Complexity, chaos and the moving D3-brane

We use the wave-function method developed in area of quantum information to investigate the quantum circuit complexity of the small quantum fluctuations around the probe $D_3$ brane moving in $AdS_5\times S^5$ bulk. In our consideration, the reference and target states are chosen as the vacuum state and the squeezed quantum state respectively. The evolution of parameters characterizing the squeezed quantum state are governed by the time-dependent $Schr\ddot{o}dinger$ equation, in which the Hamiltonian operator is derived from the perturbative action of $D_3$ brane. For a quantum chaotic system, some recent works indicate that the evolution of quantum circuit complexity could provide equivalent information like the out-of-time-order correlators. Basing on this inference, our results show that the quantum fluctuations around the non-BPS brane manifestly evolve into the chaotic regime at the late time, while the chaotic behavior is not easy to observe in case of BPS brane. In holographic viewpoint, it implies that the thermodynamic system consist of the $N=4$ supersymmetric particles in non-BPS states evolve into a chaotic system more easily than the one in BPS state.