  We study the crossover of the quantum Loschmidt echo (or fidelity) from the
golden rule regime to the perturbation-independent exponential decay regime by
using the kicked top model. It is shown that the deviation of the
perturbation-independent decay of the averaged fidelity from the Lyapunov decay
results from quantum fluctuations in individual fidelity, which are caused by
the coherence in the initial coherent states.
  With an averaging procedure suppressing the quantum fluctuations effectively,
the perturbation-independent decay is found to be close to the Lyapunov decay.
  We also show that the Fourier transform of the fidelity is determined
directly by the initial state and the eigenstates of the Floquet operators of
the two classically chaotic systems concerned.
  The absolute value part and the phase part of the Fourier transform of the
fidelity are found to be divided into several correlated parts, which is a
manifestation of the coherence of the initial coherent state.
  In the whole crossover region, some important properties of the fidelity,
such as the exponent of its exponential decay and the short initial time within
which the fidelity almost does not change, are found to be closely related to
properties of the central part of its Fourier transform.
