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where $\mathcal{T}$ denotes the time ordering operator.  Using the Baker-Campbell-Hausdorff formula, we can replace the above expression by the tensor product expression:  \begin{equation}  |\Psi_{s-c}(t_u) \rangle \approx \\  \left[ \mathcal{T}Exp\left[ \int_{0}^{t_u} dt_u' \left[ \hat{H}_{s} + \hat{V}(\hat{s}, c, t_{u}')\right] \right] |\psi_{s}(t_u = 0, t_{c}) \rangle \right] \otimes \left[ e^{-i \hat{H}_{c} t_{u}} |t_c \rangle \right], \end{equation}~\label{approx_sep}  under the condition $t_u ||[\hat{H}_c, \int_{0}^{t_u} dt_u' \hat{V}(\hat{s}, c, t_{u}')]|| \ll1$. Since $||\int_{0}^{t_u} dt_u' \hat{V}(\hat{s}, c, t_{u}')|| \leq ||\hat{V}(\hat{s}, c, t_{u})|| t_u$, we can write this condition as:  \begin{equation}