Malamala
- 348
- 28
Hello! If I have a 2 level system, with the energy splitting between the 2 levels ##\omega_{12}## and an external perturbation characterized by a frequency ##\omega_P##, if ##\omega_{12}>>\omega_P## I can use the adiabatic approximation, and assume that the initial state of the system changes slowly in time while for ##\omega_{12}<<\omega_P## I can assume that the perturbation doesn't have any effect on the system (it averages out over the relevant time scales). I was wondering if I have a 3 level system with ##E_1<E_2<E_3## such that ##\omega_{12}<<\omega_P<<\omega_{23}##. In general, the Hamiltonian of the system would look like this:
$$
\begin{pmatrix}
E_1 & f_{12}(t) & f_{13}(t) \\
f_{12}^*(t) & E_2 & f_{23}(t) \\
f_{13}^*(t) & f_{23}^*(t) & E_3
\end{pmatrix}
$$
But using the intuition from the 2 level system case, can I ignore ##f_{12}(t)##, as the system of these 2 levels (1 and 2) moves on time scales much slower than ##\omega_P##, and assume that ##f_{23}(t)## and ##f_{13}(t)## move very slow and thus use the adiabatic approximation? In practice I would basically have:
$$
\begin{pmatrix}
E_1 & 0 & f_{13}(t) \\
0 & E_2 & f_{23}(t) \\
f_{13}^*(t) & f_{23}^*(t) & E_3
\end{pmatrix}
$$
Or in this case I would need to fully solve the SE, without being able to make any approximations? Thank you!
$$
\begin{pmatrix}
E_1 & f_{12}(t) & f_{13}(t) \\
f_{12}^*(t) & E_2 & f_{23}(t) \\
f_{13}^*(t) & f_{23}^*(t) & E_3
\end{pmatrix}
$$
But using the intuition from the 2 level system case, can I ignore ##f_{12}(t)##, as the system of these 2 levels (1 and 2) moves on time scales much slower than ##\omega_P##, and assume that ##f_{23}(t)## and ##f_{13}(t)## move very slow and thus use the adiabatic approximation? In practice I would basically have:
$$
\begin{pmatrix}
E_1 & 0 & f_{13}(t) \\
0 & E_2 & f_{23}(t) \\
f_{13}^*(t) & f_{23}^*(t) & E_3
\end{pmatrix}
$$
Or in this case I would need to fully solve the SE, without being able to make any approximations? Thank you!