Information scrambling of the dilute Bose gas at low temperature

We calculate the quantum Lyapunov exponent ${\lambda }_L$ and butterfly velocity $v_B$ in the dilute Bose gas at temperature $T$ deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time ordered correlators, from which ${\lambda }_L$ and $v_B$ are extracted. At very low temperature where elementary excitations are phonon-like, we find ${\lambda }_L\propto T^5$ and $v_B\sim c$, the sound velocity. At relatively high temperature, we have ${\lambda }_L\propto T$ and $v_B\sim c(T\mathrm{/}{T}_{\ast }{)}^{0.23}$. We find ${\lambda }_L$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on $T$. The chaos diffusion constant $D_L=v_B^2\mathrm{/}{\lambda }_L$, on the other hand, differs from the energy diffusion constant $D_E$. We find $D_E\ll D_L$ at very low temperature and $D_E\gg D_L$ otherwise.