Ehrenfest-time dependence of counting statistics for chaotic ballistic systems

Transport properties of open chaotic ballistic systems and their statistics can be expressed in terms of the scattering matrix connecting incoming and outgoing wave functions. Here we calculate the dependence of correlation functions of arbitrarily many pairs of scattering matrices at different energies on the Ehrenfest time using trajectory-based semiclassical methods. This enables us to verify the prediction from effective random-matrix theory that one part of the correlation function obtains an exponential damping depending on the Ehrenfest time, while also allowing us to obtain the additional contribution that arises from bands of always correlated trajectories. The resulting Ehrenfest-time dependence, responsible, e.g., for secondary gaps in the density of states of Andreev billiards, can also be seen to have strong effects on other transport quantities, such as the distribution of delay times.

Figure 1

The trajectory sets with encounters that contribute to the third correlation function $C(\epsilon ,3)$.

Figure 2

A 3-encounter as it can be approximated in the RMT-treatment [cf. Fig. 1c]. The encounter stretches are marked by a box (shown red).

Figure 3

A 3-encounter as previously treated with Ehrenfest time.[25] The encounter stretches are marked by a box (shown red).

Figure 4

A diagram with two 2-encounters as we treat it with Ehrenfest time. The encounter stretches of the two 2-encounters are marked by boxes (shown red and blue). A possible position of the Poincaré surface of section (PSS) is marked by a black vertical line.

Figure 5

One 2-encounter is located fully inside the other, corresponding to our treatment of a generalized version of a 3-encounter. The two 2-encounters are marked by boxes (indicated by different colors).

Figure 6

The second of two 2-encounters now enters the lead so that only $t_c$ of it remains inside the system.

Figure 7

A ladder of consecutive 2-encounters. The encounter stretches are marked by boxes (shown in different colors).

Figure 8

One possible example of a mixed case: One encounter is fully contained inside another, the others form a ladder as considered before.

Figure 9

A general diagram containing also orbits with more than two stretches. The encounters are marked by boxes (shown in different colors).

Figure 10

Definition of the times ${\mathrm{t̃}}_i$ in the case of more than two encounter stretches on one orbit. The encounter stretches are shown thicker (blue).

Figure 11

An encounter diagram containing also orbits with more than two stretches. In contrast to Fig. 9, encounter stretches here are allowed to be contained fully inside others. The encounters are marked by boxes (shown in different colors).

Figure 12

Band of $n=3$ correlated trajectories. The length of the orbits is marked by a box; the duration of the encounter $t_{\mathrm{enc}}=1/\lambda \ln (c^2/{max}_i|s_i\left|{max}_j\right|u_j|)$ is marked by a dotted box.

Figure 13

An example of a band of three trajectories that possesses an encounter with another trajectory. The band is marked by a thicker box (red stretches), the encounter of the other trajectory with the band by a dotted box (blue stretch), and the links by thin (black) lines. The link durations are denoted by $t_3$ and $t_4$, and the durations of the band before and after the PSS by $t_1$ and $t_2$.

Figure 14

The probability density of the Wigner delay times for $\tau =1$.