{\mathbf{#1}} \newcommand{\vbv}{\vb{v}} \newcommand{\vbr}{\vb{r}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} $$

Slalom in complex time - Supplementary information

This page is supplementary information to the paper

Slalom in complex time: emergence of low-energy structures in tunnel ionization via complex time contours. E. Pisanty and M. Ivanov. Phys. Rev. A 93, 043408 (2016), arXiv:1507.00011.

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Figure S1

This is a 3D version of Figure 8 in the main text, which shows the classical closest-approach times as a surface in $(\omega t,p_x,p_z)$ space. Note in particular that the red parts of the surface, which represent maxima of $\vbr^2$, always face towards the left (towards negative t), whereas the green parts of the surface are $\vbr^2$ minima and always face towards the right (towards positive $t$).

Figure S2

This is a 3D version of Figure 10 in the main text, which shows the quantum, complex-valued closest-approach times on the three-dimensional space $(\Re(\omega t),p_x,p_z)$. The imaginary part $\Im(\omega t)$ is denoted by the colour: black points are mostly real, while red (blue) points have positive (negative) imaginary parts.

Figure S3

The quantum times of closest approach form a two-dimensional surface within the four-dimensional space $(p_x,p_z,\Re(\omega t),\Im(\omega t))$. Here we show a projection onto the three-dimensional space $(\Re(\omega t),\Im(\omega t),p_z)$, which yields Figure 11(a) of the main text if $p_z$ is projected out. (The colour scale follows the scheme of Figure 11(a).) Some solutions, however, are mostly real are not discerned in this view; instead, they are visible by projecting out $\Im(\omega t)$. Other features of this surface, like the internal structure of the first bounded lobe, require more complex perspectives. Soft recollisions, on the other hand, require a more detailed view, which we show in Figure S4.

Figure S4

Saddle A: Saddle B:

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.