  We obtain an initial value representation for the quantum Loschmidt echo from
the semiclassical theory of Wigner function evolution, together with classical
first-order perturbation theory. In the limit of small actions, the amplitude
of each trajectory reduces to unity, just as in the dephasing representation
introduced by Vanicek, but these trajectories are here generated by the mean
Hamiltonian for both the forward and the backward motion. This slight change of
action may substantially alter the phase. The amplitude correction depends on
the second derivative of the action. This improved dephasing approximation is
verified to work even for quadratic Hamiltonians, for which the semiclassical
evolution is exact, thus extending the range of application beyond its original
scope in quantum chaos.

Quantum Physics