Decoherence
Energy-based decoherence correction [22, 23] applies exponential damping to inactive adiabatic states, while renormalizing the active state to preserve the total wavefunction norm (\(\sum |c_i|^2 = 1\)).
\[c_i \rightarrow c_i \exp\left(-\frac{\Delta t}{\tau_{ia}}\right), \quad \forall \; i \ne a,\]
\[c_a = c_a \left[\frac{1 - \sum_{i \ne a}|c_i|^2}{|c_a|^2}\right]^{1/2},\]
with
\[\tau_{ia} = \frac{\hbar}{|E_i - E_a|}
\left( C + \frac{E_0}{T_a} \right),\]
where
\(E_i\) is the energy of inactive state \(i\),
\(E_a\) is the energy of active state \(a\),
\(T_a\) is the kinetic energy on the active surface,
\(C\) and \(E_0\) are empirical constants.