Quantum theory sets a bound on the minimal time it takes for a system to evolve from initial state to target state. This bound is called the quantum speed limit (QSL) time. The quantum speed limit time is used to quantify the maximal speed of the quantum evolution. The quantum evolution will be faster if the quantum speed limit time de- creases. In this work, we study the quantum speed limit time for an open quantum system in the presence of disturbance in an environment. We use the model which is provided by Ban [Phys. Rev. A 99, 012116 (2019)]. In this model, two quantum systems A and S interact with environment sequentially. At first, quantum system A interacts with the environment E as an auxiliary system, then quantum system S starts its inter- action with disturbed environment immediately. In this work, we consider the dephasing coupling with two types of environment that has different spectral density: Ohmic and Lorentzian. We observe that, non-Markovian effects will appear in the dynamics of the second quantum system S due to the interaction of the first quantum system A with the environment. Given the fact that the quantum speed limit time reduces due to the non- Markovian feature of quantum evolution, we show that disturbance effects will reduce the quantum speed limit time for the dynamics of the second quantum system S.
