Exact Quantum Dynamics

Adiabatic Dynamics

\begin{equation} \exp\left(-iH\Delta t\right)=\exp\left(-iV\frac{\Delta t}{2}\right)\exp\left(-iT\Delta t\right)\exp\left(-iV\frac{\Delta t}{2}\right) \end{equation}

Two-State Nonadiabatic Dynamics

\begin{equation} \exp\left(-iH\Delta t\right)=\exp\left(-i\begin{pmatrix}V_{11}&V_{12}\\V_{21}&V_{22}\end{pmatrix}\frac{\Delta t}{2}\right)\exp\left(-i\begin{pmatrix}T&0\\0&T\end{pmatrix}\Delta t\right)\exp\left(-i\begin{pmatrix}V_{11}&V_{12}\\V_{21}&V_{22}\end{pmatrix}\frac{\Delta t}{2}\right) \end{equation}

\begin{equation} \exp\left[-i\begin{pmatrix}V_{11}&V_{12}\\V_{21}&V_{22}\end{pmatrix}\frac{\Delta t}{2}\right]=\exp\left[-i(V_{11}+V_{22})\frac{\Delta t}{4}\right]\left[\cos\left(\sqrt{D}\frac{\Delta t}{4}\right)\begin{pmatrix}1&0\\0&1\end{pmatrix}+i\frac{\sin\left(\sqrt{D}\frac{\Delta t}{4}\right)}{\sqrt{D}}\begin{pmatrix}V_{22}-V_{11}&-2V_{12}\\ -2V_{21}&V_{11}-V_{22}\end{pmatrix}\right], \end{equation} where $D=4|V_{21}|^2+(V_{11}-V_{22})^2$.