Semiclassical relation between open trajectories and periodic orbits for the Wigner time delay

The Wigner time delay of a classically chaotic quantum system can be expressed semiclassically either in terms of pairs of scattering trajectories that enter and leave the system or in terms of the periodic orbits trapped inside the system. We show how these two pictures are related on the semiclassical level. We start from the semiclassical formula with the scattering trajectories and derive from it all terms in the periodic orbit formula for the time delay. The main ingredient in this calculation are correlations between scattering trajectories which are due to trajectories that approach the trapped periodic orbits closely. The equivalence between the two pictures is also demonstrated by considering correlation functions of the time delay. A corresponding calculation for the conductance gives no periodic orbit contributions in leading order.

Figure 1

(Color online) An example of a trajectory (full line) with two encounter regions and its partner trajectory (dashed line). The encounter regions are indicated by rectangular boxes.

Figure 2

(Color online) A schematic picture of a trajectory that approaches an unstable periodic orbit follows it a number of times and leaves it again. In a Poincaré map transverse to the periodic orbit the trajectory moves along the invariant hyperbola $su=\text{const}\text{.}$

Figure 3

Two trajectories which follow the periodic orbit at $O$ five and seven times, respectively, within the region of the Poincaré section bounded by the constant $c$. The action difference is $S_{\alpha }-S_{{\alpha }^{\prime }}=A-2S_p$, where $S_p$ is the action of the periodic orbit and $A$ is the sum of the areas of the two triangles $O{P}_1^{\prime }{P}_1$ and $O{P}_5{P}_7^{\prime }$.

Figure 4

(Color online) An example of a trajectory (the dots on the two full lines) with a two-encounter in the vicinity of a periodic orbit and some of its partner trajectories (the dots on the dashed lines). The partner trajectories are obtained by drawing rectangular boxes which have points of the original trajectory (full lines) in opposite corners.

Figure 5

(Color online) An example of a trajectory quadruplet where the trajectories $\alpha$ and $\beta$ (full lines) have one two-encounter. Their partner trajectories (dashed lines) cross the encounter region differently. The encounter region is indicated by a rectangular box.