Semiclassical energy transition of driven chaotic systems: phase coherence on scar disks

Abstract

A trajectory segment in an energy shell, which combines to form a closed curve with a segment in another canonically driven energy shell, adds an oscillatory semiclassical contribution to the smooth classical background of the quantum probability density for a transition between their energies. If either segment is part of a Bohr-quantized periodic orbit of either shell, the centre of its endpoints lies on a scar disk of the spectral Wigner function for single static energy shell and the contribution to the transition is reinforced by phase coherence. The exact representation of the transition density as an integral over spectral Wigner functions, which was previously derived for the special case where the system undergoes a reflection in phase space, is here generalized to arbitrary unitary transformations. If these are generated continuously by a driving Hamiltonian, there will be a finite lapse in the driving time for the transition to start, until the initially nested shells touch each other and then start to overlap. The stationary phase evaluation of the multidimensional integral for the transition density selects the pair of matching trajectory segments on each shell, which close to form a piecewise smooth compound orbit. Each compound orbit shows up as a fixed point of a product of mappings that generalize Poincar\'e maps on the intersection of the shells. Thus, the closed compound orbits are isolated if the original Hamiltonian is chaotic. The actions of the compound orbits depend on the driving time, or on any other parameter of the transformation of the original eigenstates.