Sensitivity of Quantum Motion to Perturbation in Triangle Map

We study quantum Loschmidt echo, or fidelity, in the triangle map whose classical counterpart has linear instability and weak chaos. Numerically, three regimes of fidelity decay have been found with respect to the perturbation strength $ϵ$. In the regime of weak perturbation, the fidelity decays as $\exp (-c{ϵ}^2{t}^{\gamma })$ with $\gamma \simeq 1.7$. In the regime of strong perturbation, the fidelity is approximately a function of $\epsilon t^{2.5}$, which is predicted for the classical fidelity [G. Casati, {\it et al}, Phys.Rev.Lett.{\bf 94}, 114101 (2005)], and decays slower than power-law decay for long times. In an intermediate regime, the fidelity has approximately an exponential decay $\exp (-c^{\mathrm{\prime }}ϵt)$.