Energy transition density of driven chaotic systems: A compound trace formula

Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit segments, one in the energy shell of the original Hamiltonian and the other in the energy shell of the driven Hamiltonian, with endpoints which coincide. Viewed in the time domain, the same pair of trajectory segments arises in the semiclassical evaluation of the trace of a compound propagator: the product of the complex exponentials of the original Hamiltonian and of its driven image. It is shown here that the probability density is the double Fourier transform of this trace, so that the closed compound orbits emulate the role played by periodic orbits in Gutzwiller's trace formula in its semiclassical evaluation. The phase of the oscillations with the energies or evolution parameters agree with those previously obtained, whereas the amplitude of the contribution of each closed compound orbit is more compact and independent of any feature of the Weyl-Wigner representation in which the calculation was carried out.