Semiclassical Mechanism for the Quantum Decay in Open Chaotic Systems

We address the decay in open chaotic quantum systems and calculate semiclassical corrections to the classical exponential decay. We confirm random matrix predictions and, going beyond, calculate Ehrenfest time effects. To support our results we perform extensive numerical simulations. Within our approach we show that certain (previously unnoticed) pairs of interfering, correlated classical trajectories are of vital importance. They also provide the dynamical mechanism for related phenomena such as photoionization and photodissociation, for which we compute cross-section correlations. Moreover, these orbits allow us to establish a semiclassical version of the continuity equation.

Figure 1

Pairs of correlated classical trajectories $\gamma$ (full line) and ${\gamma }^{\prime }$ (dashed) generating the leading quantum corrections to the classical decay probability. While in panel (a) the encounter region (box) connects a loop with two legs, the paths begin or end inside the encounter region (“one-leg-loops”) in (b) with and in (c) without a self-crossing in configuration space. The zoom into the encounter region in (b) depicts the position of the Poincaré surface of section used.

Figure 2

Averaged ratio $R(t)$ between numerical quantum mechanical decay ${\rho }_{\mathrm{qm}}^{\mathrm{sim}}(t)$ and classical decay ${\rho }_{\mathrm{cl}}^{\mathrm{sim}}(t)$ (red dots, the bars correspond to the standard deviation after averaging, see text) compared with corresponding semiclassical predictions based on the quadratic term in Eq. (6) (dashed line) and Eq. (10) (full line). Upper inset: ${\rho }_{\mathrm{qm}}^{\mathrm{sim}}(t)$ (red full line) and ${\rho }_{\mathrm{cl}}^{\mathrm{sim}}(t)$ (dashed) for the wave packet shown in the lower inset. Lower inset: Desymmetrized diamond billiard, defined as the fundamental domain of the area confined by four intersecting disks of radius $R$ centered at the vertices of a square of length $2L$ ($L=100$, $R=131$) with opening $w=16$. The initial Gaussian wave packet shown is of size $\sigma =3$ and ${\lambda }_B=3$. The arrow indicates the momentum direction.