# Adiabatic theorem for a 3 level system

+ 1 like - 0 dislike
288 views

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification