Effect of nonlinear nonlinear coupling to a pure dephasing model

We investigate the influence of the nonlinear coupling to the coherence of a pure dephasing model. The total system consists of a qubit and a Bosonic bath, which are coupled by an interaction $H_I=g_1\sigma_z\otimes x+g_2\sigma_z\otimes x^2$ with $x=\frac{1}{\sqrt{2}}(a+a^\dagger)$. It's shown that no matter how small $g_2$ is, the long time behavior of the coherence is significantly changed by the nonlinear coupling for free induction decay (FID), while the effect of $g_1$ can be neglected as long as $g_1$ is much smaller than the enegy splitting of the qubit. In the case that many-pulse dynamical decoupling control is exerted on the qubit, $g_2$ also modulates the oscillation of the coherence. Our results indicate that the nonlinear coupling must be taken into account for long time dynamics.